RiskMetrics TM Technical Document

.P.Morgan/Reuers RiskMerics TM Technical Documen Fourh Ediion, 1996 New York December 17, 1996.P. Morgan and Reuers have eamed up o enhance RiskMerics. Morgan will coninue o be responsible for enhancing he mehods oulined in his documen, while Reuers will conrol he producion and disribuion of he RiskMerics daa ses. Expanded secions on mehodology ouline enhanced analyical soluions for dealing wih nonlinear opions risks and inroduce mehods on how o accoun for non-normal disribuions. Enclosed diskee conains many examples used in his documen. I allows readers o experimen wih our risk measuremen echniques. All publicaions and daily daa ses are available free of charge on.p. Morgan s Web page on he Inerne a hp://www.jpmorgan.com/riskmanagemen/riskmerics/riskmerics.hml. This page is accessible direcly or hrough hird pary services such as CompuServe, America Online, or Prodigy. Morgan Guarany Trus Company Risk Managemen Advisory acques Longersaey (1-1) 648-4936 riskmerics@jpmorgan.com Reuers Ld Inernaional Markeing Marin Spencer (44-171) 54-360 marin.spencer@reuers.com This Technical Documen provides a deailed descripion of RiskMerics, a se of echniques and daa o measure marke risks in porfolios of fixed income insrumens, equiies, foreign exchange, commodiies, and heir derivaives issued in over 30 counries. This ediion has been expanded significanly from he previous release issued in May 1995. We make his mehodology and he corresponding RiskMerics daa ses available for hree reasons: 1. We are ineresed in promoing greaer ransparency of marke risks. Transparency is he key o effecive risk managemen.. Our aim has been o esablish a benchmark for marke risk measuremen. The absence of a common poin of reference for marke risks makes i difficul o compare differen approaches o and measures of marke risks. Risks are comparable only when hey are measured wih he same yardsick. 3. We inend o provide our cliens wih sound advice, including advice on managing heir marke risks. We describe he RiskMerics mehodology as an aid o cliens in undersanding and evaluaing ha advice. Boh.P. Morgan and Reuers are commied o furher he developmen of RiskMerics as a fully ransparen se of risk measuremen mehods. We look forward o coninued feedback on how o mainain he qualiy ha has made RiskMerics he benchmark for measuring marke risk. RiskMerics is based on, bu differs significanly from, he risk measuremen mehodology developed by.p. Morgan for he measuremen, managemen, and conrol of marke risks in is rading, arbirage, and own invesmen accoun aciviies. We remind our readers ha no amoun of sophisicaed analyics will replace experience and professional judgmen in managing risks. RiskMerics is nohing more han a high-qualiy ool for he professional risk manager involved in he financial markes and is no a guaranee of specific resuls.

RiskMerics Technical Documen Fourh Ediion (December 1996) Copyrigh 1996 Morgan Guarany Trus Company of New York. All righs reserved. RiskMerics is a regisered rademark of. P. Morgan in he Unied Saes and in oher counries. I is wrien wih he symbol a is firs occurrence in his publicaion, and as RiskMerics hereafer.

Preface o he fourh ediion iii This book This is he reference documen for RiskMerics. I covers all aspecs of RiskMerics and supersedes all previous ediions of he Technical Documen. I is mean o serve as a reference o he mehodology of saisical esimaion of marke risk, as well as deailed documenaion of he analyics ha generae he daa ses ha are published daily on our Inerne Web sies. This documen reviews 1. The concepual framework underlying he mehodologies for esimaing marke risks.. The saisics of financial marke reurns. 3. How o model financial insrumen exposures o a variey of marke risk facors. 4. The daa ses of saisical measures ha we esimae and disribue daily over he Inerne and shorly, he Reuers Web. Measuremen and managemen of marke risks coninues o be as much a craf as i is a science. I has evolved rapidly over he las 15 years and has coninued o evolve since we launched RiskMerics in Ocober 1994. Dozens of professionals a.p. Morgan have conribued o he developmen of his marke risk managemen echnology and he laes documen conains enries or conribuions from a significan number of our marke risk professionals. We have received numerous consrucive commens and criicisms from professionals a Cenral Banks and regulaory bodies in many counries, from our compeiors a oher financial insiuions, from a large number specialiss in academia and las, bu no leas, from our cliens. Wihou heir feedback, help, and encouragemen o pursue our sraegy of open disclosure of mehodology and free access o daa, we would no have been as successful in advancing his echnology as much as we have over he las wo years. Wha is RiskMerics? RiskMerics is a se of ools ha enable paricipans in he financial markes o esimae heir exposure o marke risk under wha has been called he Value-a-Risk framework. RiskMerics has hree basic componens: A se of marke risk measuremen mehodologies oulined in his documen. Daa ses of volailiy and correlaion daa used in he compuaion of marke risk. Sofware sysems developed by.p.morgan, subsidiaries of Reuers, and hird pary vendors ha implemen he mehodologies described herein. Wih he help of his documen and he associaed line of producs, users should be in a posiion o esimae marke risks in porfolios of foreign exchange, fixed income, equiy and commodiy producs..p. Morgan and Reuers eam up on RiskMerics In une 1996,.P. Morgan signed an agreemen wih Reuers o cooperae on he building of a new and more powerful version of RiskMerics. Since he launch of RiskMerics in Ocober 1994, we have received numerous requess o add new producs, insrumens, and markes o he daily volailiy and correlaion daa ses. We have also perceived he need in he marke for a more flexible VaR daa ool han he sandard marices ha are currenly disribued over he Inerne. The new

iv Preface o he fourh ediion parnership wih Reuers, which will be based on he precep ha boh firms will focus on heir respecive srenghs, will help us achieve hese objecives. Mehodology.P. Morgan will coninue o develop he RiskMerics se of VaR mehodologies and publish hem in he quarerly RiskMerics Monior and in he annual RiskMerics Technical Documen. RiskMerics daa ses Reuers will ake over he responsibiliy for daa sourcing as well as producion and delivery of he risk daa ses. The curren RiskMerics daa ses will coninue o be available on he Inerne free of charge and will be furher improved as a benchmark ool designed o broaden he undersanding of he principles of marke risk measuremen. When.P. Morgan firs launched RiskMerics in Ocober 1994, he objecive was o go for broad marke coverage iniially, and follow up wih more granulariy in erms of he markes and insrumens covered. This over ime, would reduce he need for proxies and would provide addiional daa o measure more accuraely he risk associaed wih non-linear insrumens. The parnership will address hese new markes and producs and will also inroduce a new cusomizable service, which will be available over he Reuers Web service. The cusomizable RiskMerics approach will give risk managers he abiliy o scale daa o mee he needs of heir individual rading profiles. Is capabiliies will range from providing cusomized covariance marices needed o run VaR calculaions, o supplying daa for hisorical simulaion and sress-esing scenarios. More deails on hese plans will be discussed in laer ediions of he RiskMerics Monior. Sysems Boh.P. Morgan and Reuers, hrough is Sailfish subsidiary, have developed clien-sie RiskMerics VaR applicaions. These producs, ogeher wih he expanding suie of hird pary applicaions will coninue o provide RiskMerics implemenaions. Wha is new in his fourh ediion? In erms of conen, he Fourh Ediion of he Technical Documen incorporaes he changes and refinemens o he mehodology ha were iniially oulined in he 1995 1996 ediions of he RiskMerics Monior: Expanded framework: We have worked exensively on refining he analyical framework for analyzing opions risk wihou having o perform relaively ime consuming simulaions and have oulined he basis for an improved mehodology which incorporaes beer informaion on he ails of disribuions relaed o financial asse price reurns; we ve also developed a daa synchronizaion algorihm o refine our volailiy and correlaion esimaes for producs which do no rade in he same ime zone; New markes: We expanded he daily daa ses o include esimaed volailiies and correlaions of addiional foreign exchange, fixed income and equiy markes, paricularly in Souh Eas Asia and Lain America. Fine-uned mehodology: We have modified he approach in a number of ways. Firs, we ve changed our definiion of price volailiy which is now based on a oal reurn concep; we ve also revised some of he algorihms used in our mapping rouines and are in he process of redefining he echniques used in esimaing equiy porfolio risk. RiskMerics Technical Documen Fourh Ediion

Preface o he fourh ediion v RiskMerics producs: While we have coninued o expand he lis of hird paries providing RiskMerics producs and suppor, his is no longer included wih his documen. Given he rapid pace of change in he availabiliy of risk managemen sofware producs, readers are advised o consul our Inerne web sie for he laes available lis of producs. This lis, which now includes FourFifeen,.P. Morgan s own VaR calculaor and repor generaing sofware, coninues o grow, aesing o he broad accepance RiskMerics has achieved. New ools o use he RiskMerics daa ses: We have published an Excel add-in funcion which enables users o impor volailiies and correlaions direcly ino a spreadshee. This ool is available from our Inerne web sie. The srucure of he documen has changed only slighly. As before, is size warrans he following noe: One need no read and undersand he enire documen in order o benefi from RiskMerics. The documen is organized in pars ha address subjecs of paricular ineres o many readers. Par I: Risk Measuremen Framework This par is for he general praciioner. I provides a pracical framework on how o hink abou marke risks, how o apply ha hinking in pracice, and how o inerpre he resuls. I reviews he differen approaches o risk esimaion, shows how he calculaions work on simple examples and discusses how he resuls can be used in limi managemen, performance evaluaion, and capial allocaion. Par II: Saisics of Financial Marke Reurns This par requires an undersanding and ineres in saisical analysis. I reviews he assumpions behind he saisics used o describe financial marke reurns and how disribuions of fuure reurns can be esimaed. Par III: Risk Modeling of Financial Insrumens This par is required reading for implemenaion of a marke risk measuremen sysem. I reviews how posiions in any asse class can be described in a sandardized fashion (foreign exchange, ineres raes, equiies, and commodiies). Special aenion is given o derivaives posiions. The purpose is o demysify derivaives in order o show ha heir marke risks can be measured in he same fashion as heir underlying. Par IV: RiskMerics Daa Ses Appendices This par should be of ineres o users of he RiskMerics daa ses. Firs i describes he sources of all daily price and rae daa. I hen discusses he aribues of each volailiy and correlaion series in he RiskMerics daa ses. And las, i provides deailed forma descripions required o decipher he daa ses ha can be downloaded from public or commercial sources. This par reviews some of he more echnical issues surrounding mehodology and regulaory requiremens for marke risk capial in banks and demonsraes he use of Risk- Merics wih he example diskee provided wih his documen. Finally, Appendix H shows you how o access he RiskMerics daa ses from he Inerne.

vi Preface o he fourh ediion RiskMerics examples diskee This diskee is locaed inside he back cover. I conains an Excel workbook ha includes some of he examples shown in his documen. Such examples are idenified by he icon shown here. Fuure plans We expec o updae his Technical Documen annually as we adap our marke risk sandards o furher improve he echniques and daa o mee he changing needs of our cliens. RiskMerics is a now an inegral par of.p. Morgan s Risk Managemen Services group which provides advisory services o a wide variey of he firm s cliens. We coninue o welcome any suggesions o enhance he mehodology and adap i furher o he needs of he marke. All suggesions, requess and inquiries should be direced o he auhors of his publicaion or o your local RiskMerics conacs lised on he back cover. Acknowledgmens The auhors would like o hank he numerous individuals who paricipaed in he wriing and ediing of his documen, paricularly Chris Finger and Chris Ahaide from.p. Morgan s risk managemen research group, and Elizabeh Frederick and ohn Maero from our risk advisory pracice. Finally, his documen could no have been produced wihou he conribuions of our consuling edior, Taiana Kolubayev. We apologize for any omissions o his lis. RiskMerics Technical Documen Fourh Ediion

vii Table of conens Par I Risk Measuremen Framework Chaper 1. Inroducion 3 1.1 An inroducion o Value-a-Risk and RiskMerics 6 1. A more advanced approach o Value-a-Risk using RiskMerics 7 1.3 Wha RiskMerics provides 16 Chaper. Hisorical perspecive of VaR 19.1 From ALM o VaR. VaR in he framework of modern financial managemen 4.3 Alernaive approaches o risk esimaion 6 Chaper 3. Applying he risk measures 31 3.1 Marke risk limis 33 3. Calibraing valuaion and risk models 34 3.3 Performance evaluaion 34 3.4 Regulaory reporing, capial requiremen 36 Par II Saisics of Financial Marke Reurns Chaper 4. Saisical and probabiliy foundaions 43 4.1 Definiion of financial price changes and reurns 45 4. Modeling financial prices and reurns 49 4.3 Invesigaing he random-walk model 54 4.4 Summary of our findings 64 4.5 A review of hisorical observaions of reurn disribuions 64 4.6 RiskMerics model of financial reurns: A modified random walk 73 4.7 Summary 74 Chaper 5. Esimaion and forecas 75 5.1 Forecass from implied versus hisorical informaion 77 5. RiskMerics forecasing mehodology 78 5.3 Esimaing he parameers of he RiskMerics model 90 5.4 Summary and concluding remarks 100 Par III Risk Modeling of Financial Insrumens Chaper 6. Marke risk mehodology 105 6.1 Sep 1 Idenifying exposures and cash flows 107 6. Sep Mapping cash flows ono RiskMerics verices 117 6.3 Sep 3 Compuing Value-a-Risk 11 6.4 Examples 134 Chaper 7. Mone Carlo 149 7.1 Scenario generaion 151 7. Porfolio valuaion 155 7.3 Summary 157 7.4 Commens 159

viii Table of conens Par IV RiskMerics Daa Ses Chaper 8. Daa and relaed saisical issues 163 8.1 Consrucing RiskMerics raes and prices 165 8. Filling in missing daa 170 8.3 The properies of correlaion (covariance) marices and VaR 176 8.4 Rebasing RiskMerics volailiies and correlaions 183 8.5 Nonsynchronous daa collecion 184 Chaper 9. Time series sources 197 9.1 Foreign exchange 199 9. Money marke raes 199 9.3 Governmen bond zero raes 00 9.4 Swap raes 0 9.5 Equiy indices 03 9.6 Commodiies 05 Chaper 10. RiskMerics volailiy and correlaion files 07 10.1 Availabiliy 09 10. File names 09 10.3 Daa series naming sandards 09 10.4 Forma of volailiy files 11 10.5 Forma of correlaion files 1 10.6 Daa series order 14 10.7 Underlying price/rae availabiliy 14 Par V Backesing Chaper 11. Performance assessmen 17 11.1 Sample porfolio 19 11. Assessing he RiskMerics model 0 11.3 Summary 3 Appendices Appendix A. Tess of condiional normaliy 7 Appendix B. Relaxing he assumpion of condiional normaliy 35 Appendix C. Mehods for deermining he opimal decay facor 43 Appendix D. Assessing he accuracy of he dela-gamma approach 47 Appendix E. Rouines o simulae correlaed normal random variables 53 Appendix F. BIS regulaory requiremens 57 Appendix G. Using he RiskMerics examples diskee 63 Appendix H. RiskMerics on he Inerne 67 Reference Glossary of erms 71 Bibliography 75 RiskMerics Technical Documen Fourh Ediion

ix Lis of chars Char 1.1 VaR saisics 6 Char 1. Simulaed porfolio changes 9 Char 1.3 Acual cash flows 9 Char 1.4 Mapping acual cash flows ono RiskMerics verices 10 Char 1.5 Value of pu opion on USD/DEM 14 Char 1.6 Hisogram and scaergram of rae disribuions 15 Char 1.7 Valuaion of insrumens in sample porfolio 15 Char 1.8 Represenaion of VaR 16 Char 1.9 Componens of RiskMerics 17 Char.1 Asse liabiliy managemen Char. Value-a-Risk managemen in rading 3 Char.3 Comparing ALM o VaR managemen 4 Char.4 Two seps beyond accouning 5 Char 3.1 Hierarchical VaR limi srucure 33 Char 3. Ex pos validaion of risk models: DEaR vs. acual daily P&L 34 Char 3.3 Performance evaluaion riangle 35 Char 3.4 Example: comparison of cumulaive rading revenues 35 Char 3.5 Example: applying he evaluaion riangle 36 Char 4.1 Absolue price change and log price change in U.S. 30-year governmen bond 47 Char 4. Simulaed saionary/mean-revering ime series 5 Char 4.3 Simulaed nonsaionary ime series 53 Char 4.4 Observed saionary ime series 53 Char 4.5 Observed nonsaionary ime series 54 Char 4.6 USD/DEM reurns 55 Char 4.7 USD/FRF reurns 55 Char 4.8 Sample auocorrelaion coefficiens for USD/DEM foreign exchange reurns 57 Char 4.9 Sample auocorrelaion coefficiens for USD S&P 500 reurns 58 Char 4.10 USD/DEM reurns squared 60 Char 4.11 S&P 500 reurns squared 60 Char 4.1 Sample auocorrelaion coefficiens of USD/DEM squared reurns 61 Char 4.13 Sample auocorrelaion coefficiens of S&P 500 squared reurns 61 Char 4.14 Cross produc of USD/DEM and USD/FRF reurns 63 Char 4.15 Correlogram of he cross produc of USD/DEM and USD/FRF reurns 63 Char 4.16 Lepokuroic vs. normal disribuion 65 Char 4.17 Normal disribuion wih differen means and variances 67 Char 4.18 Seleced percenile of sandard normal disribuion 69 Char 4.19 One-ailed confidence inerval 70 Char 4.0 Two-ailed confidence inerval 71 Char 4.1 Lognormal probabiliy densiy funcion 73 Char 5.1 DEM/GBP exchange rae 79 Char 5. Log price changes in GBP/DEM and VaR esimaes (1.65σ) 80 Char 5.3 NLG/DEM exchange rae and volailiy 87 Char 5.4 S&P 500 reurns and VaR esimaes (1.65σ) 88 Char 5.5 GARCH(1,1)-normal and EWMA esimaors 90 Char 5.6 USD/DEM foreign exchange 9 Char 5.7 Tolerance level and decay facor 94 Char 5.8 Relaionship beween hisorical observaions and decay facor 95 Char 5.9 Exponenial weighs for T = 100 95 Char 5.10 One-day volailiy forecass on USD/DEM reurns 96 Char 5.11 One-day correlaion forecass for reurns on USD/DEM FX rae and on S&P500 96 Char 5.1 Simulaed reurns from RiskMerics model 101 Char 6.1 French franc 10-year benchmark maps 109

x Lis of chars Char 6. Cash flow represenaion of a simple bond 109 Char 6.3 Cash flow represenaion of a FRN 110 Char 6.4 Esimaed cash flows of a FRN 111 Char 6.5 Cash flow represenaion of simple ineres rae swap 111 Char 6.6 Cash flow represenaion of forward saring swap 11 Char 6.7 Cash flows of he floaing paymens in a forward saring swap 113 Char 6.8 Cash flow represenaion of FRA 113 Char 6.9 Replicaing cash flows of 3-monh vs. 6-monh FRA 114 Char 6.10 Cash flow represenaion of 3-monh Eurodollar fuure 114 Char 6.11 Replicaing cash flows of a Eurodollar fuures conrac 114 Char 6.1 FX forward o buy Deusche marks wih US dollars in 6 monhs 115 Char 6.13 Replicaing cash flows of an FX forward 115 Char 6.14 Acual cash flows of currency swap 116 Char 6.15 RiskMerics cash flow mapping 118 Char 6.16 Linear and nonlinear payoff funcions 13 Char 6.17 VaR horizon and mauriy of money marke deposi 18 Char 6.18 Long and shor opion posiions 131 Char 6.19 DEM 3-year swaps in Q1-94 141 Char 7.1 Frequency disribuions for and 153 Char 7. Frequency disribuion for DEM bond price 154 Char 7.3 Frequency disribuion for USD/DEM exchange rae 154 Char 7.4 Value of pu opion on USD/DEM 157 Char 7.5 Disribuion of porfolio reurns 158 Char 8.1 Consan mauriy fuure: price calculaion 170 Char 8. Graphical represenaion 175 Char 8.3 Number of variables used in EM and parameers required 176 Char 8.4 Correlaion forecass vs. reurn inerval 185 Char 8.5 Time char 188 Char 8.6 10-year Ausralia/US governmen bond zero correlaion 190 Char 8.7 Adjusing 10-year USD/AUD bond zero correlaion 194 Char 8.8 10-year apan/us governmen bond zero correlaion 195 Char 9.1 Volailiy esimaes: daily horizon 0 Char 11.1 One-day Profi/Loss and VaR esimaes 19 Char 11. Hisogram of sandardized reurns 1 Char 11.3 Sandardized lower-ail reurns Char 11.4 Sandardized upper-ail reurns Char A.1 Sandard normal disribuion and hisogram of reurns on USD/DEM 7 Char A. Quanile-quanile plo of USD/DEM 3 Char A.3 Quanile-quanile plo of 3-monh serling 34 Char B.1 Tails of normal mixure densiies 38 Char B. GED disribuion 39 Char B.3 Lef ail of GED (ν) disribuion 40 Char D.1 Dela vs. ime o expiraion and underlying price 48 Char D. Gamma vs. ime o expiraion and underlying price 49 RiskMerics Technical Documen Fourh Ediion

xi Lis of ables Table.1 Two discriminaing facors o review VaR models 9 Table 3.1 Comparing he Basel Commiee proposal wih RiskMerics 39 Table 4.1 Absolue, relaive and log price changes 46 Table 4. Reurn aggregaion 49 Table 4.3 Box-Ljung es saisic 58 Table 4.4 Box-Ljung saisics 59 Table 4.5 Box-Ljung saisics on squared log price changes (cv = 5) 6 Table 4.6 Model classes 66 Table 4.7 VaR saisics based on RiskMerics and BIS/Basel requiremens 71 Table 5.1 Volailiy esimaors 78 Table 5. Calculaing equally and exponenially weighed volailiy 81 Table 5.3 Applying he recursive exponenial weighing scheme o compue volailiy 8 Table 5.4 Covariance esimaors 83 Table 5.5 Recursive covariance and correlaion predicor 84 Table 5.6 Mean, sandard deviaion and correlaion calculaions 91 Table 5.7 The number of hisorical observaions used by he EWMA model 94 Table 5.8 Opimal decay facors based on volailiy forecass 99 Table 5.9 Summary of RiskMerics volailiy and correlaion forecass 100 Table 6.1 Daa provided in he daily RiskMerics daa se 11 Table 6. Daa calculaed from he daily RiskMerics daa se 11 Table 6.3 Relaionship beween insrumen and underlying price/rae 13 Table 6.4 Saisical feaures of an opion and is underlying reurn 130 Table 6.5 RiskMerics daa for 7, March 1995 134 Table 6.6 RiskMerics map of single cash flow 134 Table 6.7 RiskMerics map for muliple cash flows 135 Table 6.8 Mapping a 6x1 shor FRF FRA a incepion 137 Table 6.9 Mapping a 6x1 shor FRF FRA held for one monh 137 Table 6.10 Srucured noe specificaion 139 Table 6.11 Acual cash flows of a srucured noe 139 Table 6.1 VaR calculaion of srucured noe 140 Table 6.13 VaR calculaion on srucured noe 140 Table 6.14 Cash flow mapping and VaR of ineres-rae swap 14 Table 6.15 VaR on foreign exchange forward 143 Table 6.16 Marke daa and RiskMerics esimaes as of rade dae uly 1, 1994 145 Table 6.17 Cash flow mapping and VaR of commodiy fuures conrac 145 Table 6.18 Porfolio specificaion 147 Table 6.19 Porfolio saisics 148 Table 6.0 Value-a-Risk esimaes (USD) 148 Table 7.1 Mone Carlo scenarios 155 Table 7. Mone Carlo scenarios valuaion of opion 156 Table 7.3 Value-a-Risk for example porfolio 158 Table 8.1 Consrucion of rolling nearby fuures prices for Ligh Swee Crude (WTI) 168 Table 8. Price calculaion for 1-monh CMF NY Harbor # Heaing Oil 169 Table 8.3 Belgian franc 10-year zero coupon rae 175 Table 8.4 Singular values for USD yield curve daa marix 18 Table 8.5 Singular values for equiy indices reurns 18 Table 8.6 Correlaions of daily percenage changes wih USD 10-year 184 Table 8.7 Schedule of daa collecion 186 Table 8.7 Schedule of daa collecion 187 Table 8.8 RiskMerics closing prices 191 Table 8.9 Sample saisics on RiskMerics daily covariance forecass 191 Table 8.10 RiskMerics daily covariance forecass 19

xii Lis of ables Table 8.11 Relaionship beween lagged reurns and applied weighs 193 Table 8.1 Original and adjused correlaion forecass 193 Table 8.13 Correlaions beween US and foreign insrumens 196 Table 9.1 Foreign exchange 199 Table 9. Money marke raes: sources and erm srucures 00 Table 9.3 Governmen bond zero raes: sources and erm srucures 01 Table 9.4 Swap zero raes: sources and erm srucures 03 Table 9.5 Equiy indices: sources 04 Table 9.6 Commodiies: sources and erm srucures 05 Table 9.7 Energy mauriies 05 Table 9.8 Base meal mauriies 06 Table 10.1 RiskMerics file names 09 Table 10. Currency and commodiy idenifiers 10 Table 10.3 Mauriy and asse class idenifiers 10 Table 10.4 Sample volailiy file 11 Table 10.5 Daa columns and forma in volailiy files 1 Table 10.6 Sample correlaion file 13 Table 10.7 Daa columns and forma in correlaion files 13 Table 11.1 Realized percenages of VaR violaions 0 Table 11. Realized ail reurn averages 1 Table A.1 Sample mean and sandard deviaion esimaes for USD/DEM FX 8 Table A. Tesing for univariae condiional normaliy 30 Table B.1 Parameer esimaes for he Souh African rand 40 Table B. Sample saisics on sandardized reurns 41 Table B.3 VaR saisics (in %) for he 1s and 99h perceniles 4 Table D.1 Parameers used in opion valuaion 49 Table D. MAPE (%) for call, 1-day forecas horizon 51 Table D.3 ME (%) for call, 1-day forecas horizons 51 RiskMerics Technical Documen Fourh Ediion

1 Par I Risk Measuremen Framework

RiskMerics Technical Documen Fourh Ediion

3 Chaper 1. Inroducion 1.1 An inroducion o Value-a-Risk and RiskMerics 6 1. A more advanced approach o Value-a-Risk using RiskMerics 7 1..1 Using RiskMerics o compue VaR on a porfolio of cash flows 9 1.. Measuring he risk of nonlinear posiions 11 1.3 Wha RiskMerics provides 16 1.3.1 An overview 16 1.3. Deailed specificaion 18 Par I: Risk Measuremen Framework

4 RiskMerics Technical Documen Fourh Ediion

5 Chaper 1. Inroducion acques Longersaey Morgan Guarany Trus Company Risk Managemen Advisory (1-1) 648-4936 riskmerics@jpmorgan.com This chaper serves as an inroducion o he RiskMerics produc. RiskMerics is a se of mehodologies and daa for measuring marke risk. By marke risk, we mean he poenial for changes in value of a posiion resuling from changes in marke prices. We define risk as he degree of uncerainy of fuure ne reurns. This uncerainy akes many forms, which is why mos paricipans in he financial markes are subjec o a variey of risks. A common classificaion of risks is based on he source of he underlying uncerainy: Credi risk esimaes he poenial loss because of he inabiliy of a counerpary o mee is obligaions. Operaional risk resuls from errors ha can be made in insrucing paymens or seling ransacions. Liquidiy risk is refleced in he inabiliy of a firm o fund is illiquid asses. Marke risk, he subjec of he mehodology described in his documen, involves he uncerainy of fuure earnings resuling from changes in marke condiions, (e.g., prices of asses, ineres raes). Over he las few years measures of marke risk have become synonymous wih he erm Value-a-Risk. RiskMerics has hree basic componens: The firs is a se of mehodologies oulining how risk managers can compue Value-a-Risk on a porfolio of financial insrumens. These mehodologies are explained in his Technical Documen, which is an annual publicaion, and in he RiskMerics Monior, he quarerly updae o he Technical Documen. The second is daa ha we disribue o enable marke paricipans o carry ou he mehodologies se forh in his documen. The hird is Value-a-Risk calculaion and reporing sofware designed by.p. Morgan, Reuers, and hird pary developers. These sysems apply he mehodologies se forh in his documen and will no be discussed in his publicaion. This chaper is organized as follows: Secion 1.1 presens he definiion of Value-a-Risk (VaR) and some simple examples of how RiskMerics offers he inpus necessary o compue VaR. The purpose of his secion is o offer a basic approach o VaR calculaions. Secion 1. describes more deailed examples of VaR calculaions for a more horough undersanding of how RiskMerics and VaR calculaions fi ogeher. In Secion 1.. we provide an example of how o compue VaR on a porfolio conaining opions (nonlinear risk) using wo differen mehodologies. Secion 1.3 presens he conens of RiskMerics a boh he general and deailed level. This secion provides a sep-by-sep analysis of he producion of RiskMerics volailiy and correlaion files as well as he mehods ha are necessary o compue VaR. For easy reference we provide secion numbers wihin each sep so ha ineresed readers can learn more abou ha paricular subjec. Par I: Risk Measuremen Framework

6 Chaper 1. Inroducion Reading his chaper requires a basic undersanding of saisics. For assisance, readers can refer o he glossary a he end of his documen. 1.1 An inroducion o Value-a-Risk and RiskMerics Value-a-Risk is a measure of he maximum poenial change in value of a porfolio of financial insrumens wih a given probabiliy over a pre-se horizon. VaR answers he quesion: how much can I lose wih x% probabiliy over a given ime horizon. For example, if you hink ha here is a 95% chance ha he DEM/USD exchange rae will no fall by more han 1% of is curren value over he nex day, you can calculae he maximum poenial loss on, say, a USD 100 million DEM/USD posiion by using he mehodology and daa provided by RiskMerics. The following examples describe how o compue VaR using sandard deviaions and correlaions of financial reurns (provided by RiskMerics) under he assumpion ha hese reurns are normally disribued. (RiskMerics provides alernaive mehodological choices o address he inacurracies resuling from his simplifying assumpion). Example 1: You are a USD-based corporaion and hold a DEM 140 million FX posiion. Wha is your VaR over a 1-day horizon given ha here is a 5% chance ha he realized loss will be greaer han wha VaR projeced? The choice of he 5% probabiliy is discreionary and differs across insiuions using he VaR framework. Char 1.1 VaR saisics No. of observaions 5% Wha is your exposure? Wha is your risk? r /σ The firs sep in he calculaion is o compue your exposure o marke risk (i.e., mark-o-marke your posiion). As a USDbased invesor, your exposure is equal o he marke value of he posiion in your base currency. If he foreign exchange rae is 1.40 DEM/USD, he marke value of he posiion is USD 100 million. Moving from exposure o risk requires an esimae of how much he exchange rae can poenially move. The sandard deviaion of he reurn on he DEM/USD exchange rae, measured hisorically can provide an indicaion of he size of rae movemens. In his example, we calculaed he DEM/USD daily sandard deviaion o be 0.565%. Now, under he sandard RiskMerics assumpion ha sandardized reurns ( ( r σ ) on DEM/USD are normally disribued given he value of his sandard deviaion, VaR is given by 1.65 imes he sandard deviaion (ha is, 1.65σ) or 0.93% (see Char 1.1). This means ha he DEM/USD exchange rae is no expeced o drop more han 0.93%, 95% of he ime. RiskMerics provides users wih he VaR saisics 1.65σ. In USD, he VaR of he posiion 1 is equal o he marke value of he posiion imes he esimaed volailiy or: FX Risk: $100 million 0.93% = $93,000 Wha his number means is ha 95% of he ime, you will no lose more han $93,000 over he nex 4 hours. 1 This is a simple approximaion. RiskMerics Technical Documen Fourh Ediion

Sec. 1. A more advanced approach o Value-a-Risk using RiskMerics 7 Example : Le s complicae maers somewha. You are a USD-based corporaion and hold a DEM 140 million posiion in he 10-year German governmen bond. Wha is your VaR over a 1-day horizon period, again, given ha here is a 5% chance of undersaing he realized loss? Wha is your exposure? Wha is your risk? The only difference versus he previous example is ha you now have boh ineres rae risk on he bond and FX risk resuling from he DEM exposure. The exposure is sill USD 100 million bu i is now a risk o wo marke risk facors. If you use an esimae of 10-year German bond sandard deviaion of 0.605%, you can calculae: Ineres rae risk: $100 million 1.65 0.605% = $999,000 FX Risk: $100 million 1.65 0.565% = $93,000 Now, he oal risk of he bond is no simply he sum of he ineres rae and FX risk because he correlaion beween he reurn on he DEM/USD exchange rae he reurn on he 10- year German bond is relevan. In his case, we esimaed he correlaion beween he reurns on he DEM/USD exchange rae and he 10-year German governmen bond o be 0.7. Using a formula common in sandard porfolio heory, he oal risk of he posiion is given by: [1.1] VaR = σ Ineres rae + σ FX + ( ρ Ineres rae, FX σ Ineres rae σ FX ) VaR = ( 0.999) + ( 0.93) + ( 0.7 0.999 0.93) = $ 1.168 million To compue VaR in his example, RiskMerics provides users wih he VaR of ineres rae componen (i.e., 1.65 0.605), he VaR of he foreign exchange posiion (i.e., 1.65 0.565) and he correlaion beween he wo reurn series, 0.7. 1. A more advanced approach o Value-a-Risk using RiskMerics Value-a-Risk is a number ha represens he poenial change in a porfolio s fuure value. How his change is defined depends on (1) he horizon over which he porfolio s change in value is measured and () he degree of confidence chosen by he risk manager. VaR calculaions can be performed wihou using sandard deviaion or correlaion forecass. These are simply one se of inpus ha can be used o calculae VaR, and ha RiskMerics provides for ha purpose. The principal reason for preferring o work wih sandard deviaions (volailiy) is he srong evidence ha he volailiy of financial reurns is predicable. Therefore, if volailiy is predicable, i makes sense o make forecass of i o predic fuure values of he reurn disribuion. Correlaion is a measure of how wo series move ogeher. For example, a correlaion of 1 implies ha wo series move perfecly ogeher in he same direcion. Par I: Risk Measuremen Framework

8 Chaper 1. Inroducion Suppose we wan o compue he Value-a-Risk of a porfolio over a 1-day horizon wih a 5% chance ha he acual loss in he porfolio s value is greaer han he VaR esimae. The Value-a- Risk calculaion consiss of he following seps. 1. Mark-o-marke he curren porfolio. Denoe his value by V 0.. Define he fuure value of he porfolio, V, as where 3 1 V 1 = V 0 e r r represens he reurn on he porfolio over he horizon. For a 1-day horizon, his sep is unnecessary as RiskMerics assumes a 0 reurn. 3. Make a forecas of he 1-day reurn on he porfolio and denoe his value by rˆ, such ha here is a 5% chance ha he acual reurn will be less han rˆ. Alernaively expressed, Probabiliy ( r < rˆ) = 5%. 4. Define he porfolio s fuure wors case value Vˆ 1, as Vˆ 1 = V 0 e rˆ. The Value-a-Risk esimae is simply V 0 Vˆ 1. Noice ha he VaR esimae can be wrien as V 0 1 e rˆ. In he case ha rˆ is sufficienly small, e rˆ 1 + rˆ so ha VaR is approximaely equal ov 0 rˆ. is approximaely equal o V 0 rˆ. The purpose of a risk measuremen sysem such as RiskMerics is o offer a means o compue rˆ. Wihin his more general framework we use a simple example o demonsrae how he RiskMerics mehodologies and daa enable users o compue VaR. Assume he forecas horizon over which VaR is measured is one day and he level of confidence in he forecas o 5%. Following he seps oulined above, he calculaion would proceed as follows: V 0 µ10 1. Consider a porfolio whose curren marked-o-marke value,, is USD 500 million.. To carry ou he VaR calculaion we require 1-day forecass of he mean. Wihin he RiskMerics framework, we assume ha he mean reurn over a 1-day horizon period is equal o 0. 3. We also need he sandard deviaion, σ 10, of he reurns in his porfolio. Assuming ha he reurn on his porfolio is disribued condiionally normal, rˆ= 1.65σ 10 + µ 10. The RiskMerics daa se provides he erm 1.65 σ 10. Hence, seing µ 10 = 0and σ, we ge. 4 10 = 0.031 V 1 = USD 474. million 4. This yields a Value-a-Risk of USD 5.8 million (given by V 0 Vˆ 1 ). The hisogram in Char 1. presens fuure changes in value of he porfolio. VaR reduces risk o jus one number, i.e., a loss associaed wih a given probabiliy. I is ofen useful for risk managers o focus on he oal disribuion of poenial gains and losses and we will discuss why his is so laer in his documen. (See Secion 6.3). 3 Where e is approximaely.7183 4 This number is compued from e 1.65σ V0 RiskMerics Technical Documen Fourh Ediion

Sec. 1. A more advanced approach o Value-a-Risk using RiskMerics 9 Char 1. Simulaed porfolio changes Probabiliy 0.10 0.09 0.08 95% confidence: 0.07 $5.8 million 0.06 0.05 0.04 0.03 0.0 0.01 0.00-48 -40-3 -4-16 -8 0 8 16 4 3 40 48 P/L ($million) 1..1 Using RiskMerics o compue VaR on a porfolio of cash flows Calculaing VaR usually involves more seps han he basic ones oulined in he examples above. Even before calculaing VaR, you need o esimae o which risk facors a paricular porfolio is exposed. The preferred mehodology for doing his is o decompose financial insrumens ino heir basic cash flow componens. The RiskMerics mehodology and daa allow users o compue he VaR on porfolios consising of a variey of cash flows. We use a simple example (a porfolio consising of hree cash flows) o demonsrae how o compue VaR. Sep 1. Each financial posiion in a porfolio is expressed as one or more cash flows ha are marked-o-marke a curren marke raes. For example, consider an insrumen ha gives rise o hree USD 100 cash flows each occurring in 1, 4, and 7 monhs ime as shown in Char 1.3. Char 1.3 Acual cash flows 100 100 100 1m 4m 7m Principal flows Sep. When necessary, he acual cash flows are convered o RiskMerics cash flows by mapping (redisribuing) hem ono a sandard grid of mauriy verices, known as RiskMerics verices, which are fixed a he following inervals: 1m 3m 6m 1m yr 3yr 4yr 5yr 7yr 9yr 10yr 15yr 0yr 30yr The purpose of he mapping is o sandardize he cash flow inervals of he insrumen such ha we can use he volailiies and correlaions ha are rouinely compued for he given verices in he RiskMerics daa ses. (I would be impossible o provide volailiy and correlaion esimaes on every possible mauriy so RiskMerics provides a mapping mehod- Par I: Risk Measuremen Framework

10 Chaper 1. Inroducion ology which disribues cash flows o a workable se of sandard mauriies). The mehodology for mapping cash flows is deailed in Chaper 6. To map he cash flows, we use he RiskMerics verices closes o he acual verices and redisribue he acual cash flows as shown in Char 1.4. Char 1.4 Mapping acual cash flows ono RiskMerics verices 100 100 100 1m 4m 7m Acual cashflows 100 60 40 70 30 1m 3m 6m 1m Cashflow mapping 100 60 110 30 1m 3m 6m 1m RiskMerics cashflows The RiskMerics cash flow map is used o work backwards o calculae he reurn for each of he acual cash flows from he cash flow a he associaed RiskMerics verex, or verices. For each acual cash flow, an analyical expression is used o express he relaive change in value of he acual cash flow in erms of an underlying reurn on a paricular insrumen. Coninuing wih Char 1.4, we can wrie he reurn on he acual 4-monh cash flow in erms of he combined reurns on he 3-monh (60%) and 6-monh (40%) RiskMerics cash flows: [1.] r 4m = 0.60r 3m + 0.40r 6m where r 4m r 3m r 6m = = = reurn on he acual 4-monh cash flow reurn on he 3-monh RiskMerics cash flow reurn on he 6-monh RiskMerics cash flow Similarly, he reurn on he 7-monh cash flow can be wrien as [1.3] r 7m = 0.70r 6m + 0.30r 1m Noe ha he reurn on he acual 1-monh cash flow is equal o he reurn on he 1-monh insrumen. Sep 3. VaR is calculaed a he 5h percenile of he disribuion of porfolio reurn, and for a specified ime horizon. In he example above, he disribuion of he porfolio reurn, r p, is wrien as: [1.4] r p = 0.33r 1m + 0.0r 3m + 0.37r 6m + 0.10r 1m RiskMerics Technical Documen Fourh Ediion

Sec. 1. A more advanced approach o Value-a-Risk using RiskMerics 11 where, for example he porfolio weigh 0.33 is he resul of 100 divided by he oal porfolio value 300. Now, o compue VaR a he 95h percen confidence level we need he fifh percenile of he porfolio reurn disribuion. Under he assumpion ha r p is disribued condiionally normal, he fifh percenile is 1.65 σ p where σ p is he sandard deviaion of he porfolio reurn disribuion. Applying Eq. [1.1] o a porfolio conaining more han wo insrumens requires using simple marix algebra. We can hus express his VaR calculaion as follows: [1.5] VaR = VRV T where V is a vecor of VaR esimaes per insrumen, V = [ ( 0.33 1.65σ1m), ( 0.0 1.65σ 3m ), ( 0.37 1.65σ 6m ), ( 0.10 1.65σ 1m )], and R is he correlaion marix 1 ρ 3m, 1m ρ 6m, 1m ρ 1m, 1m [1.6] R = ρ 1m, 3m 1 ρ 6m, 3m ρ 1m, 3m ρ 1m, 6m ρ 3m, 6m 1 ρ 1m, 6m ρ 1m, 1m ρ 3m, 1m ρ 6m, 1m 1 where, for example, reurns. ρ 1m, 3m is he correlaion esimae beween 1-monh and 3-monh Noe ha RiskMerics provides he vecor of informaion V = [ ( 1.65σ 1m ), ( 1.65σ 3m ), ( 1.65σ 6m ), ( 1.65σ 1m )] as well as he correlaion marix R. Wha he user has o provide are he acual porfolio weighs. 1.. Measuring he risk of nonlinear posiions When he relaionship beween posiion value and marke raes is nonlinear, hen we canno esimae changes in value by muliplying esimaed changes in raes by sensiiviy of he posiion o changing raes; he laer is no consan (i.e., he definiion of a nonlinear posiion). In our previous examples, we could easily esimae he risk of a fixed income or foreign exchange produc by assuming a linear relaionship beween he value of an insrumen and he value of is underlying. This is no a reasonable assumpion when dealing wih nonlinear producs such as opions. RiskMerics offers wo mehodologies, an analyical approximaion and a srucured Mone Carlo simulaion o compue he VaR of nonlinear posiions: 1. The firs mehod approximaes he nonlinear relaionship via a mahemaical expression ha relaes he reurn on he posiion o he reurn on he underlying raes. This is done by using wha is known as a Taylor series expansion. This approach no longer necessarily assumes ha he change in value of he insrumen is approximaed by is dela alone (he firs derivaive of he opion s value wih respec o he underlying variable) bu ha a second order erm using he opion s gamma (he second derivaive of he opion s value wih respec o he underlying price) mus be inroduced o Par I: Risk Measuremen Framework

1 Chaper 1. Inroducion measure he curvaure of changes in value around he curren value. In pracice, oher greeks such as vega (volailiy), rho (ineres rae) and hea (ime o mauriy) can also be used o improve he accuracy of he approximaion. In Secion 1...1, we presen wo ypes of analyical mehods for compuing VaR he dela and dela-gamma approximaion.. The second alernaive, srucured Mone Carlo simulaion, involves creaing a large number of possible rae scenarios and revaluing he insrumen under each of hese scenarios. VaR is hen defined as he 5h percenile of he disribuion of value changes. Due o he required revaluaions, his approach is compuaionally more inensive han he firs approach. The wo mehods differ no in erms of how marke movemens are forecas (since boh use he RiskMerics volailiy and correlaion esimaes) bu in how he value of porfolios changes as a resul of marke movemens. The analyical approach approximaes changes in value, while he srucured Mone Carlo fully revalues porfolios under various scenarios. Le us illusrae hese wo mehods using a pracical example. We will consider hroughou his secion a porfolio comprised of wo asses: Asse 1: a fuure cash flow sream of DEM 1 million o be received in one year s ime. The curren 1-year DEM rae is 10% so he curren marke value of he insrumen is DEM 909,091. Asse : an a-he-money (ATM) DEM pu/usd call opion wih conrac size of DEM 1 million and expiraion dae one monh in he fuure. The premium of he opion is 0.0105 and he spo exchange rae a which he conrac was concluded is 1.538 DEM/USD. We assume he implied volailiy a which he opion is priced is 14%. The value of his porfolio depends on he USD/DEM exchange rae and he one-year DEM bond price. Technically, he value of he opion also changes wih USD ineres raes and he implied volailiy, bu we will no consider hese effecs. Our risk horizon for he example will be five days. We ake as he daily volailiies of hese wo asses σ FX = 0.4% and σ B = 0.08% and as he correlaion beween he wo ρ = 0.17. Boh alernaives will focus on price risk exclusively and herefore ignore he risk associaed wih volailiy (vega), ineres rae (rho) and ime decay (hea risk). 1...1 Analyical mehod There are various ways o analyically approximae nonlinear VaR. This secion reviews he wo alernaives which we discussed previously. Dela approximaion The sandard VaR approach can be used o come up wih firs order approximaions of porfolios ha conain opions. (This is essenially he same simplificaion ha fixed income raders use when hey focus exclusively on he duraion of heir porfolio). The simples such approximaion is o esimae changes in he opion value via a linear model, which is commonly known as he dela approximaion. Dela is he firs derivaive of he opion price wih respec o he spo exchange rae. The value of δ for he opion in his example is 0.4919. In he analyical mehod, we mus firs wrie down he reurn on he porfolio whose VaR we are rying o calculae. The reurn on his porfolio consising of a cash flow in one year and a pu on he DEM/call on he USD is wrien as follows: [1.7] r p =r 1y + + r DEM ------------- USD δ r DEM ------------- USD RiskMerics Technical Documen Fourh Ediion

Sec. 1. A more advanced approach o Value-a-Risk using RiskMerics 13 where r 1 p r DEM ------------- USD δ = = = he price reurn on he 1-year German ineres raes he reurn on he DEM/USD exchange rae he dela of he opion Under he assumpion ha he porfolio reurn is normally disribued, VaR a he 95% confidence level is given by [1.8] VaR = 1.65 σ 1y + ( 1 + δ) σ DEM + ------------- USD 1+ δ ( )ρ1y, ------------- DEM USD σ 1 y σ DEM ------------- USD Using our volailiies and correlaions forecass for DEM/USD and he 1-year DEM rae (scaled up o he weekly horizon using he square roo of ime rule), he weekly VaR for he porfolio using he dela equivalen approach can be approximaed by: Marke value in USD VaR(1w) 1-yr DEM cash flow $591,086 $1,745 FX posiion - FX hedge $300,331 $4,654 Diversified VaR $4,684 Dela-gamma approximaion The dela approximaion is reasonably accurae when he exchange rae does no change significanly, bu less so in he more exreme cases. This is because he dela is a linear approximaion of a non linear relaionship beween he value of he exchange rae and he price of he opion as shown in Char 1.5. We may be able o improve his approximaion by including he gamma erm, which accouns for nonlinear (i.e. squared reurns) effecs of changes in he spo rae (his aemps o replicae he convex opion price o FX rae relaionship as shown in Char 1.5). The expression for he porfolio reurn is now [1.9] r p =r 1y + r DEM + δ r DEM + 0.5 ΓP DEM r DEM ------------- ------------- ------------- ------------- USD USD USD USD where P DEM ------------- USD Γ = = he value of he DEM/USD exchange rae when he VaR forecas is made he gamma of he opion. In his example, Γ = DEM/USD 15.14. Now, he gamma erm (he fourh erm in Eq. [1.9]) inroduces skewness ino he disribuion of r P (i.e., he disribuion is no longer symmerical around is mean). Therefore, since his violaes one of he assumpions of normaliy (symmery) we can no longer calculae he 95h percenile VaR as 1.65 imes he sandard deviaion of r p. Insead we mus find he appropriae muliple (he counerpar o 1.65) ha incorporaes he skewness effec. We compue he 5h percenile of r p s disribuion (Eq. [1.9]) by compuing is firs four momens, i.e., r p s mean, variance, skewness and kurosis. We hen find disribuion whose firs four momens mach hose of r p s. (See Secion 6.3 for deails.) Par I: Risk Measuremen Framework

14 Chaper 1. Inroducion Applying his mehodology o his approach we find he VaR for his porfolio o be USD 3,708. Noe ha in his example, incorporaing gamma reduces VaR relaive o he dela only approximaion (from USD 5006 o USD 3708). Char 1.5 Value of pu opion on USD/DEM srike = 0.65 USD/DEM. Value in USD/DEM. Opion value 0.06 0.05 0.04 0.03 0.0 0.01 0 Full valuaion Dela + gamma -0.01 Dela -0.0 0.60 0.61 0.6 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 USD/DEM exchange rae 1... Srucured Mone-Carlo Simulaion Given he limiaions of analyical VaR for porfolios whose P/L disribuions may no be symmerical le alone normally disribued, anoher possible roue is o use a model which insead of esimaing changes in value by he produc of a rae change (σ) and a sensiiviy (δ, Γ), focuses on revaluing posiions a changed rae levels. This approach is based on a full valuaion precep where all insrumens are marked o marke under a large number of scenarios driven by he volailiy and correlaion esimaes. The Mone Carlo mehodology consiss of hree major seps: 1. Scenario generaion Using he volailiy and correlaion esimaes for he underlying asses in our porfolio, we produce a large number of fuure price scenarios in accordance wih he lognormal models described previously. The mehodology for generaing scenarios from volailiy and correlaion esimaes is described in Appendix E.. Porfolio valuaion For each scenario, we compue a porfolio value. 3. Summary We repor he resuls of he simulaion, eiher as a porfolio disribuion or as a paricular risk measure. Using our volailiy and correlaion esimaes, we can apply our simulaion echnique o our example porfolio. We can generae a large number of scenarios (1000 in his example case) of DEM 1-year and DEM/USD exchange raes a he 1-week horizon. Char 1.6 shows he acual disribuions for boh insrumens as well as he scaergram indicaing he degree of correlaion ( 0.17) beween he wo rae series. RiskMerics Technical Documen Fourh Ediion

Sec. 1. A more advanced approach o Value-a-Risk using RiskMerics 15 Par I: Risk Measuremen Framework Char 1.6 Hisogram and scaergram of rae disribuions -yr DEM rae and DEM/USD rae Wih he se of ineres and foreign exchange raes obained under simulaion, we can revalue boh of he insrumens in our porfolio. Their respecive payous are shown in Char 1.7. Char 1.7 Valuaion of insrumens in sample porfolio Value of he cash flow sream Value of he FX opion The final ask is o analyze he disribuion of values and selec he VaR using he appropriae percenile. Char 1.8 shows he value of he componens of he porfolio a he end of he horizon period. 9.3% 9.5% 9.7% 10.0%10.%10.4%10.6% 0 0 40 60 80 100 10 1.49 1.50 1.5 1.53 1.55 1.56 1.58 0 0 40 60 80 100 10 Frequency Frequency Yields P/L 9.30 9.55 9.80 10.05 10.30 10.55 585.0 587.5 590.0 59.5 595.0 Opion value - USD housands Yield 0 5 10 15 0 5 1.48 1.5 1.5 1.54 1.56 1.58 1.6 DEM/USD Cash flow value - USD housands

16 Chaper 1. Inroducion Char 1.8 Represenaion of VaR Hisogram of porfolio values Frequency 100 90 80 70 Curren value = USD 601,388 95% percenile = USD 596,89 VaR = USD 4,599 60 50 40 30 0 10 0 595 598 601 604 607 610 614 Porfolio value Percen 100 90 80 70 60 50 40 30 0 10 0 595 598 601 604 607 610 614 Porfolio value The chars above provide a visual indicaion as o why he dela approximaion is usually no suiable for porfolios ha conain opions. The disribuion of reurns in porfolios ha include opions is ypically skewed. The sandard dela equivalen VaR approach expecs symmery around he mean and applies a basic normal disribuion approach (i.e., he 95% percenile equaes o a 1.65 sandard deviaion move). In his case, he lack of symmery in he disribuion does no allow us o apply he normal approximaion. Furhermore, he disribuion s skewness resuls in a VaR number ha is basically posiion dependen (i.e., he risk is differen wheher you are long or shor he opion). 1.3 Wha RiskMerics provides As discussed previously, RiskMerics has hree basic componens which are deailed below. 1.3.1 An overview Wih RiskMerics.P. Morgan and Reuers provide 1. A se of mehodologies for saisical marke risk measures ha are based on, bu differ significanly from, he mehodology developed and used wihin.p. Morgan. This approach was developed so as o enable oher financial insiuions, corporae reasuries, and invesors o esimae heir marke risks in a consisen and reasonable fashion. Mehodology defines how posiions are o be mapped and how poenial marke movemens are esimaed and is deailed in he following chapers.. Daily recompued daa ses which are comprehensive ses of consisenly esimaed insrumen level VaRs (i.e., 1.65 sandard deviaions) and correlaions across a large number of asse classes and insrumens. We currenly disribue hree differen daa ses over he Inerne: one for shor erm rading risks, he second for inermediae erm invesmen risks and he hird for regulaory reporing. These are made available o he marke free of charge. In he near fuure, a more cusomizable version of RiskMerics where users will be able o creae covariance marices from a large underlying daabase according o various numerical mehods will be made available over he Reuers Web. This produc will no replace he RiskMerics Technical Documen Fourh Ediion

Sec. 1.3 Wha RiskMerics provides 17 Char 1.9 Componens of RiskMerics daa ses available over he Inerne bu will provide subscribers o he Reuers services wih a more flexible ool. The four basic classes of insrumens ha RiskMerics mehodology and daa ses cover are represened as follows: Fixed income insrumens are represened by combinaions of amouns of cash flows in a given currency a specified daes in he fuure. RiskMerics applies a fixed number of daes (14 verices) and wo ypes of credi sandings: governmen and non-governmen. The daa ses associaed wih fixed income are zero coupon insrumen VaR saisics, i.e., 1.65σ, and correlaions for boh governmen and swap yield curves. Foreign exchange ransacions are represened by an amoun and wo currencies. RiskMerics allows for 30 differen currency pairs (as measured agains he USD). Equiy insrumens are represened by an amoun and currency of an equiy baske index in any of 30 differen counries. Currenly, RiskMerics does no consider he individual characerisics of a company sock bu only he weighed baske of companies as represened by he local index. Commodiies posiions are represened by amouns of seleced sandardized commodiy fuures conracs raded on commodiy exchanges 3. Sofware provided by.p. Morgan, Reuers and hird pary firms ha use he RiskMerics mehodology and daa documened herein. RiskMerics Volailiy & correlaion esimaes Posing Transacion Bloer (Invenory) Risk Projecion Mapping Posiion Evaluaion Valuaion Esimaed Risks Profis & Losses Risk /Reurn Measures RiskMerics mehodology Sysem implemenaions Since he RiskMerics mehodology and he daa ses are in he public domain and freely available, anyone is free o implemen sysems uilizing hese componens of RiskMerics. Third paries have developed risk managemen sysems for a wide range of cliens using differen mehodologies. The following paragraphs provide a axonomy comparing he differen approaches. Par I: Risk Measuremen Framework

18 Chaper 1. Inroducion 1.3. Deailed specificaion The secion below provides a brief overview of how he RiskMerics daases are produced and how he parameers we provide can be used in a VaR calculaion. 1.3..1 Producion of volailiy and correlaion daa ses RiskMerics provides he following ses of volailiy and corresponding correlaion daa files. One se is for use in esimaing VaR wih a forecas horizon of one day. The oher se is opimized for a VaR forecas horizon of one monh. The hird se is based on he quaniaive crieria se by he Bank for Inernaional Selemens on he use of VaR models o esimae he capial required o cover marke risks. The process by which hese daa files are consruced are as follows: 1. Financial prices are recorded from global daa sources. (In 1997, RiskMerics will swich o using Reuers daa exclusively). For cerain fixed income insrumens we consruc zero raes. See Chaper 9 for daa sources and RiskMerics building blocks.. Fill in missing prices by using he Expecaion Maximizaion algorihm (deailed in Secion 8.). Prices can be missing for a variey of reasons, from echnical failures o holiday schedules. 3. Compue daily price reurns on all 480 ime series (Secion 4.1). 4. Compue sandard deviaions and correlaions of financial price reurns for a 1-day VaR forecas horizon. This is done by consrucing exponenially weighed forecass. (See Secion 5.). Producion of he daily saisics also involves seing he sample daily mean o zero. (See Secion 5.3). If daa is recorded a differen imes (Sep 1), users may require an adjusmen algorihm applied o he correlaion esimaes. Such an algorihm is explained in Secion 8.5. Also, users who need o rebase he daases o accoun for a base currency oher han he USD should see Secion 8.4. 5. Compue sandard deviaions and correlaions of financial price reurns for 1-monh VaR forecas horizon. This is done by consrucing exponenially weighed forecass (Secion 5.3). Producion of he monhly saisics also involves seing he sample daily mean o zero. 1.3.. RiskMerics VaR calculaion 1. The firs sep in he VaR calculaion is for he user o define hree parameers: (1) VaR forecas horizon he ime over which VaR is calculaed, () confidence level he probabiliy ha he realized change in porfolio will be less han he VaR predicion, and (3) he base currency.. For a given porfolio, once he cash flows have been idenified and marked-o-marke (Secion 6.1) hey need o be mapped o he RiskMerics verices (Secion 6.). 3. Having mapped all he posiions, a decision mus be made as o how o compue VaR. If he user is willing o assume ha he porfolio reurn is approximaely condiionally normal, hen download he appropriae daa files (insrumen level VaRs and correlaions) and compue VaR using he sandard RiskMerics approach (Secion 6.3). 4. If he user s porfolio is subjec o nonlinear risk o he exen ha he assumpion of condiional normaliy is no longer valid, hen he user can choose beween wo mehodologies dela-gamma and srucured Mone Carlo. The former is an approximaion of he laer. See Secion 6.3 for a descripion of dela-gamma and Chaper 7for an explanaion of srucured Mone Carlo. RiskMerics Technical Documen Fourh Ediion

19 Chaper. Hisorical perspecive of VaR.1 From ALM o VaR. VaR in he framework of modern financial managemen 4..1 Valuaion 5.. Risk esimaion 5.3 Alernaive approaches o risk esimaion 6.3.1 Esimaing changes in value 6.3. Esimaing marke movemens 7 Par I: Risk Measuremen Framework

0 RiskMerics Technical Documen Fourh Ediion

1 Chaper. Hisorical perspecive of VaR acques Longersaey Morgan Guarany Trus Company Risk Managemen Advisory (1-1) 648-4936 riskmerics@jpmorgan.com Measuring he risks associaed wih being a paricipan in he financial markes has become he focus of inense sudy by banks, corporaions, invesmen managers and regulaors. Cerain risks such as counerpary defaul have always figured a he op of mos banks concerns. Ohers such as marke risk (he poenial loss associaed wih marke behavior) have only goen ino he limeligh over he pas few years. Why has he ineres in marke risk measuremen and monioring arisen? The answer lies in he significan changes ha he financial markes have undergone over he las wo decades. 1. Securiizaion: Across markes, raded securiies have replaced many illiquid insrumens, e.g., loans and morgages have been securiized o permi disinermediaion and rading. Global securiies markes have expanded and boh exchange raded and over-he-couner derivaives have become major componens of he markes. These developmens, along wih echnological breakhroughs in daa processing, have gone hand in hand wih changes in managemen pracices a movemen away from managemen based on accrual accouning oward risk managemen based on marking-o-marke of posiions. Increased liquidiy and pricing availabiliy along wih a new focus on rading led o he implemenaion of frequen revaluaion of posiions, he mark-o-marke concep. As invesmens became more liquid, he poenial for frequen and accurae reporing of invesmen gains and losses has led an increasing number of firms o manage daily earnings from a mark-o-marke perspecive. The swich from accrual accouning o mark-o-marke ofen resuls in higher swings in repored reurns, herefore increasing he need for managers o focus on he volailiy of he underlying markes. The markes have no suddenly become more volaile, bu he focus on risks hrough mark-o-marke has highlighed he poenial volailiy of earnings. Given he move o frequenly revalue posiions, managers have become more concerned wih esimaing he poenial effec of changes in marke condiions on he value of heir posiions.. Performance: Significan effors have been made o develop mehods and sysems o measure financial performance. Indices for foreign exchange, fixed income securiies, commodiies, and equiies have become commonplace and are used exensively o monior reurns wihin and/or across asse classes as well as o allocae funds. The somewha exclusive focus on reurns, however, has led o incomplee performance analysis. Reurn measuremen gives no indicaion of he cos in erms of risk (volailiy of reurns). Higher reurns can only be obained a he expense of higher risks. While his rade-off is well known, he risk measuremen componen of he analysis has no received broad aenion. Invesors and rading managers are searching for common sandards o measure marke risks and o esimae beer he risk/reurn profile of individual asses or asse classes. Nowihsanding he exernal consrains from he regulaory agencies, he managemen of financial firms have also been searching for ways o measure marke risks, given he poenially damaging effec of miscalculaed risks on company earnings. As a resul, banks, invesmen firms, and corporaions are now in he process of inegraing measures of marke risk ino heir managemen philosophy. They are designing and implemening marke risk monioring sysems ha can provide managemen wih imely informaion on posiions and he esimaed loss poenial of each posiion. Over he las few years, here have been significan developmens in concepualizing a common framework for measuring marke risk. The indusry has produced a wide variey of indices o measure reurn, bu lile has been done o sandardize he measure of risk. Over he las 15 years many marke paricipans, academics, and regulaory bodies have developed conceps for measuring Par I: Risk Measuremen Framework

Chaper. Hisorical perspecive of VaR marke risks. Over he las five years, wo approaches have evolved as a means o measure marke risk. The firs approach, which we refer o as he saisical approach, involves forecasing a porfolio s reurn disribuion using probabiliy and saisical models. The second approach is referred o as scenario analysis. This mehodology simply revalues a porfolio under differen values of marke raes and prices. Noe ha in sress scenario analysis does no necessarily require he use of a probabiliy or saisical model. Insead, he fuure raes and prices ha are used in he revaluaion can be arbirarily chosen. Risk managers should use boh approaches he saisical approach o monior risks coninuously in all risk-aking unis and he scenario approach on a case-by-case basis o esimae risks in unique circumsances. This documen explains, in deail, he saisical approach RiskMerics o measure marke risk. This chaper is organized as follows: Secion.1 reviews how VaR was developed o suppor he risk managemen needs of rading aciviies as opposed o invesmen books. Though he disincion o dae has been an accouning one no an economic one, VaR conceps are now being used across he board. Secion. looks a he basic seps of he risk monioring process. Secion.3 reviews he alernaive VaR models currenly being used and how RiskMerics provides end-users wih he basic building blocks o es differen approaches..1 From ALM o VaR A well esablished mehod of looking a marke risks in he banking indusry is o forecas earnings under predeermined price/rae marke condiions (or scenarios). Earnings here are defined as earnings repored in a firm s Financial Saemens using generally acceped accouning principles. For many insiuions he bulk of aciviies are repored on an accrual basis, i.e., ransacions are booked a hisorical coss +/- accruals. Only a limied number of rading iems are marked o marke. Because changes in marke raes manifes hemselves only slowly when earnings are repored on an accrual basis, he simulaion of income has o be done over exended periods, i.e., unil mos of he ransacions on he books maure. Char.1 illusraes his convenional Asse/Liabiliy Managemen approach. Char.1 Asse liabiliy managemen Invenory of financial ransacions Accrual iems Trading iems Income simulaion Inermediae erm rae forecass Projeced income saemen There are wo major drawbacks o his mehodology: I requires projecing marke rae developmens over exended periods ino he fuure. RiskMerics Technical Documen Fourh Ediion

Sec..1 From ALM o VaR 3 I suppors he illusion ha gains and losses occur a he ime hey show up in he accrual accouns (i.e., when hey are realized following accouning principles). Wha his means is ha reurn is only defined as ne ineres earnings, a framework which ignores he change in price componen of he reurn funcion. Every invesor would agree ha he oal reurn on a bond posiion is he sum of he ineres earned and he change in he value of he bond over a given ime horizon. Tradiional ALM, as a resul of accouning convenions, ignores he change in value of he insrumen since posiions are no marked o marke. This has ofen lead crafy ALM managers o creae posiions which look aracive on paper because of high ne ineres earnings, bu which would no perform as well if heir change in marke value were considered. The marke risk in rading posiions is usually measured differenly and managed separaely. Trading posiions are marked-o-marke and he marke value is hen subjeced o projecions of changes in shor erm in raes and prices. This is much less hazardous as rae forecass are usually limied o shor horizons, i.e., he ime i should ake o close ou or hedge he rading posiion. Char. Value-a-Risk managemen in rading Invenory of financial ransacions Accrual iems Trading iems mark o marke Marke values Value simulaion Projeced marke value changes Curren marke raes & prices Shor erm price forecass The disincion beween accrual iems and rading iems and heir separae reamen for marke risk managemen has led o significan complicaions paricularly when ransacions classified as rading iems under generally acceped accouning principles are used o hedge ransacions classified as accrual iems. In an effor o overcome his difficuly, many firms paricularly hose wih relaively large rading books have expanded he marke risk approach o also include accrual iems, a leas for inernal risk managemen reporing. This is done by esimaing he fair marke value of he accrual iems and he changes in heir fair value under differen shor erm scenarios. Thus we are winessing he evoluion of an alernaive o he convenional approach of Asse/Liabiliy Managemen, he Value-a-Risk approach. I sared in pure rading operaions, bu is now gaining increased following in he financial indusry. Par I: Risk Measuremen Framework

4 Chaper. Hisorical perspecive of VaR Char.3 Comparing ALM o VaR managemen Projeced income saemen Convenional Asse/Liabiliy Managemen Income simulaion Invenory of financial ransacions Accrual iems Trading iems mark o marke New Value a Risk Managemen Proxy values Marke values Risk facors Projeced marke value changes Inermediae erm rae forecass Curren marke raes & prices Shor erm price forecass The advanages of VaR Managemen are ha i Incorporaes he mark-o-marke approach uniformly. Relies on a much shorer horizon forecas of marke variables. This improves he risk esimae as shor horizon forecass end o be more accurae han long horizon forecass. Of course, drawbacks exis. One of hem is ha i may no be rivial o mark cerain ransacions o marke or even undersand heir behavior under cerain rae environmens. This is paricularly rue for insrumens such as demand deposis in a reail banking environmen for example. Whaever he difficulies, he aim of geing an inegraed picure of a firm s exposure o marke risks is worh a number of assumpions, some of which may be reasonable represenaions of realiy. In he case of demand deposis, a recen aricle by Professor Rober arrow oulines how power swaps could be modelled o represen a reail bank s core deposi base risks (RISK, February 1996). Some criics also argue ha marking-o-marke all ransacions over shor ime periods creaes oo much earnings or volailiy. Looking a risks in his fashion may be misleading. This is he direcion of he indusry and is accouning regulaors however and i will be up o financial analyss o adap o he new environmen. The volailiy of earnings will no jus appear ou of he blue. The changes in accouning pracices will ulimaely show economic realiy as i really is. Marke risk can be absolue or relaive. In is absolue form, wha is measured is he loss in he value of a posiion or a porfolio resuling from changes in marke condiions. Absolue marke risk is wha managers of rading operaions measure. Corporaes who wish o esimae real poenial losses from heir reasury operaions also focus on absolue marke risk. Regulaors are ineresed in absolue marke risks in relaion o a firm s capial. When invesmen performance is measured agains an index, he inheren marke risk is relaive in he sense ha i measures he poenial underperformance agains a benchmark.. VaR in he framework of modern financial managemen As discussed before here are wo seps o VaR measuremen. Firs, all posiions need o be marked o marke (valuaion). Second we need o esimae he fuure variabiliy of he marke value. Char.4 illusraes his poin. RiskMerics Technical Documen Fourh Ediion

Sec.. VaR in he framework of modern financial managemen 5 Char.4 Two seps beyond accouning Curren marke raes & prices Projeced scenarios or esimaed volailiies & correlaions Accouning Valuaion Risk Projecion Accrual iems Trading iems Mapping Equivalen Posiion Trading iems Mapping Toal Posiion Balance Shee Economic values Marke Risks..1 Valuaion Trading iems are valued a heir curren prices/raes as quoed in liquid secondary markes. To value ransacions for which, in he absence of a liquid secondary marke, no marke value exiss, we firs map hem ino equivalen posiions, or decompose hem ino pars for which secondary marke prices exis. The mos basic such par is a single cash flow wih a given mauriy and currency of he payor. Mos ransacions can be described as a combinaion of such cash flows and hus can be valued approximaely as he sum of marke values of heir componen cash flows. Only non-markeable iems ha conain opions canno be valued in his simple manner. For heir valuaion we also need expeced volailiies and correlaions of he prices and raes ha affec heir value, and we need an opions pricing model. Volailiies describe poenial movemens in raes wih a given probabiliy; correlaions describe he inerdependencies beween differen raes and prices. Thus, for some valuaions, we require volailiies and correlaions... Risk esimaion Here we esimae value changes as a consequence of expeced changes in prices and raes. The poenial changes in prices are defined by eiher specific scenarios or a se of volailiy and correlaion esimaes. If he value of a posiion depends on a single rae, hen he poenial change in value is a funcion of he raes in he scenarios or volailiy of ha rae. If he value of a posiion depends on muliple raes, hen he poenial change in is value is a funcion of he combinaion of raes in each scenario or of each volailiy and of each correlaion beween all pairs of raes. Generaing equivalen posiions on an aggregae basis faciliaes he simulaion. As will be shown laer, he simulaion can be done algebraically (using saisics and marix algebra), or exhausively by compuing esimaed value changes for many combinaions of rae changes. In he RiskMerics framework, forecass of volailiies and correlaions play a cenral role. They are required for valuaions in he case of derivaives, he criical inpus for risk esimaion. Par I: Risk Measuremen Framework

6 Chaper. Hisorical perspecive of VaR.3 Alernaive approaches o risk esimaion More han one VaR model is currenly being used and mos praciioners have seleced an approach based on heir specific needs, he ypes of posiions hey hold, heir willingness o rade off accuracy for speed (or vice versa), and oher consideraions. The differen models used oday differ on basically wo frons: How he changes in he values of financial insrumens are esimaed as a resul of marke movemens. How he poenial marke movemens are esimaed. Wha makes he variey of models currenly employed is he fac ha he choices made on he wo frons menioned above can be mixed and mached in differen ways..3.1 Esimaing changes in value There are basically wo approaches o esimaing how he value of a porfolio changes as a resul of marke movemens: analyical mehods and simulaion mehods..3.1.1 Analyical mehods One such mehod is he analyical sensiiviy approach based on he following equaion: esimaed value change = f (posiion sensiiviy, esimaed rae/price change) where he posiion sensiiviy facor esablishes he relaionship beween he value of he insrumen and of he underlying rae or price, and deermines he accuracy of he risk approximaion. In is simples form, he analyical sensiiviy approach looks like his: esimaed value change = posiion sensiiviy esimaed rae change For example, he value change of a fixed income insrumen can be esimaed by using he insrumen s duraion. Alhough his linear approximaion simplifies he convex price/yield relaionship of a bond, i is exensively used in pracice because duraion ofen accouns for he mos significan percenage of he risk profile of a fixed income insrumen. Similar simplificaions can be made for opions where he esimaed change in value is approximaed by he opion s dela. The iniial versions of RiskMerics basically used an analyical VaR approach ha assumed ha marke risk could be esimaed by using a simple firs-order approximaion such as he one oulined above. We have since exended he analyical approach o accoun for nonlinear relaionships beween marke value and rae changes (e.g., opions), which requires accouning for gamma risk in addiion o dela risk. The more refined version of he analyical approach looks like his: esimaed value change = (posiion sensiiviy 1 esimaed rae change) + 1/ (posiion sensiiviy ) (esimaed rae change) +... In he case of an opion, he firs-order posiion sensiiviy is he dela, while he second-order erm is he gamma. Higher order effecs can also be esimaed using an analyical approach, bu he mah ypically ges more complex. The analyical approach requires ha posiions be summarized in some fashion so ha he esimaed rae changes can be applied. This process of aggregaing posiions is called mapping and is described in Chaper 6. RiskMerics Technical Documen Fourh Ediion

Sec..3 Alernaive approaches o risk esimaion 7 The advanages of analyical models is ha hey are compuaionally efficien and enable users o esimae risk in a imely fashion..3.1. Simulaion mehods The second se of approaches, ypically referred o as Full Valuaion models rely on revaluing a porfolio of insrumens under differen scenarios. How hese scenarios are generaed varies across models, from basic hisorical simulaion o disribuions of reurns generaed from a se of volailiy and correlaion esimaes such as RiskMerics. Some models include user-defined scenarios ha are based off of major marke evens and which are aimed a esimaing risk in crisis condiions. This process is ofen referred o a sress esing. Full Valuaion models ypically provide a richer se of risk measures since users are able o focus on he enire disribuion of reurns insead of a single VaR number. Their main drawback is he fac ha he full valuaion of large porfolios under a significan number of scenarios is compuaionally inensive and akes ime. I may no be he preferred approach when he goal is o provide senior managemen wih a imely snapsho of risks across a large organizaion..3. Esimaing marke movemens The second discriminan beween VaR approaches is how marke movemens are esimaed. There is much more variey here and he following lis is no an exhausive lis of curren pracice. RiskMerics RiskMerics uses hisorical ime series analysis o derive esimaes of volailiies and correlaions on a large se of financial insrumens. I assumes ha he disribuion of pas reurns can be modelled o provide us wih a reasonable forecas of fuure reurns over differen horizons. While RiskMerics assumes condiional normaliy of reurns, we have refined he esimaion process o incorporae he fac ha mos markes show kurosis and lepokurosis. We will be publishing facors o adjus for his effec once he RiskMerics cusomizable daa engine becomes available on he Reuers Web. These volailiy and correlaion esimaes can be used as inpus o: Analyical VaR models Full valuaion models. In Appendix E we ouline how he RiskMerics volailiy and correlaion daa ses can be used o drive simulaions of fuure reurns. Hisorical Simulaion The hisorical simulaion approach, which is usually applied under a full valuaion model, makes no explici assumpions abou he disribuion of asse reurns. Under hisorical simulaion, porfolios are valued under a number of differen hisorical ime windows which are user defined. These lookback periods ypically range from 6 monhs o years. Once he RiskMerics cusomizable daa engine becomes available on he ReuersWeb, users will be able o access he underlying hisorical daa needed o perform his ype of simulaion. Mone Carlo Simulaion While hisorical simulaion quanifies risk by replicaing one specific hisorical pah of marke evoluion, sochasic simulaion approaches aemp o generae many more pahs of marke reurns. These reurns are generaed using a defined sochasic process (for example, assume ha ineres raes follow a random walk) and saisical parameers ha drive he process (for example, he mean and variance of he random variable).the RiskMerics daa ses can be used as inpus o his process. Par I: Risk Measuremen Framework

8 Chaper. Hisorical perspecive of VaR In addiion, he following VaR models add refinemens o he resuls generaed by he approaches lised above. Implied volailiies Some praciioners will also look o he marke o provide hem wih an indicaion of fuure poenial reurn disribuions. Implied volailiy as exraced from a paricular opion pricing model is he marke s forecas of fuure volailiy. Implied volailiies are ofen used in comparison o hisory o refine he risk analysis. Implied volailiies are no currenly used o drive global VaR models as his would require observable opions prices on all insrumens ha compose a porfolio. Unforunaely, he universe of consisenly observable opions prices is no ye large enough; generally only exchange raded opions are reliable sources of prices. In paricular, he number of implied correlaions ha can be derived from raded opions prices is insignifican compared o he number of correlaions required o esimae risks in porfolios conaining many asse ypes. User-defined scenarios Mos risk managemen models add user-defined rae and price movemens o he sandard VaR number, if only o es he effec of wha could happen if hisorical paerns do no repea hemselves. Some scenarios are subjecively chosen while ohers recreae pas crises evens. The laer is referred o as sress esing and is an inegral par of a well designed risk managemen process. Selecing he appropriae measuremen mehod is no, however, sraighforward. udgmen in he choice of mehodologies will always be imporan. Cos benefi rade-offs are differen for each user, depending on his posiion in he markes, he number and ypes of insrumens raded, and he echnology available. Differen choices can be made even a differen levels of an organizaion, depending on he objecives. While rading desks of a bank may require precise risk esimaion involving simulaion on relaively small porfolios, senior managemen may op for an analyical approach ha is cos efficien and imely. I is imporan for senior managemen o know wheher he risk of he insiuion is $10 million or $50 million. I is irrelevan for hem o make he disincion beween $10 million and $11 million. Achieving his level of accuracy a he senior managemen level is no only irrelevan, bu can also be unachievable operaionally, or a a cos which is no consisen wih shareholder value. Since is inroducion, RiskMerics has become he umbrella name for a series of VaR mehodologies, from he simple analyical esimaion based on he precep ha all insrumens are linear (he so-called dela approximaion) o he srucured Mone Carlo simulaion. No all paricipans wih exposure o he financial and commodiies markes will have he resources o perform exensive simulaions. Tha is why we have srived in his updae of he RiskMerics Technical Documen o refine analyical approximaions of risk for non-linear insrumens (he dela-gamma approximaions). During 1997, he availabiliy of hisorical raes and prices under he RiskMerics cusomizable daa engine will make hisorical simulaion an opion for users of our producs. RiskMerics Technical Documen Fourh Ediion

Sec..3 Alernaive approaches o risk esimaion 9 Table.1 Two discriminaing facors o review VaR models How o esimae he change in he value of insrumens Analyical Full Valuaion How o esimae rae and price changes Full VaR model RiskMerics Covariance marices applied o sandard insrumen maps. Covariance marices used o define scenarios for srucured Mone Carlo. Hisorical simulaion No applicable. Porfolios revalued under hisorical reurn disribuions (lookback period varies. Mone Carlo No applicable. Saisical parameers deermine sochasic processes. Sources of daa vary (can include RiskMerics covariance marices). Parial VaR model Implied volailiies Covariance marices applied o sandard insrumen maps. Covariance marices used o define scenarios for srucured Mone Carlo. User defined Sensiiviy analysis on single insrumens. Limied number of scenarios. Par I: Risk Measuremen Framework

30 Chaper. Hisorical perspecive of VaR RiskMerics Technical Documen Fourh Ediion

31 Chaper 3. Applying he risk measures 3.1 Marke risk limis 33 3. Calibraing valuaion and risk models 34 3.3 Performance evaluaion 34 3.4 Regulaory reporing, capial requiremen 36 3.4.1 Capial Adequacy Direcive 36 3.4. Basel Commiee Proposal 37 Par I: Risk Measuremen Framework

3 RiskMerics Technical Documen Fourh Ediion

33 Chaper 3. Applying he risk measures acques Longersaey Morgan Guarany Trus Company Risk Managemen Advisory (1-1) 648-4936 riskmerics@jpmorgan.com The measures of marke risk oulined in he preceding secions can have a variey of applicaions. We will highligh jus a few: To measure and compare marke risks. To check he valuaion/risk models. To evaluae he performance of risk akers on a reurn/risk basis. To esimae capial levels required o suppor risk aking. 3.1 Marke risk limis Posiion limis have radiionally been expressed in nominal erms, fuures equivalens or oher denominaors unrelaed o he amoun of risk effecively incurred. The manager of a USD bond porfolio will be old for example ha he canno hold more han 100 million USD worh of U.S. Treasury bonds. In mos cases, he measure conains some risk consrain expressed in a paricular mauriy or duraion equivalen (if he 100 million limi is in -year equivalens, he manager will no be able o inves 100 million in 30-year bonds). Seing limis in erms of Value-a-Risk has significan advanages: posiion benchmarks become a funcion of risk and posiions in differen markes while producs can be compared hrough his common measure. A common denominaor rids he sandard limis manuals of a muliude of measures which are differen for every asse class. Limis become meaningful for managemen as hey represen a reasonable esimae of how much could be los. A furher advanage of Value-a-Risk limis comes from he fac ha VaR measures incorporae porfolio or risk diversificaion effecs. This leads o hierarchical limi srucures in which he risk limi a higher levels can be lower han he sum of risk limis of unis reporing o i. Char 3.1 Hierarchical VaR limi srucure Business Area VaR-Limi: $0MM Business Group A VaR-Limi: $10MM Business Group B VaR-Limi: $1MM Business Group C VaR-Limi: $8MM Uni A1 VaR-Limi: $8MM Uni A VaR-Limi: $7MM Uni C1 VaR-Limi: $6MM Uni C VaR-Limi: $5MM Uni C3 VaR-Limi: $3MM Seing limis in erms of risk helps business managers o allocae risk o hose areas which hey feel offer he mos poenial, or in which heir firms experise is greaes. This moivaes managers of muliple risk aciviies o favor risk reducing diversificaion sraegies. Par I: Risk Measuremen Framework

34 Chaper 3. Applying he risk measures 3. Calibraing valuaion and risk models An effecive mehod o check he validiy of he underlying valuaion and risk models is o compare DEaR esimaes wih realized daily profis and losses over ime. Char 3. illusraes he concep. The sars show he daily P&L of a global rading business during he firs 7 monhs of 1993, he wo lines show he Daily Earnings a Risk, plus and minus. Char 3. Ex pos validaion of risk models: DEaR vs. acual daily P&L 7.5 5.0.5 0 -.5 + DEaR -5.0 - DEaR -7.5 an Feb Mar Apr May un ul By definiion, he cone delimied by he +/ DEaR lines should conain 90% of all he sars, because DEaR is defined as he maximum amoun of expeced profi or losses 90% of he ime. If here are subsanially more han 10% of he sars ouside he DEaR-cone, hen he underlying models underesimae he risks. If here are no sars ouside he DEaR cone and no even close o he lines, hen he underlying models overesimae he risks. This ype of char is only a reasonable reflecion of he risk saisics if he daily profi and losses are derived solely from overnigh risk aking and no inraday rading and oher aciviies. Ofen his is no he case. Then insead of he daily P&L you should plo wha is ofen referred o as he no-acion-p&l ; i describes he hypoheical P&L on he posiion ha would have been incurred if he previous day s closing posiion had been kep for he nex 4 hours and hen revalued. This daa is ofen difficul o collec. 3.3 Performance evaluaion To dae, rading and posiion aking alen have been rewarded o a significan exen on he basis of oal reurns. Given he high rewards besowed on ousanding rading alen his may bias he rading professionals owards aking excessive risks.i is ofen referred o as giving raders a free opion on he capial of your firm. The ineres of he firm or capial provider may be geing ou of line wih he ineres of he risk aking individual unless he risks are properly measured and reurns are adjused for he amoun of risk effecively aken. RiskMerics Technical Documen Fourh Ediion

Sec. 3.3 Performance evaluaion 35 To do his correcly one needs a sandard measure of risks. Ideally risk aking should be evaluaed on he basis of hree inerlinked measures: revenues, volailiy of revenues, and risks. This is illusraed by Char 3.3: Char 3.3 Performance evaluaion riangle Risks Risk Raio Efficiency Raio Revenues Sharpe Raio Volailiy of revenues Including esimaed (ex ane) and realized (ex pos) volailiy of profis adds an exra dimension o performance evaluaion. The raio of P&L over risk (risk raio) and of P&L over volailiy (Sharpe raio) can be combined ino wha we define as a rader s efficiency raio (esimaed risk/realized volailiy) ha measures an individual s capaciy o ranslae esimaed risk ino low realized volailiy of revenues. Consider an example o illusrae he issue. Assume you have o evaluae Trader #1 relaive o Trader # and he only informaion on hand is he hisory of heir respecive cumulaive rading revenues (i.e., rading profis). This informaion allows you o compare heir profis and volailiy of heir profis, bu says nohing abou heir risks. Char 3.4 Example: comparison of cumulaive rading revenues cumulaive revenues Trader #1 Trader # 6 6 5 5 4 4 3 3 1 1 0 0-1 199 1993-1 199 1993 Par I: Risk Measuremen Framework

36 Chaper 3. Applying he risk measures Char 3.5 Example: applying he evaluaion riangle Wih risk informaion you can compare he raders more effecively. Char 3.5 shows, for he wo raders he risk raio, sharpe raio, and efficiency raio over ime. Trader #1 Trader # Sharpe Raio 0.6 0.6 P&L vol(p&l) 0-0.6 0-0.6 Risk Raio P&L DEaR 0.4 0. 0 0.4 0. 0-0. -0. Efficiency Raio DEaR vol(p&l) 1 0 199 1993 1 0 199 1993 Noe, you have no informaion on he ype of marke hese raders operae in or he size of posiions hey have aken. Neverheless Char 3.5 provides ineresing comparaive informaion which lead o a richer evaluaion. 3.4 Regulaory reporing, capial requiremen Financial insiuions such as banks and invesmen firms will soon have o mee capial requiremens o cover he marke risks ha hey incur as a resul of heir normal operaions. Currenly he driving forces developing inernaional sandards for marke risk based capial requiremens are he European Communiy which issued a binding Capial Adequacy Direcive (EC-CAD) and he Basel Commiee on Banking Supervision a he Bank for Inernaional Selemens (Basel Commiee) which has recenly come ou wih revised proposals on he use of banks inernal models. (See Appendix F for more informaion.) 3.4.1 Capial Adequacy Direcive The European Union s EEC 93/6 direcive mandaes banks and invesmen firms o se capial aside o cover marke risks. In a nushell he EC-CAD compues he capial requiremen as a sum of capial requiremens on posiions of differen ypes in differen markes. I does no ake ino accoun he risk reducing effec of diversificaion. As a resul, he sric applicaion of he curren recommendaions will lead o financial insiuions, paricularly he ones which are acive inernaionally in many differen markes, o overesimae heir marke risks and consequenly be required o mainain very high capial levels. While here may be some prudenial advanages o his, i is RiskMerics Technical Documen Fourh Ediion

Sec. 3.4 Regulaory reporing, capial requiremen 37 no an efficien allocaion of financial resources and could lead cerain aciviies o be moved ouside he jurisdicion of he financial regulaory auhoriies. 3.4. Basel Commiee Proposal In anuary 1996, he Basel Commiee on Banking Supervision of he BIS issued a revised consulaive proposal on an Inernal Model-Based Approach o Marke Risk Capial Requiremens ha represens a big sep forward in recognizing he new quaniaive risk esimaion echniques used by he banking indusry. These proposals recognize ha curren pracice among many financial insiuions has superseded he original guidelines in erms of sophisicaion, and ha banks should be given he flexibiliy o use more advanced mehodologies. This so-called inernal models approach addresses a number of issues ha were raised when banks commened on he original proposal daed April 1993. Table 3.1 compares he mehodologies for esimaing marke risks as recenly proposed by he Basel Commiee wih he RiskMerics mehodology covered in his documen. This comparison focuses exclusively on he so-called quaniaive facors ha he BIS guidelines will require banks o use. I does no address he qualiaive ones relaed o he risk managemen process and which are beyond he scope of his documen. While he mehodologies oulined in he BIS proposals have come a long way in overcoming imporan objecions o he firs se of proposals, here are sill a number of issues ha will be debaed furher. In order o faciliae he discussion beween regulaors and regulaed, we have published since mid-1995 in parallel wih he exising volailiy and correlaion daa ses, a RiskMerics Regulaory Daa Se. The disribuion of his regulaory daa se is no an endorsemen of he Basel commiee proposals and he following paragraphs which explain how he daa se can be used do no consiue.p. Morgan s official posiion on he conen and scope of he Basel commiee proposal. Consisen wih he oher RiskMerics daa ses, he Regulaory Daa Se conains volailiy esimaes for a 1-day holding period. Given ha he BIS rules require marke risk esimaes o be calculaed over a 10-day holding period and a 99% confidence inerval (i.e.,.33 sandard deviaions), users will need o rescale he 1-day volailiy (see Eq. [3.1]). The Basel proposals allow for his adjusmen of daa (hey acually refer o scaling up VaR esimaes bu exclude his pracice in he case of opions since i only works for insrumens whose pricing formulae are linear). Scaling up volailiy esimaes is perfecly legiimae, assuming no auocorrelaion in he daa. Scaling up Value-a-Risk does no work for opions, hough using scaled up volailiies o esimae he marke risks of opions wih adequae pricing algorihms poses no problem. As in he oher daa ses, volailiies and correlaions are measured as daily log changes in raes and prices. However, conrary o he exponenial weighing schemes used for he oher daa ses, esimaes in he Regulaory Daa Se are based on simple moving averages of 1 year of hisorical daa, sampled daily. To make i comparable o he sandard daa ses, he RiskMerics Regulaory Daa Se is based on 95% confidence. Including he adjusmen for he holding period, users downloading he daa ses will need o rescale he volailiy esimaes according o he following equaion, in order o mee he requiremens se forh in he Basel proposals (his adjusmen assumes a normal disribuion. More refined mehods incorporaing he characerisics of fa ailed disribuions are oulined in he saisics secion of his documen): [3.1].33 V Basel = --------- V 1.65 RiskMerics RD = 4.45 V RiskMerics RD 10 Par I: Risk Measuremen Framework

38 Chaper 3. Applying he risk measures where V RiskMerics RD V Basel = volailiies provided in RiskMerics Regulaory Daase = volailiies suggesed by Basel Commiee for use in inernal models Correlaions across asse classes (i.e., foreign exchange o governmen bonds for example) are supplied in he RiskMerics Regulaory Daa Se, despie he fac ha acual use of empirical correlaions in he VaR esimaes is subjec o regulaory approval. The BIS has saed ha he use of correlaions across asse classes would be based on wheher he supervisory auhoriy was saisfied wih he inegriy of he esimaion mehodology. RiskMerics Technical Documen Fourh Ediion

Sec. 3.4 Regulaory reporing, capial requiremen 39 Table 3.1 Comparing he Basel Commiee proposal wih RiskMerics Issue Basel Commiee proposal RiskMerics Mapping: how posiions are described in summary form Fixed Income: a leas 6 ime buckes, differeniae governmen yield curves and spread curves. Fixed Income: daa for 7 10 buckes of governmen yield curves in 16 markes, 4 buckes money marke raes in 7 markes, 4 6 buckes in swap raes in 18 markes. Equiies: counry indices, individual socks on basis of bea equivalen. Commodiies: o be included, no specified how. Equiies: counry indices in 7 markes, individual socks on bea (correcion for non-sysemaic risk). Commodiies: 80 volailiy series in 11 commodiies (spo and erm). Volailiy: how saisics of fuure price movemen are esimaed Volailiy expressed in sandard deviaion of normal disribuion proxy for daily hisorical observaions year or more back. Equal weighs or alernaive weighing scheme provided effecive observaion period is a leas one year. Esimae updaed a leas quarerly. Volailiy expressed in sandard deviaion of normal disribuion proxy for exponenially weighed daily hisorical observaions wih decay facors of.94 (for rading, 74 day cuoff 1%) and.97 (for invesing, 151 day cuoff a 1%). Special Regulaory Daa Se, incorporaing Basel Commiee 1-year moving average assumpion. Esimaes updaed daily. Adversiy: size of adverse move in erms of normal disribuion Minimum adverse move expeced o happen wih probabiliy of 1% (.3 sandard deviaions) over 10 business days. Permission o use daily saisics scaled up wih square roo of 10 (3.1). Equivalen o 7.3 daily sandard deviaions. For rading: minimum adverse move expeced o happen wih probabiliy of 5% (1.65 sandard deviaion) over 1 business day. For invesmen: minimum adverse move expeced o happen wih probabiliy of 5% (1.65 sandard deviaion) over 5 business days. Opions: reamen of ime value and non-lineariy Risk esimae mus consider effec of non-linear price movemen (gamma-effec). Risk esimae mus include effec of changes in implied volailiies (vega-effec). Non-linear price movemen can be esimaed analyically (delagamma) or under simulaion approach. Simulaion scenarios o be generaed from esimaed volailiies and correlaions. Esimaes of volailiies of implied volailiies currenly no provided, hus limied coverage of opions risk. Correlaion: how risks are aggregaed Porfolio effec can be considered wihin asse classes (Fixed Income, Equiy, Commodiy, FX). Use of correlaions across asse classes subjec o regulaory approval. Correlaions esimaed wih equally weighed daily daa for more han one year. Full porfolio effec considered across all possible parameer combinaions. Correlaions esimaed using exponenially weighed daily hisorical observaions wih decay facors of 0.94 (for rading, 74 day cuoff 1%) and 0.97 (for invesing, 151 day cuoff a 1%). Residuals: reamen of insrumen specific risks Insrumen specific risks no covered by sandard maps should be esimaed. Capial requiremens a leas equal o 50% of charge calculaed under sandard mehodology. Does no deal wih specific risks no covered in sandard maps. Par I: Risk Measuremen Framework

40 Chaper 3. Applying he risk measures RiskMerics Technical Documen Fourh Ediion

41 Par II Saisics of Financial Marke Reurns

4 RiskMerics Technical Documen Fourh Ediion

43 Chaper 4. Saisical and probabiliy foundaions 4.1 Definiion of financial price changes and reurns 45 4.1.1 One-day (single period) horizon 45 4.1. Muliple-day (muli-period) horizon 47 4.1.3 Percen and coninuous compounding in aggregaing reurns 48 4. Modeling financial prices and reurns 49 4..1 Random walk model for single-price asses 50 4.. Random walk model for fixed income insrumens 51 4..3 Time-dependen properies of he random walk model 51 4.3 Invesigaing he random-walk model 54 4.3.1 Is he disribuion of reurns consan over ime? 54 4.3. Are reurns saisically independen over ime? 56 4.3.3 Mulivariae exensions 6 4.4 Summary of our findings 64 4.5 A review of hisorical observaions of reurn disribuions 64 4.5.1 Modeling mehods 65 4.5. Properies of he normal disribuion 66 4.5.3 The lognormal disribuion 7 4.6 RiskMerics model of financial reurns: A modified random walk 73 4.7 Summary 74 Par II: Saisics of Financial Marke Reurns

44 RiskMerics Technical Documen Fourh Ediion

45 Chaper 4. Saisical and probabiliy foundaions Peer Zangari Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-8641 zangari_peer@jpmorgan.com This chaper presens he saisical and probabiliy underpinnings of he RiskMerics model. I explains he assumpions commonly applied o forecas he disribuion of porfolio reurns and invesigaes he empirical validiy of hese assumpions. While we have ried o make his chaper self-conained, is subjec maer does require a horough grasp of elemenary saisics. We have included many up-o-dae references on specific opics so ha he ineresed reader may pursue furher sudy in hese areas. This chaper is organized as follows: Secion 4.1 presens definiions of financial price reurns and explains he ype of reurns applied in RiskMerics. Secion 4. describes he basic random walk model for financial prices o serve as background o inroducing he RiskMerics model of reurns. Secion 4.3 looks a some observed ime series properies of financial reurns in he conex of he random walk model. Secion 4.4 summarizes he resuls presened in Secions 4.1 hrough 4.3. Secion 4.5 reviews some popular models of financial reurns and presens a review of he normal and lognormal disribuions. Secion 4.6 presens he RiskMerics model as a modified random walk. This secion liss he assumpions of he RiskMerics model ha is, wha RiskMerics assumes abou he evoluion of financial reurns over ime and he disribuion of reurns a any poin in ime. Secion 4.7 is a chaper summary. 4.1 Definiion of financial price changes and reurns 1 Risk is ofen measured in erms of price changes. These changes can ake a variey of forms such as absolue price change, relaive price change, and log price change. When a price change is defined relaive o some iniial price, i is known as a reurn. RiskMerics measures change in value of a porfolio (ofen referred o as he adverse price move) in erms of log price changes also known as coninuously-compounded reurns. Nex, we explain differen definiions of price reurns. 4.1.1 One-day (single period) horizon Denoe by day. P he price of a securiy a dae. In his documen, is aken o represen one business The absolue price change on a securiy beween daes and 1 (i.e., one day) is defined as [4.1] D = P P 1 1 References for his secion are, Campbell, Lo and MacKinley (1995) and Taylor, S.. (1987). Par II: Saisics of Financial Marke Reurns

46 Chaper 4. Saisical and probabiliy foundaions The relaive price change, or percen reurn, R, for he same period is [4.] P R P 1 = ---------------------- P 1 If he gross reurn on a securiy is jus 1 + R, hen he log price change (or coninuously-compounded reurn),, of a securiy is defined o be he naural logarihm of is gross reurn. Tha is, r [4.3] r = ln ( 1 + R ) = P ln ----------- P 1 = ( p p 1 ) where p = ln ( P ) is he naural logarihm of P. In pracice, he main reason for working wih reurns raher han prices is ha reurns have more aracive saisical properies han prices, as will be shown below. Furher, reurns (relaive and log price changes) are ofen preferred o absolue price changes because he laer do no measure change in erms of he given price level. To illusrae he differen resuls ha differen price changes can yield, Table 4.1 presens daily USD/DEM exchange raes for he period 8-Mar-96 hrough 1-Apr-96 and he corresponding daily absolue, relaive, and log price changes. Table 4.1 Absolue, relaive and log price changes* Dae Price (USD/DEM), P Absolue price change (%), D Relaive price change (%), R Log price change (%,) r 8-Mar-96 0.67654 0.47 0.635 0.633 9-Mar-96 0.6773 0.078 0.115 0.115 1-Apr-96 0.674 0.310 0.458 0.459 -Apr-96 0.67485 0.063 0.093 0.093 3-Apr-96 0.67604 0.119 0.176 0.176 4-Apr-96 0.67545 0.059 0.087 0.087 5-Apr-96 0.67449 0.096 0.14-0.14 8-Apr-96 0.67668 0.19 0.35 0.34 9-Apr-96 0.67033 0.635 0.938 0.943 10-Apr-96 0.66680 0.353 0.57 0.58 11-Apr-96 0.66609 0.071 0.106 0.107 1-Apr-96 0.66503 0.106 0.159 0.159 * RiskMerics foreign exchange series are quoed as USD per uni foreign currency given ha he daases are sandardized for users whose base currency is he USD. This is he inverse of marke quoaion sandards for mos currency pairs. As expeced, all hree series of price changes have he same sign for any given day. Also, noice he similariy beween he log and relaive price changes. In fac, we should expec hese wo reurn series o be similar o one anoher for small changes in he underlying prices. In conras, he absolue change series is quie differen from he oher wo series. Alhough i is called percen reurn, he relaive price change is expressed as a decimal number. RiskMerics Technical Documen Fourh Ediion

Sec. 4.1 Definiion of financial price changes and reurns 47 To furher illusrae he poenial differences beween absolue and log price changes, Char 4.1 shows daily absolue and log price changes for he U.S. 30-year governmen bond over he firs quarer of 1996. Char 4.1 Absolue price change and log price change in U.S. 30-year governmen bond Absolue price change 0.006 0.004 0.00 0-0.00-0.004-0.006-0.008 absolue price change log price change -0.010 1-an 19-an 8-Feb 8-Feb 19-Mar Log price change 0.06 0.04 0.0 0-0.0-0.04-0.06-0.08-0.10 Char 4.1 shows ha movemens of he wo changes over ime are quie similar alhough he magniude of heir variaion is differen. This laer poin and he resuls presened in Table 4.1 should make i clear ha i is imporan o undersand he convenion chosen for measuring price changes. 4.1. Muliple-day (muli-period) horizon The reurns R and r described above are 1-day reurns. We now show how o use hem o compue reurns for horizons greaer han one day. Muliple-day percen reurns over he mos recen k days, R ( k), are defined simply as [4.4] R ( k) = P P ---------------------- k P k In erms of 1-day reurns, he muliple-day gross reurn 1-day gross reurns. 1 + R ( k ) is given by he produc of 1+ R ( k) = ( 1+ R ) ( 1 + R 1 ) ( 1 + R k 1 ) [4.5] = = P ----------- P 1 P ---------- P k P 1 P ----------- P k ----------------- 1 P k Noe ha in Eq. [4.5] he k-day reurn is a discreely compounded reurn. For coninuously compounded reurns, he muliple-day reurn ( k ) is defined as r [4.6] r ( k) P ---------- = ln P k Par II: Saisics of Financial Marke Reurns

48 Chaper 4. Saisical and probabiliy foundaions The coninuously-compounded reurn r ( k) is he sum of k coninuously-compounded 1-day reurns. To see his we use he relaion r ( k) = ln [ 1+ R ( k) ]. The reurn r ( k) can hen be wrien as [4.7] r ( k) = ln [ 1+ R ( k) ] = ln [ ( 1 + R ) ( 1+ R ) 1 R 1 ( + k )] 1 = r + r 1 + + r k + 1 Noice from Eq. [4.7] ha compounding, a muliplicaive operaion, is convered o an addiive operaion by aking logarihms. Therefore, muliple day reurns based on coninuous compounding are simple sums of one-day reurns. As an example of how 1-day reurns are used o generae a muliple-day reurn, we use a 1-monh period, defined by RiskMerics as having 5 business days. Working wih log price changes, he coninuously compounded reurn over one monh is given by [4.8] r ( 5) = r + r 1 + + r 4 Tha is, he 1-monh reurn is he sum of he las 5 1-day reurns. 4.1.3 Percen and coninuous compounding in aggregaing reurns When deciding wheher o work wih percen or coninuously compounded reurns i is imporan o undersand how such reurns aggregae boh across ime and across individual reurns a any poin in ime. In he preceding secion we showed how muliple-day reurns can be consruced from 1-day reurns by aggregaing he laer across ime. This is known as emporal aggregaion. However, here is anoher ype of aggregaion known as cross-secion aggregaion. In he laer approach, aggregaion is across individual reurns (each corresponding o a specific insrumen) a a paricular poin in ime. For example, consider a porfolio ha consiss of hree insrumens. Le r i and R i ( i = 13,, ) be he coninuously compounded and percen reurns, respecively and le w i represen he porfolio weighs. (The parameer w i represens he fracion of he oal porfolio value allocaed o he ih insrumen wih he condiion ha assuming no shor posiions w 1 + w + w 3 = 1). If he iniial value of his porfolio is P 0 he price of he porfolio one period laer wih coninuously compounded reurns is [4.9] P 1 w 1 P 0 e r 1 w P 0 e r w 3 P 0 e r 3 = + + P 1 Solving Eq. [4.9] for he porfolio reurn, r p = ln -----, we ge [4.10] r p w 1 e r 1 w e r w 3 e r 3 = ln + + The price of he porfolio one period laer wih discree compounding, i.e., using percen reurns, is P 0 [4.11] P 1 = w 1 P 0 ( 1 + r 1 ) + w P 0 ( 1+ r ) + w 3 P 0 ( 1+ r 3 ) ( P The percen porfolio reurn, R 1 P 0 ) p = ------------------------, is given by P 0 [4.1] R p = w 1 r 1 + w r + w 3 r 3 RiskMerics Technical Documen Fourh Ediion

Sec. 4. Modeling financial prices and reurns 49 Equaion [4.1] is he expression ofen used o describe a porfolio reurn as a weighed sum of individual reurns. Table 4. presens expressions for reurns ha are consruced from emporal and cross-secion aggregaion for percen and coninuously compounded reurns. Table 4. Reurn aggregaion Aggregaion Temporal Cross-secion Percen reurns Coninuously compounded reurns R i r i T ( k) = ( 1+ R i ) 1 R p = w i R i = 1 T N i = 1 ( k) = r i r p w i e r = ln i = 1 N i = 1 The able shows ha when aggregaion is done across ime, i is more convenien o work wih coninuously compounded reurns whereas when aggregaion is across asses, percen reurns offer a simpler expression. As previously saed, log price changes (coninuously compounded reurns) are used in RiskMerics as he basis for all compuaions. In pracice, RiskMerics assumes ha a porfolio reurn is a weighed average of coninuously compounded reurns. Tha is, a porfolio reurn is defined as follows [4.13] r p N i = 1 w i r i As will be discussed in deail in he nex secion, when 1-day reurns are compued using, hen a model describing he disribuion of 1-day reurns exends sraighforwardly o reurns greaer han one day. 3 In he nex wo secions (4. and 4.3) we describe a class of ime series models and invesigae he empirical properies of financial reurns. These secions serve as imporan background o undersanding he assumpions RiskMerics applies o financial reurns. r 4. Modeling financial prices and reurns A risk measuremen model aemps o characerize he fuure change in a porfolio s value. Ofen, i does so by making forecass of each of a porfolio s underlying insrumen s fuure price changes, using only pas changes o consruc hese forecass. This ask of describing fuure price changes requires ha we model he following; (1) he emporal dynamics of reurns, i.e., model he evoluion of reurns over ime, and () he disribuion of reurns a any poin in ime. A widely used class of models ha describes he evoluion of price reurns is based on he noion ha financial prices follow a random walk. 3 There are wo oher reasons for using log price changes. The firs relaes o Siegel s paradox, Meese, R.A. and Rogoff, K. (1983). The second relaes o preserving normaliy for FX cross raes. Simply pu, when using log price changes, FX cross raes can be wrien as differences of base currency raes. (See Secion 8.4 for deails.) Par II: Saisics of Financial Marke Reurns

50 Chaper 4. Saisical and probabiliy foundaions 4..1 Random walk model for single-price asses In his secion we presen a model for a securiy wih a single price. Such a model applies naurally o asses such as foreign exchange raes, commodiies, and equiies where only one price exiss per asse. The fundamenal model of asse price dynamics is he random walk model, [4.14] P = µ + P 1 + σε P P 1 = µ + σε, ε IID N ( 0, 1) where IID sands for idenically and independenly disribued 4, and N ( 01, ) sands for he normal disribuion wih mean 0 and variance 1. Eq. [4.14] posis he evoluion of prices and heir disribuion by noing ha a any poin in ime, he curren price P depends on a fixed parameer µ, las period s price P 1, and a normally disribued random variable, ε. Simply pu, µ and σ affec he mean and variance of s disribuion, respecively. P The condiional disribuion of P, given, is normally disribued. 5 P 1 An obvious drawback of his model is ha here will always be a non-zero probabiliy ha prices are negaive. 6 One way o guaranee ha prices will be non-negaive is o model he log price p as a random walk wih normally disribued changes. [4.15] p = µ + p 1 + σε ε IID N ( 0, 1) Noice ha since we are modeling log prices, Eq. [4.15] is a model for coninuously compounded reurns, i.e., r = µ + σε. Now, we can derive an expression for prices, P given las period s price from Eq. [4.15]: P 1 [4.16] P = P 1 exp ( µ + σε ) where exp ( x) and e.718. P 1 e x Since boh and exp ( µ + σε ) are non-negaive, we are guaraneed ha P will never be negaive. Also, when is normally disribued, follows a lognormal disribuion. 7 ε Noice ha boh versions of he random walk model above assume ha he change in (log) prices has a consan variance (i.e., σ does no change wih ime). We can relax his (unrealisic) assumpion, hus allowing he variance of price changes o vary wih ime. Furher, he variance could be modeled as a funcion of pas informaion such as pas variances. By allowing he variance o vary over ime we have he model P [4.17] p = µ + p 1 + σ ε ε N ( 01, ) 4 See Secion 4.3 for he meaning of hese assumpions. 5 The uncondiional disribuion of P is undefined in ha is mean and variance are infinie. This can easily be seen by solving Eq. [4.14] for P as a funcion of pas ε s. 6 This is because he normal disribuion places a posiive probabiliy on all poins from negaive o posiive infiniy. See Secion 4.5. for a discussion of he normal disribuion. 7 See Secion 4.5.3 for a complee descripion of he lognormal disribuion. RiskMerics Technical Documen Fourh Ediion

Sec. 4. Modeling financial prices and reurns 51 This version of he random walk model is imporan since i will be shown below ha RiskMerics assumes ha log prices evolve according o Eq. [4.17] wih he parameer µ se o zero. 4.. Random walk model for fixed income insrumens Wih fixed income insrumens we observe boh prices and yields. When prices and yields exis, we mus decide wheher o model he log changes in he yields or in he prices. For example, for bonds, a well documened shorcoming of modeling price reurns according o Eq. [4.15] is ha he mehod ignores a bond s price pull o par phenomenon. Tha is, a bond has he disinc feaure ha as i approaches mauriy, is price converges o is face value. Consequenly, he bond price volailiy will converge o zero. Therefore, when modeling he dynamic behavior of bonds (and oher fixed income insrumens), he bond yields raher han he bond prices are ofen modeled according o he lognormal disribuion. Tha is, if Y denoes he yield on a bond a period, hen y = ln ( Y ) is modeled as [4.18] y = µ + y 1 + σε ε IID N ( 0, 1) (Noe ha similar o Eq. [4.17] we can incorporae a ime-varying variance ino Eq. [4.18]). In addiion o accouning for he pull o par phenomenon, anoher imporan reason for modeling he yield raher han he price according o Eq. [4.18] is ha posiive yields are guaraneed. In he conex of bond opion pricing, a srong case can ofen be made for modeling yields as lognormal. 8 4..3 Time-dependen properies of he random walk model Each of he random walk models presened in Secions 4..1 and 4.. imply a cerain movemen in financial prices over ime. In his secion we use Eq. [4.15] he random walk model in log prices, p o explain some imporan properies of price dynamics implied by he random walk model. Specifically, we discuss he properies of saionary (mean-revering) and nonsaionary ime series. A saionary process is one where he mean and variance are consan and finie over ime. 9 In order o inroduce he properies of a saionary ime series we mus firs generalize Eq. [4.15] o he following model. [4.19] p = µ + c p 1+ ε ε IID N ( 0, 1), p 0 = 0 where c is a parameer. Here, a saionary ime series is generaed when 1< c< 1. For example, if we se c = 0.5, we can simulae a saionary ime series using [4.0] p = 0.01 + 0.5 p 1 + ε ε IID N ( 0, 1), p 0 = 0 8 For a discussion on he poenial advanages of modeling yield levels as lognormal, see Fabozzi (1989, Chaper 3). 9 Saionariy also requires ha he (auo-)covariance of reurns a differen imes is only a funcion of he ime beween he reurns, and no he imes a which hey occur. This definiion of saionariy is known as weak or covariance saionariy. Par II: Saisics of Financial Marke Reurns

5 Chaper 4. Saisical and probabiliy foundaions Char 4. shows he simulaed saionary ime series based on 500 simulaions. Char 4. Simulaed saionary/mean-revering ime series Log price 4 3 1 0-1 - -3-4 1 91 181 71 361 451 Time () Char 4. shows how a saionary series flucuaes around is mean, which in his model is 0.0. Hence, saionary series are mean-revering since, regardless of he flucuaions ampliudes, he series revers o is mean. Unlike a mean-revering ime series, a nonsaionary ime series does no flucuae around a fixed mean. For example, in Eq. [4.15] he mean and variance of he log price p condiional on some original observed price, say, are given by he following expressions p 0 [4.1] E 0 [ p p 0 ] = p 0 + µ (mean) V 0 [ p p 0 ] = σ (variance) where E 0 [ ] and V 0 [ ] are he expecaion and variance operaors aken a ime 0. Eq. [4.1] shows ha boh he mean and variance of he log price are a funcion of ime such ha, as ime increases, so does p s condiional mean and variance. The fac ha is mean and variance change wih ime and blow-up as ime increases is a characerisic of a nonsaionary ime series. To illusrae he properies of a nonsaionary ime series, we use he random walk model, Eq. [4.15], o simulae 500 daa poins. Specifically, we simulae a series based on he following model, [4.] p = 0.01 + p 1 + ε ε IID N ( 0, 1), p 0 = 0 RiskMerics Technical Documen Fourh Ediion

Sec. 4. Modeling financial prices and reurns 53 The simulaed series is shown in Char 4.3. Char 4.3 Simulaed nonsaionary ime series Log price 30 5 0 15 10 5 0 µ = 1% -5-10 -15 1 91 181 71 361 451 Time () Noice how his series has a posiive drif ha grows wih ime, represening he erm µ in Eq. [4.1]. This is a ypical feaure of a nonsaionary ime series. In he preceding examples, noice ha he difference beween hese saionary and nonsaionary series is driven by he coefficien on las period s log price p 1. When his coefficien is 1, as in Eq. [4.], he process generaing log prices is known o have a uni roo. As should be expeced, given he differences beween saionary and non-saionary imes series and heir implicaions for saisical analysis, here is a large body of lieraure devoed o esing for he presence of a uni roo. 10 Real world examples of saionary and nonsaionary series are shown in Chars 4.4 and 4.5. For he same period, Char 4.4 plos he USD 30-year rae, a saionary ime series. Char 4.4 Observed saionary ime series USD 30-year yield USD 30-year zero yield (%) 8.5 8.0 7.5 7.30% 7.0 6.5 6.0 1993 1994 1995 1996 10 A common saisical es for a uni roo is known as he augmened Dickey-Fuller es. See Greene, (1993). Par II: Saisics of Financial Marke Reurns

54 Chaper 4. Saisical and probabiliy foundaions Noice how he 30-year raes flucuae around he sample average of 7.30%, signifying ha he ime series for his period is mean-revering. Char 4.5 plos he S&P 500 index for he period anuary 4, 1993 hrough une 8, 1996. Char 4.5 Observed nonsaionary ime series S&P 500 index S&P 500 700 650 600 550 500 504 450 400 1993 1994 1995 1996 Noice ha he S&P 500 index does no flucuae around he sample mean of 504, bu raher has a disinc rend upwards. Comparing he S&P 500 series o he simulaed nonsaionary daa in Char 4.3, we see ha i has all he markings of a nonsaionary process. 4.3 Invesigaing he random-walk model Thus far we have focused on a simple version of he random walk model (Eq. [4.15]) o demonsrae some imporan ime series properies of financial (log) prices. Recall ha his model describes how he prices of financial asses evolve over ime, assuming ha logarihmic price changes are idenically and independenly disribued (IID). These assumpions imply: 1. A each poin in ime,, log price changes are disribued wih a mean 0 and variance (idenically disribued). This implies ha he mean and variance of he log price changes are homoskedasic, or unchanging over ime.. Log price changes are saisically independen of each oher over ime (independenly disribued). Tha is o say, he values of reurns sampled a differen poins are compleely unrelaed In his secion we invesigae he validiy of hese assumpions by analyzing real-world daa. We find evidence ha he IID assumpions do no hold. 11 σ 11 Recen (nonparameric) ess o deermine wheher a ime series is IID are presened in Campbell and Dufour (1995). RiskMerics Technical Documen Fourh Ediion

Sec. 4.3 Invesigaing he random-walk model 55 4.3.1 Is he disribuion of reurns consan over ime? Visual inspecion of real-world daa can be a useful way o help undersand wheher he assumpions of IID reurns hold. Using a ime series of reurns, we invesigae wheher he firs assumpion of IID, idenically disribued reurns, is indeed valid. We find ha i is violaed and presen he following daa as evidence. Chars 4.6 and 4.7 show ime series plos of coninuously compounded reurns for he USD/DEM and USD/FRF exchange raes, respecively. 1 Char 4.6 USD/DEM reurns USD/DEM reurns 0.04 0.03 0.0 low volailiy high volailiy 0.01 0-0.01-0.0-0.03 an-93 un-93 Dec-93 un-94 Dec-94 May-95 Nov-95 Char 4.7 USD/FRF reurns USD/FRF reurns 0.03 high volailiy 0.0 low volailiy 0.01 0-0.01-0.0-0.03 an-93 un-93 Dec-93 un-94 Dec-94 May-95 Nov-95 These ime series show clear evidence of volailiy clusering. Tha is, periods of large reurns are clusered and disinc from periods of small reurns, which are also clusered. If we measure such volailiy in erms of variance (or is square roo, i.e., he sandard deviaion), hen i is fair o hink ha variance changes wih ime, reflecing he clusers of large and small reurns. In erms of he model in Eq. [4.15], his means ha σ is changing wih ime (). In saisics, changing variances are ofen denoed by he erm heeroscedasiciy. 1 This noaion (i.e., USD per DEM) is no necessarily marke convenion. Par II: Saisics of Financial Marke Reurns

56 Chaper 4. Saisical and probabiliy foundaions In Chars 4.6 and 4.7 we also noice no only he individual volailiy clusering, bu he correlaion of he clusers beween reurn series. For example, noe ha periods of high volailiy in USD/DEM reurns coincide wih high volailiy in USD/FRF reurns. Such correlaion beween reurns series moivaes he developmen of mulivariae models, ha is, models of reurns ha measure no only individual series variance (volailiy), bu also he correlaion beween reurn series. 4.3. Are reurns saisically independen over ime? Having esablished, albei informally, he possibiliy of ime-varying variances, and consequenly, a violaion of he idenically disribued assumpion, we now invesigae he validiy of he independence assumpion, i.e., he second assumpion of IID. From our mehods and he daa ha we presen in he following secions (4.3..1 hrough 4.3..3), we conclude ha reurns in a given series are no independen of each oher. In Chars 4.6 and 4.7, he persisence displayed by he volailiy clusers shows some evidence of auocorrelaion in variances. Tha is, he variances of he series are correlaed across ime. If reurns are saisically independen over ime, hen hey are no auocorrelaed. Therefore, a naural mehod for deermining if reurns are saisically independen is o es wheher or no hey are auocorrelaed. In order o do so, we begin by defining correlaion and a mehod of esing for auocorrelaion. 4.3..1 Auocorrelaion of daily log price changes For a given ime series of reurns, he auocorrelaion coefficien measures he correlaion of reurns across ime. In general, he sandard correlaion coefficien beween wo random variables X and Y is given by he covariance beween X and Y divided by heir sandard deviaions: [4.3] ρ xy = σ xy ----------- σ x σ y σ xy where represens he covariance beween X and Y. A simple way o undersand wha covariance measures is o begin wih he definiion of variance. The variance of a random variable X is a measure of he variaion of X around is mean,. The mahemaical expression for variance is µ X [4.4] E ( X µ X ) where he erm E[ ] is he mahemaical expecaion or more simply, he average. Whereas he variance measures he magniude of variaion of one random variable (in his case X), covariance measures he covariaion of wo random variables (say, X and Y). I follows ha if he variance of X is he expeced value of ( X µ X ) imes ( X µ X ), hen he covariance of X and Y is he expeced value of ( X µ X ) imes ( Y µ Y ), or [4.5] E[ ( X µ X ) ( Y µ Y )] Now, for a ime series of observaions r, = 1 T, he kh order auocorrelaion coefficien ρ(k) is defined as: [4.6], σ σ σ ρ k k = ---------------- = --------------, k σ σ k RiskMerics Technical Documen Fourh Ediion

Sec. 4.3 Invesigaing he random-walk model 57 Noice ha since ρ(k) operaes on jus one series he subscrips on he covariance and sandard deviaion refer o he ime index on he reurn series. For a given sample of reurns, r, = 1 T, we can esimae Eq. [4.6] using he sample auocorrelaion coefficien which is given by: [4.7] ρˆ k = T = k + 1 { ( )} [ T ( k 1) ] ( r r) r k r ------------------------------------------------------------------------------------------------------------ T = 1 { ( r) } [ T 1] r where k = number of lags (days), and r = T 1 -- r, is he sample mean. T = 1 If a ime series is no auocorrelaed hen esimaes of ρˆ k will no be significanly differen from 0. In fac, when here is a large amoun of hisorical reurns available, we can calculae a 95% confidence band around 0 for each auocorrelaion coefficien 13 1.96 as ± ---------. T Chars 4.8 and 4.9 show he sample auocorrelaion coefficien ρˆ k ploed agains differen lags k (measured in days), along wih he 95% confidence band around zero for USD/DEM foreign exchange and S&P 500 log price changes, respecively, for he period anuary 4, 1990 o une 4, 1996. These chars are known as correlograms. The dashed lines represen he upper and lower 95% confidence bands ±4.7%. If here is no auocorrelaion, ha is, if he series are purely random, hen we expec only one in weny of he sample auocorrelaion coefficiens o lie ouside he confidence bands. Char 4.8 Sample auocorrelaion coefficiens for USD/DEM foreign exchange reurns Auocorrelaion 0.08 0.06 0.04 0.0 0-0.0-0.04-0.06-0.08 1 1 3 34 45 56 67 78 89 100 Lag (days) 13 This an asympoic es saisic since i relies on a large value of T, say, T > 1000. See Harvey (p. 43, 1993). Par II: Saisics of Financial Marke Reurns

58 Chaper 4. Saisical and probabiliy foundaions Char 4.9 Sample auocorrelaion coefficiens for USD S&P 500 reurns Auocorrelaion 0.08 0.06 0.04 0.0 0-0.0-0.04-0.06-0.08-0.10 1 1 3 34 45 56 67 78 89 100 Lag (days) Overall, boh chars show very lile evidence of auocorrelaion in daily log price changes. Even in he cases where he auocorrelaions are ouside he confidence bands, he auocorrelaion coefficiens are quie small (less han 10%). 4.3.. Box-Ljung saisic for daily log price changes While he above chars are useful for geing a general idea abou he level of auocorrelaion of log price changes, here are more formal mehods of esing for auocorrelaion. An ofen cied mehod is he Box-Ljung (BL) es saisic, 14 defined as [4.8] BL ( p) = T ( T + ) p k = 1 ρ k ----------- T k Under he null hypohesis ha a ime series is no auocorrelaed, BL ( p ), is disribued chisquared wih p degrees of freedom. In Eq. [4.8], p denoes he number of auocorrelaions used o esimae he saisic. We applied his es o he USD/DEM and S&P 500 reurns for p = 15. In his case, he 5% chi-squared criical value is 5. Therefore, values of he BL(10) saisic greaer han 5 implies ha here is saisical evidence of auocorrelaion. The resuls are shown in Table 4.3. Table 4.3 Box-Ljung es saisic Series BL ˆ ( 15) USD/DEM 15 S&P 500 5 14 See Wes and Cho (1995) for modificaions o his saisic. RiskMerics Technical Documen Fourh Ediion

Sec. 4.3 Invesigaing he random-walk model 59 We also applied his es o he daily log price changes of a seleced series of commodiy fuures conracs because, when ploed agains ime, hese series appear auocorrelaed. In hese ess we chose p = 10 which implies a criical value of 18.31 a he 95% confidence level. Table 4.4 presens he resuls along wih he firs order auocorrelaion coefficien,. Table 4.4 Box-Ljung saisics ρ 1 Conrac* Mauriy (mhs.) ρˆ 1 BL ˆ ( 10) WTI 1 0.0338 5.4 3 0.0586 7.60 6 0.097 13.6 1 0.133 5.70 LME Copper 3 0.075 8.48 15 0.0900 19.04 7 0.151 16.11 * Noe ha he higher auocorrelaion associaed wih conracs wih longer mauriies may be due o he fac ha such conracs are less liquid han conracs wih shor mauriies. The preceding ess show lile evidence of auocorrelaion for some daily log price change series. The fac ha he auocorrelaion is no srong agrees wih previous research. I is ofen found ha financial reurns over he shor-run (daily) are auocorrelaed bu he magniudes of he auocorrelaion are oo small (close o zero) o be economically significan. 15 For longer reurn horizons (i.e., beyond a year), however, here is evidence of significan negaive auocorrelaion (Fama and French, 1988). 4.3..3 Auocorrelaion of squared daily log price changes (reurns) As previously saed, alhough reurns (log price changes) are uncorrelaed, hey may no be independen. In he academic lieraure, such dependence is demonsraed by he auocorrelaion of he variances of reurns. Alernaively expressed, while he reurns are no auocorrelaed, heir squares are auocorrelaed. And since he expeced value of he squared reurns are variances 16, auocorrelaion in he squared reurns implies auocorrelaion in variances. The relaionship beween squared reurns and variances is eviden from he definiion of variance, σ. [4.9] σ = = E[ r E( r )] E r [ E( r )] Assuming ha he mean of he reurns is zero, i.e., E( r ) = 0, we ge σ = E r. 15 In oher words, i would be very difficul o form profiable rading rules based on auocorrelaion in daily log price changes (Tucker, 199). Also, more recen work has shown ha over shor horizons, auocorrelaion in daily reurns may be he resul of insiuional facors raher han purely inefficien markes (Boudoukh, Richardson and Whielaw, 1994). 16 This is rue if he expeced values of reurns are zero.the plausibiliy of assuming a mean of zero for daily reurns will be discussed in Secion 5.3.1.1. Par II: Saisics of Financial Marke Reurns

60 Chaper 4. Saisical and probabiliy foundaions Chars 4.10 and 4.11 show ime series of squared reurns for he USD/DEM exchange rae and for he S&P 500 index. Char 4.10 USD/DEM reurns squared USD/DEM log reurns squared (%) 18 16 14 1 10 8 6 4 0 1990 1991 199 1993 1994 1995 1996 Char 4.11 S&P 500 reurns squared S&P 500 log reurns squared (%) 14 1 10 8 6 4 0 1990 1991 199 1993 1994 1995 1996 Noice he clusers of large and small spikes in boh series. These clusers represen periods of high and low volailiy recognized in Secion 4..1. To analyze he auocorrelaion srucure of he squared reurns, as in he case of log price changes, we compue sample auocorrelaion coefficiens and he Box-Ljung saisic. Chars 4.1 and 4.13 presen correlograms for he squared reurn series of USD/DEM foreign exchange and S&P 500, respecively. RiskMerics Technical Documen Fourh Ediion

Sec. 4.3 Invesigaing he random-walk model 61 Char 4.1 Sample auocorrelaion coefficiens of USD/DEM squared reurns Auocorrelaion 0.14 0.1 0.10 0.08 0.06 0.04 0.0 0.00-0.0-0.04 1 1 3 34 45 56 67 78 89 100 Lag (days) Char 4.13 Sample auocorrelaion coefficiens of S&P 500 squared reurns Auocorrelaion 0.16 0.14 0.1 0.10 0.08 0.06 0.04 0.0 0.00-0.0 1 1 3 34 45 56 67 78 89 100 Lag (days) Par II: Saisics of Financial Marke Reurns

6 Chaper 4. Saisical and probabiliy foundaions Comparing he correlograms (Chars 4.8 and 4.9) based on daily log price changes o hose based on he squared daily log price changes (Chars 4.1 and 4.13), we find he auocorrelaion coefficiens of he squared log price changes are larger and more persisen han hose for log price changes. In fac, much of he significan auocorrelaion in he squared log price changes is posiive and well above he asympoic 95% confidence band of 4.7%. 17 The Box-Ljung saisics for he squared log price change series are presened in Table 4.5. Table 4.5 Box-Ljung saisics on squared log price changes (cv = 5) Series BL ˆ ( 15) USD/DEM 153 S&P 500 07 This able shows he dramaic effec ha he squared log price changes has on he BL es. For all hree series we rejec he null hypohesis ha he variances of daily reurns are no auocorrelaed. 18 4.3.3 Mulivariae exensions Thus far, we have focused our aenion on he empirical properies of individual reurns ime series. I appears ha he variances of reurns ha were analyzed vary wih ime and are auocorrelaed. As saed in Secion 4.3.1, reurns appear correlaed (hrough heir variances, a leas) no only across ime bu also across securiies. The laer finding moivaes a sudy of he empirical properies of correlaion, or more precisely, covariance beween wo reurn series. We invesigae wheher covariances are auocorrelaed by using he same logic applied o variances. Recall ha we deermined wheher variances are auocorrelaed by checking wheher observed squared reurns are auocorrelaed. We used Eq. [4.9] o show he relaion beween variances and squared reurns. Now, suppose we are ineresed in he covariance beween wo reurn series r 1, and r,. We can derive a relaionship beween he covariance, σ 1,, and observed reurns as follows. We begin wih a definiion of covariance beween and. r 1, r, [4.30] σ 1, = E{ [ r 1, E( r 1, )] [ r, E( r, )]} = E ( r ) E r 1 ( 1, )E( r, ), r, Assuming ha he mean of he reurns is zero for boh reurn series, we ge [4.31] σ 1, = E ( r ) 1, r, In words, Eq. [4.31] saes ha he covariance beween, and r, is he expecaion of he cross-produc of reurns minus he produc of he expecaions. In models explaining variances, he focus is ofen on squared reurns because of he presumpion ha for daily reurns, squared expeced reurns are small. Focusing on cross-producs of reurns can be jusified in he same way. r 1 17 Noe ha his confidence band may no be appropriae due o he fac ha he underlying daa are no reurns, bu squared reurns. 18 For a discussion on ess of auocorrelaion on squared reurns (residuals) see McLeod and Li (1983) and Li and Mak (1994). RiskMerics Technical Documen Fourh Ediion

Sec. 4.3 Invesigaing he random-walk model 63 Char 4.14 presens a ime series of he cross produc ( r 1, imes r, ) of he reurns on USD/DEM and USD/FRF exchange raes. This series is a proxy for he covariance beween he reurns on he wo exchange raes. Char 4.14 Cross produc of USD/DEM and USD/FRF reurns Cross produc 0.10 0.08 0.06 0.04 0.0 0-0.0 1993 1994 1995 1996 Char 4.14 shows ha he covariance (correlaion) beween he reurns on he wo exchange raes is posiive over a large segmen of he sample period. Time series generaed from he cross produc of wo reurn series no only offers insigh ino he emporal dynamics of correlaion bu also can be used in a regression conex o deermine he sabiliy of correlaions over ime. Similar o he correlogram of squared reurns, he correlogram of he cross produc of reurns on he wo exchange raes can be used o deermine wheher he covariance of hese wo series are auocorrelaed. Char 4.15 shows he auocorrelaions of he cross-producs of reurns on USD/ DEM and USD/FRF exchange raes ploed agains 50 daily lags. Char 4.15 Correlogram of he cross produc of USD/DEM and USD/FRF reurns Auocorrelaion 0.14 0.1 0.10 0.08 0.06 0.04 0.0 0-0.0-0.04-0.06 13 4 35 46 Lag (days) Par II: Saisics of Financial Marke Reurns

64 Chaper 4. Saisical and probabiliy foundaions The BL(10) es associaed wih he cross produc of reurns on he wo exchange rae series is 37, which is saisically significan (i.e., here is evidence of auocorrelaion) a he 95% confidence level. 4.4 Summary of our findings Up o his poin, Chaper 4 focused on he dynamic feaures of daily coninuously compounded reurns, oherwise known as log price changes, and developed he opic as follows: We inroduced hree versions of he random walk model o describe how financial prices evolve over ime. We used a paricular version of his model (Eq. [4.15]) o highligh he differences beween saionary (mean-revering) and nonsaionary ime series. We invesigaed he assumpions ha log price changes are idenically and independenly disribued. To deermine wheher he disribuion ha generaes reurns is idenical over ime, we ploed log price changes agains ime. From ime series plos of reurns and heir squares we observed he well documened phenomenon of volailiy clusering which implies ha he variance of daily log price changes vary over ime (i.e., hey are heeroscedasic), hus violaing he idenical assumpion. 19 To es independence, we analyzed he auocorrelaion coefficiens of boh log price changes and squared log price changes. We found ha while daily log price changes have small auocorrelaions, heir squares ofen have significan auocorrelaions. Much of his analysis has focused on shor-horizon (daily) reurns. In general, however, observed disribuions of reurns wih longer horizons, such as a monh or a quarer, are ofen differen from disribuions of daily reurns. 0 From his poin, Chaper 4 reviews how reurns are assumed o be disribued a each poin in ime. Specifically, we describe he normal disribuion in deail. In RiskMerics, i is assumed ha reurns are disribued according o he condiional normal disribuion. 4.5 A review of hisorical observaions of reurn disribuions As shown in Eq. [4.15] and Eq. [4.17], reurns were assumed o follow, respecively, an uncondiional and condiional normal disribuion. The implicaions of he assumpion ha financial reurns are normally disribued, a leas uncondiionally, has a long hisory in finance. Since he early work of Mandelbro (1963) and Fama (1965), researchers have documened cerain sylized facs abou he saisical properies of financial reurns. A large percenage of hese sudies focus on high frequency or daily log price changes. Their conclusions can be summarized in four basic observaions: Financial reurn disribuions have fa ails. This means ha exreme price movemens occur more frequenly han implied by a normal disribuion. The peak of he reurn disribuion is higher and narrower han ha prediced by he normal disribuion. Noe ha his characerisic (ofen referred o as he hin wais ) along wih fa ails is a characerisic of a lepokuroic disribuion. 19 See for example, Engle and Bollerslev (1986). 0 See, for example, Richardson and Smih (1993) RiskMerics Technical Documen Fourh Ediion

Sec. 4.5 A review of hisorical observaions of reurn disribuions 65 Reurns have small auocorrelaions. Squared reurns ofen have significan auocorrelaions. Char 4.16 illusraes a lepokuroic disribuion of log price changes in USD/DEM exchange raes for he period 8-Mar-96 hrough 1-Apr-96 and compares i o a normal disribuion. In his char, he lepokuroic disribuion can be hough of as a smoohed hisogram, since i is obained hrough a smoohing process known as kernel densiy esimaion. 1 A kernel densiy esimae of he hisogram, raher han he hisogram iself, is ofen used since i produces a smooh line ha is easier o compare o he rue densiy funcion (normal, in his example). Char 4.16 Lepokuroic vs. normal disribuion PDF 0.8 0.7 USD/DEM log reurns 0.6 0.5 0.4 0.3 Normal 0. 0.1 hin wais fa ails 0-3.4 -.3-1.1 0.0 1.1.3 3.4 Reurns (%) 4.5.1 Modeling mehods Having documened he failure of he normal disribuion o accuraely model reurns, researchers sared looking for alernaive modeling mehods, which have since evolved ino wo classes: uncondiional (ime-independen) and condiional disribuions (ime-dependen) of reurns. Models in he class of uncondiional disribuion of reurns assume ha reurns are independen of each oher and ha he reurn-generaing process is linear wih parameers ha are independen of pas realizaions. An example of a model ha falls ino his class is he sandard normal disribuion wih mean µ and variance σ (noe here is no ime subscrip). Oher examples of uncondiional disribuion models include infinie-variance symmeric and asymmeric sable Pareian disribuions, and finie variance disribuions including he -disribuion, mixed-diffusion-jump model, and he compound normal model. 1 See Silverman (1986). Par II: Saisics of Financial Marke Reurns

66 Chaper 4. Saisical and probabiliy foundaions The second class of models, he condiional disribuion of reurns, arises from evidence ha refues he idenically and independenly disribued assumpions (as presened in Secions 4.3.1 and 4.3.). Models in his caegory, such as he GARCH and Sochasic Volailiy, rea volailiy as a ime-dependen, persisen process. These models are imporan because hey accoun for volailiy clusering, a frequenly observed phenomenon among reurn series. The models for characerizing reurns are presened in Table 4.6 along wih supporing references. Table 4.6 Model classes Disribuion Model Reference Uncondiional (ime independen) Infinie variance: symmeric sable Pareian Mandelbro (1963) asymmeric sable Pareian Tucker (199) Finie variance: Normal Bachelier (1900) Suden Blaberg & Gonedes (1974) Mixed diffusion jump orion (1988) Compound normal Kon (1988) Condiional (ime dependen) GARCH: Normal Bollerslev (1986) Suden Bollerslev (1987) Sochasic Volailiy: Normal Ruiz (1994) Suden Harvey e. al (1994) Generalized error disribuion Ruiz (1994) I is imporan o remember ha while condiional and uncondiional processes are based on differen assumpions, excep for he uncondiional normal model, models from boh classes generae daa ha possess fa ails. 4.5. Properies of he normal disribuion All of he models presened in Table 4.6 are parameric in ha he underlying disribuions depend on various parameers. One of he mos widely applied parameric probabiliy disribuion is he normal disribuion, represened by is bell shaped curve. This secion reviews he properies of he normal disribuion as hey apply o he RiskMerics mehod of calculaing VaR. Recall ha he VaR of a single asse (a ime ) can be wrien as follows: [4.3] VaR = [ 1 exp ( 1.65σ 1 )]V 1 or, using he common approximaion [4.33] VaR 1.65σ 1 V 1 V 1 where is he marked-o-marke value of he insrumen and σ 1 is he sandard deviaion of coninuously compounded reurns for ime made a ime 1. For a specific comparison beween ime-dependen and ime-independen processes, see Ghose and Kroner (1993). RiskMerics Technical Documen Fourh Ediion

Sec. 4.5 A review of hisorical observaions of reurn disribuions 67 4.5..1 Mean and variance If i is assumed ha reurns are generaed according o he normal disribuion, hen i is believed ha he enire disribuion of reurns can be characerized by wo parameers: is mean and variance. Mahemaically, he normal probabiliy densiy funcion for a random variable is 3 r [4.34] f ( r ) = 1 1 ---------------- exp -------- πσ σ ( r µ ) where µ = mean of he random variable, which affecs he locaion of he disribuion s peak σ = variance of he random variable, which affecs he disribuion s widh π 3.1416 Noe ha he normal disribuion as shown in Eq. [4.34] is an uncondiional disribuion since he mean and variance parameers are no ime-dependen and, herefore, do no have ime subscrips. Char 4.17 shows how he mean and variance affec he shape of he normal disribuion. Char 4.17 Normal disribuion wih differen means and variances Sandard normal PDF 0.40 0.35 0.30 0.5 0.0 0.15 0.10 (µ=0, σ=1) (µ=5, σ=.) 0.05 (µ=0, σ=.5) 0-5 -4-3 - -1 0 1 3 4 6 Sandard deviaion Now ha we have an undersanding of he role of he mean and variance in he normal disribuion we can presen heir formulae. The mahemaical expression for he mean and variance of some random variable r, are as follows: [4.35] µ = E [ r ] (mean) σ = E ( r µ ) (variance) 3 Noe ha we are abusing noaion since r represens boh a random variable and observed reurn. We hope ha by he conex in which r is used i will be clear wha we are referring o. Par II: Saisics of Financial Marke Reurns

68 Chaper 4. Saisical and probabiliy foundaions where E[ ] denoes he mahemaical expecaion. Two addiional measures ha we will make reference o wihin his documen are known as skewness and kurosis. Skewness characerizes he asymmery of a disribuion around is mean. The expression for skewness is given by [4.36] s 3 = E ( r µ ) 3 (skewness) For he normal disribuion skewness is zero. In pracice, i is more convenien o work wih he skewness coefficien which is defined as [4.37] γ = E ( r µ ) 3 -------------------------------- (skewness coefficien) σ 3 Kurosis measures he relaive peakedness or flaness of a given disribuion. The expression for kurosis is given by [4.38] s 4 = E ( r µ ) 4 (kurosis) As in he case of skewness, in pracice, researchers frequenly work wih he kurosis coefficien defined as [4.39] κ= E ( r µ ) 4 -------------------------------- (kurosis coefficen) σ 4 For he normal disribuion, kurosis is 3. This fac leads o he definiion of excess kurosis which is defined as kurosis minus 3. 4.5.. Using perceniles o measure marke risk Marke risk is ofen measured in erms of a percenile (also referred o as quanile) of a porfolio s reurn disribuion. The araciveness of working wih a percenile raher han say, he variance of a disribuion, is ha a percenile corresponds o boh a magniude (e.g., he dollar amoun a risk) and an exac probabiliy (e.g., he probabiliy ha he magniude will no be exceeded). The ph percenile of a disribuion of reurns is defined as he value ha exceeds p percen of he reurns. Mahemaically, he ph percenile (denoed by α) of a coninuous probabiliy disribuion, is given by he following formula [4.40] α p = f ( r) dr where f (r) represens he PDF (e.g., Eq. [4.34]) So for example, he 5h percenile is he value (poin on he disribuion curve) such ha 95 percen of he observaions lie above i (see Char 4.18). When we speak of perceniles hey are ofen of he perceniles of a sandardized disribuion, which is simply a disribuion of mean-cenered variables scaled by heir sandard deviaion. For example, suppose he log price change r is normally disribued wih mean µ and variance σ. The sandardized reurn r is defined as RiskMerics Technical Documen Fourh Ediion

Sec. 4.5 A review of hisorical observaions of reurn disribuions 69 [4.41] r = r µ -------------- σ Therefore, he disribuion of r is normal wih mean 0 and variance 1. An example of a sandardized disribuion is presened above (µ = 0, σ = 1). Char 4.18 illusraes he posiions of some seleced perceniles of he sandard normal disribuion. 4 Char 4.18 Seleced percenile of sandard normal disribuion Sandard normal PDF 0.40 0.30 0.0 0.10 0-5 -4-3 - -1 0 1 3 4 6-1.8 (10h percenile) -1.65 (5h percenile) -.33 (1s percenile) Sandard deviaion We can use he perceniles of he sandard disribuion along wih Eq. [4.41] o derive he perceniles of observed reurns. For example, suppose ha we wan o find he 5h percenile of r, under he assumpion ha reurns are normally disribued. We know, by definiion, ha [4.4a] Probabiliy ( r < 1.65) = 5% [4.4b] Probabiliy [ ( r µ ) σ < 1.65] = 5% From Eq. [4.4b], re-arranging erms yields [4.43] Probabiliy ( r < 1.65σ + µ ) = 5% According o Eq. [4.43], here is a 5% probabiliy ha an observed reurn a ime is less han 1.65 imes is sandard deviaion plus is mean. Noice ha when µ = 0, we are lef wih he sandard resul ha is he basis for shor-erm horizon VaR calculaion, i.e., [4.44] Probabiliy ( r < 1.65σ ) = 5% 4 Noe ha he seleced perceniles above (1%, 5%, and 10%) reside in he ails of he disribuion. Roughly, he ails of a disribuion are he areas where less hen, say, 10% of he observaions fall. Par II: Saisics of Financial Marke Reurns

70 Chaper 4. Saisical and probabiliy foundaions 4.5..3 One-ailed and wo-ailed confidence inervals Equaion [4.44] is very imporan as he basis of VaR calculaions in RiskMerics. I should be recognized, however, ha here are differen ways of saing he confidence inerval associaed wih he same risk olerance. For example, since he normal disribuion is symmeric, hen [4.45] Probabiliy ( r < 1.65σ + µ ) = Probabiliy ( r > 1.65σ + µ ) = 5% Therefore, since he enire area under he probabiliy curve in Char 4.18 is 100%, i follows ha [4.46a] Probabiliy ( 1.65σ + µ < r < 1.65σ + µ ) = 90% [4.46b] Probabiliy ( 1.65σ + µ < r ) = 95% Chars 4.19 and 4.0 show he relaionship beween a one-ailed 95% confidence inerval and a wo-ailed 90% confidence inerval. Noice ha he saemens in Eqs. [4.46a] and [4.46b] are consisen wih Eq. [4.45], a 5% probabiliy ha he reurn being less han 1.65 sandard deviaions. 5 Char 4.19 One-ailed confidence inerval Sandard normal PDF 0.40 0.30 0.0 95% 0.10 0.00 5% -5-4 -3 - -1 0 1 3 4-1.65 Sandard deviaion 5 The wo saemens are no equivalen in he conex of formal hypohesis esing. See DeGroo (1989, chaper 8). RiskMerics Technical Documen Fourh Ediion

Sec. 4.5 A review of hisorical observaions of reurn disribuions 71 Char 4.0 Two-ailed confidence inerval Sandard normal PDF 0.40 0.30 0.0 90% 0.10 5% 0 5% -5-4 -3 - -1 0 1 3 4-1.65 1.65 Sandard deviaion Table 4.7 shows he confidence inervals ha are prescribed by sandard and BIS-complian versions of RiskMerics, and a which he one-ailed and wo-ailed ess yield he same VaR figures. 6 Table 4.7 VaR saisics based on RiskMerics and BIS/Basel requiremens Confidence inerval RiskMerics mehod One-ailed Two-ailed Sandard 95% ( 1.65σ) BIS/Basel Regulaory 99% (.33σ) 90% ( /+1.65σ ) 98% ( /+.33σ) 4.5..4 Aggregaion in he normal model An imporan propery of he normal disribuion is ha he sum of normal random variables is iself normally disribued. 7 This propery is useful since porfolio reurns are he weighed sum of individual securiy reurns. As previously saed (p. 49) RiskMerics assumes ha he reurn on a porfolio,,, is he weighed sum of N underlying reurns (see Eq. [4.1]). For pracical purposes we require a model of reurns ha no only relaes he underlying reurns o one anoher bu also relaes he disribuion of he weighed sum of he underlying reurns o he porfolio reurn disribuion. To ake an example, consider he case when N = 3, ha is, he porfolio reurn depends on hree underlying reurns. The porfolio reurn is given by r p [4.47] r p = w 1 r 1, + w r, + w 3 r 3, 6 For ease of exposiion we ignore ime subscrips. 7 These random variables mus be drawn from a mulivariae disribuion. Par II: Saisics of Financial Marke Reurns

7 Chaper 4. Saisical and probabiliy foundaions We can model each underlying reurn as a random walk ha is similar o Eq. [4.17]. This yields [4.48a] [4.48b] [4.48c], = µ 1 + σ 1 r 1 r, = µ + σ r 3, = µ + σ 3, ε 1,, ε,, ε 3, Now, since we have hree variables we mus accoun for heir movemens relaive o one anoher. These movemens are capured by pairwise correlaions. Tha is, we define measures ha quanify he linear associaion beween each pair of reurns. Assuming ha he ε s are mulivariae normally (MVN) disribued we have he model ε 1, 0 1 ρ 1, ρ 13, [4.49] ε, MVN 0, ρ 1, 1 ρ 3,, or more succincly, ε MVN ( µ, R ) ε 3, 0 ρ 31, ρ 3, 1 where parameer marix R represens he correlaion marix of ( ε 1,, ε,, ε 3, ). Therefore, if we apply he assumpions behind Eq. [4.49] (ha he sum of MVN random variables is normal) o he porfolio reurn Eq. [4.47], we know ha r p is normally disribued wih mean µ p, and variance. The formulae for he mean and variance are σ p, [4.50a] µ p, = w 1 µ 1 + w µ + w 3 µ 3 [4.50b] σ p, = w 1σp, w σp, w 3σp, + + + w 1 w σ 1, + w 1 w 3 σ 13, + w w 3 σ 3, σ ij where he erms, represen he covariance beween reurns i and j. In general, hese resuls hold for ( N 1 ) underlying reurns. Since he underlying reurns are disribued condiionally mulivariae normal, he porfolio reurn is univariae normal wih a mean and variance ha are simple funcions of he underlying porfolio weighs, variances and covariances. 4.5.3 The lognormal disribuion In Secion 4..1 we claimed ha if log price changes are normally disribued, hen price,, condiional on P 1 is lognormally disribued. This saemen implies ha P, given P 1, is drawn from he probabiliy densiy funcion P [4.51] f ( P ) = 1 ( lnp --------------------------- 1 µ ) exp -------------------------------------- P 1 > P 1 σ π σ 0 where [4.5] [4.53] P follows a lognormal disribuion wih a mean and variance given by E[ P ] = exp µ + 5σ V ( P ) = expµ exp σ exp σ RiskMerics Technical Documen Fourh Ediion

Sec. 4.6 RiskMerics model of financial reurns: A modified random walk 73 Char 4.1 shows he probabiliy densiy funcion for he lognormal random variable µ = 0, σ = 1 and P 1 = 1. P when Char 4.1 Lognormal probabiliy densiy funcion PDF 0.40 0.35 0.30 0.5 0.0 0.15 0.10 0.05 0.00 0 4 8 1 16 0 Price Unlike he normal probabiliy densiy funcion, he lognormal PDF has a lower bound greaer han zero and is skewed o he righ. 4.6 RiskMerics model of financial reurns: A modified random walk We can now use he resuls of he las four secions o wrie down a model of how reurns are generaed over ime. Our analysis has shown ha: Reurn variances are heeroscedasic (change over ime) and auocorrelaed. Reurn covariances are auocorrelaed and possess dynamic feaures. The assumpion ha reurns are normally disribued is useful because of he following: (i) only he mean and variance are required o describe he enire shape of he disribuion 8 (ii) he sum of mulivariae normal reurns is normally disribued. This fac faciliaes he descripion of porfolio reurns, which are he weighed sum of underlying reurns. Given hese poins, we can now sae he assumpions underlying he RiskMerics variance/covariance mehodology. Consider a se of N securiies, i = 1, N. The RiskMerics model assumes ha reurns are generaed according o he following model [4.54] r i, = σ i, ε i ε, ε i N ( 01, ) MVN ( 0, R ) ε = [ ε 1, ε, ε, N ] 8 The covariances are also required when here is more han one reurn series. Par II: Saisics of Financial Marke Reurns

74 Chaper 4. Saisical and probabiliy foundaions where R is an NxN ime-dependen correlaion marix. The variance of each reurn, σ i, and he correlaion beween reurns, ρ ij,, are a funcion of ime. The propery ha he disribuion of reurns is normal given a ime dependen mean and correlaion marix assumes ha reurns follow a condiional normal disribuion condiional on ime. Noice ha in Eq. [4.54] we excluded erm µ i. As will be discussed in more deail in Secion 5.3.1.1, he mean reurn represened by µ i is se o zero. In Appendix A we propose a se of saisical ess o assess wheher observed financial reurns follow a condiional normal disribuion. In Appendix B we discuss alernaive disribuions ha relax he normaliy assumpion. 4.7 Summary In his chaper, we presened he saisical and probabiliy assumpions on he evoluion and disribuion of financial reurns in some simple models. This discussion served as background o he specificaion of he assumpions behind he RiskMerics VaR mehodology. In review, his chaper covered he following subjecs. The chaper began by oulining a simple version of he VaR calculaion. We hen: Defined absolue price change, relaive price change, log price change, and reurns. Showed he imporance of undersanding he use of differen price change definiions. Esablished ha RiskMerics measures changes in porfolio value in erms of coninuouslycompounded reurns. Inroduced emporal aggregaion and cross-secion aggregaion o show he implicaions of working wih relaive and log reurns. Inroduced he random walk model for: 9 Single-price asses Fixed income insrumens Found evidence ha conradics he assumpion ha reurns are IID (idenically and independenly) normal. In realiy, coninuously compounded reurns are: No idenical over ime. (The variance of he reurn disribuion changes over ime) No saisically independen of each oher over ime. (Evidence of auocorrelaion beween reurn series and wihin a reurn series.) Explained he properies of he normal disribuion, and, lasly, Presened he RiskMerics model as a modified random walk ha assumes ha reurns are condiionally normally disribued. 9 While he random walk model serves as he basis for many popular models of reurns in finance, anoher class of models ha has received considerable aenion laely is based on he phenomenon of long-range dependence. Briefly, such models are buil on he noion ha observaions recorded in he disan pas are correlaed o observaions in he disan fuure. (See Campbell, e. al (1995) for a review of long-range dependence models.) RiskMerics Technical Documen Fourh Ediion

75 Chaper 5. Esimaion and forecas 5.1 Forecass from implied versus hisorical informaion 77 5. RiskMerics forecasing mehodology 78 5..1 Volailiy esimaion and forecasing 78 5.. Muliple day forecass 84 5..3 More recen echniques 88 5.3 Esimaing he parameers of he RiskMerics model 90 5.3.1 Sample size and esimaion issues 90 5.3. Choosing he decay facor 96 5.4 Summary and concluding remarks 100 Par II: Saisics of Financial Marke Reurns

76 RiskMerics Technical Documen Fourh Ediion

77 Chaper 5. Esimaion and forecas Peer Zangari Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-8641 zangari_peer@jpmorgan.com In his chaper we presen a mehodology for forecasing he parameers of he mulivariae condiional normal disribuion, i.e., variances and covariances of reurns whose empirical properies were examined in Chaper 4, Saisical and probabiliy foundaions. The reason for forecasing variances and covariances of reurns is o use hem o forecas a porfolio s change in value over a given horizon, which can run over one day o several monhs. This chaper is organized as follows: Secion 5.1 briefly explains why RiskMerics forecass of variances and covariances are generaed from hisorical daa raher han derived from opion prices. Secion 5. describes he RiskMerics forecasing mehodology, i.e., Use of he exponenially weighed moving average (EWMA) model o produce forecass of variances and covariances. This includes an explanaion as o why he EWMA is preferred o he simple moving average model. How o compue forecass over longer ime horizons, such as one monh. Secion 5. also discusses alernaive, more advanced mehods for forecasing variances and covariances. Secion 5.3 explains wo imporan implemenaion issues involving he RiskMerics forecass: (1) he reliabiliy of he forecass in relaion o he number of hisorical daa poins used o produce hem, and () he choice of he decay facor used in he EWMA model. Secion 5.4 concludes he chaper wih a review of he RiskMerics forecasing model. Finally, praciioners ofen refer o he erm volailiy when speaking of movemens in financial prices and raes. In wha follows we use he erm volailiy o mean he sandard deviaion of coninuously compounded financial reurns. 5.1 Forecass from implied versus hisorical informaion RiskMerics forecass are based on hisorical price daa, alhough in heory, hey may be derived from opion prices. From a pracical poin of view, implied forecass inroduce a number of problems. For example, an implied volailiy (IV) is based enirely on expecaions given a paricular opion pricing model. Therefore, as noed in Kroner, Kneafsey and Claessens (1995), since mos opion pricing models assume ha he sandard deviaion is consan, he IV becomes difficul o inerpre and will no lead o good forecass if he opion formula used o derive i is no correcly specified. Moreover, IV forecass are associaed wih a fixed forecas horizon. For example, he implied volailiy derived from a 3 monh USD/DEM opion is exclusively for a 3 monh forecas horizon. However, a risk manager may be ineresed in he VaR of his opion over he nex day. If RiskMerics were o use implied saisics, i would require observable opions prices on all insrumens ha compose a porfolio. Currenly, he universe of consisenly observable opions prices is no large enough o provide a complee se of implied saisics; generally only exchangeraded opions are reliable sources of prices. In paricular, he number of implied correlaions ha can be derived from raded opion prices is insignifican compared o he number of correlaions required o esimae risks in porfolios consising of many ypes of asses. Par II: Saisics of Financial Marke Reurns

78 Chaper 5. Esimaion and forecas Academic research has compared he forecasing abiliy of implied and hisorical volailiy models. The evidence of he superior forecasing abiliy of hisorical volailiy over implied volailiy is mixed, depending on he ime series considered. For example, Xu and Taylor (1995, p. 804) noe ha, prior research concludes ha volailiy predicors calculaed from opions prices are beer predicors of fuure volailiy han sandard deviaions calculaed from hisorical asse price daa. Kroner, Kneafsey and Claessens (1995, p. 9), on he oher hand, noe ha researchers are beginning o conclude ha GARCH (hisorical based) forecass ouperform implied volailiy forecass. Since implied sandard deviaion capures marke expecaions and pure ime series models rely solely on pas informaion, hese models can be combined o forecas he sandard deviaion of reurns. 5. RiskMerics forecasing mehodology RiskMerics uses he exponenially weighed moving average model (EWMA) o forecas variances and covariances (volailiies and correlaions) of he mulivariae normal disribuion. This approach is jus as simple, ye an improvemen over he radiional volailiy forecasing mehod ha relies on moving averages wih fixed, equal weighs. This laer mehod is referred o as he simple moving average (SMA) model. 5..1 Volailiy esimaion and forecasing 1 One way o capure he dynamic feaures of volailiy is o use an exponenial moving average of hisorical observaions where he laes observaions carry he highes weigh in he volailiy esimae. This approach has wo imporan advanages over he equally weighed model. Firs, volailiy reacs faser o shocks in he marke as recen daa carry more weigh han daa in he disan pas. Second, following a shock (a large reurn), he volailiy declines exponenially as he weigh of he shock observaion falls. In conras, he use of a simple moving average leads o relaively abrup changes in he sandard deviaion once he shock falls ou of he measuremen sample, which, in mos cases, can be several monhs afer i occurs. For a given se of T reurns, Table 5.1 presens he formulae used o compue he equally and exponenially weighed (sandard deviaion) volailiy. Table 5.1 Volailiy esimaors* Equally weighed Exponenially weighed T 1 σ= -- ( r T r ) σ = ( 1 λ) λ 1 ( r r) = 1 * In wriing he volailiy esimaors we inenionally do no use ime subscrips. T = 1 In comparing he wo esimaors (equal and exponenial), noice ha he exponenially weighed moving average model depends on he parameer λ (0 < λ <1) which is ofen referred o as he decay facor. This parameer deermines he relaive weighs ha are applied o he observaions (reurns) and he effecive amoun of daa used in esimaing volailiy. Ways of esimaing λ are discussed in deail in Secion 5.3.. 1 In his secion we refer loosely o he erms esimaion and forecas. The reader should noe, however, ha hese erms do have disinc meanings. RiskMerics Technical Documen Fourh Ediion

Sec. 5. RiskMerics forecasing mehodology 79 We poin ou ha in wriing he EWMA esimaor in Table 5.1 we applied he approximaion [5.1] T λ j 1 j = 1 ------------------ 1 ( 1 λ) These wo expressions are equivalen in he limi, i.e., as T. Moreover, for purpose of comparison o he equally weighed facor 1/T, he more appropriae version of he EWMA is [5.] T λ 1 λ j 1 j = 1 raher han ( 1 λ )λ 1. Also, noice ha when λ = 1, Eq. [5.] collapses o 1/T. Chars 5.1 and 5. highligh an imporan difference beween equally and exponenially weighed volailiy forecass using as an example he GBP/DEM exchange rae in he fall of 199. In lae Augus of ha year, he foreign exchange markes wen ino a urmoil ha led a number of Europe s currencies o leave he ERM and be devalued. The sandard deviaion esimae using an exponenial moving average rapidly refleced his sae of evens, bu also incorporaed he decline in volailiy over subsequen monhs. The simple 6-monh moving average esimae of volailiy ook longer o regiser he shock o he marke and remained higher in spie of he fac ha he foreign exchange markes calmed down over he res of he year. Char 5.1 DEM/GBP exchange rae DEM/GBP.95.85.75.65.55.45.35.5 199 1993 1994 Par II: Saisics of Financial Marke Reurns

80 Chaper 5. Esimaion and forecas Char 5. Log price changes in GBP/DEM and VaR esimaes (1.65σ) Log price change 3.00.00 1.00 0.00-1.00 -.00-3.00-4.00-5.00 199 1993 1994 VaR esimae 1.80 1.60 1.40 1.0 1.00 0.80 0.60 0.40 0.0 0 Exponenially weighed moving average Simple moving average 199 1993 1994 This example would sugges ha EWMA is more saisfacory, given ha when combined wih frequen updaes, i incorporaes exernal shocks beer han equally weighed moving averages, hus providing a more realisic measure of curren volailiy. Alhough he exponenially weighed moving average esimaion ranks a level above simple moving averages in erms of sophisicaion, i is no complex o implemen. To suppor his poin, Table 5. presens an example of he compuaion required o esimae equally and exponenially weighed moving average volailiies. Volailiy esimaes are based on 0 daily reurns on he USD/DEM exchange rae. We arbirarily choose λ = 0.94 and keep maers simple by seing he sample mean, r, o zero. RiskMerics Technical Documen Fourh Ediion

Sec. 5. RiskMerics forecasing mehodology 81 Table 5. Calculaing equally and exponenially weighed volailiy Dae A B C D Volailiy Reurn USD/DEM (%) Reurn squared (%) Equal weigh (T = 0) Exponenial weigh (λ = 0.94) Equally weighed, B C Exponenially weighed, B D 8-Mar-96 0.634 0.40 0.05 0.019 0.00 0.007 9-Mar-96 0.115 0.013 0.05 0.00 0.001 0.000 1-Apr-96-0.460 0.11 0.05 0.01 0.011 0.004 -Apr-96 0.094 0.009 0.05 0.0 0.000 0.000 3-Apr-96 0.176 0.031 0.05 0.04 0.00 0.001 4-Apr-96-0.088 0.008 0.05 0.05 0.000 0.000 5-Apr-96-0.14 0.00 0.05 0.07 0.001 0.001 8-Apr-96 0.34 0.105 0.05 0.09 0.005 0.003 9-Apr-96-0.943 0.889 0.05 0.030 0.044 0.07 10-Apr-96-0.58 0.79 0.05 0.03 0.014 0.009 11-Apr-96-0.107 0.011 0.05 0.034 0.001 0.000 1-Apr-96-0.160 0.06 0.05 0.037 0.001 0.001 15-Apr-96-0.445 0.198 0.05 0.039 0.010 0.008 16-Apr-96 0.053 0.003 0.05 0.041 0.000 0.000 17-Apr-96 0.15 0.03 0.05 0.044 0.001 0.001 18-Apr-96-0.318 0.101 0.05 0.047 0.005 0.005 19-Apr-96 0.44 0.180 0.05 0.050 0.009 0.009 -Apr-96-0.708 0.501 0.05 0.053 0.05 0.07 3-Apr-96-0.105 0.011 0.05 0.056 0.001 0.001 4-Apr-96-0.57 0.066 0.05 0.060 0.003 0.004 Sandard deviaion: Equally weighed 0.393 Exponenially weighed 0.333 Noice ha he difference beween he wo esimaed sandard deviaions resuls from he differen weighing schemes. Whereas he equally weighed approach weighs each squared reurn by 5%, he exponenially weighed scheme applies a 6% weigh o he mos recen squared reurn and 1.9% weigh o he mos disan observaion. An aracive feaure of he exponenially weighed esimaor is ha i can be wrien in recursive form which, in urn, will be used as a basis for making volailiy forecass. In order o derive he recursive form, i is assumed ha an infinie amoun of daa are available. For example, assuming again ha he sample mean is zero, we can derive he period + 1 variance forecas, given daa available a ime (one day earlier) as [5.3] σ 1, + 1 = λσ 1, 1 + ( 1 λ)r 1, The 1-day RiskMerics volailiy forecas is given by he expression [5.4] σ 1, + 1 = λσ 1, 1 + ( 1 λ)r 1, Par II: Saisics of Financial Marke Reurns

8 Chaper 5. Esimaion and forecas The subscrip + 1 is read he ime + 1 forecas given informaion up o and including ime. The subscrip 1 is read in a similar fashion. This noaion underscores he fac ha we are reaing he variance (volailiy) as ime-dependen. The fac ha his period s variance forecas depends on las period s variance is consisen wih he observed auocorrelaion in squared reurns discussed in Secion 4.3. We derive Eq. [5.3] as follows. [5.5] σ 1, + 1 = = = = ( 1 λ ) λ i r 1, i (1-λ ) i = 0 r 1, (1-λ )r 1, λσ 1, 1 λr 1, 1 λ + + r 1, + + λ( 1 λ) r 1, 1 + λr 1, + r 1 + ( 1 λ)r 1,, 3 Using daily reurns, Table 5.3 presens an example of how Eq. [5.3] can be used in pracice o produce a 1-day volailiy forecas on USD/DEM reurns for he period March 8 hrough April 4, 1996. Table 5.3 Applying he recursive exponenial weighing scheme o compue volailiy Daily reurns on USD/DEM Dae A Reurn USD/DEM B Recursive variance Dae The volailiy forecas made on April 4 for he following day is he square roo of 0.4% (he variance) which is 0.473%. 5..1.1 Covariance and correlaion esimaion and forecass We use he EWMA model o consruc covariance and correlaion forecass in he same manner as we did volailiy forecass excep ha insead of working wih he square of one series, we work wih he produc of wo differen series. Table 5.4 presens covariance esimaors based on equally and exponenially weighed mehods. A Reurn USD/DEM B Recursive variance 8-Mar-96 0.633 0.401 11-Apr-96 0.107 0.96 9-Mar-96 0.115 0.378 1-Apr-96 0.159 0.80 1-Apr-96 0.459 0.368 15-Apr-96 0.445 0.75 -Apr-96 0.093 0.346 16-Apr-96 0.053 0.58 3-Apr-96 0.176 0.37 17-Apr-96 0.15 0.44 4-Apr-96 0.087 0.308 18-Apr-96 0.318 0.36 5-Apr-96 0.14 0.91 19-Apr-96 0.44 0.3 8-Apr-96 0.34 0.80 -Apr-96 0.708 0.48 9-Apr-96 0.943 0.316 3-Apr-96 0.105 0.34 10-Apr-96 0.58 0.314 4-Apr-96 0.57 0.4 *Iniial variance forecas = iniial reurn squared. Figures following his number are obained by applying he recursive formula. RiskMerics Technical Documen Fourh Ediion

Sec. 5. RiskMerics forecasing mehodology 83 Table 5.4 Covariance esimaors Equally weighed Exponenially weighed T 1 σ 1 j 1 = -- ( r T 1 r 1 ) ( r 1 r ) σ 1 = (1-λ ) λ ( r 1 r 1 ) ( r 1 r ) = 1 T j = 1 Analogous o he expression for a variance forecas (Eq. [5.3]), he covariance forecas can also be wrien in recursive form. For example, he 1-day covariance forecas beween any wo reurn series, r 1, and r, made a ime is [5.6] σ 1, + 1 = λσ 1, 1 + ( 1 λ)r 1 r We can derive Eq. [5.6] as follows. [5.7] σ 1, + 1 = = = = ( 1 λ) λ i r 1 i i = 0 ( 1 λ) r 1 ( 1 λ)r 1 + λ ( 1 λ) r 1 λσ 1, 1, r, i +, r, λr 1, 1 r 1, r, +, 1 r, 1 ( 1 λ)r 1 +, 1 r, 1 +, λ r r, + λr 1, r, + λ r 1,, 3 r, 3 In order o derive correlaion forecass we apply he corresponding covariance and volailiy forecas. Recall ha correlaion is he covariance beween he wo reurn series, say, r 1, and r,, divided by he produc of heir sandard deviaions. Mahemaically, he one-day RiskMerics predicion of correlaion is given by he expression [5.8] σ 1, + 1 ρ 1, + 1 = -------------------------------------- σ 1, + 1 σ, + 1 Table 5.5 presens an example of how o compue recursive covariance and correlaion forecass applied o he USD/DEM exchange rae and S&P 500 reurn series. Par II: Saisics of Financial Marke Reurns

84 Chaper 5. Esimaion and forecas Table 5.5 Recursive covariance and correlaion predicor Dae Reurns USD/DEM (%) Reurns S&P 500 (%) Recursive variance USD/DEM Recursive variance S&P 500 Recursive covariance (λ = 0.94) Recursive correlaion (λ = 0.94) 8-Mar-96 0.634 0.005 0.40 0.000 0.003 1.000 9-Mar-96 0.115 0.53 0.379 0.017 0.001 0.011 1-Apr-96-0.460 1.67 0.369 0.11 0.036 0.176 -Apr-96 0.094 0.34 0.347 0.109 0.03 0.166 3-Apr-96 0.176 0.095 0.38 0.103 0.09 0.160 4-Apr-96-0.088 0.003 0.309 0.097 0.08 0.160 5-Apr-96-0.14 0.144 0.91 0.09 0.05 0.151 8-Apr-96 0.34 1.643 0.80 0.49 0.055 0.09 9-Apr-96-0.943 0.319 0.317 0.40 0.034 0.13 10-Apr-96-0.58 1.36 0.315 0.337 0.011 0.035 11-Apr-96-0.107 0.367 0.96 0.35 0.013 0.04 1-Apr-96-0.160 0.87 0.80 0.351 0.004 0.01 15-Apr-96-0.445 0.904 0.75 0.379 0.00 0.063 16-Apr-96 0.053 0.390 0.59 0.365 0.018 0.059 17-Apr-96 0.15 0.57 0.45 0.360 0.0 0.073 18-Apr-96-0.318 0.311 0.36 0.344 0.06 0.093 19-Apr-96 0.44 0.7 0.33 0.37 0.019 0.069 -Apr-96-0.708 0.436 0.49 0.318 0.036 0.19 3-Apr-96-0.105 0.568 0.35 0.319 0.038 0.138 4-Apr-96-0.57 0.17 0.4 0.30 0.03 0.14 Noe ha he saring poins for recursion for he covariance is 0.634 0.005. From Table 5.5 we can see ha he correlaion predicion for he period 4-Apr-96 hrough 5-Apr-96 is 1.4%. 5.. Muliple day forecass Thus far, we have presened 1-day forecass which were defined over he period hrough + 1, where each represens one business day. Risk managers, however, are ofen ineresed in forecas horizons greaer han one-day. We now demonsrae how o consruc variance (sandard deviaion) and covariance (correlaion) forecass using he EWMA model over longer ime horizons. Generally speaking, he T-period (i.e., over T days) forecass of he variance and covariance are, respecively, [5.9] σ 1, + T = Tσ 1, + 1 or σ 1 =, + T Tσ 1, + 1 and [5.10] σ 1, + T = Tσ 1, + 1 Equaions [5.9] and [5.10] imply ha he correlaion forecass remain unchanged regardless of he forecas horizon. Tha is, RiskMerics Technical Documen Fourh Ediion

Sec. 5. RiskMerics forecasing mehodology 85 [5.11] Tσ 1, + 1 ρ + T Tσ 1, + 1 Tσ, + 1 = ------------------------------------------------------- = ρ + 1 Noice ha muliple day forecass are simple muliples of one-day forecass. For example, if we define one monh o be equivalen o 5 days, hen he 1-monh variance and covariance forecass are 5 imes he respecive 1-day forecass and he 1-monh correlaion is he same as he one-day correlaion. We now show how we arrive a Eq. [5.9] and Eq. [5.10]. Recall ha RiskMerics assumes ha log prices p are generaed according o he model [5.1] p 1, = p 1, 1 + σ 1, ε 1, ε 1, IID N ( 01, ) Recursively solving Eq. [5.1] and wriing he model in erms of reurns, we ge [5.13] = T r 1, + T σ 1 s = 1, + s ε 1, + s Taking he variance of Eq. [5.13] as of ime implies he following expression for he forecas variance [5.14] σ 1, + T, + T = E r 1 = T s = 1 E σ 1, + s Similar seps can be used o find he T days-ahead covariance forecas, i.e., [5.15] σ 1, + T = E [ r 1, + T r ] = E σ 1, + T T s = 1, + s Now, we need o evaluae he righ-hand side of Eq. [5.14] and Eq. [5.15]. To do so, we work wih he recursive form of he EWMA model for he variance and covariance. To make maers concree, consider he case where we have wo (correlaed) reurn series, r 1, and r,. In vecor form 3, le s wrie he 1-day forecas of he wo variances and covariance as follows: In RiskMerics, 1-day and 1-monh forecass differ because we use differen decay facors when making he forecass. 3 We use he vec represenaion as presened in Engle and Kroner (1995). Par II: Saisics of Financial Marke Reurns

86 Chaper 5. Esimaion and forecas [5.16] σ + 1 = = σ 1, + 1 σ 1, + 1 σ, + 1 λ 00 0λ0 00λ σ 1, 1 σ 1, 1 σ, 1 + 1 λ 0 0 0 1 λ 0 0 0 1 λ r 1, r 1, r, r, Using he expecaion operaor a ime, wrie he forecas over S days as [5.17] E σ + s = λ 00 0λ0 00λ E σ 1, + s 1 E σ 1, + s 1 E σ, + s 1 + 1 λ 0 0 0 1 λ 0 0 0 1 λ E r 1, + s 1 E [ r 1, + s 1 r ] E r, + s 1, + s 1 Evaluaing he expecaions of he squared reurns and heir cross produc yields [5.18] E σ + s = λ 00 1 λ 0 0 E σ 1, + s 1 0λ0 + 0 1 λ 0 E σ 1, + s 1 00λ 0 0 1 λ E σ, + s 1 = E σ + s 1 Tha is, he variance forecass for wo consecuive periods are he same. Consequenly, he T-period forecas is defined as [5.19] σ + T = = T s = 1 E σ + s T E σ + 1 so ha he T-period forecas of he variance/covariance vecor is [5.0] σ + T = T σ + 1 This leads o he square roo of ime relaionship for he sandard deviaion forecas RiskMerics Technical Documen Fourh Ediion

Sec. 5. RiskMerics forecasing mehodology 87 [5.1] σ 1, T + = T σ 1, + 1 Having found ha volailiy and covariance forecass scale wih ime, a few poins are worh noing abou Eq. [5.1]. Typically, he square roo of ime rule resuls from he assumpion ha variances are consan. Obviously, in he above derivaion, volailiies and covariances vary wih ime. Implicily, wha we are assuming in modeling he variances and covariances as exponenially weighed moving averages is ha he variance process is nonsaionary. Such a model has been sudied exensively in he academic lieraure (Nelson 1990, Lumsdaine, 1995) and is referred o as he IGARCH model. 4 In pracice, scaling up volailiy forecass may someimes lead o resuls ha do no make much sense. Three insances when scaling up volailiy esimaes prove problemaic are: When raes/prices are mean-revering (see Secion 4..3) When boundaries limi he poenial movemens in raes and prices When esimaes of volailiies opimized o forecas changes over a paricular horizon are used for anoher horizon (jumping from daily o annual forecass, for example). Take he simple example of he Duch guilder o Deusche mark exchange rae. On March, 1995, he cross rae as quoed a London close of business was 1.1048 NLG/DEM. The RiskMerics daily volailiy esimae was 0.1648%, which mean ha over he nex 4 hours, he rae was likely o move wihin a 1.1186 o 1.13 range wih 90% probabiliy (he nex day s rae was 1.113 NLG/DEM). The Neherlands and Germany have mainained bilaeral.5% bands wihin he ERM so scaling up a daily volailiy esimae can quickly lead o exchange rae esimaes which are exremely unlikely o occur in realiy. An example of his is shown by Char 5.3: Char 5.3 NLG/DEM exchange rae and volailiy NLG/DEM 1.16 1.15 1.14 1.13 1.1 1.11 1.1 1.09 cenral rae.5% -.5% 1.08 Mar-1995 un-1995 Sep-1995 Dec-1995 Mar-1996 4 Noe ha whereas we essenially arrive a a model ha reflecs an IGARCH (wihou an inercep), our moivaion behind is derivaion was more boom up in he sense ha we waned o derive a model ha is generally consisen wih observed reurns while being simple o implemen in pracice. The formal approach o IGARCH is more op down in ha a formal saisical model is wrien down which hen maximum likelihood esimaion is used o esimae is parameers. Par II: Saisics of Financial Marke Reurns

88 Chaper 5. Esimaion and forecas Applying he square roo of ime rule wih cauion does no apply exclusively o exchange raes ha are consrained by poliical arrangemens. Suppose you had been rying o forecas he S&P 500 s poenial annual volailiy on April 5, 1994. The index sood a 448.3 and is previous declines had increased he daily volailiy esimae o 1.39%. Char 5.4 exends his daily volailiy esimae ou o he end of he firs quarer of 1995 using he square roo of ime rule. The char shows how a shor erm increase in daily volailiy would bias an esimae of volailiy over any oher ime horizon, for example, a year. Char 5.4 S&P 500 reurns and VaR esimaes (1.65σ) S&P 500 600 550 500 450 400 1.39 Daily volailiy 1.6 1.4 1. 1.0 0.8 350 Cone of poenial S&P 500 levels (90% confidence) 300 an-1994 May-1994 Sep-1994 an-1995 0.6 0.4 The preceding wo examples underscore he imporance of undersanding how volailiy esimaes for horizons longer han a day are calculaed. When daily volailiy forecass are scaled, nonsensical resuls may occur because he scale facor does no accoun for real-world resricions. 5..3 More recen echniques Research in finance and economerics has devoed significan effors in recen years o come up wih more formal mehods o esimae sandard deviaions and correlaions. These are ofen referred o as volailiy models. The mehods range from exreme value echniques (Parkinson, 1980) and wo sep regression analysis (Davidian and Carroll, 1987), o more complicaed nonlinear modelling such as GARCH (Bollerslev, 1986), sochasic volailiy (Harvey e. al, 1994) and applicaions of chaoic dynamics (LeBaron, 1994). Among academics, and increasingly among praciioners, GARCH-ype models have gained he mos aenion. This is due o he evidence ha ime series realizaions of reurns ofen exhibi ime-dependen volailiy. This idea was firs formalized in Engle s (198) ARCH (Auo Regressive Condiional Heeroscedasiciy) model which is based on he specificaion of condiional densiies a successive periods of ime wih a imedependen volailiy process. Of he mehods jus menioned, he leas compuaionally demanding procedures for esimaing volailiy are he exreme value and regression mehods. Exreme value esimaors use various ypes of daa such as high, low, opening and closing prices and ransacion volume. While his approach is known for is relaive efficiency (i.e., small variance), i is subjec o bias. On he oher hand, he wo sep regression mehod reas he underlying volailiy model as a regression involving he absolue value of reurns on lagged values. Applicaions of his mehod o monhly volailiy can be found in Schwer (1989) and Pagan and Schwer (1990). Since he inroducion of he basic ARCH model, exensions include generalized ARCH (GARCH), Inegraed GARCH (IGARCH), Exponenial GARCH (EGARCH) and Swiching RiskMerics Technical Documen Fourh Ediion

Sec. 5. RiskMerics forecasing mehodology 89 Regime ARCH (SWARCH), jus o name a few. Numerous ess of GARCH-ype models o foreign exchange and sock markes have demonsraed ha hese relaively sophisicaed approaches can provide somewha beer esimaes of volailiy han simple moving averages, paricularly over shor ime horizons such as a day or a week. More recen research on modeling volailiy involves Sochasic Volailiy (SV) models. In his approach, volailiy may be reaed as an unobserved variable, he logarihm of which is modeled as a linear sochasic process, such as an auoregression. Since hese models are quie new, heir empirical properies have ye o be esablished. However, from a pracical poin of view, an appealing feaure of he SV models is ha heir esimaion is less dauning han heir counerpar EGARCH models. 5 In a recen sudy, Wes and Cho (1995) found ha GARCH models did no significanly ouperform he equally weighed sandard deviaion esimaes in ou-of-sample forecass, excep for very shor ime horizons. In anoher sudy on foreign exchange raes and equiy reurns, Heynen and Ka (1993) showed ha while GARCH models have beer predicive abiliy for foreign exchange, he advanage over a simple random walk esimaor disappears when he oulook period chosen is more han 0 days. We have eleced o calculae he volailiies and correlaions in he RiskMerics daa se using exponenial moving averages. This choice is viewed as an opimal balance given he consrains under which mos risk managemen praciioners work. Since he GARCH models are becoming more popular among praciioners, we demonsrae he behavior of he daily volailiy esimaor by comparing is forecass o hose produced by a GARCH(1,1) volailiy model wih normal disurbances. If r represens he ime daily reurn, hen he reurn generaing process for he GARCH(1,1) volailiy model is given by [5.] r = σ ε ε IID N ( 01, ) σ = 0.0147 + 0.881σ 1 + 0.088r 1 This model is parameerized according o he resuls produced in Ruiz (1993). They were esimaed from daily reurn daa for he Briish pound. The following graph shows variance forecass produced by his model and he exponenial esimaor wih he decay facor se o 0.94. The forecass from he EWMA are based on he following equaion: [5.3] σ + 1 = 0.94σ 1 + 0.06r 5 Bayesian SV models, on he oher hand, are compuaionally inensive. Par II: Saisics of Financial Marke Reurns

90 Chaper 5. Esimaion and forecas Char 5.5 GARCH(1,1)-normal and EWMA esimaors GBP parameers Variance (%) 1.8 1.6 1.4 1. 1.0 0.8 0.6 0.4 0. 0 Exponenial GARCH (1,1) Noice from Char 5.5, he dynamics of he exponenial model's forecass closely mimic hose produced by he GARCH(1,1) model. This should no be surprising given our findings ha he exponenial model is similar in form o he IGARCH model. A naural exension of univariae GARCH and Sochasic Volailiy models has been o model condiional covariances and correlaions. Wih he abiliy o esimae more parameers of he reurn generaing process comes growing compuaional complexiy. 6 Ofen, o make models racable, resricions are placed on eiher he process describing he condiional covariance marix or he facors ha explain covariance dynamics. Recen discussion and applicaions of mulivariae GARCH models include Engle and Kroner (1995), Karolyi (1995), King, Senena and Wadhwani (1994). Harvey (1993) presens work on mulivariae exensions o he sochasic volailiy models. 5.3 Esimaing he parameers of he RiskMerics model In his secion we address wo imporan issues ha arise when we esimae RiskMerics volailiies and correlaions. The firs issue concerns he esimaion of he sample mean. In pracice, when we make volailiy and correlaion forecass we se he sample mean o zero. The second issue involves he esimaion of he exponenial decay facor which is used in volailiy and correlaion forecass. 5.3.1 Sample size and esimaion issues Whenever we mus esimae and/or forecas means, sandard deviaions and correlaions, we would like o be reasonably confiden in he resuls. Here, confidence is measured by he sandard error of he esimae or forecas; in general, he smaller he sandard error, he more confiden we are abou is value. I is imporan, herefore, o use he larges samples available when compuing hese saisics. We illusrae he relaionship beween sample size and confidence inervals nex. For ease of exposiion we use equally weighed saisics. The resuls presened below carry over o he case of exponenially weighed saisics as well. 6 Wih respec o he required compuaion of he bivariae EGARCH model, Braun, Nelson and Sunier (1991) noe ha, ease of compuaion is, alas, no a feaure even of he bivariae model. For, example, he FORTRAN code for compuing he analyic derivaives ran o fory pages. RiskMerics Technical Documen Fourh Ediion

Sec. 5.3 Esimaing he parameers of he RiskMerics model 91 5.3.1.1 The sample mean Table 5.6 shows ha he mean esimaes for USD/DEM foreign exchange reurns and S&P 500 reurns are 0.114 and 0.010 percen, respecively. To show he variabiliy of he sample mean, Char 5.6 presens hisorical esimaes of he sample mean for USD/DEM exchange rae reurns. Each esimae of he mean is based on a 74-day rolling window, ha is, every day in he sample period we esimae a mean based on reurns over he las 74 days. Table 5.6 Mean, sandard deviaion and correlaion calculaions USD/DEM and S&P500 reurns Reurns Dae USD/DEM S&P 500 8-Mar-96 0.634 0.005 9-Mar-96 0.115 0.53 1-Apr-96 0.460 1.67 -Apr-96 0.094 0.34 3-Apr-96 0.176 0.095 4-Apr-96 0.088 0.003 5-Apr-96 0.14 0.144 8-Apr-96 0.34 1.643 9-Apr-96 0.943 0.319 10-Apr-96 0.58 1.36 11-Apr-96 0.107 0.367 1-Apr-96 0.160 0.87 15-Apr-96 0.445 0.904 16-Apr-96 0.053 0.390 17-Apr-96 0.15 0.57 18-Apr-96 0.318 0.311 19-Apr-96 0.44 0.7 -Apr-96 0.708 0.436 3-Apr-96 0.105 0.568 4-Apr-96 0.57 0.17 Mean 0.114 0.010 Sandard deviaion 0.393 0.688 Correlaion 0.180 Par II: Saisics of Financial Marke Reurns

9 Chaper 5. Esimaion and forecas Char 5.6 USD/DEM foreign exchange Sample mean 0.5 0.0 0.15 0.10 0.05 0-0.05-0.10-0.15-0.0-0.5 anuary 1987 hrough March 1993 Char 5.6 shows how he esimaes of he mean of reurns on USD/DEM flucuae around zero. An ineresing feaure of he equally weighed sample mean esimaor is ha he mean esimae does no depend direcly on he number of observaions used o consruc i. For example, recall ha he 1-day log reurn is defined as r = ln ( P P 1 ) = p p 1. Now, he sample mean of reurns for he period = 1,, T is [5.4] r T 1 = -- p T p 1 = = 1 1 -- ( p T T p 0 ) Hence, we see ha he sample mean esimaor depends only on he firs and las observed prices; all oher prices drop ou of he calculaion. Since his esimaor does no depend on he number of observed prices beween = 0 and = T bu raher on he lengh of he sample period, neiher does is sandard error. The implicaion of his effec can bes be demonsraed wih a simple example. 7 Suppose a price reurn has a sandard deviaion of 10 percen and we have 4 years of hisorical price daa. The sandard deviaion of he sample mean is 10 4 = 5 percen. So, if he average annual reurn were 0 percen over he 4-year sample (which consiss of over 1000 daa poins), a 95 percen confidence region for he rue mean would range from 10 percen o 30 percen. In addiion, recall ha he variance of a reurns series,, can be wrien as σ E r [ E( r )] =. orion (1995) noes ha wih daily daa he average erm E r dominaes he erm [ E( r )] by a ypical facor of 700 o one. Therefore, ignoring expeced reurns is unlikely o cause a percepible bias in he volailiy esimae. To reduce he uncerainy and imprecision of he esimaed mean, i may be more accurae o se he mean o some value which is consisen wih financial heory. In RiskMerics, we assume ha he mean value of daily reurns is zero. Tha is, sandard deviaion esimaes are cen- r 7 This example is adaped from Figlewski, (1994). RiskMerics Technical Documen Fourh Ediion

Sec. 5.3 Esimaing he parameers of he RiskMerics model 93 ered around zero, raher han he sample mean. Similarly, when compuing he covariance, deviaions of reurns are aken around zero raher han he sample mean. 5.3.1. Volailiy and correlaion Volailiy and correlaion forecass based on he EWMA model requires ha we choose an appropriae value of he decay facor λ. As a pracical maer, i is imporan o deermine he effecive number of hisorical observaions ha are used in he volailiy and correlaion forecass. We can compue he number of effecive days used by he variance (volailiy) and covariance (correlaion) forecass. To do so, we use he meric [5.5] Ω K = ( 1 λ) λ = K Ω K Seing equal o a value he olerance level ( ) we can solve for K, he effecive number of days of daa used by he EWMA. The formula for deermining K is ϒ L [5.6] K = lnϒ ---------- L lnλ Equaion [5.6] is derived as follows [5.7] Ω K = ( 1 λ ) λ = = K ϒ L which implies [5.8] λ K ( 1 λ) 1+ λ+ λ + = ϒ L Solving Eq. [5.8] for K we ge Eq. [5.6]. Table 5.7 shows he relaionship beween he olerance level, he decay facor, and he effecive amoun of daa required by he EWMA. Par II: Saisics of Financial Marke Reurns

94 Chaper 5. Esimaion and forecas Table 5.7 The number of hisorical observaions used by he EWMA model daily reurns Decay facor 0.001% 0.01% 0.1% 1 % 0.85 71 57 43 8 0.86 76 61 46 31 0.87 83 66 50 33 0.88 90 7 54 36 0.89 99 79 59 40 0.9 109 87 66 44 0.91 1 98 73 49 0.9 138 110 83 55 0.93 159 17 95 63 0.94 186 149 11 74 0.95 4 180 135 90 0.96 8 6 169 113 0.97 378 30 7 151 0.98 570 456 34 8 0.99 1146 916 687 458 For example, seing a olerance level o 1% and he decay facor o 0.97, we see he EWMA uses approximaely 151 days of hisorical daa o forecas fuure volailiy/correlaion. Char 5.7 depics he relaionship beween he olerance level and he amoun of hisorical daa implied by he decay facor Char 5.7 Tolerance level and decay facor Days of hisorical daa a olerance level: Daa poin weighs 3.0.5 λ = 0.97.0 1.5 1.0 Σ weighs = 1% 0.5 0 151 days Days of hisorical daa RiskMerics Technical Documen Fourh Ediion

Sec. 5.3 Esimaing he parameers of he RiskMerics model 95 Char 5.8 shows he relaionship beween he number of days of daa required by EWMA and various values of he decay facor. Char 5.8 Relaionship beween hisorical observaions and decay facor Number of hisorical daa poins 1000 900 800 700 600 500 400 151 days 74 days 300 00 100 0 0.85 0.87 0.88 0.90 0.91 0.93 0.94 0.96 0.97 0.99 Exponenial decay facor For a differen perspecive on he relaionship beween he number of daa poins used and differen values of he decay facor, consider Char 5.9. I shows he weighs for differen decay facors over a fixed window size of T = 100 (approximaely 6 monhs of daa). Char 5.9 Exponenial weighs for T = 100 decay facors = 1,.99,.97,.95,.93 a exp() 0.07 0.93 0.06 0.05 0.95 0.04 0.03 0.97 0.0 0.01 1.00 0.99 0 100 75 50 5 Days of hisorical daa Noe ha while he decay facor of 0.93 weighs he mos recen daa more han he facor 0.99, afer 40 days, he weigh associaed wih he decay facor of 0.93 is below he weigh of 0.99. Hence, he closer he decay facor is o 1, he less responsive i is o he mos recen daa. Par II: Saisics of Financial Marke Reurns

96 Chaper 5. Esimaion and forecas Now we consider he effec of sample size on volailiy and correlaion forecass. Char 5.10 presens wo hisorical ime series of 1-day volailiy forecass on he reurns series in USD/DEM exchange rae. One volailiy series was consruced wih a decay facor of 0.85, he oher used 0.98. (Refer o Table 5.7 for he relaionship beween he decay facor and he amoun of daa used). Char 5.10 One-day volailiy forecass on USD/DEM reurns λ = 0.85 (black line), λ = 0.98 (gray line) Sandard deviaion 1.6 1.4 1. 1.0 0.8 0.6 0.4 0. 0 1993 1994 1995 1996 As expeced, he volailiy forecass based on more hisorical observaions are smooher han hose ha rely on much less daa. One-day forecass of correlaion beween he reurns on he USD/DEM foreign exchange rae and S&P 500 for wo differen decay facors are presened in Char 5.11. Char 5.11 One-day correlaion forecass for reurns on USD/DEM FX rae and on S&P500 λ = 0.85 (black line), λ = 0.98 (gray line) Correlaion 0.6 0.4 0. 0-0. -0.4-0.6 1993 1994 1995 1996 Again, he ime series wih he higher decay facor produces more sable (hough no necessarily more accurae) forecass. RiskMerics Technical Documen Fourh Ediion

Sec. 5.3 Esimaing he parameers of he RiskMerics model 97 5.3. Choosing he decay facor In his secion we explain how we deermine he decay facors (λ s) ha are used o produce he RiskMerics volailiy and correlaion forecass. We begin by describing he general problem of choosing opimal λ s for volailiies and correlaions ha are consisen wih heir respecive covariance marix. We hen discuss how RiskMerics chooses is wo opimal decay facors; one for he daily daa se (λ = 0.94), and he oher for he monhly daa se (λ = 0.97). RiskMerics produces volailiy and correlaion forecass on over 480 ime series. This requires 480 variance forecass and 114,960 covariance forecass. Since hese parameers comprise a covariance marix, he opimal decay facors for each variance and covariance forecas are no independen of one anoher. We explain his concep wih a simple example ha consiss of wo reurn series, and. The covariance marix associaed wih hese reurns is given by r 1, r, [5.9] Σ = σ 1 λ1 σ 1 λ 3 ( ) σ 1 ( λ 3 ) ( ) σ ( λ ) We wrie each parameer explicily as a funcion of is decay facor. As we can see from Eq. [5.9], he covariance marix, Σ, is a funcion of 3 decay facors, λ 1, λ and λ 3. Now, Σ, o be properly defined mus conain cerain properies. For example, Σ mus be such ha he following hree condiions are me: σ 1 σ The variances, and, canno be negaive σ 1 σ 1 The covariances and mus be equal (i.e., Σ is symmeric) r 1 The correlaion beween, and r, has he range 1 ρ 1. (Recall he definiion of correlaion, ρ, ρ = σ 1 ( σ 1 σ ). I follows hen ha decay facors mus be chosen such ha hey no only produce good forecass of fuure variances and covariances, bu ha he values of hese decay facors are consisen wih he properies of he covariance marix o which hey belong. In heory, while i is possible o choose opimal decays facors ha are consisen wih heir respecive covariance marix, in pracice his ask is exceedingly complex for large covariance marices (such as he kind ha RiskMerics produces ha has 140,000 elemens). Therefore, i becomes necessary o pu some srucure (resricions) on he opimal λ s. RiskMerics applies one opimal decay facor o he enire covariance marix. Tha is, we use one decay facor for he daily volailiy and correlaion marix and one for he monhly volailiy and correlaion marix. This decay facor is deermined from individual variance forecass across 450 ime series (his process will be discussed in Secion 5.3..). Recenly, Crnkovic and Drachman (1995) 8 have shown ha while i is possible o consruc a covariance marix wih differen decay facors ha is posiive semi-definie, his marix is subjec o subsanial bias. 9 We now describe a measure applied by RiskMerics o deermine he opimal decay facor, i.e., ha decay facor ha provides superior forecas accuracy. 8 From personal communicaion. 9 See Secion 8.3 for an explanaion of posiive semi-definie and is relaionship o covariance marices. Par II: Saisics of Financial Marke Reurns

98 Chaper 5. Esimaion and forecas 5.3..1 Roo mean squared error (RMSE) crierion 10 The definiion of he ime + 1 forecas of he variance of he reurn, r + 1, made one period earlier is simply E r + 1 = σ + 1, i.e., he expeced value of he squared reurn one-period earlier. Similarly, he definiion of he ime + 1 forecas of he covariance beween wo reurn series, r 1, + 1 and r, + 1 made one period earlier is E [ r 1, + 1 r, + 1 ] = σ 1, + 1. In general, hese resuls hold for any forecas made a ime + j, j 1. Now, if we define he variance forecas error as ε + 1 = r σ i hen follows ha he + 1 + 1 expeced value of he forecas error is zero, i.e., E [ ε + 1 ] = E r. Based on + 1 σ + 1 = 0 his relaion a naural requiremen for choosing λ is o minimize average squared errors. When applied o daily forecass of variance, his leads o he (daily) roo mean squared predicion error which is given by T 1 [5.30] RMSE v = ---- r (variance) T + 1 σˆ + 1 ( λ ) = 1 where he forecas value of he variance is wrien explicily as a funcion of λ. In pracice we find he opimal decay facor λ* by searching for he smalles RMSE over differen values of λ. Tha is, we search for he decay facor ha produces he bes forecass (i.e., minimizes he forecas measures). Alhough RiskMerics does no assess he accuracy of covariance forecass, similar resuls o hose for he variance can be derived for covariance forecass, i.e., he covariance forecas error is ε 1, + 1 = r 1, + 1 r σ such ha, + 1 1, + 1 E [ ε ] 1, + 1 = E [ r 1, + 1 r ] σ, + 1 1, + 1 = 0 and T 1 [5.31] RMSE c ---- = r (covariance) T 1, + 1 r, + 1 σˆ 1, + 1 ( λ) = 1 The measures presened above are purely saisical in naure. For risk managemen purposes, his may no be opimal since oher facors come ino play ha deermine he bes forecas. For example, he decay facor should allow enough sabiliy in he variance and covariance forecass so ha hese forecass are useful for risk managers who do no updae heir sysems on a daily basis. 11 Nex, we explain how we deermine he wo RiskMerics opimal decay facors, one for daily and one for monhly forecass. 10 See Appendix C for alernaive measures o assess forecas accuracy. 11 Wes, Edison and Cho (1993) suggesed ha an ineresing alernaive basis for comparing forecass is o calculae he uiliy of an invesor wih a paricular uiliy funcion invesing on he basis of differen variance forecass. We plan o pursue his idea from a risk managemen perspecive in fuure research. RiskMerics Technical Documen Fourh Ediion

Sec. 5.3 Esimaing he parameers of he RiskMerics model 99 5.3.. How RiskMerics chooses is opimal decay facor RiskMerics currenly processes 480 ime series, and associaed wih each series is an opimal decay facor ha minimizes he roo mean squared error of he variance forecas (i.e., Eq. [5.30]). We choose RMSE as he forecas error measure crierion. 1 Table 5.8 presens opimal decay facors for reurn series in five series. Table 5.8 Opimal decay facors based on volailiy forecass based on RMSE crierion Counry Foreign exchange 5-year swaps 10-year zero prices Equiy indices 1-year money marke raes Ausria 0.945 Ausralia 0.980 0.955 0.975 0.975 0.970 Belgium 0.945 0.935 0.935 0.965 0.850 Canada 0.960 0.965 0.960 0.990 Swizerland 0.955 0.835 0.970 0.980 Germany 0.955 0.940 0.960 0.980 0.970 Denmark 0.950 0.905 0.90 0.985 0.850 Spain 0.90 0.95 0.935 0.980 0.945 France 0.955 0.945 0.945 0.985 Finland 0.995 0.960 Grea Briain 0.960 0.950 0.960 0.975 0.990 Hong Kong 0.980 Ireland 0.990 0.95 Ialy 0.940 0.960 0.935 0.970 0.990 apan 0.965 0.965 0.950 0.955 0.985 Neherlands 0.960 0.945 0.950 0.975 0.970 Norway 0.975 New Zealand 0.975 0.980 Porugal 0.940 0.895 Sweden 0.985 0.980 0.885 Singapore 0.950 0.935 Unied Saes 0.970 0.980 0.980 0.965 ECU 0.950 For he daily and monhly daa ses we compue one opimal decay facor from he 480+ ime series. Denoe he ih opimal decay facor by λˆ i and le N (i = 1,,, N) denoe he number of ime series in he RiskMerics daabase. Also, le τ i denoe he ih RMSE associaed wih λˆ i, i.e., τ i is he minimum RMSE for he ih ime series. We derive he one opimal decay facor as follows: 1. Find Π, he sum of all N minimal RMSE s, s: [5.3] Π =. τ i i = 1. Define he relaive error measure: N τ i 1 We have chosen his crierion because i penalizes large forecas errors more severely, and provides more useful resuls han oher common accuracy saisics. Par II: Saisics of Financial Marke Reurns

100 Chaper 5. Esimaion and forecas [5.33] N θ i = τ i τ i i = 1 3. Define he weigh : φ i [5.34] 1 φ i = θ i N i = 1 1 θ i where N i = 1 φ i = 1 4. The opimal decay facor λ is defined as [5.35] λ = N i = 1 φ i λˆi Tha is, he opimal decay facor applied by RiskMerics is a weighed average of individual opimal decay facors where he weighs are a measure of individual forecas accuracy. Applying his mehodology o boh daily and monhly reurns we find ha he decay facor for he daily daa se is 0.94, and he decay facor for he monhly daa se is 0.97. Table 5.9 Summary of RiskMerics volailiy and correlaion forecass 5.4 Summary and concluding remarks In his chaper we explained he mehodology and pracical issues surrounding he esimaion of he RiskMerics volailiies and correlaions. Table 5.9 summarizes he imporan resuls abou he RiskMerics volailiy and correlaion forecass. Forecas Expression* Decay facor # of daily reurns used in producion Effecive # of daily reurns used in esimaion 1-day volailiy σ 1, + 1 = λσ 1, 1 + ( 1 λ)r 1, 0.94 550 75 σ 1, + 1 ρ 1, + 1 σ 1, + 1 σ, + 1 1-day correlaion = -------------------------------------- 0.94 550 75 1-monh volailiy σ 1, + 5 = 5 σ 1, + 1 0.97 550 150 1-monh correlaion = 0.97 550 150 ρ 1, + 5 ρ 1, + 1 * Noe ha in all calculaions he sample mean of daily reurns is se o zero. This number is a dependen of he decay facor explained in Secion 5.3.1.. RiskMerics Technical Documen Fourh Ediion

Sec. 5.4 Summary and concluding remarks 101 Lasly, recall from Chaper 4 ha RiskMerics assumes ha reurns are generaed according o he model [5.36] r = σ ε ε IID N ( 01, ) Now, given he recursive form of he EWMA model, a more complee version of he RiskMerics model for any individual ime series is [5.37] r = σ ε ε IID N ( 01, ) σ = λσ 1 + ( 1 λ)r 1 Since Eq. [5.37] describes a process by which reurns are generaed, we can deermine wheher his model (evaluaed a he opimal decay facor) can replicae he disincive feaures of he observed daa as presened in Chaper 4. We do so by generaing a ime series of daily reurns from Eq. [5.37] for a given value of λ. A simulaed ime series from Eq. [5.37] wih λ = 0.94 is shown in Char 5.1. Char 5.1 Simulaed reurns from RiskMerics model Daily reurns (r ) 0.3 0. 0.1 0-0.1-0. -0.3-0.4 00 400 600 800 1000 Time () Char 5.1 shows ha he RiskMerics model can replicae he volailiy clusering feaure noed in Chaper 4 (compare Char 5.1 o Chars 4.6 and 4.7). Par II: Saisics of Financial Marke Reurns

10 Chaper 5. Esimaion and forecas RiskMerics Technical Documen Fourh Ediion

103 Par III Risk Modeling of Financial Insrumens

104 RiskMerics Technical Documen Fourh Ediion

105 Chaper 6. Marke risk mehodology 6.1 Sep 1 Idenifying exposures and cash flows 107 6.1.1 Fixed Income 107 6.1. Foreign exchange 115 6.1.3 Equiies 117 6.1.4 Commodiies 117 6. Sep Mapping cash flows ono RiskMerics verices 117 6..1 RiskMerics verices 117 6.. Compuing RiskMerics cash flows 119 6.3 Sep 3 Compuing Value-a-Risk 11 6.3.1 Relaing changes in posiion values o underlying reurns 1 6.3. Simple VaR calculaion 15 6.3.3 Dela-gamma VaR mehodology (for porfolios conaining opions) 19 6.4 Examples 134 Par III: Risk Modeling of Financial Insrumens

106 RiskMerics Technical Documen Fourh Ediion

107 Chaper 6. Marke risk mehodology Peer Zangari Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-8641 zangari_peer@jpmorgan.com This chaper explains he mehodology RiskMerics uses o calculae VaR for porfolios ha include muliple insrumens such as simple bonds, swaps, foreign exchange, equiy and oher posiions. The chaper is organized as follows: Secion 6.1 describes how o decompose various posiions ino cash flows. Secion 6. covers how o conver or map he acual cash flows ono he corresponding RiskMerics verices. Secion 6.3 explains wo analyical approaches o measuring VaR. Secion 6.4 presens a number of examples o illusrae he applicaion of he RiskMerics mehodology. 6.1 Sep 1 Idenifying exposures and cash flows The RiskMerics building block for describing any posiion is a cash flow. A cash flow is defined by an amoun of a currency, a paymen dae and he credi sanding of he payor. Once deermined, hese cash flows are marked-o-marke. Marking-o-marke a posiion s cash flows means deermining he presen value of he cash flows given curren marke raes and prices. This procedure requires curren marke raes, including he curren on-he-run yield curve for newly issued deb, and a zero-coupon yield curve on insrumens ha pay no cash flow unil mauriy. 1 The zero coupon rae is he relevan rae for discouning cash flows received in a paricular fuure period. We now describe how o express posiions in fixed income, foreign exchange, equiy, and commodiies in erms of cash flows. The general process of describing a posiion in erms of cash flows is known as mapping. 6.1.1 Fixed Income Ineres rae posiions describe he disribuion of cash flows over ime. Praciioners have applied various mehods o express, or map, he cash flows of ineres rae posiions, he mos common hree being (1) duraion map, () principal map, and (3) cash flow map. In his book we use he cash flow map mehod, bu for comparison, presen he wo oher mehods as viable alernaives. Duraion map The firs and mos common mehod o characerize a posiion s cash flows is by is duraion (he weighed average life of a posiion s ineres and principal paymens). Macaulay duraion is a measure of he weighed average mauriy of an insrumen s cash flows. Modified duraion is a measure of a bond s price sensiiviy o changes in ineres raes. In general, duraion provides risk managers wih a simplified view of a porfolio s marke risk. Is main drawback is ha i assumes a linear relaionship beween price changes and yield changes. Moreover, 1 See The. P. Morgan/Arhur Andersen Guide o Corporae Exposure Managemen (p. 54, 1994). I is ofen suggesed ha implied forward raes are required o esimae he floaing raes o be paid in fuure periods. In his documen, however, we will show why forward raes are no necessarily required. Par III: Risk Modeling of Financial Insrumens

108 Chaper 6. Marke risk mehodology his approach works well when here are so-called parallel shifs in he yield curve bu poorly when yield curves wis. Duraion maps are used exensively in fixed income invesmen managemen. Many invesmen managers aciviies are consrained by risk limis expressed in erms of porfolio duraion. Principal map A second mehod, used exensively over he las wo decades by commercial banks, is o describe a global posiion in erms of when principal paymens occur. These principal maps form he basis for asse/liabiliy managemen. ARBLs (Asses Repricing Before Liabiliies) are used by banks o quanify ineres rae risk in erms of cumulaive asses mauring before liabiliies. This mehod is employed mos ofen when risks are expressed and earnings are accouned for on an accrual basis. The main problem wih principal maps is ha hey assume ha all ineres paymens occur a curren marke raes. This is ofen no a good assumpion paricularly when posiions include fixed rae insrumens wih long mauriies and when ineres raes are volaile. Principal maps describe an insrumen only as a funcion of he value and iming of redempion. Cash flow map The hird mehod, and he one RiskMerics applies is known as cash flow mapping. Fixed income securiies can be easily represened as cash flows given heir sandard fuure sream of paymens. In pracice, his is equivalen o decomposing a bond ino a sream of zero-coupon insrumens. Complicaions in applying his echnique can arise, however, when some of hese cash flows are uncerain, as wih callable or puable bonds. The following example shows how each of he mapping mehodologies can be applied in pracice. Char 6.1 shows how a 10-year French OAT (FRF 100,000 francs nominal, 7.5% of April 005) can be mapped under he approaches lised above: The duraion map associaes he marke value of he insrumen agains he bond s Macaulay duraion of 6.88 years. The principal map allocaes he marke value of he bond o he 10-year mauriy verex. The cash flow map shows he disribuion over ime of he curren marke value of all fuure sreams (coupons + principals). As shown in Char 6.1, he cash flow map (presen valued) reas all cash flows separaely and does no group hem ogeher as do he duraion and principal maps. Cash flow mapping is he preferred alernaive because i reas cash flows as being disinc and separae, enabling us o model he risk of he fixed income posiion beer han if he cash flows were simply represened by a grouped cash flow as in he duraion and principal maps. RiskMerics Technical Documen Fourh Ediion

Sec. 6.1 Sep 1 Idenifying exposures and cash flows 109 Char 6.1 French franc 10-year benchmark maps amouns in housands of marke value Duraion Principal flows Cash flows 100 90 80 70 60 50 40 30 0 10 0 95 97 99 01 03 05 100 90 80 70 60 50 40 30 0 10 0 95 97 99 01 03 05 50 45 40 35 30 5 0 15 10 5 0 95 97 99 01 03 05 6.1.1.1 Simple bonds Consider a hypoheical bond wih a par value of 100, a mauriy of 4 years and a coupon rae of 5%. Assume ha he bond is purchased a ime 0 and ha coupon paymens are payed on an annual basis a he beginning of each year. Char 6. shows he bond s cash flows. 3 In general, arrows poining upwards signify cash inflows and arrows poining downwards represen ouflows. Also, a cash flow s magniude is proporional o he lengh of he arrow; he aller (shorer) he arrow he greaer (lower) he cash flow. Char 6. Cash flow represenaion of a simple bond 105 5 5 5 0 1 3 4 years We can represen he cash flows of he simple bond in our example as cash flows from four zerocoupon bonds wih mauriies of 1,,3 and 4 years. This implies ha on a risk basis, here is no difference beween holding he simple bond or he corresponding four zero-coupon bonds. 6.1.1. Floaing rae noes (FRN) A floaing rae noe (FRN) is an insrumen ha is based on a principal, P, ha pays floaing coupons. A FRN s coupon paymen is defined as he produc of he principal and a floaing rae ha is se some ime in advance of he acual coupon paymen. For example, if coupon paymens are paid on a semiannual basis, he 6-monh LIBOR rae would be used o deermine he paymen in 6 monh s ime. The coupon paymens would adjus accordingly depending on he curren 6-monh LIBOR rae when he floaing rae is rese. The principal is exchanged a boh he beginning and end of he FRN s life. 3 We ignore he paymen for he bond. Tha is, we do no accoun for he iniial (negaive) cash flow a ime 0. Par III: Risk Modeling of Financial Insrumens

110 Chaper 6. Marke risk mehodology Char 6.3 shows he cash flows for a hypoheical FRN lasing 4 years. The floaing paymens are represened by he gray shaded arrows. The black arrows represen fixed paymens. All paymens are assumed o occur on a yearly basis. Char 6.3 Cash flow represenaion of a FRN 0 1 3 4 years Noice ha he firs paymen (a year 1) is known, and herefore, fixed. Also, he las paymen represens he fac ha he principal is known a he fourh year, bu he final coupon paymen is unknown. We now show how o evaluae he fuure floaing paymens. Suppose ha a ime (beween 0 and 1 year), a risk manager is ineresed in analyzing he floaing paymen ha will be received in year 3. The rae ha deermines his value is se in he second year and lass one year. Now, implied forward raes are ofen used o forecas floaing raes. The fundamenal arbirage relaionship beween curren and fuure raes implies ha he 1-year rae, as of year saisfies he expression [6.1] ( 1 + r, ) ( 1+ f 1, ) = ( 1+ r 3, ) where r i,j is he i-year rae se a ime j and f i,j is he i period forward rae se a ime j. So, for example, f 1, is he 1-year rae, beginning a he second year. I follows ha he cash flow implied by his rae occurs in year 3. Since we know a ime boh r -, (he - year rae) and r 3-, (he 3- year rae), we can solve for he implied forward rae as a funcion of observed raes. i.e., [6.] ( 1+ r 3, ), = ----------------------------- ( 1 + r, ) 1 f 1 We can apply same echnique o all oher implied forward raes so ha we can solve for f 1,1, f 1,, f 1,3 and deermine he expeced fuure paymens. The forecas coupon paymen, for example, a ime 3 is P f. The presen value of his paymen a ime, is simply ( 1, P f ) 1, ( 1 + r. 3, ) Subsiuing Eq. [6.] ino he expression for he discouned coupon paymen yields, [6.3] P f1, ----------------------------- = ( 1+ r 3, ) P P ----------------------------- ----------------------------- ( 1 + r, ) ( 1 + r 3, ) Equaion [6.3] shows ha he expeced coupon paymen can be wrien in erms of known zero coupon raes. We can apply similar mehods o he oher coupon paymens so ha we can wrie he cash flows of he FRN as [6.4] P r P 10, P f FRN = ----------------------------- 11, ( 1+ r 1, ) + P f ----------------------------- 1, ( 1+ r, ) + P f ----------------------------- 13, ( 1+ r 3, ) + P ----------------------------- ( 1+ r 4, ) + ----------------------------- ( 1 + r 4, ) The righ-hand side of Eq. [6.4] is equal o RiskMerics Technical Documen Fourh Ediion

Sec. 6.1 Sep 1 Idenifying exposures and cash flows 111 [6.5] P r 10, P ----------------------------- ( 1+ r 1, ) + P ----------------------------- ----------------------------- P ( 1 + r 1, ) ( 1 + r, ) + P ----------------------------- 1 r ----------------------------- ( +, ) ( 1 + r 3, ) P + P ----------------------------- ----------------------------- P ( 1+ r 3, ) ( 1 + r 4, ) + ----------------------------- ( 1 + r 4, ) Equaion [6.5] collapses o he presen value [6.6] P ( 1 + r 10, ) -------------------------------- ( 1+ r 1, ) Char 6.4 shows ha he cash flow of he FRN from he ime perspecive, is P ( 1 + r 10, ). Therefore, we would rea he FRN s cash flows as a cash flow from a zero coupon bond wih mauriy 1- period. Char 6.4 Esimaed cash flows of a FRN P(1 + r 1,0 ) 0 1 3 4 years Noice ha if he cash flows in Char 6.3 were compued relaive o ime zero (he sar of he FRN), raher han o ime, he cash flow would be simply P a = 0, represening he par value of he FRN a is sar. 6.1.1.3 Simple ineres-rae swaps Invesors ener ino ineres-rae swaps o change heir exposure o ineres rae uncerainy by exchanging ineres flows. In order o undersand how o idenify a simple ineres-rae swap s cash flows, a swap should be hough of as a porfolio consising of one fixed and one floaing rae insrumen. Specifically, he fixed leg is represened by a simple bond wihou an exchange of principal. The floaing leg is a FRN wih he cavea ha he principal is used only o deermine coupon paymens, and is no exchanged. Char 6.5 shows he cash flows of an ineres-rae swap ha receives fixed rae and pays he floaing rae. Char 6.5 Cash flow represenaion of simple ineres rae swap Receive Fixed side Pay Floaing side 0 1 3 4 years 0 1 3 4 Par III: Risk Modeling of Financial Insrumens

11 Chaper 6. Marke risk mehodology We compue he cash flows relaive o ime, (again, beween 0 and 1 year) afer he sar of he swap. The cash flows on he fixed side are simply he fixed coupon paymens over he nex 4 years which, as already explained in Secion 6.1.1.1, are reaed as holding four zero-coupon bonds. The cash flows on he floaing side are derived in he exac manner as he paymens for he FRN (excep now we are shor he floaing paymens). The presen value of he cash flow map of he floaing side of he swap is given by Eq. [6.7] [6.7] -------------------------------- P ( 1 + r 10, ), ( 1+ r 1, ) where P is he principal of he swap. Noice he similariy beween his cash flow and ha given by Eq. [6.6] for he FRN. Hence, we can represen he cash flows on he floaing side of he swap as being shor a zero coupon bond wih mauriy 1-. 6.1.1.4 Forward saring swap A forward saring swap is an insrumen where one eners ino an agreemen o swap ineres paymens a some fuure dae. Unlike a simple swap none of he floaing raes are fixed in advance. Char 6.6 shows he cash flows of a forward saring swap. Char 6.6 Cash flow represenaion of forward saring swap Receive Fixed side Pay Floaing side 0 1 3 4 5 years 0 1 3 4 5 Suppose ha an invesor eners ino a forward saring swap wih 5 years o mauriy a some ime (he rade dae), and he sar dae of he swap, (i.e., he dae when he floaing raes are fixed) is year. Saring in year 3, paymens are made every year unil year 5. The cash flows for his insrumen are essenially he same as a simple ineres-rae swap, bu now he firs floaing paymen is unknown. The cash flows on he fixed side are simply he cash flows discouned back o ime. On he floaing side, he cash flows are, again, deermined by he implied forward raes. The cash flow map for he (shor) floaing paymens is represened by Eq. [6.8]. [6.8] ----------------------------- P ( 1 + r, ) RiskMerics Technical Documen Fourh Ediion

Sec. 6.1 Sep 1 Idenifying exposures and cash flows 113 Char 6.7 depics his cash flow. Char 6.7 Cash flows of he floaing paymens in a forward saring swap 0 1 3 4 5 -P years Noice ha his cash flow map is equivalen o being shor a - zero coupon bond. 6.1.1.5 Forward rae agreemen (FRA) A forward rae agreemen (FRA) is an ineres rae conrac. I locks in an ineres rae, eiher a borrowing rae (buying a FRA) or a lending rae (selling a FRA) for a specific period in he fuure. FRAs are similar o fuures bu are over-he-couner insrumens and can be cusomized for any mauriy. A FRA is a noional conrac. Therefore, here is no exchange of principal a he expiry dae (i.e., he fixing dae). In effec, FRAs allow marke paricipans o lock in a forward rae ha equals he implied break even rae beween money marke and erm deposis. 4 To undersand how o map he cash flows of a FRA, le s consider a simple, hypoheical example of a purchase of a 3 vs. 6 FRA a r% on a noional amoun P. This is equivalen o locking in a borrowing rae for 3 monhs saring in 3 monhs. The noaion 3 vs. 6 hus refers o he sar dae of he underlying versus he end dae of he underlying, wih he sar dae being he delivery dae of he conrac. Char 6.8 depics he cash flows of his FRA. Char 6.8 Cash flow represenaion of FRA i 0 3m 6m We can replicae hese cash flows by going long he curren 3-monh rae and shor he 6-monh rae as shown in Char 6.9. 4 For more deails on FRAs, refer o Valuing and Using FRAs (Hakim Mamoni, Ocober, 1994, P Morgan publicaion). Par III: Risk Modeling of Financial Insrumens

114 Chaper 6. Marke risk mehodology Char 6.9 Replicaing cash flows of 3-monh vs. 6-monh FRA 0 3m 6m Noe ha he gray arrows no longer represen floaing paymens. The gray and black arrows represen he cash flows associaed wih going shor a 6-monh zero coupon bond and long a 3-monh zero coupon bond, respecively. The benefi of working wih he cash flows in Char 6.9 raher han in Char 6.8, is ha he laer requires informaion on forward raes whereas he former does no. 6.1.1.6 Ineres rae fuure We now consider he cash flow map of a 3-monh Eurodollar fuure conrac ha expires in 3 monhs. Taking ime 0 o represen he curren dae, we represen he fuure s cash flows by an ouflow in 3 monhs and an inflow in 6 monhs, as shown in Char 6.10. Char 6.10 Cash flow represenaion of 3-monh Eurodollar fuure 0 3m 6m To be more specific, if he curren USD 3-monh Eurodollar deposi rae is 7.0%, a purchaser of his fuures conrac would face a cash ouflow of USD 981,800 in 3 monhs and a cash inflow of USD 1,000,000 in 6 monhs. We can hen represen hese cash flows as being shor he curren 3- monh rae and invesing his money in he curren 6-monh rae. Hence, he cash flows of his Eurodollar fuures conrac can be replicaed by a shor 3-monh posiion and a long 6-monh posiion as shown in Char 6.11. Char 6.11 Replicaing cash flows of a Eurodollar fuures conrac 0 3m 6m RiskMerics Technical Documen Fourh Ediion

Sec. 6.1 Sep 1 Idenifying exposures and cash flows 115 6.1. Foreign exchange Financial posiions are described in erms of a base or home currency. For example, American insiuions repor risks in U.S. dollars, while German insiuions use Deusche marks. A risk manager s risk profile is no independen of he currency in which risk is repored. For example, consider wo invesors. One invesor is based in US dollars, he oher in Ialian lira. Boh invesors purchase an Ialian governmen bond. Whereas he USD based invesor is exposed o boh ineres rae and exchange rae risk (by way of he ITL/USD exchange rae), he lira based invesor is exposed only o ineres rae risk. Therefore, an imporan sep o measure foreign exchange risk is o undersand how cash flows are generaed by foreign exchange posiions. 6.1..1 Spo posiions Describing cash flows of spo foreign exchange posiions is rivial. Graphically, up and down arrows represen long and shor posiions in foreign exchange, respecively. 6.1.. Forward foreign exchange posiions A foreign exchange (FX) forward is an agreemen o exchange a a fuure dae, an amoun of one currency for anoher a a specified forward rae. Mapping a forward foreign exchange posiion is faciliaed by he abiliy o express he forward as a funcion of wo ineres raes and a spo foreign exchange rae. 5 For example, Char 6.1 shows he cash flows of an FX forward ha allows an invesor o buy Deusche marks wih US dollars in 6 monhs ime a a prespecified forward rae. Char 6.1 FX forward o buy Deusche marks wih US dollars in 6 monhs Buy DEM 0 6m Sell USD We can replicae hese cash flows by borrowing dollars a ime 0 a he 6-monh ineres rae r USD, and immediaely invesing hese dollars in Germany a a rae r DEM,, This scenario would generae he cash flows which, a he 6-monh mark, are idenical o hose of he forward conrac. These cash flows are shown in Char 6.13. Char 6.13 Replicaing cash flows of an FX forward Borrow USD Lend DEM 0 6m Buy DEM Sell USD The abiliy o replicae fuure foreign exchange cash flows wih ineres rae posiions resuls from wha is known as ineres rae pariy (IRP). We now demonsrae his condiion. Le he spo rae, S, of he home currency expressed in unis of foreign currency, (e.g., if he home currency is he US dollars and he foreign currency is Deusche marks, is expressed in US dollars per Deusche S 5 For simpliciy, we ignore oher facors such as ransacion coss and possible risk premia. Par III: Risk Modeling of Financial Insrumens

116 Chaper 6. Marke risk mehodology marks (USD/DEM)). The forward rae, F, is he exchange rae observed a ime, which guaranees a spo rae a some fuure ime T. Under ineres rae pariy he following condiion holds [6.9] F ( 1 + r USD, ) = S -------------------------------- ( 1 r ) + DEM, I follows from IRP ha he abiliy o conver cash flows of an FX forward ino equivalen borrowing and lending posiions implies ha holding an FX forward involves cash flows ha are exposed o boh foreign exchange and ineres risk. 6.1..3 Currency swaps Currency swaps are swaps for which he wo legs of he swap are each denominaed in a differen currency. For example, one pary migh receive fixed rae Deusche marks, he oher floaing rae US dollars. Unlike an ineres-rae swap, he noional principal in a currency swap is exchanged a he beginning and end of he swap. 6 Char 6.14 shows he cash flows for a hypoheical currency swap wih a mauriy of 4 years and paying fixed rae Deusche marks and floaing rae US dollars on an annual basis. For compleeness, we presen he cash flows associaed wih he iniial exchange of principal. Char 6.14 Acual cash flows of currency swap USD DEM + final coupon DEM 0 1 3 4 5 years 0 1 3 4 5 USD + final coupon From he perspecive of holding he swap a ime beween 0 and year 1, he fixed leg of he swap has he same cash flows as he simple bond presened in Secion 6.1.1.1. The cash flows of he floaing leg are he same as ha as a shor posiion in a FRN. 6.1.3 Equiies The cash flows of equiy are simple spo posiions expressed in home currency equivalens. Equiy posiions held in foreign counries are subjec o foreign exchange risk in addiion o he risk from holding equiy. 6 There are currency swaps where one or boh of he noional amouns are no exchanged. RiskMerics Technical Documen Fourh Ediion

Sec. 6. Sep Mapping cash flows ono RiskMerics verices 117 6.1.4 Commodiies Exposures o commodiies can be explained using a framework similar o ha of ineres raes. Risks arise in boh he spo marke (you purchase a produc oday and sore i over ime) and from ransacions ha ake place in he fuure (e.g., physical delivery of a produc in one monh s ime). 6.1.4.1 Commodiy fuures conrac Commodiy fuures conracs enable invesors o rade producs for fuure delivery wih relaive ease and also serve as a price seing and risk ransferring mechanisms for commodiy producers. These conracs provide marke paricipans wih valuable informaion abou he erm srucure of commodiies prices. 6.1.4. Commodiy swap Insiuions do no have o limi hemselves o fuures conracs when hey paricipae in he commodiy markes. They can ener ino swaps o change heir exposure o ineres raes, currency, and/or commodiy risks. A ypical commodiy swap enails an insiuion o paying (receiving) fixed amouns in exchange for receiving (paying) variable amouns wih respec o an index (e.g., an average of he daily price of he nearby naural gas fuures conrac). In many respecs, commodiy swaps are similar o ineres-rae swaps. Unlike an ineres-rae swap he underlying insrumen of a commodiy swap can be of variable qualiy hereby making he erms of he ransacion more complex. 6. Sep Mapping cash flows ono RiskMerics verices In he las secion we described cash flows generaed by paricular classes of insrumens. Financial insrumens, in general, can generae numerous cash flows, each one occurring a a unique ime. This gives rise o an unwieldy number of combinaions of cash flow daes when many insrumens are considered. As a resul, we are faced wih he impracical ask of having o compue an inracable number of volailiies and correlaions for he VaR calculaion. To more easily esimae he risks associaed wih insrumens cash flows, we need o simplify he ime srucure of hese cash flows. The RiskMerics mehod of simplifying ime srucure involves cash flow mapping, i.e., redisribuing (mapping) he observed cash flows ono so-called RiskMerics verices, o produce RiskMerics cash flows. 6..1 RiskMerics verices All RiskMerics cash flows use one or more of he 14 RiskMerics verices shown below (and on page 107). 1m 3m 6m 1m yr 3yr 4yr 5yr 7yr 9yr 10yr 15yr 0yr 30yr These verices have wo imporan properies: They are fixed and hold a any ime now and in he fuure for all insrumens, linear and nonlinear. (.P. Morgan can occasionally redefine hese verices o keep up wih marke rends.) RiskMerics daa ses provide volailiies and correlaions for each of hese verices (and only for hese verices). Mapping an acual cash flow involves spliing i beween he wo closes RiskMerics verices (unless he cash flow happens o coincide wih a RiskMerics verex). For example, a cash flow occurring in 6 years is represened as a combinaion of a 5-year and a 7-year cash flow. Char 6.15 Par III: Risk Modeling of Financial Insrumens

118 Chaper 6. Marke risk mehodology shows how he acual cash flow occurring a year 6 is spli ino he synheic (RiskMerics) cash flows occurring a he 5- and 7-year verices. Char 6.15 RiskMerics cash flow mapping Acual Cashflows 5 6 7 years RiskMerics Cashflows 5 6 7 years The wo fracions of he cash flow are weighed such ha he following hree condiions hold: 1. Marke value is preserved. The oal marke value of he wo RiskMerics cash flows mus be equal o he marke value of he original cash flow.. Marke risk is preserved. The marke risk of he porfolio of he RiskMerics cash flows mus also be equal o he marke risk of he original cash flow. 3. Sign is preserved. The RiskMerics cash flows have he same sign as he original cash flow. In he rivial case ha he acual verex and RiskMerics verex coincide, 100% of he acual cash flow is allocaed o he RiskMerics verex. I is imporan o undersand ha RiskMerics cash flow mapping differs from convenional mapping mehods in he hree condiions ha i sipulaes. A common pracice used o dae hroughou he financial indusry has been o follow wo sandard rules when allocaing cash flows beween verices: 1. Mainain presen value. For example, he sum of he cash flows mauring in 5 and 7 years mus be equal o he original cash flow occurring in year 6.. Mainain duraion. The duraion of he combinaion of 5- and 7-year cash flows mus also be equal o he duraion of he 6-year cash flow. Cash flow maps like hese are similar o a barbell ype rade, where an exising posiion is replaced by a combinaion of wo insrumens disribued along he yield curve under he condiion ha he rade remains duraion neural. Barbell rades are enered ino by invesors who are duraion-consrained bu have a view on a shif in he yield curve. Wha is a perfecly defensible invesmen sraegy, however, canno be simply applied o risk esimaion. 6.. Compuing RiskMerics cash flows For allocaing acual cash flows o RiskMerics verices, RiskMerics proposes a mehodology ha is based on he variance (σ ) of financial reurns. The advanage of working wih he variance is ha i is a risk measure closely associaed wih one of he ways RiskMerics compues VaR, namely he simple VaR mehod as opposed o he dela-gamma or Mone Carlo mehods. RiskMerics Technical Documen Fourh Ediion

Sec. 6. Sep Mapping cash flows ono RiskMerics verices 119 In order o faciliae he necessary mapping, he RiskMerics daa ses provide users wih volailiies on, and correlaions across many insrumens in 33 markes. For example, in he US governmen bond marke, RiskMerics daa ses provide volailiies and correlaions on he -, 3-, 4-, 5-, 7-, 9-, 10-, 15-, 0-, and 30-year zero coupon bonds. We now demonsrae how o conver acual cash flows o RiskMerics cash flows, coninuing wih he example of allocaing a cash flow in year 6 o he 5- and 7-year verices (Char 6.15). We denoe he allocaions o he 5- and 7-year verices by α and (1-α), respecively. The procedure presened below is no resriced o fixed income insrumens, bu applies o all fuure cash flows. 1. Calculae he acual cash flow s inerpolaed yield: We obain he 6-year yield, y 6, from a linear inerpolaion of he 5- and 7-year yields provided in he RiskMerics daa ses. Using he following equaion, [6.10] y 6 = ây 5 + ( 1 â) y 7 0 â 1 where y 6 = inerpolaed 6-year zero yield â = linear weighing coefficien, â = 0.5 in his example y 5 = 5-year zero yield = 7-year zero yield y 7 If an acual cash flow verex is no equidisan beween he wo RiskMerics verices, hen he greaer of he wo values, â and (1 â ), is assigned o he closer RiskMerics verex.. Deermine he acual cash flow s presen value: From he 6-year zero yield, y 6, we deermine he presen value, P 6, of he cash flow occurring a he 6-year verex. (In general, P i denoes he presen value of a cash flow occurring in i years.) 3. Calculae he sandard deviaion of he price reurn on he acual cash flow: We obain he sandard deviaion, σ 6, of he reurn on he 6-year zero coupon bond, by a linear inerpolaion of he sandard deviaions of he 5- and 7-year price reurns, i.e., σ 5 and σ 7, respecively. Noe ha σ5 and σ7 are provided in he RiskMerics daa ses as he VaR saisics 1.65σ 5 and 1.65σ 7, respecively. Hence, 1.65σ 6 is he inerpolaed VaR. To obain σ 6, we use he following equaion: [6.11] σ 6 = âσ 5 + ( 1 â)σ 7 0 â 1 where â = linear weighing coefficien from Eq. [6.10] σ 5 σ 7 = sandard deviaion of he 5-year reurn = sandard deviaion of he 7-year reurn Par III: Risk Modeling of Financial Insrumens

10 Chaper 6. Marke risk mehodology 4. Compue he allocaion, α and (1-α), from he following equaion: [6.1] Variance ( r 6yr ) = Variance [αr 5yr + ( 1 α)r 7yr ], or he equivalen σ 6 α σ 5 α ( 1 α)ρ 57, σ 5 σ 7 ( 1 α) = + + σ 7 is pro- where ρ 5,7, is he correlaion beween he 5- and 7- year reurns. (Noe ha vided in he correlaion marix in RiskMerics daa ses). ρ 57, Equaion [6.1] can be wrien in he quadraic form [6.13] aα + bα+ c= 0 where a = σ 5 + σ 7 ρ 57, σ 5 σ 7 b= ρ 57, σ 5 σ 7 σ 7 c= σ 7 σ 6 The soluion o α is given by [6.14] b± b 4ac α= -------------------------------------- a Noice ha Eq. [6.14] yields wo soluions (roos). We choose he soluion ha saisfies he hree condiions lised on page 118. 5. Disribue he acual cash flow ono he RiskMerics verices: Spli he acual cash flow a year 6 ino wo componens, α and (1-α), where you allocae α o he 5-year RiskMerics verex and (1-α) o he 7-year RiskMerics verex. Using he seps above, we compue a RiskMerics cash flow map from he following real-world daa. Suppose ha on uly 31, 1996, he cash flow occurring in 6 years is USD 100. The RiskMerics daily daa ses provide he saisics shown in Table 6.1, from which we calculae he daa shown in Table 6.. 7 7 Recall ha RiskMerics provides VaR saisics ha is, 1.65 imes he sandard deviaion. RiskMerics Technical Documen Fourh Ediion

Sec. 6.3 Sep 3 Compuing Value-a-Risk 11 Table 6.1 Daa provided in he daily RiskMerics daa se y 5 5-year yield 6.605% y 7 7-year yield 6.745% 1.65σ 5 1.65σ 7 ρ 57, volailiy on he 5-year bond price reurn 0.5770% volailiy on he 7-year bond price reurn 0.8095% correlaion beween he 5- and 7-year bond reurns 0.9975 Table 6. Daa calculaed from he daily RiskMerics daa se y 6 σ 6 σ 6 σ 5 σ 7 6-year yield 6.675% (from Eq. [6.10], where â= 0.5 ) sandard deviaion on he 6-year bond price reurn 0.40% variance on he 6-year bond price reurn 1.765 10 3 % (from Eq. [6.11]) variance on he 5-year bond price reurn 1.3 10 3 % variance on he 7-year bond price reurn.406 10 3 % To solve for α, we subsiue he values in Tables 6.1 and 6. ino Eq. [6.13] o find he following values: a =.14 10 6 b = 1.39 10 5 c = 6.41 10 6 which in Eq. [6.14], yields he soluions α = 5.999 and α = 0.489. We disqualify he firs soluion since (1 α) violaes he sign preservaion condiion (page 118). From Sep on page 119, he presen value of USD 100 in 6 years is USD 93.74, i.e., he 6-year cash flow. We allocae 49.66% (USD 46.55) of i o he 5-year verex and 50.33% (USD 47.17) o he 7-year verex. We hus obain he RiskMerics cash flow map shown in Char 6.15. The preceding example demonsraed how o map a single cash flow o RiskMerics verices. In pracice porfolios ofen conain many cash flows, each of which has o be mapped o he RiskMerics verices. In such insances, cash flow mapping simply requires a repeaed applicaion of he mehodology explained in his secion. 6.3 Sep 3 Compuing Value-a-Risk This secion explains wo analyical approaches o measuring Value-a-Risk: simple VaR for linear insrumens, and dela-gamma VaR for nonlinear insrumens, where he erms linear and nonlinear describe he relaionship of a posiion s underlying reurns o he posiion s relaive change in value. (For more informaion abou simple VaR mehodology, see Secion 6.3.. For more informaion abou dela-gamma mehodology, see Secion 6.3.3.) In he simple VaR approach, we assume ha reurns on securiies follow a condiionally mulivariae normal disribuion (see Chaper 4) and ha he relaive change in a posiion s value is a linear funcion of he underlying reurn. Defining VaR as he 5h percenile of he disribuion of a porfo- Par III: Risk Modeling of Financial Insrumens

1 Chaper 6. Marke risk mehodology lio s relaive changes, we compue VaR as 1.65 imes he porfolio s sandard deviaion, where he muliple 1.65 is derived from he normal disribuion. This sandard deviaion depends on he volailiies and correlaions of he underlying reurns and on he presen value of cash flows. In he dela-gamma approach, we coninue o assume ha reurns on securiies are normally disribued, bu allow for a nonlinear relaionship beween he posiion s value and he underlying reurns. Specifically, we allow for a second-order or gamma effec, which implies ha he disribuion of he porfolio s relaive change is no longer normal. Therefore, we canno define VaR as 1.65 imes he porfolio s sandard deviaion. Insead, we compue VaR in wo basic seps. Firs, we calculae he firs four momens of he porfolio s reurn disribuion, i.e., he mean, sandard deviaion, skewness and kurosis. Second, we find a disribuion ha has he same four momens and hen calculae he fifh percenile of his disribuion, from which we finally compue he VaR. The choice of approach depends on he ype of posiions ha are a risk, i.e., linear or non-linear posiions, as defined above. 6.3.1 Relaing changes in posiion values o underlying reurns This secion explains he lineariy and nonlineariy of insrumens in he conex of RiskMerics mehodology. Value-a-Risk measures he marke risk of a porfolio. We define a porfolio as a se of posiions, each of which is composed of some underlying securiy. In order o calculae he risk of he porfolio, we mus be able o compue he risks of he posiions ha compose he porfolio. This requires an undersanding of how a posiion s value changes when he value on is underlying securiy changes. Thus, we classify posiions ino simple posiions, which are linear, and ino derivaive posiions, which can be furher broken down ino linear and nonlinear derivaive posiions. As an example of a simple posiion, he relaive change in value of a USD 100 million dollar posiion in DEM is a linear funcion of he relaive change in value in he USD/DEM exchange rae (i.e., he reurn on he USD/DEM exchange rae). The value of derivaive posiions depends on he value of some oher securiy. For example, he value of a forward rae agreemen, a linear derivaive, depends on he value of some fuure ineres rae. In conras, oher derivaives may have a nonlinear relaionship beween he relaive change in value of he derivaive posiion and he value of he underlying securiy. For example, he relaive change in value of an opion on he USD/FRF exchange rae is a nonlinear funcion of he reurn on ha rae. RiskMerics Technical Documen Fourh Ediion

Sec. 6.3 Sep 3 Compuing Value-a-Risk 13 Char 6.16 shows how he reurn on a posiion varies wih he reurn on he underlying securiy. Char 6.16 Linear and nonlinear payoff funcions Posiion value Underlying securiy The sraigh lines in Char 6.16 signify a consan relaionship beween he posiion and underlying securiy. The black line represens a one-o-one relaionship beween he posiion value and he underlying securiy. Noe ha for a payoff o be linear, he movemen beween he posiion value and he underlying securiy s value does no have o be one-o-one. For example, he change in value of a simple opion can be expressed in erms of he dela (slope) of he underlying securiy, where he dela varies beween 1 and +1. Char 6.16 shows a payoff funcion where dela is 0.5 (gray line). When payoffs are nonlinear here is no longer a sraigh line relaionship beween he posiion value and he underlying securiy s value. Char 6.16 shows ha he payoff line is curved such ha he posiion value can change dramaically as he underlying securiy value increases. The convexiy of he line is quanified by he parameer gamma. In summary, linear payoffs are characerized by a consan slope, dela. Their convexiy, gamma, is always equal o zero. VaR for such insrumens is calculaed from he simple VaR mehodology (Secion 6.3.). For nonlinear payoffs, dela can ake on any value beween 1 and +1, while gamma is always non-zero, accouning for he observed curvaure of he payoff funcion. Nonlinear insrumens are hus reaed by he dela-gamma mehodology (alhough he same mehodology can also be used o handle linear insrumens. See Secion 6.3.3 on page 19). Table 6.3 liss seleced posiions (insrumens), heir underlying reurns, and he relaionship beween he wo. Table 6.3 Relaionship beween insrumen and underlying price/rae Type of posiion Insrumen* Underlying price/rae Simple (linear): Bond Bond price Sock Local marke index Foreign exchange FX rae Commodiy Commodiy price IR swap Swap price Linear derivaive: Floaing rae noe Money marke price Par III: Risk Modeling of Financial Insrumens

14 Chaper 6. Marke risk mehodology Table 6.3 (coninued) Relaionship beween insrumen and underlying price/rae Type of posiion Insrumen* Underlying price/rae FX forward FX rae/money marke price Forward rae agreemen Money marke price Currency swap Swap price/fx rae Nonlinear derivaive: Sock Opion Sock price Bond Opion Bond price FX Opion FX rae * Treaed by r. See he remainder of Secion 6.3. Treaed by r. See he remainder of Secion 6.3. Noe, however, he relaionship beween a bond price and is yield is nonlinear. ) 6.3.1.1 Linear posiions Using he qualiaive informaion in he preceding secion, we now formally derive he relaionship beween he relaive change in he value of a posiion and an underlying reurn for linear insrumens. We denoe he relaive change in value of he ih posiion, a ime, as. In he simple case where here is a linear one-o-one correspondence beween he relaive change in value of his posiion and some underlying reurn r, we have. 8 j, r i, = r j, In general, we denoe a posiion ha is linearly relaed o an underlying reurn as =, where δ is a scalar. Noice ha in he case of fixed income insrumens, he underlying value is defined in erms of prices on zero equivalen bonds (Table 6.3). Alernaively, underlying reurns could have been defined in erms of yields. For example, in he case of bonds, here is no longer a one-o-one correspondence beween a change in he underlying yield and he change in he price of he insrumen. In fac, he relaionship beween he change in price of he bond and yield is nonlinear. Since we only deal wih zero-coupon bonds we focus on hese. Furher, we work wih coninuous compounding. Assuming coninuous compounding, he price of an N-period zero-coupon bond a ime, P, wih yield y is ) r i, δr j, ) ) r i, [6.15] P = e y N A second order approximaion o he relaive change in P yields [6.16] 1 r = y N ( y y ) + -- ( y N ) ( y y ) ) Now, if we define he reurn r in erms of relaive yield changes, i.e., r = ( y y ), hen we have [6.17] r ) = y N ( r ) 1 + -- ( y N ) ( r ) 8 Technically, his resuls from he fac ha he derivaive of he price of he securiy wih respec o he underlying price is 1. RiskMerics Technical Documen Fourh Ediion

Sec. 6.3 Sep 3 Compuing Value-a-Risk 15 Equaion [6.17] reveals wo properies: (1) If we ignore he second erm on he righ-hand side we find ha he relaive price change is linearly, bu no in one-o-one correspondence, relaed o he reurn on yield. () If we include he second erm, hen here is a nonlinear relaionship beween reurn, r, and relaive price change. 6.3.1. Nonlinear posiions (opions) In opions posiions here is a nonlinear relaionship beween he change in value of he posiion and he underlying reurn. We explain his relaionship wih a simple sock opion. For a given se of parameers denoe he opion s price by V ( P, K, τρσ,, ) where P is he spo price on he underlying sock a ime, K is he opion s exercise price, τ is he ime o mauriy of he opion in erms of a year, ρ is he riskless rae of a securiy ha maures when he opion does, and σ is he sandard deviaion of he log sock price change over he horizon of he opion. In order o obain an expression for he reurn on he opion,, we approximae he fuure value of he opion V ( P + 1, K, τρσ,, ) wih a second-order Taylor series expansion around he curren values (spo raes), P, K, τρ,, and σ. This yields, ) r i, [6.18] V 1 V V ( P + 1, K, τρσ,, ) V 0 ( P, K, τρσ,, ) + P ( P + 1 P ) + -- P ( P + 1 P ) which can be rewrien in he more concise form [6.19] dv δ ( dp) 1 + -- Γ ( dp) Noice ha dv, he change in value of he opion, is in unis of price P hus δ is uniless and Γ is in unis of 1/P. Wriing Eq. [6.19] in erms of relaive changes, we ge [6.0] 1 r η δr + -- Γ P r ) where η measures he leverage effec of holding he opion, δ measures he relaive change in he value of he opion given a change in he value of he price P, Γ measures he relaive change in he value of he opion given a change in he value of δ. r As Eq. [6.0] shows, he relaive change,, in he opion is a nonlinear funcion of, he reurn on he underlying sock price, since i involves he erm r. ) r 6.3. Simple VaR calculaion In his secion we provide he general formula o compue VaR for linear insrumens. (These insrumens include he firs nine lised in Table 6.3.) The example provided below deals exclusively wih he VaR calculaion a he 95% confidence inerval using he daa provided by Risk- Merics. Consider a porfolio ha consiss of N posiions and ha each of he posiions consiss of one cash flow on which we have volailiy and correlaion forecass. Denoe he relaive change in value of he nh posiion by. We can wrie he change in value of he porfolio,, as r n, ) r p, ) [6.1] ) r p N, = ω n r n, = n= 1 ) N n= 1 ω n δ n r n, Par III: Risk Modeling of Financial Insrumens

16 Chaper 6. Marke risk mehodology where ω n is he oal (nominal) amoun invesed in he nh posiion. For example, suppose ha he oal curren marke value of a porfolio is $100 and ha $10 is allocaed o he firs posiion. I follows ha = $10. Now, suppose ha he VaR forecas horizon is one day. In RiskMerics, he VaR on a porfolio of simple linear insrumens can be compued by 1.65 imes he sandard deviaion of r p, he porfolio reurn, one day ahead.the expression of VaR is given as follows. [6.] VaR = σ 1 (Value-a-Risk esimae) where ω 1 T R 1 σ 1 ) [6.3] σ 1 = 1.65σ 1 1 [, ωδ 1 1 1.65σ, 1 ωδ 1.65σ N, 1 ω δ ] N N is he individual VaR vecor (1xN) and [6.4] R 1 = 1 ρ 1 1, ρ 1N, 1 ρ 1, 1 1 ρ N1, 1 1 is he NxN correlaion marix of he reurns on he underlying cash flows. The above compuaions are for porfolios whose reurns can be reasonably approximaed by he condiional normal disribuion. In oher words, i is assumed ha he porfolio reurn follows a condiional normal disribuion. 6.3..1 Fixed income insrumens In his secion we address wo imporan issues relaed o calculaing he VaR on a porfolio of fixed income insrumens. The firs issue relaes o wha variable should be used o measure volailiy and correlaion. In oher words, should we compue volailiies and correlaions on prices or on yields? The second issue deals wih incorporaing he roll down and pull-o-par effecs of bonds ino VaR calculaions. We discussed in Secion 6.3.1.1 ha one may choose o model eiher he yield (ineres rae) or he price of a fixed income insrumen. RiskMerics compues he price volailiies and correlaions on all fixed income insrumens. This is done by firs compuing zero raes for all insrumens wih a mauriy of over one year, and hen consrucing prices from hese series using he expression (coninuous compounding). [6.5] P = e y N where y is he curren yield on he N-period zero-coupon bond. For money marke raes, i.e., insrumens wih a mauriy of less han one-year, prices are consruced from he formula [6.6] 1 P = ----------------------- ( 1+ y ) N RiskMerics Technical Documen Fourh Ediion

Sec. 6.3 Sep 3 Compuing Value-a-Risk 17 Since praciioners ofen hink of volailiies on fixed income insrumens in erms of yield, we presen he price volailiy in erms of yield volailiy. Saring wih Eq. [6.5], we find he price reurn o be [6.7] r = ln ( P P 1 ) = N ( y 1 y ) Therefore, he sandard deviaion of price reurns under coninuous compounding is given by he expression [6.8] σ = Nσ( y 1 y ) where σ ( y 1 y ) is he sandard deviaion of y. Wha Eq. [6.8] saes is ha price y 1 reurn volailiy is he mauriy on he underlying insrumen imes he sandard deviaion of he absolue changes in yields. Performing he same exercise on Eq. [6.6] we find he price reurn o be [6.9] r = ln ( P P 1 ) 1 + y 1 = N ln ------------------- 1 + y In his case (discree compounding) he sandard deviaion of price reurns is [6.30] σ = Nσ 1 + y ln ------------------- 1 + y 1 1 + y 1 + y where σ ------------------- ln is he sandard deviaion of ln -------------------. 1 + y 1 1 + y 1 We now explain how o incorporae he unique feaures of fixed income insrumens in VaR calculaions. 9 Tradiionally, RiskMerics reas a cash flow as a zero coupon bond and subjecs i o wo assumpions: (1) There is no expeced change in he marke value of such a bond, and () he volailiy of he bond s marke value scales up wih he square roo of he ime horizon. In realiy, he bond s marke value sysemaically increases oward is par value (he pull o par effec), and is daily volailiy decreases as i moves closer o par (he roll down effec). The wo assumpions imply ha he cash flow is reaed as a generic bond (a bond whose mauriy is always he same) raher han as an insrumen whose mauriy decreases wih ime. While his leads o an accurae depicion of he risk of he fuure cash flow for shor forecas horizons, for longer horizons, i can resul in a significan oversaemen of risk. Suppose ha as of oday, a USD based invesor currenly holds a one-year USD money marke deposi and is ineresed in compuing a Value-a-Risk esimae of his insrumen over a 3-monh forecas horizon. Tha is, he invesor would like o know he maximum loss on his deposi (a a 95% confidence level) if he held he deposi for 3 monhs. To compue he risk of his posiion we compue he VaR of holding 9-monh deposi wih a forecas horizon 3-monhs. In oher words, we are measuring he volailiy on he 9-monh deposi over a 3-monh forecas horizon. To do his we use he curren 9-monh money marke raes. This addresses he roll down effec. In addiion, he expeced value of holding a one-year deposi for 3 monhs is no zero. Insead, he mean reurn is 9 This secion is based on he aricle by Chrisopher C. Finger, Accouning for he pull o par and roll down for RiskMerics cash flows, RiskMerics Monior, Sepember 16, 1996. Par III: Risk Modeling of Financial Insrumens

18 Chaper 6. Marke risk mehodology non-zero reflecing he pull-o-par phenomenon. Char 6.17 presens a visual descripion of he siuaion. Char 6.17 VaR horizon and mauriy of money marke deposi Today VaR forecas horizon MM deposi expires 0 90 days 360 days In general, he mehodology o measure he VaR of a fuure cash flow(s) ha occurs in T days over a forecas horizon of days ( < T) is as follows. y T 1. Use he T- rae,, o discoun he cash flow occurring in T days ime. Denoe he presen value of his cash flow by V T. Compue VaR as V T σ T. Noe ha in he preceding example, T = 360, = 90, is he 70-day rae and σ T is he sandard deviaion of he disribuion of reurns on he 70-day rae. 6.3.. Equiy posiions The marke risk of he sock, VaR, is defined as he marke value of he invesmen in ha sock, V, muliplied by he price volailiy esimae of ha sock s reurns, 1.65σ. y T [6.31] VaR = V 1.65σ Since RiskMerics does no publish volailiy esimaes for individual socks, equiy posiions are mapped o heir respecive local indices. This mehodology is based upon he principles of singleindex models (he Capial Asse Pricing Model is one example) ha relae he reurn of a sock o he reurn of a sock (marke) index in order o aemp o forecas he correlaion srucure beween securiies. Le he reurn of a sock,, be defined as r [6.3] r = β r m, + α + ε where r = reurn of he sock r m, = he reurn of he marke index β = bea, a measure of he expeced change in r m, = he expeced value of a sock's reurn ha is firm-specific α ε = he random elemen of he firm-specific reurn wih E [ ε ] = 0 and E [ ε ] = σ ε As such, he reurns of an asse are explained by marke-specific componens ( β R m, ) 3 and by sock-specific componens ( α + ε ). Similarly, he oal variance of a sock is a funcion of he marke- and firm-specific variances. RiskMerics Technical Documen Fourh Ediion

Sec. 6.3 Sep 3 Compuing Value-a-Risk 19 Since he firm-specific componen can be diversified away by increasing he number of differen equiies ha comprise a given porfolio, he marke risk, VaR, of he sock can be expressed as a funcion of he sock index [6.33] = β σm +σ ε. Subsiuing Eq. [6.33] ino Eq. [6.31] yields [6.34] VaR = V β 1.65σ m,, where σ, 1.65σ m, = he RiskMerics VaR esimae for he appropriae sock index. As wih individual socks, Eq. [6.34] should also be used o calculae he VaR for posiions ha consis of issue ha hemselves are par of he EMBI+. 6.3.3 Dela-gamma VaR mehodology (for porfolios conaining opions) In his secion we describe a mehodology known as dela-gamma ha allows users o compue he Value-a-Risk of a porfolio ha conains opions. Specifically, we provide a mehodology o incorporae he dela, gamma and hea of individual opions in he VaR calculaion. We explain his mehodology by firs showing how i applies o a single opion and hen o a porfolio ha conains hree opions. To keep our examples simple, we assume ha each opion is a funcion of one cash flow. In oher words, we can wrie he reurn on each opion as [6.35] ) r 1, = δ 1 r 1, + 0.5Γ 1r 1, +θ 1n where δ 1 = η 1 δ 1 Γ 1 = η 1 P 1, Γ 1 θ 1 = θ 1 V 1 n = VaR forecas horizon = opion s premium V 1 For a complee derivaion of Eq. [6.35], see Appendix D. Similarly, we can wrie he reurns on he oher wo opions as [6.36] r, = δ r, + 0.5Γ r, +θ n and ) ) Le s begin by demonsraing he effec of incorporaing gamma and hea componens on he reurn disribuion of he opion. We do so by comparing he saisical feaures on he reurn on opion 1, r 1,, and he reurn of is underlying cash flow, r 1,. Recall ha RiskMerics assumes ha he reurns on he underlying asses,, are normally disribued wih mean 0 and variance ) r 1, r 3, = δ 3 r 3, + 0.5Γ 3r 3, +θ 3n Par III: Risk Modeling of Financial Insrumens

130 Chaper 6. Marke risk mehodology σ 1, r 1, ). Table 6.4 shows he firs four momens 10 he mean, variance, skewness, and kurosis for and. r 1, Table 6.4 Saisical feaures of an opion and is underlying reurn Saisical parameer Opion Underlying Reurn ) r 1, r 1, Mean 0.5Γ σ 1, + θ 1n 0 Variance σ1, + 0.5Γ σ 4 1, σ 1, δ 1 4 3 6 Skewness 3δ 1 Γ 1σ 1, + Γ 1 σ 1, 0 6 Γ 1σ1 4 σ 8 Kurosis 1δ 1, + 3Γ 1 1, + 3σ 1, 3 σ 1, The resuls presened in Table 6.4 poin o hree ineresing findings. 4 Firs, even hough i is assumed ha he reurn on he underlying has a zero mean reurn, his is no rue for he opion s reurn unless boh gamma and hea are zero. Also, he sign of he opion s mean will be deermined by he relaive magniudes and signs of boh gamma and hea and wheher one is long or shor he opion. Second, he variance of he reurn on he opion differs from he variance of he reurn on he underlying insrumen by he facor δ 1 + 0.5Γ σ 1,. And hird, depending on wheher one is long or shor he opion deermines wheher he reurn on he opion disribuion is negaively or posiively skewed. To see his, on a shor opion posiion, V 1 < 0 and herefore Γ 1 < 0. Consequenly, he erm Γ 3 in he skewness expression will be negaive. As an example of his poin, Char 6.18 shows he probabiliy densiy funcions for long and shor opions posiions (along wih he normal probabiliy curve). 4 10 See Secion 4.5..1 for he definiion of hese momens. RiskMerics Technical Documen Fourh Ediion

Sec. 6.3 Sep 3 Compuing Value-a-Risk 131 Char 6.18 Long and shor opion posiions negaive and posiive skewness PDF 0.45 0.40 0.35 0.30 0.5 0.0 0.15 0.10 0.05 Negaive skew Normal 0-6 -5-4 -3 - -1 0 1 3 4 5 6 Sandard deviaion Posiive skew Noe ha in variance and kurosis, he sign of Γ is immaerial since in hese expressions Γ is raised o an even power. Now, o deermine he numerical values of he momens presened in Table 6.4 we need esimaes of δ 1, Γ 1, θ 1 and σ 1,. Esimaes of he firs hree parameers are easily found by applying a Black-Scholes ype valuaion model. The variance,, is given in he RiskMerics daa ses. r 1, Having obained he firs four momens of s disribuion, we find a disribuion ha has he same momens bu whose disribuion we know exacly. In oher words, we mach he momens of r 1, s disribuion o one of a se of possible disribuions known as ohnson disribuions. Here, maching momens simply means finding a disribuion ha has he same mean, sandard deviaion, skewness and kurosis as r 1, s. The name ohnson comes from he saisician Norman ohnson who described a process of maching a paricular disribuion o a given se of momens. ) ) ) Maching momens o a family of disribuions requires ha we specify a ransformaion from he opion s reurn r 1, o a reurn, r 1,, ha has a sandard normal disribuion. For example, ohnson (1949) suggesed he general ransformaion ) σ 1, [6.37], c, = a + bf ------------------- d r 1 ) r 1 where f( ) is a monoonic funcion and a, b, c and d are parameers whose values are deermined by r 1, s firs four momens. In addiion o he normal disribuion, he ohnson family of disribuions consiss of hree ypes of ransformaions. ) [6.38], c, = a + blog ------------------- d r 1 ) r 1 (Lognormal ) [6.39], a b h 1 r 1, c = + sin ------------------- d (Unbounded ) r 1 ) and Par III: Risk Modeling of Financial Insrumens

13 Chaper 6. Marke risk mehodology [6.40] r 1, c, = a + blog ----------------------------- c+ d r 1, r 1 ) ) (Bounded ) wih he resricion ( c< r 1, < c+ d). ) To find esimaes of a, b, c and d, we apply a modified version of Hill, Hill and Holder s (1976) algorihm. 11 Given hese esimaes we can calculae any percenile of r 1, s disribuion given he corresponding sandard normal percenile (e.g., 1.65). This approximae percenile is hen used in he VaR calculaion. For example, suppose ha we have esimaes of δ 1, Γ 1, θ 1 and σ 1, and ha hey resul in he following momens: mean = 0., variance = 1, skewness coefficien = 0.75 and kurosis coefficien = 7. Noe ha hese numbers would be derived from he formulae presened in Table 6.3. Applying he Hill e. al. algorihm we find ha he seleced disribuion is Unbounded wih parameer esimaes: a = 0.58, b = 1.768, c = 0.353, and d = 1.406. Therefore, he percenile of ) r 1, ) s disribuion is based on he ransformaion [6.41] r 1, = sinh ) ( r 1, a) ----------------------- b d + c r 1 Seing, = 1.65, he esimaed 5h percenile of r 1, s disribuion is 1.45. Tha is, he opion s fifh percenile is increased by 0.0. In his hypoheical example, he incorporaion of gamma and hea reduces he risk relaive o holding he underlying. We now show ha i is sraighforward o compue he VaR of a porfolio of opions. In paricular, we show his for he case of a porfolio ha conains hree opions. We begin by wriing he porfolio reurn as ) [6.4] ) r p, = ω 1 r 1, + ω r, + ω 3 r 3, ) ) ) where ω i = V i 3 Vi -------------- i = 1 r p To compue he momens of, s disribuion we need he RiskMerics covariance marix, Σ, of he underlying reurns { r 1,, r, r }, and he dela, gamma and hea cash flow vecors ha are, 3, defined as follows: ) δ 1 Γ 1 0 0 [6.43] δ = δ, Γ =, and 0 Γ 0 θ δ 3 0 0 Γ 3 = θ 1 θ θ 3 r p We find he 5h percenile, s disribuion he same way we found he 5h percenile of, s disribuion, as shown previously. The only difference is ha now he expressions for he four momens are more complicaed. For example, he mean and variance of he porfolio reurn are ) ) r 1 11 These original algorihms (numbers 99 and 100) are available in heir enirey on he Web a he SaLib Griffihs and Hill Archive. The URL is hp://lib.sa.cmu.edu/griffihs-hill/. RiskMerics Technical Documen Fourh Ediion

Sec. 6.3 Sep 3 Compuing Value-a-Risk 133 [6.44] µ p, = 0.5 race [ Γ Σ] + θ i and 3 i = 1 [6.45] σ p, = δ Σδ + 0.5 race ( Γ Σ) where race is an operaor ha sums he diagonal elemens of a marix. The dela-gamma mehodology described in his secion exends o opions ha have more han one underlying cash flow (e.g., bond opion). We have presened a simple example purposely o faciliae our exposiion of he mehodology. See, Appendix D for an assessmen of he mehodology. Finally, he mehodologies presened in Secion 6.3 do no require simulaion. All ha is necessary for compuing VaR is a covariance marix, financial parameers (such as dela, gamma and hea) and posiion values. In he nex chaper we presen a mehodology known as srucured Mone Carlo ha compues VaR by firs simulaing fuure pahs of financial prices and/or raes. Par III: Risk Modeling of Financial Insrumens

134 Chaper 6. Marke risk mehodology Table 6.6 RiskMerics map of single cash flow 6.4 Examples In his secion we presen nine examples of VaR calculaions for he various insrumens discussed in his chaper. Noe ha he diskee symbol placed o he lef of each example means ha he example appears on he enclosed diskee a he back of he book. Ex. 6.1 Governmen bond mapping of a single cash flow Suppose ha on March 7, 1995, an invesor owns FRF 100,000 of he French OAT benchmark 7.5% mauring in April 005. This bond pays coupon flows of FRF 7,500 each over he nex 10 years and reurns he principal invesmen a mauriy. One of hese flows occurs in 6.08 years, beween he sandard verices of 5 and 7 years (for which volailiies and correlaions are available). All he daa required o compue he cash flow map is readily available in he RiskMerics daa ses excep for he price volailiy (1.65σ 6.08 ) of he original cash flow s presen value. This mus be inerpolaed from he price volailiies already deermined for he RiskMerics verices. Applying he hree condiions on page 118 and using Eqs. [6.10] [6.14] wih he RiskMerics daa in Table 6.5, we solve for he allocaion α (and (1-α)), and obain he values α = 4.30 and α = 0.4073. Given he consrain ha boh of he allocaed cash flows mus have he same sign as he original cash flow, we rejec he firs soluion, which would lead o a shor posiion in he second proxy cash flow. As a resul, our original cash flow of FRF 7,500, whose presen value is FRF 4,774, mus be mapped as a combinaion of a 5-year mauriy cash flow of FRF 1,944 (40.73% of he original cash flow s PV) and a 7-year mauriy cash flow of FRF,89 (59.7% of he original cash flow s PV). The cash flow map is shown in Table 6.6. Table 6.5 RiskMerics daa for 7, March 1995 RiskMerics Verex Yield,% P. Vol, * (1.65σ ) Yield Vol, (1.65σ ) Correlaion Marix, ρ ij 5yr 7yr 5yr 7.68 0.533 1.50 1.000 0.963 7yr 7.794 0.696 1.37 0.963 1.000 * P. Vol = price volailiy, also called he VaR saisic. While his daa is provided in he daa se, i is no used in his calculaion. Coupon Flow Term Yield,% (Acual) Sep 1 * Sep * Sep 3 * Sep 4 * Sep 5 * Yield,% (y 6.08 ) P. Vol, (1.65σ ) P. Vol, (1.65σ 6.08 ) RiskMerics RiskMerics (Inerpolaed) (PV) 6.08 (RiskMerics) (Inerpolaed) Verex Allocaion Cash flow 7.68 0.533 5yr 0.4073 1,950 7,500 6.08yr 7.717 4,774 0.64 7.794 0.696 7yr 0.397,84 * Sep from he mapping procedure on pages 119 11. Also, daa in his column is calculaed from he daa in Table 6.5. Noe ha in Sep 3 he price volailiy, 1.65σ 6.08, raher han he sandard deviaion alone, is compued. In his example â = 0.46. PV = presen value. P. Vol = price volailiy, also called he VaR saisic. RiskMerics Technical Documen Fourh Ediion

Sec. 6.4 Examples 135 Ex. 6. Governmen bond mapping of muliple cash flows A full se of posiions can easily be mapped in he same fashion as he single cash flow in he las example. The example below akes he insrumen in Ex. 6.1, i.e., he 10-year French OAT benchmark on March 7, 1995, and decomposes all of he componen cash flows according o he mehod described on pages 119 11, o creae a deailed RiskMerics cash flow map. Table 6.7 shows how he 100,000 French franc nominal posiion whose marke value is FRF 97,400, is decomposed ino nine represenaive presen value cash flows. The able also shows he VaR for he cash flow a each RiskMerics verex and he diversified Value-a-Risk. In his example, noe ha he firs cash flow (on 5-Apr-95) occurs in less han one monh s ime relaive o March 7, bu is allocaed a 100% weigh o he 1-monh RiskMerics verex. The reason for 100% allocaion is ha verices shorer han one monh are no defined in he RiskMerics daa ses. Table 6.7 RiskMerics map for muliple cash flows Bond daa RiskMerics verices 1m 1y y 3y 4y 5y 7y 10y 15y Yield volailiy 7.00 3.16.10 1.74 1.63 1.50 1.37 1.36 1.9 Selemen 30-Mar Curren yield 8.5 7.04 7.8 7.39 7.54 7.63 7.79 7.9 8.15 Principal 100,000 Price volailiy 0.04 0.1 0.9 0.36 0.46 0.53 0.70 1.00 1.46 Price 97.4 Correlaion Marix 1m 1.00 0.75 0.53 0.48 0.45 0.4 0.33 0.33 0.33 Coupon 7.50 1y 0.75 1.00 0.88 0.81 0.78 0.74 0.61 0.63 0.58 Basis 365 y 0.53 0.88 1.00 0.99 0.96 0.9 0.80 0.8 0.76 3y 0.48 0.81 0.99 1.00 0.98 0.95 0.85 0.87 0.81 4y 0.45 0.78 0.96 0.98 1.00 0.99 0.91 0.93 0.88 5y 0.4 0.74 0.9 0.95 0.99 1.00 0.96 0.96 0.93 7y 0.33 0.61 0.80 0.85 0.91 0.96 1.00 1.00 0.99 10y 0.33 0.63 0.8 0.87 0.93 0.96 1.00 1.00 0.99 15y 0.33 0.58 0.76 0.81 0.88 0.93 0.99 0.99 1.00 Dae Flow Term Yield PV Md. Dur P.Vol 5-Apr-95 7,500 0.071 8.04 7,456 0.066 0.03 7,456 5-Apr-96 7,500 1.074 7.056 6,970 1.003 0.18 5,594 1,376 5-Apr-97 7,500.074 7.84 6,48 1.933 0.9 5,780 703 5-Apr-98 7,500 3.074 7.40 6,0.86 0.366 5,505 517 5-Apr-99 7,500 4.074 7.543 5,577 3.788 0.463 5,105 47 5-Apr-00 7,500 5.077 7.635 5,16 4.717 0.539 4,93 40 5-Apr-01 7,500 6.077 7.70 4,773 5.641 0.64 1,944,89 5-Apr-0 7,500 7.077 7.798 4,408 6.565 0.703 4,30 107 5-Apr-03 7,500 8.077 7.855 4,07 7.488 0.805,589 1,483 5-Apr-04 7,500 9.079 7.895 3,76 8.415 0.905 1,131,631 5-Apr-05 107,500 10.079 7.919 49,863 9.340 1.004 49,019 844 RiskMerics verices 1m 1y y 3y 4y 5y 7y 10y 15y Toal Verex Mapping 7,456 5,594 7,156 6,07 5,6 7,339 11,091 53,39 844 RiskMerics Verex VaR 3 1 0 6 39 77 530 1 Diversified Value a Risk 77 FRF over he nex 4 hours % of marke value 0.7% Money marke rae volailiies are used for verices below years. Governmen bond zero volailiies are used for -year and oher verices. Par III: Risk Modeling of Financial Insrumens

136 Chaper 6. Marke risk mehodology Ex. 6.3 Forward rae agreemen cash flow mapping and VaR A forward rae agreemen is an ineres-rae conrac. I locks in an ineres rae, eiher a borrowing rae (buying a FRA) or a lending rae (selling a FRA) for a specific period in he fuure. FRAs are similar o fuures, bu are over-he-couner insrumens and can be cusomized for any mauriy. Because a FRA is a noional conrac, here is no exchange of principal a he expiry dae (i.e., he fixing dae). If he rae is higher a selemen han he FRA rae agreed by he counerparies when hey raded, hen he seller of a FRA agrees o pay he buyer he presen value of he ineres rae differenial applied o he nominal amoun agreed upon a he ime of he rade. The ineres rae differenial is beween he FRA rae and he observed fixing rae for he period. In mos cases his is he LIBOR rae for any given currency. The general FRA pricing equaion is given by [6.46] Y TS T L FRA = T L 1 + Y L ----- B --------------------------------- 1 T S 1+ Y S ----- B B ----------------- T L T S where Y L T L = yield of he longer mauriy leg = mauriy of he longer mauriy leg in days B = Basis (360 or 365) = yield of he shorer mauriy leg = mauriy of he shorer mauriy leg in days Y S T S In effec, FRAs allow marke paricipans o lock in a forward rae ha equals he implied breakeven rae beween money marke erm deposis. Given ha a FRA is a linear combinaion of money marke raes, i is simple o express is degree of risk as a funcion of he combinaion of hese raes. Suppose ha on anuary 6, 1995 you sold a 6x1 FRA on a noional 1 million French francs a 7.4%. This is equivalen o locking in an invesmen rae for 6 monhs saring in 6 monhs ime. The rae of 7.4% is calculaed by combining he 6- and 1-monh money marke raes using he general pricing equaion, Eq. [6.46], which can be rewrien as follows o reflec he no-arbirage condiion: [6.47] 1 + Y 1m -------- 365 = 181 1+ Y 6m -------- 365 1+ Y 6 1F 365 184 RA -------- where Y 6m, Y 1m = 6- and 1-monh French franc yields, respecively Y 6m, Y 1m = 6 x 1 FRA rae This FRA ransacion is equivalen o borrowing FRF 1 million for 6 monhs on a discoun basis (i.e., oal liabiliy of FRF 1 million in 6 monhs ime) and invesing he proceeds (FRF 969,11) for 1 monhs. This combinaion can be mapped easily ino he RiskMerics verices as shown in Table 6.8. The curren presen value of hese wo posiions is shown in column (6). The Value-a- RiskMerics Technical Documen Fourh Ediion

Sec. 6.4 Examples 137 Example 6.3 (coninued) Table 6.8 Mapping a 6x1 shor FRF FRA a incepion Risk of each leg of he FRA over a 1-monh horizon period, shown in column (9), is obained by muliplying he absolue presen value of he posiion by he monhly price volailiy of he equivalen mauriy deposi. The porfolio VaR is obained by applying he 6- o 1-monh correlaion o he VaR esimae from column (9). Observed daa RiskMerics daa se Calculaed values (1) () (3) (4) (5) (6) (7) (8) (9) Cash flow Term (mhs.) Yield,% Volailiies Correlaion marix Yield Price 6m 1m Presen value RiskMerics verex RiskMerics cash flow VaR esimae 1,000,000 6 6.39 6.94 0.1 1.00 0.70-969,11 6m 969,11,081 1,036,317 1 6.93 7.4 0.48 0.70 1.00 969,11 1m 969,11 4,66 Toal 0 0 Porfolio VaR 3,530 Table 6.9 Mapping a 6x1 shor FRF FRA held for one monh One monh ino he rade, he mapping becomes somewha more complex as he cash flows have now shorer mauriies (he insrumen is now in fac a 5x11 FRA). The 5-monh cash flow mus be mapped as a combinaion of 3-monh and 6-monh RiskMerics verices (Table 6.9), while he 11-monh cash flow mus be spli beween he 6-monh and 1-monh verices. Observed daa RiskMerics daa se Calculaed values Cash flow Term (mhs.) Volailiies Correlaion marix Yield Price 3m 6m 1m Yield,% Presen value RiskMerics verex RiskMerics cash flow VaR esimae 6.77 0.1 1 0.81 0.67 3m 30,3 96 1,000,000 5 6.1 975,30 7.91 0.19 0.81 1.00 0.68 1,048 6m 549,300 1,036,317 11 6.65 976,894 7.14 0.41 0.67 0.68 1.00 1m 853,14 3,533 Toal 1,59 1,59 Porfolio VaR,777 Par III: Risk Modeling of Financial Insrumens

138 Chaper 6. Marke risk mehodology Example 6.3 (coninued) One monh ino he rade, he change in marke value of he conrac is a posiive FRF 1,59. This is well wihin he range of possible gains or losses prediced (wih a 95% probabiliy) by he previous monh s Value-a-Risk esimae of FRF 3,530. Unwinding a FRA, i.e., hedging ou he ineres rae risk beween he FRA rae and he marke rae, before mauriy requires enering ino a conrac of opposie sign a daes ha no longer qualify as sandard mauriies. If you waned o unwind he posiion in his example one monh afer he dealing dae, you would have o ask a quoe o buy a 5x11 FRA, a broken daed insrumen ha is less liquid and herefore is quoed a higher bid-offer spreads. The raes in column (1) above do no ake his ino accoun. They were derived by inerpolaing raes beween sandard mauriies. Acual marke quoes would have been slighly less favorable, reducing he profi on he ransacion. This risk is liquidiy relaed and is no idenified in he VaR calculaions. RiskMerics Technical Documen Fourh Ediion

Sec. 6.4 Examples 139 Ex. 6.4 Srucured noe The basic conceps used o esimae he marke risk of simple derivaives can be exended o more complex insrumens. Suppose ha in early 1994, when he marke consensus was ha German raes were o coninue o decline, you had purchased a one year index noe linked o wo year raes. This 1-year insrumen leveraged a view ha he DEM -year swap rae in 1-year s ime would be below he forward rae measured a he ime he ransacion was enered ino. The characerisics of he insrumen are shown in Table 6.10. Table 6.10 Srucured noe specificaion Issuer Company A Forma Euro Medium Term Noe Issue dae 9 February 94 Mauriy dae 9 February 95 Issue price 100% Amoun DEM 35,000,000 Coupon 5.10% Srike 5.10% Redempion value 100%+0*(Srike- -year DEM swap rae) * Alhough hese deails are hypoheical, similar producs were markeed in 1994. While seemingly complex, his ransacion is in fac lile more (disregarding minor convexiy issues) han a bond o which a leveraged long-daed FRA had been aached. As a holder of he noe, you were long he 3-year swap rae and shor he 1-year rae, wih significan leverage aached o he difference. Table 6.11 shows how he leveraged noe can be decomposed ino he cash flows of he wo underlying building blocks: The 1-year DEM 35 million bond wih a 5.10% coupon. The forward swap (-year swap saring in one year). The forward principal cash flows of he swap are equal o 0 imes he noional amoun of he noe divided by he PVBP (price value of a basis poin) of a -year insrumen, or 1.86 in his case. The forward coupons are equal o he forward principal imes he coupon rae of 5.10%. Table 6.11 Acual cash flows of a srucured noe Term (years) Bond Swap Toal Principal Coupon Principal Coupon cash flow 1 35,000,000 1,785,000 376,996,98 340,11,98 19,6,843 19,6,843 3 376,996,98 19,6,843 396,3,771 Combining he bond and he swap creaes hree annual cash flows where he invesor is shor DEM 340 million in he 1-year, and long DEM 19 and DEM 396 million in he - and 3-year mauriies. A issue, he marke value of hese cash flows is equal o DEM 35 million, he insrumen s issue price. Par III: Risk Modeling of Financial Insrumens

140 Chaper 6. Marke risk mehodology Example 6.4 (coninued) Table 6.1 VaR calculaion of srucured noe One monh forecas horizon Each of he hree cash flows is mapped o RiskMerics verices o produce he cash flow map shown in Table 6.1. Observed daa RiskMerics daa se Calculaed values Cash flow Term (years) Yield,% Price volailiy Correlaion marix 1y y 3y Presen value RiskMerics verex RiskMerics cash flow VaR esimae 340,11,98 1 5.48 0.33 1.00 0.46 0.43 3,536,906 1y 3,536,906 1,067,597 19,6,843 5.15 0.46 0.46 1.00 0.95 17,389,594 y 17,389,594 79,644 396,3,77 3 5. 0.68 0.43 0.95 1.00 340,147,31 3y 340,147,31,309,600 Toal 35,000,000 Porfolio VaR,155,108 Table 6.13 VaR calculaion on srucured noe One-monh ino life of insrumen Using he appropriae volailiies and correlaions as of February 9, 1994, he Value-a-Risk of such a posiion over a 1-monh horizon was around DEM.15 million. One monh ino he life of he insrumen on March 9, he mapping and risk esimaion could have been repeaed using updaed ineres raes as well as updaed RiskMerics volailiies and correlaions. Table 6.13 shows he resul. Observed daa RiskMerics daa se Calculaed values Cash flow Term (years) Yield,% Price volailiy Correlaion marix 6m 1y y 3y Presen value RiskMerics verex RiskMerics cash flow VaR esimae 0.16 1.00 0.83 0.58 0.54 6m 43,18,017 68,03 340,11,98 0.9 5.53 975,30 0.35 0.83 1.00 0.58 0.54 1y 77,979,83 960,50 19,6,843 1.9 5.53 17,399,085 0.65 0.58 0.58 0.94 1.00 y 39,059,679 5180 19,6,843.9 5.68 337,30,77 1.03 0.54 0.54 0.94 1.00 3y 31,934,965 3,18,971 Toal 30,976,801 4,03,199 Porfolio VaR 3,018,143 The movemen in marke raes led he marke value of he noe o fall by over DEM 4 million, wice he maximum amoun expeced o happen wih a 95% probabiliy using he previous monh s RiskMerics volailiies and correlaions. Why? RiskMerics Technical Documen Fourh Ediion

Sec. 6.4 Examples 141 Example 6.4 (coninued) Char 6.19 shows he 3-year DEM swap rae moved 45+bp from 5.% o slighly above 5.68% during he monh wice he maximum amoun expeced wih a 95% probabiliy (4.56% 5.%; i.e., 3 basis poins). This was clearly a large rae move. The RiskMerics volailiy esimae increased from 4.56% o 6.37% as of March 9. This reflecs he rapid adjusmen o recen observaions resuling from he use of an exponenial moving average esimaion mehod. Correspondingly he VaR of he srucured noe increased 44% over he period o jus over DEM 3 million. Char 6.19 DEM 3-year swaps in Q1-94 Percen 10.5 9.5 8.5 Monhly yield volailiy 7.5 6.5 5.5 4.5 3.5 3-year DEM zero swap yield an-1994 Feb-1994 Mar-1994 Apr-1994 May-1994 The message in hese examples is ha wih proper cash flow mapping, he risks in complex derivaives can be easily esimaed using he RiskMerics mehodology and daa ses. Par III: Risk Modeling of Financial Insrumens

14 Chaper 6. Marke risk mehodology Ex. 6.5 Ineres-rae swap Invesors ener ino swaps o change heir exposure o ineres rae uncerainy by exchanging ineres flows. In order o undersand how o map is cash flows, a swap should be hough of as a porfolio of one fixed- and one floaing-rae ransacion. Specifically, he fixed leg of a swap is mapped as if i were a bond, while he floaing leg is considered o be a FRN. Marke risk esimaion is sraighforward if he value of each leg is considered separaely. The fixed leg exposes an invesor o ineres rae variabiliy as would a bond. Since he floaing leg is valued as if i were a FRN, if ineres raes change, hen forward raes used o value he leg change and i will revalue o par. Once a floaing paymen is se, he remaining porion of he floaing leg will revalue o par, and we need only consider ineres rae exposure wih respec o ha se cash flow. The deails of his will be provided in a forhcoming ediion of he RiskMerics Monior. Consider he following example. A company ha eners ino a 5-year USD ineres-rae swap pays 9.379% fixed and receives floaing cash flows indexed off of 1-year USD LIBOR fla on a noional amoun of USD 1,000,000. For simpliciy, he rese/paymen daes are annual. Table 6.14 presens he daa used o esimae he marke risk of his ransacion. Table 6.14 Cash flow mapping and VaR of ineres-rae swap Term Observed daa RiskMerics daa se Calculaed values Zero rae Cash flow Correlaion marix Price Fixed Floaing volailiy, % 1yr yr 3yr 4yr 5yr Ne presen value, USD 1yr 8.75 86,47 1,000,000 0.07 1.000 0.949 0.933 0.93 0.911 913,753 yr 9.08 85,986 0.067 0.949 1.000 0.98 0.978 0.964 85,986 3yr 9.4 85,860 0.11 0.933 0.98 1.0000 0.995 0.984 85,860 4yr 9.34 85,78 0.149 0.93 0.978 0.995 1.000 0.986 85,78 5yr 9.4 999,69 0.190 0.911 0.964 0.984 0.986 1.000 999,69 Toal 343,505 Porfolio VaR 1,958 RiskMerics Technical Documen Fourh Ediion

Sec. 6.4 Examples 143 Table 6.15 VaR on foreign exchange forward Ex. 6.6 Foreign exchange forward Below is an example of how o calculae he marke risk of buying a 1-year 153,000,000 DEM/ USD foreign exchange forward. Noe ha buying a DEM/USD foreign exchange forward is equivalen o borrowing US dollars for 1-year (shor money marke posiion) and using hem o purchase Deusche marks in one year s ime (shor foreign exchange posiion). We ake he holding period o be one day. Based on a 1-day volailiy forecas, he foreign exchange risk, in USD, is $904,9 (94,004,163 0.963%) as shown in Table 6.15. The ineres rae risk is calculaed by muliplying he curren marke value of each 1-monh leg (he shor in USD and he long in DEM) imes is respecive ineres rae volailiy. Therefore, he Value-a-Risk for a 1-day holding period is $91,880. Insrumen Observed daa RiskMerics daa se Calculaed values Cash flow Term (years) Yield,% Price volailiy Correlaion marix DEM FX DEM 1y USD 1y Presen value, USD VaR esimae DEM Spo FX 0.963 1.0000 0.0035 0.004 94,004,163 904,9 DEM 1y 153,000,000 1 6.1 0.074 0.0035 1.0000 0.140 94,004,163 45,855 USD 1y 99,80,670 1 6.65 0.116 0.004 0.140 1.0000 94,004,163 108,64 Toal 94,004,163 Porfolio VaR 3,530 Par III: Risk Modeling of Financial Insrumens

144 Chaper 6. Marke risk mehodology Ex. 6.7 Equiy Consider a hree-asse porfolio in which an invesor holds socks ABC and XYZ (boh U.S. companies) as well as a baske of socks ha replicae he S & P 500 index. The marke risk of his porfolio, VaR p, is [6.48] VaR p = ( V ABC 1.65σ RABC ) + ( V XYZ 1.65σ RXYZ ) + V SP500 1.65σ RSP500 Rewriing his equaion in erms of Eq. [6.48] [6.49] VaR s = V s β s 1.65σ RM where 1.65σ RM = he RiskMerics volailiy esimae for he appropriae sock index, yields [6.50] VaR p = ( V ABC β ABC 1.65σ RSP500 ) + ( V XYZ β XYZ 1.65σ RSP500 ) + ( V SP500 1.65σ RSP500 ) Facoring he common erm and solving for he porfolio VaR resuls in VaR p = 1.65σ RSP500 [ ( V ABC β ABC ) + ( V XYZ β XYZ ) + V SP500 ] [6.51] = 4.83% [ 1, 000, 000 ( 0.5) + ( 1, 000, 000) ( 1.5) + 1, 000, 000] = 4.83% ( 3, 000, 000) = USD 144, 960 The mehodology for esimaing he marke risk of a muli-index porfolio is similar o he process above and akes ino accoun correlaion among indices as well as foreign exchange raes. Since all posiions mus be described in a base or home currency, you need o accoun for foreign exchange risk. RiskMerics Technical Documen Fourh Ediion

Sec. 6.4 Examples 145 Ex. 6.8 Commodiy fuures conrac Suppose on uly 1, 1994 you bough a 6-monh WTI fuure on a noional USD 18.3 million (1 million barrels muliplied by a price of USD 18.30 per barrel). The marke and RiskMerics daa for ha dae are presened in Table 6.16. Table 6.16 Marke daa and RiskMerics esimaes as of rade dae uly 1, 1994 WTI fuure Correlaion marix Verex LIBOR Term Price Volailiy 3m 6m 3m 5.563 0.50 10.5 1.000 6m 5.813 0.500 18.30 9.47 0.99 1.000 The iniial Value-a-Risk for a 1-monh horizon is approximaely USD 1.7 million. This represens he maximum amoun, wih 95% confidence, ha one can expec o lose from his ransacion over he nex 5 business days. Since he flow occurs in 6 monhs, he enire posiion is mapped o he 6-monh WTI verex, herefore calculaing he Value-a-Risk of his ransacion on he rade dae is simply VaR 6m Fuure = PV of cash flow RiskMerics volailiy esimae [6.5] = 18, 300, 000 ---------------------------------------------- 9.47% 5.813% 1+ ------------------ 100 0.5 = USD 1.7 million One monh ino he rade, he cash flow mapping becomes slighly more complex. Table 6.17 shows he new VaR of his ransacion. Table 6.17 Cash flow mapping and VaR of commodiy fuures conrac Term Zero rae Cash flow (PV) Price volailiy, % Correlaion marix 3m 6m RiskMerics verex RiskMerics cash flow 4.810 6.1 1.00 0.99 3m 13,084,859 4m 5.068 17,94,465 5.190 5.739 0.99 1.00 6m 4,839,605 Porfolio VaR 1,417,343 Par III: Risk Modeling of Financial Insrumens

146 Chaper 6. Marke risk mehodology Ex. 6.9 Dela-gamma mehodology Consider he siuaion where a USD (US dollar) based invesor currenly holds a USD 1million equivalen posiion in a French governmen bond ha maures in 6 years and a call opion on Deusche marks ha expires in 3 monhs. Since marke risk is measured in erms of a porfolio s reurn disribuion, he firs sep o compuing VaR is o wrie down an expression for his porfolio s reurn, R P, which consiss of one French governmen bond and one foreign exchange opion. Here, reurn is defined as he one-day relaive price change in he porfolio s value. The reurn on he porfolio is given by he expression [6.53] r p = r o + r B ) ) where r o is he reurn on he opion, and r B is he reurn on he French governmen bond. We now provide a more deailed expression for he reurns on he bond and opion. Since he cash flow generaed by he bond does no coincide wih a specific RiskMerics verex, we mus map i o he wo neares RiskMerics verices. Suppose we map 49% of he cash flow ha arrives in 6 years ime o he 5-year verex and 51 percen of he cash flow o he 7-year verex. If we denoe he reurns on he 5 and 7-year bonds by r 5 and r 7, respecively, we can wrie he reurn on he French governmen bond as [6.54] r B = 0.49r 5 + 0.51r 7 Wriing he reurn on he opion is more involved. We wrie he reurn on he opion as a funcion of is dela, gamma and hea componens. The one-day reurn on he opion is given by he expression 1 [6.55] r 0 = αδr USD/DEM + 0.5αΓP USD/DEM r USD/DEM + ) V 1 θn where r USD/DEM P USD/DEM is he one-day reurn on he DEM/USD exchange rae is he spo posiion in USD/DEM V is he price of he opion, or premium. α is he raio of P USD/DEM o V. The parameer α measures he leverage from holding he opion. δ is he dela of he opion. Dela measures he change in he value of he opion given a change in he underlying exchange rae. Γ is he gamma of he opion. Gamma measures he change in δ he underlying exchange rae. given a change in θ is he hea of he opion. Thea measures he change in he value of he opion for a given change in he opion s ime o expiry. n is he forecas horizon over which VaR is measured. In his example n is 1 for one day. 1 We derive his expression in Appendix D. RiskMerics Technical Documen Fourh Ediion

Sec. 6.4 Examples 147 Example 6.9 (coninued) We can now wrie he reurn on he porfolio as [6.56] r p = 0.49r 5 + 0.51R 7 + αδr USD/DEM + 0.5αΓP USD/DEM r USD/DEM + ) V 1 θn In paricular, we find he firs four momens of r p s disribuion ha correspond o he mean, variance, skewness and kurosis (a measure of ail hickness). These momens depend only on he price of he opion, he curren marke prices of he underlying securiies, he opion s greeks δ, Γ, θ, and he RiskMerics covariance marix, Σ. In his example, Σ is he covariance marix of reurns r 5, r 7 and r USD/DEM. Le s ake a simple hypoheical example o describe he dela-gamma mehodology. Table 6.18 presens he necessary saisics on he bond and opion posiions o apply dela-gamma. Table 6.18 Porfolio specificaion Bond Opion σ 5 = 0.95% σ USD/DEM = 1% σ 7 = 1% δ = 0.903 ρ 57, = 0.85 Γ = 0.0566 P = B USD 100 θ = 0.9156 V = USD 3.7191 Porfolio PV = USD 103.719 P USd/DEM = USD 346.3 To compue VaR we require he covariance marix [6.57] Σ = σ 6y σ 6y, USD/DEM σ 6y, USD/DEM σ USD/DEM which, when using he informaion in Table 6.18 yields [6.58] Σ = 0.00905 0.008075 0.008075 0.01000 (in percen) Also, we need he cash flows corresponding o he dela componens of he porfolio, [6.59] δ = 100 81.35 Dela cash flow on bond Dela cash flow on FX opion he cash flows corresponding o he gamma componens of he porfolio, Par III: Risk Modeling of Financial Insrumens

148 Chaper 6. Marke risk mehodology Example 6.9 (coninued) [6.60] Γ = 0 1708.47 Zero gamma cash flow on bond Gamma cash flow for FX opion and, he cash flows corresponding o he hea componens of he porfolio. [6.61] θ = 0 0.46 Zero hea cashflow on bond Thea cashflow for FX opion The implied momens of his porfolio s disribuion are presened in Table 6.19. Table 6.19 Porfolio saisics Momens Including hea Excluding hea mean 0.1608 0.0854 variance.897.897 skewness coefficien 0.747 0.747 kurosis coefficien 3.0997 3.0997 Based on he informaion presened above, VaR esimaes of his porfolio over a one-day forecas horizon are presened in Table 6.0 for hree confidence levels. For comparison we also presen VaR based on he normal model and VaR ha excludes he hea effec. Table 6.0 Value-a-Risk esimaes (USD) one-day forecas horizon: oal porfolio value is 103.719 VaR percenile Normal Dela-gamma (excluding hea) Dela-gamma (including hea) 5.0%.799.579.86.5% 3.35 3.018 3.65 1.0% 3.953 3.53 3.953 RiskMerics Technical Documen Fourh Ediion

149 Chaper 7. Mone Carlo 7.1 Scenario generaion 151 7. Porfolio valuaion 155 7..1 Full valuaion 155 7.. Linear approximaion 156 7..3 Higher order approximaions 156 7.3 Summary 157 7.4 Commens 159 Par III: Risk Modeling of Financial Insrumens

150 RiskMerics Technical Documen Fourh Ediion

151 Chaper 7. Chrisopher C. Finger Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-4657 finger_chrisopher@jpmorgan.com Mone Carlo In he previous chaper, we illusraed how o combine cash flows, volailiies, and correlaions analyically o compue he Value-a-Risk for a porfolio. We have seen ha his mehodology is applicable o linear insrumens, as well as o non-linear insrumens whose values can be well approximaed by a Taylor series expansion (ha is, by is greeks ). In his chaper, we ouline a Mone Carlo framework under which i is possible o compue VaR for porfolios whose insrumens may no be amenable o he analyic reamen. We will see ha his mehodology produces an esimae for he enire probabiliy disribuion of porfolio values, and no jus one risk measure. The Mone Carlo mehodology consiss of hree major seps: 1. Scenario generaion Using he volailiy and correlaion esimaes for he underlying asses in our porfolio, we produce a large number of fuure price scenarios in accordance wih he lognormal models described previously.. Porfolio valuaion For each scenario, we compue a porfolio value. 3. Summary We repor he resuls of he simulaion, eiher as a porfolio disribuion or as a paricular risk measure. We devoe one secion of his chaper o each of he hree seps above. To beer demonsrae he mehodology, we will consider hroughou his secion a porfolio comprised of wo asses: a fuure cash flow sream of DEM 1M o be received in one year s ime and an a he money pu opion wih conrac size of DEM 1M and expiraion dae one monh in he fuure. Assume he implied volailiy a which he opion is priced is 14%. We see ha our porfolio value is dependen on he USD/DEM exchange rae and he one year DEM bond price. (Technically, he value of he opion also changes wih USD ineres raes and he implied volailiy, bu we will no consider hese effecs.) Our risk horizon for he example will be five days. 7.1 Scenario generaion We firs recall he lognormal model which we assume for all underlying insrumens. Consider a forecas horizon of days. If an insrumen s price oday is P 0, and our esimae for he one day volailiy of his insrumen is σ, hen we model he price of he insrumen in days by [7.1] P = P 0 e σ Y where Y is a sandard normal random variable. Thus, he procedure o generae scenarios is o generae sandard normal variaes and use Eq. [7.1] o produce fuure prices. The procedure for he mulivariae case is similar, wih he added complicaion ha he Y s corresponding o each insrumen mus be correlaed according o our correlaion esimaes. In pracice, i is sraighforward o generae independen normal variaes; generaing arbirarily correlaed variaes is more involved, however. Suppose we wish o generae n normal variaes wih uni variance and correlaions given by he nx n marix Λ. The general idea is o generae n independen variaes, and hen combine hese variaes is such a way o achieve he desired correlaions. To be more precise, he procedure is as follows: Decompose Λ using he Cholesky facorizaion, yielding a lower riangular marix A such ha Λ = AA. We provide deails on his facorizaion below and in Appendix E. Par III: Risk Modeling of Financial Insrumens

15 Chaper 7. Mone Carlo Generae an n x 1 vecor Z of independen sandard normal variaes. Le Z = AY. The elemens of Z will each have uni variance and will be correlaed according o Λ. To illusrae he inuiion behind using he Cholesky decomposiion, consider he case where we wish o generae wo variaes wih correlaion marix [7.] Λ = 1 ρ ρ 1 The Cholesky facorizaion of Λ is given by [7.3] A = 1 0 ρ 1 ρ (I is easy o check ha AA = Λ.) Now say ha Y is a x 1 vecor conaining independen sandard normal random variables Y 1 and Y. If we le Z = AY, hen he elemens of Z are given by [7.4a] Z 1 = Y 1 and [7.4b] Z = ρy 1 + 1 ρ Y Clearly, Z 1 has uni variance, and since Y 1 and Y are independen, he variance of Z is given by [7.5] ρ Var ( Y 1 ) 1 ρ + Var ( Y ) = 1 Again using he fac ha Y 1 and Y are independen, we see ha he expeced value of Z 1 Z is jus ρ, and so he correlaion is as desired. Noe ha i is no necessary o use he Cholesky facorizaion, since any marix A which saisfies Λ = AA will serve in he procedure above. A singular value or eigenvalue decomposiion would yield he same resuls. The Cholesky approach is advanageous since he lower riangular srucure of A means ha fewer operaions are necessary in he AZ muliplicaion. Furher, here exis recursive algorihms o compue he Cholesky facorizaion; we provide deails on his in Appendix E. On he oher hand, he Cholesky decomposiion requires a posiive-definie correlaion marix; large marices obained from he RiskMerics daa do no always have his propery. Using he procedure above o generae random variaes wih arbirary correlaions, we may generae scenarios of asse prices. For example, suppose we wish o model he prices of wo asses ( 1) ( ) days ino he fuure. Le P 0 and P 0 indicae he prices of he asses oday, le σ 1 and σ indicae he daily volailiies of he asses, and le ρ indicae he correlaion beween he wo asses. To generae a fuure price scenario, we generae correlaed sandard normal variaes Z 1 and as oulined above and compue he fuure prices by Z ( 1) P ( 1) σ 1 Z 1 P 0 e [7.6a] = and RiskMerics Technical Documen Fourh Ediion

Sec. 7.1 Scenario generaion 153 [7.6b] ( ) P = ( ) σ Z P 0 e To generae a collecion of scenarios, we simply repea his procedure he required number of imes. For our example porfolio, he wo underlying asses o be simulaed are he USD/DEM exchange rae and he one year DEM bond price. Suppose ha he curren one year German ineres rae is 10% (meaning he presen value of a one year 1M DEM noional bond is DEM 909,091) and ha he curren USD/DEM exchange rae is 0.65. We ake as he daily volailiies of hese wo asses σ FX = 0.004 and σ B = 0.0008 and as he correlaion beween he wo ρ = 0.17. To generae one housand scenarios for values of he wo underlying asses in five days, we firs use he approach above o generae one housand pairs of sandard normal variaes whose correlaion is ρ. Label each pair Z FX and Z B. We presen hisograms for he resuls in Char 7.1. Noe ha he disribuions are essenially he same. Char 7.1 Frequency disribuions for 1000 rials Z FX and Z B Frequency 10 100 DEM Bond 80 60 USD/DEM 40 0 0-3.5 -.8 -.0-1.3-0.5 0.3 1.0 1.8.5 3.3 Sandard deviaion The nex sep is o apply Eq. [7.6a] and Eq. [7.6b]. This will creae he acual scenarios for our asses. Thus, for each pair Z FX and Z B, we creae fuure prices P FX and P B by applying [7.7a] P FX 0.65e 0.004 5 Z FX = and [7.7b] P B 909, 091e 0.0008 5 Z B = Of course, o express he bond price in USD (accouning for boh he exchange rae and ineres rae risk for he bond), i is necessary o muliply he bond price by he exchange rae in each scenario. Chars 7. and 7.3 show he disribuions of fuure prices, P B and P FX, respecively, obained by one housand simulaions. Noe ha he disribuions are no longer bell shaped, and Par III: Risk Modeling of Financial Insrumens

154 Chaper 7. Mone Carlo for he bond price, he disribuion shows a marked asymmery. This is due o he ransformaion we make from normal o lognormal variaes by applying Eq. [7.7a] and Eq. [7.7b]. Char 7. Frequency disribuion for DEM bond price 1000 rials Frequency 10 100 80 60 40 0 0 904 905 906 907 908 909 910 911 91 914 915 Bond price Char 7.3 Frequency disribuion for USD/DEM exchange rae 1000 rials Frequency 00 180 160 140 10 100 80 60 40 0 0 0.631 0.637 0.64 0.648 0.654 0.659 0.665 0.670 USD/DEM exchange rae In Table 7.1, we presen he firs en scenarios ha we generae. RiskMerics Technical Documen Fourh Ediion

Sec. 7.1 Scenario generaion 155 Table 7.1 Mone Carlo scenarios 1000 rials USD/DEM Porfolio valuaion PV of cash flow (DEM) PV of cash flow (USD) 0.6500 906,663 589,350 0.6540 907,898 593,74 0.6606 911,14 601,935 0.6513 908,004 591,399 0.6707 910,074 610,430 0.6444 908,478 585,460 0.6569 908,860 597,053 0.6559 906,797 594,789 0.6530 906,931 59,67 0.665 90,768 603,348 In he previous secion, we illusraed how o generae scenarios of fuure prices for he underlying insrumens in a porfolio. Here, we ake up he nex sep, how o value he porfolio for each of hese scenarios. We will examine hree alernaives: full valuaion, linear approximaion, and higher order approximaion. Each of he alernaives is parameric, ha is, an approach in which he value of all securiies in he porfolio is obained hrough he values of is underlying asses, and differ only in heir mehods for valuing non-linear insrumens given underlying prices. Recall ha a he curren ime, he presen value of our cash flow is DEM 909,091, or USD 590,909. The value of he opion is USD 10,479. 7.1.1 Full valuaion This is he mos sraighforward and mos accurae alernaive, bu also he mos inense compuaionally. We assume some ype of pricing formula, in our case he Black-Scholes opion pricing formula, wih which we may value our opion in each of he scenarios which we have generaed. Say V ( S, K, τ) gives he premium (in USD) associaed wih he opion of selling one DEM given spo USD/DEM rae of S, srike rae of K, and expiraion dae τ years ino he fuure. (Again, his funcion will also depend on ineres raes and he implied volailiy, bu we will no model changes in hese variables, and so will suppress hem in he noaion.) In our example, for a scenario in which he USD/DEM rae has moved o R afer five days, our opion s value (in USD) moves from 1, 000, 000 V 0.65, 0.65, ----- 1 o 1 1 000 000 V R 0.65 1 5,,,, ----- --------. The resuls of applying his mehod o our scenarios are dis- 1 365 Par III: Risk Modeling of Financial Insrumens

156 Chaper 7. Mone Carlo played in Table 7.. Noe ha scenarios wih moderae changes in he bond price can display significan changes in he value of he opion. Table 7. Mone Carlo scenarios valuaion of opion 1000 rials USD/DEM Value of opion (USD) PV of cash flow (USD) Full Dela Dela/Gamma Dela/Gamma/ Thea 0.6500 589,350 9,558 10,458 10,458 9,597 0.6540 593,74 7,75 8,54 8,644 7,783 0.6606 601,935 5,73 5,7 6,1 5,61 0.6513 591,399 8,945 9,831 9,844 8,893 0.6707 610,430,680 73 3,541,680 0.6444 585,460 1,575 13,14 13,449 1,588 0.6569 597,053 6,56 7,073 7,437 6,576 0.6559 594,789 6,950 7,565 7,83 6,971 0.6530 59,67 8,156 8,981 9,05 8,190 0.665 603,348 4,691 4,349 5,58 4,667 7.1. Linear approximaion Because uilizing he Black-Scholes formula can be inensive compuaionally, paricularly for a large number of scenarios, i is ofen desirable o use a simple approximaion o he formula. The simples such approximaion is o esimae changes in he opion value via a linear model, which is commonly known as he dela approximaion. In his case, given an iniial opion value V 0 and an iniial exchange rae R 0, we approximae a fuure opion value V 1 a a fuure exchange rae by R 1 [7.8] V 1 = V 0 + δ ( P 1 P) where [7.9] δ = ------ V ( P, S, τ) R P0 is he firs derivaive of he opion price wih respec o he spo exchange rae. For our example, V 0 is USD/DEM 0.0105 and R 0 is USD/DEM 0.65. (To compue he price of our paricular opion conrac, we muliply V 0 by DEM 1M, he noional amoun of he conrac.) The value of δ for our opion is 0.4919. Table 7. illusraes he resuls of he dela approximaion for valuing he opion s price. Noe ha for he dela approximaion, i is sill possible o uilize he sandard RiskMerics mehodology wihou resoring o simulaions. 7.1.3 Higher order approximaions I can be seen in Table 7. ha he dela approximaion is reasonably accurae for scenarios where he exchange rae does no change significanly, bu less so in he more exreme cases. I is possible o improve his approximaion by including he gamma effec, which accouns for second RiskMerics Technical Documen Fourh Ediion

Sec. 7. Summary 157 order effecs of changes in he spo rae, and he hea effec, which accouns for changes in he ime o mauriy of he opion. The wo formulas associaed wih hese added effecs are 1 [7.10] V 1 = V 0 + δ ( P 1 P 0 ) + -- Γ ( P and 1 P 0 ) [7.11] 1 V 1 = V 0 + δ( P 1 P 0 ) + -- Γ ( P 1 P 0 ) θ where is he lengh of he forecas horizon and Γ and θ are defined by [7.1a] Γ = -------- and R V ( P, S, τ) P0 [7.1b] θ = ----- V ( P, S, τ) τ τ0 Using he values Γ= DEM/USD 15.14 and θ= USD/DEM 0.069 per year, we value our porfolio for each of our scenarios. The resuls of hese approximaions are displayed in Table 7.. A plo illusraing he differences beween he various mehods of valuaion is displayed in Char 7.4; he dela/gamma/hea approximaion is no ploed since for he values considered, i almos perfecly duplicaes he full valuaion case. Noe ha even for hese higher order approximaions, analyical mehods exis for compuing perceniles of he porfolio disribuion. See, for example, he mehod oulined in Chaper 6. Char 7.4 Value of pu opion on USD/DEM srike = 0.65 USD/DEM; Value in USD/DEM Opion value 0.06 0.05 0.04 0.03 0.0 0.01 0-0.01 Full valuaion Dela + gamma Dela -0.0 0.60 0.61 0.6 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 USD/DEM exchange rae 7. Summary Finally, afer generaing a large number of scenarios and valuing he porfolio under each of hem, i is necessary o make some conclusions based on he resuls. Clearly, one measure which we would like o repor is he porfolio s Value-a-Risk. This is done simply by ordering he porfolio reurn scenarios and picking ou he resul corresponding o he desired confidence level. Par III: Risk Modeling of Financial Insrumens

158 Chaper 7. Mone Carlo For example, o compue he 5% wors case loss using 1000 rials, we order he scenarios and choose he 50h (5% 1000) lowes absolue reurn. The perceniles compued for our example under he various mehods for porfolio valuaion are repored in Table 7.3. Table 7.3 Value-a-Risk for example porfolio 1000 rials Porfolio reurn (USD) Percenile, % Full Dela Dela/Gamma Dela/Gamma/Thea 1.0 (5,750) (5,949) (4,945) (5,806).5 (5,079) (5,006) (4,45) (5,106) 5.0 (4,559) (4,39) (3,708) (4,569) 10.0 (3,66) (3,99) (,85) (3,686) 5.0 (,496) (1,808) (1,606) (,467) 50.0 (840) () 50 (81) 75.0 915 1,689 1,813 951 90.0,801 3,15 3,666,805 95.0 4,311 4,331 5,165 4,304 97.5 5,509 5,317 6,350 5,489 99.0 6,65 6,4 7,489 6,68 Thus, a he 5% confidence level and in he full valuaion case, we obain a Value-a-Risk of USD 4,559, or abou 0.75% of he curren porfolio value. A paricularly nice feaure of he Mone Carlo approach is ha we obain an esimae for he enire disribuion of porfolio reurns. This allows us o compue oher risk measures if we desire, and also o examine he shape of he disribuion. Char 7.5 illusraes he reurn disribuion for our example. Noe ha he disribuion is significanly more skewed han he disribuions for he underlying asses (see Char 7.5), which is a resul of he non-lineariy of he opion posiion. Char 7.5 Disribuion of porfolio reurns 1000 rials Frequency 175 150 Full( ) 15 Dela( ) 100 75 50 Dela + gamma ( ) 5 0 594,469 599,61 604,77 609,93 615,074 Porfolio reurn (USD) RiskMerics Technical Documen Fourh Ediion

Sec. 7.3 Commens 159 7.3 Commens In our example, we reaed he bond by assuming a lognormal process for is price. While his is convenien compuaionally, i can lead o unrealisic resuls since he model does no insure posiive discoun raes. In his case, i is possible o generae a scenario where he individual bond prices are realisic, bu where he forward rae implied by he wo simulaed prices is negaive. We have examined a number of mehods o recify his problem, including decomposing yield curve moves ino principal componens. In he end, we have concluded ha since regularly observed bond prices and volailiies make he problems above quie rare, and since he mehods we have invesigaed only improve he siuaion slighly, i is no worh he effor o implemen a more sophisicaed mehod han wha we have oulined in his chaper. We sugges a sraigh Mone Carlo simulaion wih our mehodology coupled wih a check for unrealisic discoun or forward raes. Scenarios which yield hese unrealisic raes should be rejeced from consideraion. Par III: Risk Modeling of Financial Insrumens

160 Chaper 7. Mone Carlo RiskMerics Technical Documen Fourh Ediion

161 Par IV RiskMerics Daa Ses

16 RiskMerics Technical Documen Fourh Ediion

163 Chaper 8. Daa and relaed saisical issues 8.1 Consrucing RiskMerics raes and prices 165 8.1.1 Foreign exchange 165 8.1. Ineres raes 165 8.1.3 Equiies 167 8.1.4 Commodiies 167 8. Filling in missing daa 170 8..1 Naure of missing daa 171 8.. Maximum likelihood esimaion 171 8..3 Esimaing he sample mean and covariance marix for missing daa 17 8..4 An illusraive example 174 8..5 Pracical consideraions 176 8.3 The properies of correlaion (covariance) marices and VaR 176 8.3.1 Covariance and correlaion calculaions 177 8.3. Useful linear algebra resuls as applied o he VaR calculaion 180 8.3.3 How o deermine if a covariance marix is posiive semi-definie 181 8.4 Rebasing RiskMerics volailiies and correlaions 183 8.5 Nonsynchronous daa collecion 184 8.5.1 Esimaing correlaions when he daa are nonsynchronous 188 8.5. Using he algorihm in a mulivariae framework 195 Par IV: RiskMerics Daa Ses

164 RiskMerics Technical Documen Fourh Ediion

165 Chaper 8. Daa and relaed saisical issues Peer Zangari Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-8641 zangari_peer@jpmorgan.com This chaper covers he RiskMerics underlying yields and prices ha are used in he volailiy and correlaion calculaions. I also discusses he relaionship beween he number of ime series and he amoun of hisorical daa available on hese series as i relaes o he volailiy and correlaions. This chaper is organized as follows: Secion 8.1 explains he basis or consrucion of he underlying yields and prices for each insrumen ype. Secion 8. describes he filling in of missing daa poins, i.e., expecaion maximizaion. Secion 8.3 invesigaes he properies of a generic correlaion marix since hese deermine wheher a porfolio s sandard deviaion is meaningful. Secion 8.4 provides an algorihm for recompuing he volailiies and correlaions when a porfolio is based in a currency oher han USD. Secion 8.5 presens a mehodology o calculae correlaions when he yields or prices are sampled a differen imes, i.e., daa recording is nonsynchronous. 8.1 Consrucing RiskMerics raes and prices In his secion we explain he consrucion of he underlying raes and prices ha are used in he RiskMerics calculaions. Since he daa represen only a subse of he mos liquid insrumens available in he markes, proxies should be used for he ohers. Recommendaions on how o apply RiskMerics o specific insrumens are oulined in he paragraphs below. 8.1.1 Foreign exchange RiskMerics provides esimaes of VaR saisics for reurns on 31 currencies as measured agains he US dollar (e.g., USD/DEM, USD/FRF) as well as correlaions beween reurns. The daases provided are herefore suied for esimaing foreign exchange risk from a US dollar perspecive. The mehodology for using he daa o measure foreign exchange risk from a currency perspecive oher han he US dollar is idenical o he one described (Secion 6.1.) above bu requires he inpu of revised volailiies and correlaions. These modified volailiies and correlaions can easily be derived from he original RiskMerics daases as described in Secion 8.4. Also refer o he examples diskee. Finally, measuring marke exposure o currencies currenly no included in he RiskMerics daa se will involve accessing underlying foreign exchange daa from oher sources or using one of he 31 currencies as a proxy. 8.1. Ineres raes In RiskMerics we describe he fixed income markes in erms of he price dynamics of zero coupon consan mauriy insrumens. In he ineres rae swap marke here are quoes for consan mauriies (e.g., 10-year swap rae). In he bond markes, consan mauriy raes do no exis herefore we mus consruc hem wih he aid of a erm srucure model. Par IV: RiskMerics Daa Ses

166 Chaper 8. Daa and relaed saisical issues The curren daa se provides volailiies and correlaions for reurns on money marke deposis, swaps, and zero coupon governmen bonds in 33 markes. These parameers allow direc calculaion of he volailiy of cash flows. Correlaions are provided beween all RiskMerics verices and markes. 8.1..1 Money marke deposis The volailiies of price reurns on money marke deposis are o be used o esimae he marke risk of all shor-erm cash flows (from one monh o one year). Though hey only cover one insrumen ype a he shor end of he yield curve, money marke price reurn volailiies can be applied o measure he marke risk of insrumens ha are highly correlaed wih money marke deposis, such as Treasury bills or insrumens ha reprice off of raes such as he prime rae in he US or commercial paper raes. 1 8.1.. Swaps The volailiies of price reurns on zero coupon swaps are o be used o esimae he marke risk of ineres rae swaps. We consruc zero coupon swap prices and raes because hey are required for he cashflow mapping mehodology described in Secion 6.. We now explain how RiskMerics consrucs zero coupon swap prices (raes) from observed swap prices and raes by he mehod known as boosrapping. Suppose one knows he zero-coupon erm srucure, i.e., he prices of zero-coupon swaps i P 1,, P n, where each P i = 1 ( 1+ z i ) i = 1,, n and z i is he zero-coupon rae for he swap wih mauriy i. Then i is sraighforward o find he price of a coupon swap as [8.1] P cn = P 1 S n + P S n + + P n ( 1 + S n ) S n where denoes he curren swap rae on he n period swap. Now, in pracice we observe he coupon erm srucure, P c1,, P cn mauring a each coupon paymen dae. Using he coupon swap prices we can apply Eq. [8.1] o solve for he implied zero coupon erm srucure, i.e., zero coupon swap prices and raes. Saring wih a 1-period zero coupon swap, P c1 = P 1 ( 1+ S 1 ) so ha P 1 = P c1 ( 1+ S 1 ) or z 1 = ( 1+ S 1 ) P 1. Proceeding in an ieraive manner, given he c1 discoun prices P 1,, P n 1, we can solve for P n and z n using he formula [8.] P P cn P n 1 S n P 1 S n n = ------------------------------------------------------- 1 + S n The curren RiskMerics daases do no allow differeniaion beween ineres rae risks of insrumens of differen credi qualiy; all marke risk due o credi of equal mauriy and currency is reaed he same. 8.1..3 Zero coupon governmen bonds The volailiies of price reurns on zero coupon governmen bonds are o be used o esimae he marke risk in governmen bond posiions. Zero coupon prices (raes) are used because hey are consisen wih he cash flow mapping mehodology described in Secion 6.. Zero coupon governmen bond prices can also be used as proxies for esimaing he volailiy of oher securiies when he appropriae volailiy measure does no exis (corporae issues wih mauriies longer han 10 years, for example). 1 See he fourh quarer, 1995 RiskMerics Monior for deails. RiskMerics Technical Documen Fourh Ediion

Sec. 8.1 Consrucing RiskMerics raes and prices 167 Zero coupon governmen bond yield curves canno be direcly observed, hey can only be implied from prices of a collecion of liquid bonds in he respecive marke. Consequenly, a erm srucure model mus be used o esimae a synheic zero coupon yield curve which bes fis he collecion of observed prices. Such a model generaes zero coupon yields for arbirary poins along he yield curve. 8.1..4 EMBI+ The. P. Morgan Emerging Markes Bond Index Plus racks oal reurns for raded exernal deb insrumens in he emerging markes. I is consruced as a composie of is four markes: Brady bonds, Eurobonds, U.S. dollar local markes, and loans. The EMBI+ provides invesors wih a definiion of he marke for emerging markes exernal-currency deb, a lis of he raded insrumens, and a compilaion of heir erms. U.S dollar issues currenly make up more han 95% of he index and sovereign issues make up 98%. A fuller descripion of he EMBI+ can be found in he. P. Morgan publicaion Inroducing he Emerging Markes Bond Index Plus (EMBI+) daed uly 1, 1995. 8.1.3 Equiies According o he curren RiskMerics mehodology, equiies are mapped o heir domesic marke indices (for example, S&P500 for he US, DAX for Germany, and CAC40 for Canada). Tha is o say, individual sock beas, along wih volailiies on price reurns of local marke indices are used o consruc VaR esimaes (see Secion 6.3..) of individual socks. The reason for applying he bea coefficien is ha i measures he covariaion beween he reurn on he individual sock and he reurn on he local marke index whose volailiy and correlaion are provided by RiskMerics. 8.1.4 Commodiies A commodiy fuures conrac is a sandardized agreemen o buy or sell a commodiy. The price o a buyer of a commodiy fuures conrac depends on hree facors: 1. he curren spo price of he commodiy,. he carrying coss of he commodiy. Money ied up by purchasing and carrying a commodiy could have been invesed in some risk-free, ineres bearing insrumen. There may be coss associaed wih purchasing a produc in he spo marke (ransacion coss) and holding i unil or consuming i a some laer dae (sorage coss), and 3. he expeced supply and demand for he commodiy. The fuure price of a commodiy differs from is curren spo price in a way ha is analogous o he difference beween 1-year and overnigh ineres raes for a paricular currency. From his perspecive we esablish a erm srucure of commodiy prices similar o ha of ineres raes. The mos efficien and liquid markes for mos commodiies are he fuures markes. These markes have he advanage of bringing ogeher no only producers and consumers, bu also invesors who view commodiies as hey do any oher asse class. Because of he superior liquidiy and he ransparency of he fuures markes, we have decided o use fuures prices as he foundaion for modeling commodiy risk. This applies o all commodiies excep bullion, as described below. 8.1.4.1 The need for commodiy erm srucures Fuures conracs represen sandard erms and condiions for delivery of a produc a fuure daes. Recorded over ime, heir prices represen insrumens wih decreasing mauriies. Tha is o say, if he price series of a conrac is a sequence of expeced values of a single price a a specific dae in he fuure, hen each consecuive price implies ha he insrumen is one day close o expiring. Par IV: RiskMerics Daa Ses

168 Chaper 8. Daa and relaed saisical issues RiskMerics consrucs consan mauriy conracs in he same spiri ha i consrucs consan mauriy insrumens for he fixed income marke. Compared o he fixed income markes, however, commodiy markes are significanly less liquid. This is paricularly rue for very shor and very long mauriies. Frequenly, volailiy of he fron monh conrac may decline when he conrac is very close o expiraion as i becomes unineresing o rade for a small absolue gain, difficul o rade (a hin marke may exis due o his limied poenial gain) and, dangerous o rade because of physical delivery concerns. A he long end of he curve, rading liquidiy is limied. Whenever possible, we have seleced he mauriies of commodiy conracs wih he highes liquidiy as he verices for volailiy and correlaion esimaes. These mauriies are indicaed in Table 9.6 in Secion 9.6. In order o consruc consan mauriy conracs, we have defined wo algorihms o conver observed prices ino prices from consan mauriy conracs: Rolling nearby: we simply use he price of he fuures conrac ha expires closes o a fixed mauriy. Linear inerpolaion: we linearly inerpolae beween he prices of he wo fuures conracs ha sraddle he fixed mauriy. 8.1.4. Rolling nearby fuures conracs Rolling nearby conracs are consruced by concaenaing conracs ha expire, approximaely 1, 6, and 1 monhs (for insance) in he fuure. An example of his mehod is shown in Table 8.1. Table 8.1 Consrucion of rolling nearby fuures prices for Ligh Swee Crude (WTI) Rolling nearby Acual conracs 1s 6h 1h Mar-94 Apr-94 Aug-94 Sep-94 Feb-95 Mar-95 17-Feb-94 13.93 15.08 16.17 13.93 14.13 15.08 15.8 16.17 16.3 18-Feb-94 14.3 15.11 16.17 14.3 14.3 15.11 15.3 16.17 16.3 19-Feb-94 14.1 15.06 16.13 14.1 14.4 15.06 15.5 16.13 16.7 3-Feb-94 14.4 15.3 16.33 14.4 14.39 15.3 15.43 16.33 16.47 4-Feb-94 14.41 15.44 16.46 14.41 15.4 15.44 16.3 16.46 Noe ha he price of he fron monh conrac changes from he price of he March o he April conrac when he March conrac expires. (To conserve space cerain acive conracs were omied). The principal problem wih he rolling nearby mehod is ha i may creae disconinuous price series when he underlying conrac changes: for insance, from February 3 (he March conrac) o February 4 (he April conrac) in he example above. This disconinuiy usually is he larges for very shor erm conracs and when he erm srucure of prices is seep. RiskMerics Technical Documen Fourh Ediion

Sec. 8.1 Consrucing RiskMerics raes and prices 169 8.1.4.3 Inerpolaed fuures prices To address he issue of disconinuous price series, we use he simple rule of linear inerpolaion o define consan mauriy fuures prices,, from quoed fuures prices: P cmf [8.3] P cmf = ω NB1 P NB1 + ω NB P NB where P cmf ω NB1 P NB1 = consan mauriy fuures prices δ = -- = raio of P cmf made up by P NB1 δ = days o expiraion of NB1 = days o expiraion of consan mauriy conrac = price of NB1 ω NB = 1 ω NB1 = raio of P cmf made up by P NB P NB = price of NB NB1 = nearby conrac wih a mauriy < consan mauriy conrac NB = firs conrac wih a mauriy < consan mauriy conrac The following example illusraes his mehod using he daa for he heaing oil fuures conrac. On April 6, 1994 he 1-monh consan mauriy equivalen heaing oil price is calculaed as follows: [8.4] 1day, ------------------ 9 = 30 days Price April + ------------------ 30 days Price May 1 ----- 9 = 30 47.37 + ----- 39 47.38 P 1m April 6 = 47.379 Table 8. Price calculaion for 1-monh CMF NY Harbor # Heaing Oil Table 8. illusraes he calculaion over successive days. Noe ha he acual resuls may vary slighly from he daa represened in he able because of numerical rounding. Conrac expiraion Days o expiraion Weighs (%) Conrac prices cmf Dae 1 nb* 1m cmf nb* 1 nb* 1m cmf nb* 1 nb* nb* Apr May un -Apr-94 9-Apr 3-May 31-May 7 30 39 3.33 76.67 47.87 47.86 48.15 47.86 5-Apr-94 9-Apr 5-May 31-May 4 30 36 13.33 86.67 48.3 48.18 48.48 48.187 6-Apr-94 9-Apr 6-May 31-May 3 30 35 10.00 90.00 47.37 47.38 47.78 47.379 8-Apr-94 9-Apr 30-May 31-May 1 30 33 3.33 96.67 46.5 46.57 47.0 47.005 9-Apr-94 9-Apr 31-May 31-May 0 30 3 0.00 100.00 47.05 47.09 47.49 47.490 -May-94 31-May 1-un 30-un 9 30 59 96.67 3.33 47.57 47.95 47.583 3-May-94 31-May -un 30-un 8 30 58 93.33 6.67 46.89 47.9 46.917 4-May-94 31-May 3-un 30-un 7 30 57 90.00 10.00 46.66 47.03 46.697 * 1 nb and nb indicae firs and second nearby conracs, respecively. cmf means consan mauriy fuure. Par IV: RiskMerics Daa Ses

170 Chaper 8. Daa and relaed saisical issues Char 8.1 illusraes he linear inerpolaion rule graphically. Char 8.1 Consan mauriy fuure: price calculaion 47.38 April conrac expiry cmf conrac expiry May conrac expiry P - May: 47.38 P - cmf: 47.379 47.37 P - April: 47.37 47.36 :30 δ:3 47.35 April April May May 6 9 5 31 8. Filling in missing daa The preceding secion described he ypes of raes and prices ha RiskMerics uses in is calculaions. Throughou he presenaion i was implicily assumed ha here were no missing prices. In pracice, however, his is ofen no he case. Because of marke closings in a specific locaion, daily prices are occasionally unavailable for individual insrumens and counries. Reasons for he missing daa include he occurrence of significan poliical or social evens and echnical problems (e.g., machine down ime). Very ofen, missing daa are simply replaced by he preceding day s value. This is frequenly he case in he daa obained from specialized vendors. Anoher common pracice has simply been o exclude an enire dae from which daa were missing from he sample. This resuls in valuable daa being discarded. Simply because one marke is closed on a given day should no imply ha daa from he oher counries are no useful. A large number of nonconcurren missing daa poins across markes may reduce he validiy of a risk measuremen process. Accuraely replacing missing daa is paramoun in obaining reasonable esimaes of volailiy and correlaion. In his secion we describe how missing daa poins are filled-in by a process known as he EM algorihm so ha we can underake he analysis se forh in his documen. In brief, RiskMerics applies he following seps o fill in missing raes and prices: Assume a any poin in ime ha a daa se consising of a cross-secion of reurns (ha may conain missing daa) are mulivariae normally disribued wih mean µ and covariance marix Σ. Esimae he mean and covariance marix of his daa se using he available, observed daa. Replace he missing daa poins by heir respecive condiional expecaions, i.e., use he missing daa s expeced values given curren esimaes of µ, Σ and he observed daa. RiskMerics Technical Documen Fourh Ediion

Sec. 8. Filling in missing daa 171 8..1 Naure of missing daa We assume hroughou he analysis ha he presence of missing daa occur randomly. Suppose ha a a paricular poin in ime, we have K reurn series and for each of he series we have T hisorical observaions. Le Z denoe he marix of raw, observed reurns. Z has T rows and K columns. Each row of Z is a Kx1 vecor of reurns, observed a any poin in ime, spanning all K securiies. Denoe he h row of Z by z for = 1,,...T. The marix Z may have missing daa poins. Define a complee daa marix R ha consiss of all he daa poins Z plus he filled-in reurns for he missing observaions. The h row of R is denoed r. Noe ha if here are no missing observaions hen z =r for all =1,...,T. In he case where we have wo asses (K=) and hree hisorical observaions (T=3) on each asse, R akes he form: r 11 r 1 r 1 T [8.5] R = r 1 r = r T r 31 r 3 r 3 T where T denoes ranspose. 8.. Maximum likelihood esimaion For he purpose of filling in missing daa i is assumed ha a any period, he reurn vecor r (Kx1) follows a mulivariae normal disribuion wih mean vecor µ and covariance marix Σ. The probabiliy densiy funcion of r is [8.6] p( r ) = ( π) k -- Σ k -- exp 1 -- ( r µ ) T Σ 1 ( r µ ) I is assumed ha his densiy funcion holds for all ime periods, = 1,,...,T. Nex, under he assumpion of saisical independence beween ime periods, we can wrie he join probabiliy densiy funcion of reurns given he mean and covariance marix as follows [8.7] p( r 1, r T µσ, ) = p( r ) T = 1 = ( π) Tk ----- Σ T -- exp T 1 -- ( r µ ) T Σ 1 ( r µ ) = 1 The join probabiliy densiy funcion p( r 1, r T µσ, ) describes he probabiliy densiy for he daa given a se of parameer values (i.e., µ and Σ). Define he oal parameer vecor θ = (µ,σ). Our ask is o esimae θ given he daa marix ha conains missing daa. To do so, we mus derive he likelihood funcion of θ given he daa. The likelihood funcion L ( µσ, r 1, r T ) is similar in all respecs o p( r 1, r T µσ, ) excep ha i considers he parameers as random variables and akes he daa as given. Mahemaically, he likelihood funcion is equivalen o he probabiliy densiy funcion. Inuiively, he likelihood funcion embodies he enire se of parameer values for an observed daa se. Now, for a realized sample of, say, exchange raes, we would wan o know wha se of parameer values mos likely generaed he observed daa se. The soluion o his quesion lies in maximum Par IV: RiskMerics Daa Ses

17 Chaper 8. Daa and relaed saisical issues likelihood esimaion. In essence, he maximum likelihood esimaes (MLE) θ MLE are he parameer values ha mos likely generaed he observed daa marix. θ MLE is found by maximizing he likelihood funcion L ( µσ, r 1, r T ). In pracice i is ofen easier o maximize naural logarihm of he likelihood funcion l ( µσ, r 1, r T ) which is given by [8.8] l ( µσr, ) = 1 T -- TKln ( π) -- ln Σ T 1 -- ( r µ ) T Σ 1 ( r µ ) = 1 wih respec o θ. This ranslaes ino finding soluions o he following firs order condiions: [8.9] l ( µ µσr1,, r ) = 0, T l ( Σ µσr1,, r ) = 0 T The maximum likelihood esimaors for he mean vecor, µˆ and covariance marix Σˆ are [8.10] µˆ = [ r 1, r,, r k ] T [8.11] where r i T 1 Σˆ = -- ( r T µˆ ) ( r µˆ) T = 1 represens he sample mean aken over T ime periods. 8..3 Esimaing he sample mean and covariance marix for missing daa When some observaions of r are missing, he maximum likelihood esimaes θ MLE are no available. This is eviden from he fac ha he likelihood funcion is no defined (i.e., i has no value) when i is evaluaed a he missing daa poins. To overcome his problem, we mus implemen wha is known as he EM algorihm. Since is formal exposiion (Dempser, Laird and Rubin, 1977) he expecaion maximizaion or EM algorihm (hereafer referred o as EM) has been on of he mos successful mehods of esimaion when he daa under sudy are incomplee (e.g., when some of he observaions are missing). Among is exensive applicaions, he EM algorihm has been used o resolve missing daa problems involving financial ime series (Warga, 199). For a deailed exposiion of he EM algorihm and is applicaion in finance see Kemphorne and Vyas (1994). Inuiively, EM is an ieraive algorihm ha operaes as follows. For a given se of (iniial) parameer values, insead of evaluaing he log likelihood funcion, (which is impossible, anyway) EM evaluaes he condiional expecaion of he laen (underlying) log likelihood funcion. The mahemaical condiional expecaion of he log-likelihood is aken over he observed daa poins. The expeced log likelihood is maximized o yield parameer esimaes. (The superscrip 0 sands for he iniial parameer esimae). This value is hen subsiued ino he log likelihood funcion and expecaions are aken again, and new parameer esimaes θ EM 1 are found. This ieraive process is coninued unil he algorihm converges a which ime final parameer esimaes have been generaed. For example, if he algorihm is ieraed N+1 imes hen he 0 1 N sequence of parameer esimaes θ EM, θ EM, θ, EM is generaed. The algorihm sops 0 θ EM RiskMerics Technical Documen Fourh Ediion

Sec. 8. Filling in missing daa 173 when adjacen parameer esimaes are sufficienly close o one anoher, i.e., when N sufficienly close o. θ EM N 1 θ EM The firs sep in EM is referred o as he expecaion or E-Sep. The second sep is he maximizaion or M-sep. EM ieraes beween hese wo seps, updaing he E-Sep from he parameer esimaes generaed in he M-Sep. For example, a he ih ieraion of he algorihm, he following equaions are solved in he M-Sep: is T [8.1a] µˆi + 1 = 1 (he sample mean) T -- E r z, θ i = 1 T [8.1b] Σˆ i + 1 1 -- E (he sample covariance marix) T r r T z, θ i µˆ i + 1 µˆ i + 1 T = = 1 To evaluae he expecaions in hese expressions ( E r z,θ i T and E r r z,θ i ), we make use of sandard properies for pariioning a mulivariae normal random vecor. [8.13] R R NID µ R Σ RR Σ, RR µ R Σ RR Σ RR Here, one can hink of R as he sample daa wih missing values removed and R as he vecor of he underlying complee se of observaions. Assuming ha reurns are disribued mulivariae normal, he disribuion of R condiional on R is mulivariae normal wih mean [8.14] 1 E[ R R]=µ R + Σ RR Σ RR ( R µ R ) and covariance marix [8.15] 1 Covariance ( R R)=Σ RR Σ RR Σ RR Σ RR Using Eq. [8.14] and Eq. [8.15] we can evaluae he E- and M- seps. The E -Sep is given by [8.16] E Sep 1 E[ r z, θ] = µ r + Σ rz Σ zz ( z µ z ) T T E r r z, θ Covariance r z, θ E[ r z, θ]e[ r z, θ] T = + where [8.17] T Covariance r z, θ = Σ rz Σ zz 1 Σ rz Σzr Noice ha he expressions in Eq. [8.17] are easily evaluaed since hey depend on parameers ha describe he observed and missing daa. Given he values compued in he E-Sep, he M-Sep yields updaes of he mean vecor and covariance marix. Par IV: RiskMerics Daa Ses

174 Chaper 8. Daa and relaed saisical issues [8.18] µˆ i + 1 = 1 T -- r + E r z,θ i Complee Incomplee T Σˆi+ 1 1 -- E T r r T z, θ i µˆ i + 1 µˆ i + 1 T = M-Sep = 1 T r r + 1 Complee -- = T T Covariance r z, θ + E[ r z, θ]e[ r z, θ] T Incomplee Noice ha summing over implies ha we are adding down he columns of he daa marix R. For a pracical, deailed example of he EM algorihm see ohnson and Wichern (199, pp. 03 06). A powerful resul of EM is ha when a global opimum exiss, he parameer esimaes from he EM algorihm converge o he ML esimaes. Tha is, for a sufficienly large number of ieraions, EM converges o θ MLE. Thus, he EM algorihm provides a way o calculae he ML esimaes of he unknown parameer even if all of he observaions are no available. The assumpion ha he ime series are generaed from a mulivariae normal disribuion is innocuous. Even if he rue underlying disribuion is no normal, i follows from he heory of pseudomaximum likelihood esimaion ha he parameer esimaes are asympoically consisen (Whie, 198) alhough no necessarily asympoically efficien. Tha is, i has been shown ha he pseudo- MLE obained by maximizing he unspecified log likelihood as if i were correc produces a consisen esimaor despie he misspecificaion. 8..4 An illusraive example A ypical applicaion of he EM algorihm is filling in missing values resuling from a holiday in a given marke. We applied he algorihm oulined in he secion above o he Augus 15 Assumpion holiday in he Belgian governmen bond marke. While mos European bond markes were open on ha dae, including Germany and he Neherlands which show significan correlaion wih Belgium, no daa was available for Belgium. A missing daa poin in an underlying ime series generaes wo missing poins in he log change series as shown below (from 1 o as well as from o + 1). Even hough i would be more sraighforward o calculae he underlying missing value hrough he EM algorihm and hen generae he wo missing log changes, his would be saisically inconsisen wih our basic assumpions on he disribuion of daa. In order o mainain consisency beween he underlying rae daa and he reurn series, he adjusmen for missing daa is performed in hree seps. 1. Firs he EM algorihm generaes he firs missing percenage change, or 0.419% in he example below.. From ha number, we can back ou he missing underlying yield from he previous day s level, which gives us he 8.445% in he example below. 3. Finally, he second missing log change can be calculaed from he revised underlying yield series. RiskMerics Technical Documen Fourh Ediion

Sec. 8. Filling in missing daa 175 Table 8.3 presens he underlying raes on he Belgian franc 10-year zero coupon bond, he corresponding EM forecas, and he adjused filled-in raes and reurns. Table 8.3 Belgian franc 10-year zero coupon rae applicaion of he EM algorihm o he 1994 Assumpion holiday in Belgium Observed Adjused Collecion dae 10-year rae Reurn (%) EM forecas 10-year rae Reurn (%) 11-Aug-94 8.400.411 8.410.411 1-Aug-94 8.481 0.844 8.481 0.844 15-Aug-94 missing missing 0.419 8.445* 0.419 16-Aug-94 8.44 missing 8.44 0.54 17-Aug-94 8.444 0.37 8.444 0.37 18-Aug-94 8.541 1.149 8.541 1.149 * Filled-in rae based on EM forecas. From EM. Reurn now available because prior rae (*) has been filled in. Char 8. presens a ime series of he Belgian franc 10-year rae before and afer he missing observaion was filled in by he EM algorihm. Char 8. Graphical represenaion 10-year zero coupon raes; daily % change Daily percen change 5.0 4.0 3.0.0 1.0 0.0-1.0 Belgium Germany -.0 1-Aug 3-Aug 5-Aug 9-aug 11-Aug 15-Aug 17-Aug Yield 9.0 8.5 8.0 Belgium 7.5 7.0 Germany 6.5 1-Aug 3-Aug 5-Aug 9-aug 11-Aug 15-Aug 17-Aug Par IV: RiskMerics Daa Ses

176 Chaper 8. Daa and relaed saisical issues 8..5 Pracical consideraions A major par of implemening he EM algorihm is o devise he appropriae inpu daa marices for he EM. From boh a saisical and pracical perspecive we do no run EM on our enire ime series daa se simulaneously. Insead we mus pariion he original daa series ino non-overlapping sub-marices. Our reasons for doing so are highlighed in he following example. Consider a TxK daa marix where T is he number of observaions and K is he number of price vecors. Given his daa marix, he EM mus esimae K+K(K+1)/ parameers. Consequenly, o keep he esimaion pracical K canno be oo large. To ge a beer undersanding of his issue consider Char 8.3, which plos he number of parameers esimaed by EM (K +K(K+1)/) agains he number of variables. As shown, he number of esimaed parameers grows rapidly wih he number of variables. Char 8.3 Number of variables used in EM and parameers required number of parameers (Y-axis) versus number of variables (X-axis) Number of parameers 1400 100 1000 800 600 400 00 0 10 0 30 40 50 Number of variables The submarices mus be chosen so ha vecors wihin a paricular submarix are highly correlaed while hose vecors beween submarices are no significanly correlaed. If we are allowed o choose he submarices in his way hen EM will perform as if i had he enire original daa marix. This follows from he fac ha he accuracy of parameer esimaes are no improved by adding uncorrelaed vecors. In order o achieve a logical choice of submarices, we classify reurns ino he following caegories: (1) foreign exchange, () money marke, (3) swap, (4) governmen bond, (5) equiy, and (6) commodiy. We furher decompose caegories, 3, 4, and 6 as follows. Each inpu daa marix corresponds o a paricular counry or commodiy marke. The rows of his marix correspond o ime while he columns idenify he mauriy of he asse. Foreign exchange, equiy indices, and bullion are he excepions: all exchange raes, equiy indices, and bullion are grouped ino hree separae marices. 8.3 The properies of correlaion (covariance) marices and VaR In Secion 6.3. i was shown how RiskMerics applies a correlaion marix o compue he VaR of an arbirary porfolio. In paricular, he correlaion marix was used o compue he porfolio s sandard deviaion. VaR was hen compued as a muliple of ha sandard deviaion. In his secion we invesigae he properies of a generic correlaion marix since i is hese properies ha will RiskMerics Technical Documen Fourh Ediion

Sec. 8.3 The properies of correlaion (covariance) marices and VaR 177 deermine wheher he porfolio s sandard deviaion forecas is meaningful. Specifically, we will esablish condiions 3 ha guaranee he non-negaiviy of he porfolio s variance, i.e., σ 0. A firs glance i may no seem obvious why i is necessary o undersand he condiions under which he variance is non-negaive. However, he poenial sign of he variance, and consequenly he VaR number, is direcly relaed o he relaionship beween (1) he number of individual price reurn series (i.e., cashflows) per porfolio and () he number of hisorical observaions on each of hese reurn series. In pracice here is ofen a rade-off beween he wo since, on he one hand, large porfolios require he use of many ime series, while on he oher hand, large amouns of hisorical daa are no available for many ime series. Below, we esablish condiions ha ensure he non-negaiviy of a variance ha is consruced from correlaion marices based on equally and exponenially weighed schemes. We begin wih some basic definiions of covariance and correlaion marices. 8.3.1 Covariance and correlaion calculaions In his secion we briefly review he covariance and correlaion calculaions based on equal and exponenial moving averages. We do so in order o esablish a relaionship beween he underlying reurn daa marix and he properies of he corresponding covariance (correlaion) marix. 8.3.1.1 Equal weighing scheme Le X denoe a T x K daa marix, i.e., marix of reurns. X has T rows and K columns. [8.19] X = r 11 r 1K r r T1 r TK Each column of X is a reurn series corresponding o a paricular price/rae (e.g., USD/DEM FX reurn) while each row corresponds o he ime ( = 1,...,T) a which he reurn was recorded. If we compue sandard deviaions and covariances around a zero mean, and weigh each observaion wih probabiliy 1/T, we can define he covariance marix simply by [8.0] Σ = X T X ---------- T X T where is he ranspose of X. Consider an example when T = 4 and K =. By properies, we mean specifically wheher he correlaion marix is posiive definie, posiive semidefinie or oherwise (hese erms will be defined explicily below) 3 All linear algebra proposiions saed below can be found in ohnson,. (1984). Par IV: RiskMerics Daa Ses

178 Chaper 8. Daa and relaed saisical issues r 11 r 1 [8.1] X = r 1 r r 31 r 3 X T r 11 r 1 r 31 r = 41 r 1 r r 3 r 4 r 41 r 4 An esimae of he covariance marix is given by [8.] Σ X T X = ---------- = T 4 1 -- r 4 i 1 1 4 -- r i 1 r i i = 1 i = 1 = 4 1 -- r 4 i 1 r i i = 1 4 4 1 -- r 4 i i = 1 σ 1 σ 1 σ 1 σ Nex, we show how o compue he correlaion marix R. Suppose we divide each elemen of he marix X by he sandard deviaion of he series o which i belongs; i.e., we normalize each series of X o have a sandard deviaion of 1. Call his new marix wih he sandardized values Y. The correlaion marix is r 11 ------ r 1K ------- σ 1 σ K [8.3] Y = r ------ σ r T1 ------- r TK ------- σ 1 σ K where T 1 σ j = -- r ij T j = 1 k,, i = 1 [8.4] R = Y T Y --------- T As in he previous example, if we se T = 4 and K =, he correlaion marix is [8.5] Y T Y R = --------- = T 1 -- 4 4 r i 1 1 -- ----- 4 1 4 -- σ 1 i = 1 4 i = 1 r i 1 r ------------ i σ 1 σ 4 i = 1 4 1 -- 4 r i 1 r ------------ i σ 1 σ r i ----- σ i = 1 = 1 ρ 1 ρ 1 1 RiskMerics Technical Documen Fourh Ediion

Sec. 8.3 The properies of correlaion (covariance) marices and VaR 179 8.3.1. Exponenial weighing scheme We now show how similar resuls are obained by using exponenial weighing raher han equal weighing. When compuing he covariance and correlaion marices, use, insead of he daa marix X, he augmened daa marix X shown in Eq. [8.6]. r 11 r 1K [8.6] X = λr 1 λr 1 λ 1 r λ T 1 rt1 λ T 1 rtk Now, we can define he covariance marix simply as [8.7] T Σ λ i 1 1 = X TX i = 1 To see his, consider he example when T = 4 and K =. r 11 r 1 X = λr 1 λr X T x 11 λr 1 λ r31 λ 3 r41 = λ r31 λ r3 x 1 λr λ r3 λ 3 r4 [8.8] λ 3 r41 λ 3 r4 T Σ λ i 1 1 = X TX i = 1 = 4 T λ i 1 r i1 λ i 1 i = 1 4 i = 1 λ i 1 1 4 λ i 1 r i1 r i i = 1 = r i1 r i λ i 1 i = 1 4 i = 1 r i σ 1 σ 1 σ1 σ The exponenially weighed correlaion marix is compued jus like he simple correlaion marix. The sandardized daa marix and he correlaion marix are given by he following expressions. r 11 ------ r 1K ------- σ 1 σ K [8.9] Ỹ = r ------ σ r T1 ------- r TK ------- σ 1 σ K Par IV: RiskMerics Daa Ses

180 Chaper 8. Daa and relaed saisical issues where σ j λ i 1 1 = λ i 1 r ij j = 1 K,, i = 1 and he correlaion marix is T T i = 1 [8.30] T R λ i 1 1 = Ỹ T Ỹ i = 1 which is he exac analogue o Eq. [8.5]. Therefore, all resuls for he simple correlaion marix carry over o he exponenial weighed marix. Having shown how o compue he covariance and correlaion marices, he nex sep is o show how he properies of hese marices relae o he VaR calculaions. We begin wih he definiion of posiive definie and posiive semidefinie marices. [8.31] If z T Cz > ( < ) 0 for all nonzero vecors z, hen C is said o be posiive (negaive) definie. [8.3] If z T Cz ( ) 0 for all nonzero vecors z, hen C is said o be posiive semidefinie (nonposiive definie). Now, referring back o he VaR calculaion presened in Secion 6.3., if we replace he vecor z by he weigh vecor σ 1 and C by he correlaion marix, R 1, hen i should be obvious why we seek o deermine wheher he correlaion marix is posiive definie or no. Specifically, If he correlaion marix R is posiive definie, hen VaR will always be posiive. If R is posiive semidefinie, hen VaR could be zero or posiive. If R is negaive definie, 4 hen VaR will be negaive. 8.3. Useful linear algebra resuls as applied o he VaR calculaion In order o define a relaionship beween he dimensions of he daa marix X (or X ) (i.e., he number of rows and columns of he daa marix) and he poenial values of he VaR esimaes, we mus define he rank of X. The rank of a marix X, denoed r(x), is he maximum number of linearly independen rows (and columns) of ha marix. The rank of a marix can be no greaer han he minimum number of rows or columns. Therefore, if X is T x K wih T > K (i.e., more rows han columns) hen r(x) K. In general, for an T x K marix X, r(x) min(t,k). 4 We will show below ha his is no possible. RiskMerics Technical Documen Fourh Ediion

Sec. 8.3 The properies of correlaion (covariance) marices and VaR 181 A useful resul which equaes he ranks of differen marices is: [8.33] r( X) r X T = X = r XX T As applied o he VaR calculaion, he rank of he covariance marix Σ = X T X is he same as he rank of X. We now refer o wo linear algebra resuls which esablish a relaionship beween he rank of he daa marix and he range of VaR values. [8.34] If X is T x K wih rank K < T, hen X T X is posiive definie and XX T is posiive semidefinie. [8.35] If X is T x K wih rank < min(t,k) hen X T X and XX T is posiive semidefinie. Therefore, wheher Σ is posiive definie or no will depend on he rank of he daa marix X. Based on he previous discussion, we can provide he following resuls for RiskMerics VaR calculaions. Following from Eq. [8.33], we can deduce he rank of R simply by knowing he rank of Y, he sandardized daa marix. The rank of he correlaion marix R can be no greaer han he number of hisorical daa poins used o compue he correlaion marix, and Following from Eq. [8.34], if he daa marix of reurns has more rows han columns and he columns are independen, hen R is posiive definie and VaR > 0. If no, hen Eq. [8.35] applies, and R is posiive semidefinie and VaR 0. In summary, a covariance marix, by definiion, is a leas posiive semidefinie. Simply pu, posiive semidefinie is he muli-dimensional analogue o he definiion, σ 0. 8.3.3 How o deermine if a covariance marix is posiive semi-definie 5 Finally, we explain a echnique o deermine wheher a correlaion marix is posiive (semi) definie. We would like o noe a he beginning ha due o a variey of echnical issues ha are beyond he scope of his documen, he suggesed approach described below known as he singular value decomposiion (SVD) is o serve as a general guideline raher han a sric se of rules for deermining he definieness of a correlaion marix. The singular value decomposiion (SVD) The T x K sandardized daa marix Y ( T K) may be decomposed as 6 Y = UDV where U U = V V = I K and D is diagonal wih non-negaive diagonal elemens ( ι 1, ι, ι, K ), called he singular values of Y. All of he singular values are ( 0). 5 This secion is based on Belsley (1981), Chaper 3. 6 In his secion we work wih he mean cenered and sandardized marix Y insead of X since Y is he daa marix on which an SVD should be applied. Par IV: RiskMerics Daa Ses

18 Chaper 8. Daa and relaed saisical issues A useful resul is ha he number of non-zero singular values is a funcion by he rank of Y. Specifically, if Y is full rank, hen all K singular values will be non zero. If he rank of Y is =K-, hen here will be posiive singular values and wo zero singular values. In pracice, i is difficul o deermine he number of zero singular values. This is due o ha fac ha compuers deal wih finie, no exac arihmeic. In oher words, i is difficul for a compuer o know when a singular value is really zero. To avoid having o deermine he number of zero singular values, i is recommended ha praciioners should focus on he condiion number of Y which is he raio of he larges o smalles singular values, i.e., ι max [8.36] υ= --------- (condiion number) ι min Large condiion numbers poin oward ill-condiion marices, i.e., marices ha are nearly no full rank. In oher words, a large υ implies ha here is a srong degree of collineariy beween he columns of Y. More elaborae ess of collineariy can be found in Belsley (1981). We now apply he SVD o wo daa marices. The firs daa marix consiss of ime series of price reurns on 10 USD governmen bonds for he period anuary 4, 1993 Ocober 14, 1996 (986 observaions). The columns of he daa marix correspond o he price reurns on he yr, 3yr, 4yr, 5yr, 7yr, 9yr, 10yr, 15yr, 0yr, and 30yr USD governmen bonds. The singular values for his daa marix are given in Table 8.4. Table 8.4 Singular values for USD yield curve daa marix 3.045 0.051 0.785 0.043 0.71 0.00 0.131 0.017 0.117 0.006 The condiion number, υ, is 497.4. We conduc a similar experimen on a daa marix ha consiss of 14 equiy indices. 7 The singular values are shown in Table 8.5. The daa se consiss of a oal number of 790 observaions for he period Ocober 5, 1996 hrough Ocober 14, 1996. Table 8.5 Singular values for equiy indices reurns.39 0.873 0.696 1.149 0.855 0.639 0.948 0.789 0.553 0.936 0.743 0.554 0.894 0.71 For his daa marix, he condiion number, υ, is 4.8. Noice how much lower he condiion number is for equiies han i is for he US yield curve. This resul should no be surprising since we expec he reurns on differen bonds along he yield curve o move in a similar fashion o one anoher relaive o equiy reurns. Alernaively expressed, he relaively large condiion number for he USD yield curve is indicaive of he near collineariy ha exiss among reurns on US governmen bonds. 7 For he counries Ausria, Ausralia, Belgium, Canada, Swizerland, Spain, France, Finland, Grea Briain, Hong Kong, Ireland, Ialy, apan and he Neherlands. RiskMerics Technical Documen Fourh Ediion

Sec. 8.4 Rebasing RiskMerics volailiies and correlaions 183 The purpose of he preceding exercise was o demonsrae how he inerrelaedness of individual ime series affecs he condiion of he resuling correlaion marix. As we have shown wih a simple example, highly correlaed daa (USD yield curve daa) leads o high condiion numbers relaive o less correlaed daa (equiy indices). In concluding, due o numerical rounding errors i is no unlikely for he heoreical properies of a marix o differ from is esimaed counerpar. For example, covariance marices are real, symmeric and non-posiive definie. However, when esimaing a covariance marix we may find ha he posiive definie propery is violaed. More specifically, he marix may no inver. Singulariy may arise because cerain prices included in a covariance marix form linear combinaions of oher prices. Therefore, if covariance marices fail o inver hey should be checked o deermine wheher cerain prices are linear funcions of ohers. Also, he scale of he marix elemens may be such ha i will no inver. While poor scaling may be a source of problems, i should rarely be he case. 8.4 Rebasing RiskMerics volailiies and correlaions A user s base currency will dicae how RiskMerics sandard deviaions and correlaions will be used. For example, a DEM-based invesor wih US dollar exposure is ineresed in flucuaions in he currency USD/DEM whereas he same invesor wih an exposure in Belgium francs is ineresed in flucuaions in BEF/DEM. Currenly, RiskMerics volailiy forecass are expressed in US dollars per foreign currency such as USD/DEM for all currencies. To compue volailiies on cross raes such as BEF/DEM, users mus make use of he RiskMerics provided USD/DEM and USD/ BEF volailiies as well as correlaions beween he wo. We now show how o derive he variance (sandard deviaion) of he BEF/DEM posiion. Le r 1, and r, represen he ime reurns on USD/DEM and USD/BEF, respecively, i.e., ( USD DEM) ( USD BEF) [8.37] r 1 -------------------------------------------- = ln and r ( USD DEM) = ln ------------------------------------------ 1 ( USD BEF) 1 The cross rae BEF/DEM is defined as [8.38] r 3 = ln -------------------------------------------- ( BEF DEM) = ( BEF DEM) r r 1 1 The variance of he cross rae r 3 is given by [8.39] σ 3, = σ 1, + σ, σ 1, Equaion [8.39] holds for any cross rae ha can be defined as he arihmeic difference in wo oher raes. We can find he correlaion beween wo cross raes as follows. Suppose we wan o find he correlaion beween he currencies BEF/DEM and FRF/DEM. I follows from Eq. [8.38] ha we firs need o define hese cross raes in erms of he reurns used in RiskMerics. ( USD DEM) ( USD BEF) [8.40a] r 1, -------------------------------------------- = ln, r, ( USD DEM), = ln ------------------------------------------ 1 ( USD BEF) 1 [8.40b] r -------------------------------------------- ( BEF DEM) 3,, ( BEF DEM) r ( USD FRF) = ln = 1, r, r 4, = ln ------------------------------------------ 1 ( USD FRF) 1 Par IV: RiskMerics Daa Ses

184 Chaper 8. Daa and relaed saisical issues and [8.40c] ( FRF DEM), = ln -------------------------------------------- = ( FRF DEM) r 1 1 r 5, r 4, The correlaion beween BEF/DEM and USD/FRF (r 3, and r 4, ) is he covariance of r 3, and r 4, divided by heir respecive sandard deviaions, mahemaically, [8.41] σ ρ 34, 34, = ------------------ σ 4, σ 3, σ = 1, σ 1, σ 4, σ 1, σ σ 14, + -------------------------------------------------------------------------------------------------------- 4, + σ 14, σ 1, + σ, σ 1, Analogously, he correlaion beween USD/DEM and FRF/DEM is [8.4] σ ρ 15, 35, = ------------------ σ 5, σ 1, σ = 1, σ 1, σ 4, σ 14, ------------------------------------------------------------------- + σ 14, σ 1, 8.5 Nonsynchronous daa collecion Esimaing how financial insrumens move in relaion o each oher requires daa ha are collaed, as much as possible, consisenly across markes. The poin in ime when daa are recorded is a maerial issue, paricularly when esimaing correlaions. When daa are observed (recorded) a differen imes hey are known o be nonsynchronous. Table 8.7 (pages 186 187) oulines how he daa underlying he ime series used by RiskMerics are recorded during he day. I shows ha mos of he daa are aken around 16:00 GMT. From he asse class perspecive, we see ha poenial problems will mos likely lie in saisics relaing o he governmen bond and equiy markes. To demonsrae he effec of nonsynchronous daa on correlaion forecass, we esimaed he 1-year correlaion of daily movemens beween USD 10-year zero yields colleced every day a he close of business in N.Y. wih wo series of 3-monh money marke raes, one colleced by he Briish Bankers Associaion a 11:00 a.m. in London and he oher colleced by.p. Morgan a he close of business in London (4:00 p.m.). This daa is presened in Table 8.6. Table 8.6 Correlaions of daily percenage changes wih USD 10-year Augus 1993 o une 1994 10-year USD raes collaed a N.Y. close Correlaion a London ime: LIBOR 11 a.m. 4 p.m. 1-monh 0.01 0.153 3-monh 0.13 0.396 6-monh 0.119 0.386 1-monh 0.118 0.6 RiskMerics Technical Documen Fourh Ediion

Sec. 8.5 Nonsynchronous daa collecion 185 None of he daa series are synchronous, bu he resuls show ha he money marke raes colleced a he London close have higher correlaion o he USD 10-year raes han hose colleced in he morning. Geing a consisen view of how a paricular yield curve behaves depends on addressing he iming issue correcly. While his is an imporan facor in measuring correlaions, he effec of iming diminishes as he ime horizon becomes longer. Correlaing monhly percenage changes may no be dependen on he condiion ha raes be colleced a he same ime of day. Char 8.4 shows how he correlaion esimaes agains USD 10-year zeros evolve for he wo money marke series menioned above when he horizon moves from daily changes o monhly changes. Once pas he 10- day ime inerval, he effec of iming differences beween he wo series becomes negligible. Char 8.4 Correlaion forecass vs. reurn inerval 3-monh USD LIBOR vs. 10-year USD governmen bond zero raes 0.6 0.5 0.4 0.3 3m LIBOR London p.m. 3m LIBOR London a.m. 0. 0.1 0 1 3 4 5 10 0 Reurn inerval (number of days) In a perfec world, all raes would be colleced simulaneously as all markes would rade a he same ime. One may be able o adap o nonsynchronously recorded daa by adjusing eiher he underlying reurn series or he forecass ha were compued from he nonsynchronous reurns. In his conex, daa adjusmen involves exensive research. The remaining secions of his documen presen an algorihm o adjus correlaions when he daa are nonsynchronous. Par IV: RiskMerics Daa Ses

186 Chaper 8. Daa and relaed saisical issues Table 8.7 Schedule of daa collecion London ime, a.m. Counry Insrumen summary 1:00 :00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:00 11:00 1:00 Ausralia FX/Eq/LI/Sw/Gv Eq Gv Hong Kong FX/Eq/LI/Sw LI Eq Sw Indonesia FX/Eq/LI/Sw Eq LI/Sw apan FX/Eq/LI/Sw/Gv Gv Eq Korea FX/Eq Eq Malaysia FX/Eq/LI/Sw Eq LI/Sw New Zealand FX/Eq/LI/Sw/Gv Eq LI/Gv Sw Philippines FX/Eq Eq Singapore FX/Eq/LI/Sw/Gv LI/Eq Taiwan FX/Eq/ Thailand FX/Eq/LI/Sw Eq LI/Sw Ausria FX/Eq/LI Eq Belgium FX/Eq/LI/Sw/Gv Denmark FX/Eq/LI/Sw/Gv Finland FX/Eq/LI/Sw/Gv France FX/Eq/LI/Sw/Gv Germany FX/Eq/LI/Sw/Gv Ireland FX/Eq/LI/Sw/Gv Ialy FX/Eq/LI/Sw/Gv Neherlands FX/Eq/LI/Sw/Gv Norway FX/Eq/LI/Sw/Gv Porugal FX/Eq/LI/Sw/Gv Souh Africa FX/Eq/LI//Gv Spain FX/Eq/LI/Sw/Gv Sweden FX/Eq/LI/Sw/Gv Swizerland FX/Eq/LI/Sw/Gv U.K. FX/Eq/LI/Sw/Gv ECU FX/ /LI/Sw/Gv Argenina FX/Eq Canada FX/Eq/LI/Sw/Gv Mexico FX/Eq/LI U.S. FX/Eq/LI/Sw/Gv FX = Foreign Exchange, Eq = Equiy Index, LI = LIBOR, Sw = Swap, Gv = Governmen RiskMerics Technical Documen Fourh Ediion

Sec. 8.5 Nonsynchronous daa collecion 187 Table 8.7 (coninued) Schedule of daa collecion London ime, p.m. 1:00 :00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:00 11:00 1:00 Insrumen summary Counry FX/LI/Sw FX/Eq/LI/Sw/Gv Ausralia FX FX/Eq/LI/Sw Hong Kong FX FX/Eq/LI/Sw Indonesia FX/LI/Sw FX/Eq/LI/Sw/Gv apan FX FX/Eq Korea FX FX/Eq/LI/Sw Malaysia FX FX/Eq/LI/Sw/Gv New Zealand FX FX/Eq Philippines FX FX/Eq/LI/Sw/Gv Singapore FX FX/Eq Taiwan FX FX/Eq/LI/Sw Thailand FX/LI FX/Eq/LI Ausria Eq FX/LI/Sw/Gv FX/Eq/LI/Sw/Gv Belgium Eq Gv FX/LI/Sw FX/Eq/LI/Sw/Gv Denmark Eq FX/LI FX/Eq/LI/Sw/Gv Finland Gv FX/LI/Sw/Eq FX/Eq/LI/Sw/Gv France FX/LI/Sw/Gv/Eq FX/Eq/LI/Sw/Gv Germany FX/LI/Sw/Gv Eq FX/Eq/LI/Sw/Gv Ireland FX/LI/Sw/Gv/Eq FX/Eq/LI/Sw/Gv Ialy FX/LI/Sw/Gv/Eq FX/Eq/LI/Sw/Gv Neherlands Eq FX/LI FX/Eq/LI/Sw/Gv Norway FX/LI/Eq FX/Eq/LI/Sw/Gv Porugal Eq Gv FX/LI FX/Eq/LI//Gv Souh Africa FX/LI/Sw Gv/Eq FX/Eq/LI/Sw/Gv Spain Gv FX/LI/Sw/Eq FX/Eq/LI/Sw/Gv Sweden FX/LI/Sw/Eq FX/Eq/LI/Sw/Gv Swizerland FX/LI/Sw/Eq Gv FX/Eq/LI/Sw/Gv U.K. FX/LI/Sw Gv FX/ /LI/Sw/Gv ECU FX Eq FX/Eq Argenina FX/LI/Sw Gv Eq FX/Eq/LI/Sw/Gv Canada FX/LI Eq FX/Eq/LI Mexico FX/LI/Sw Gv Eq FX/Eq/LI/Sw/Gv U.S. FX = Foreign Exchange, Eq = Equiy Index, LI = LIBOR, Sw = Swap, Gv = Governmen Par IV: RiskMerics Daa Ses

188 Chaper 8. Daa and relaed saisical issues 8.5.1 Esimaing correlaions when he daa are nonsynchronous The expansion of he RiskMerics daa se has increased he amoun of underlying prices and raes colleced in differen ime zones. The fundamenal problem wih nonsynchronous daa collecion is ha correlaion esimaes based on hese prices will be underesimaed. And esimaing correlaions accuraely is an imporan par of he RiskMerics VaR calculaion because sandard deviaion forecass used in he VaR calculaion depends on correlaion esimaes. Inernaionally diversified porfolios are ofen composed of asses ha rade in differen calendar imes in differen markes. Consider a simple example of a wo sock porfolio. Sock 1 rades only on he New York Sock Exchange (NYSE 9:30 am o 4:00 pm EST) while sock rades exclusively on he Tokyo sock exchange (TSE 7:00 pm o 1:00 am EST). Because hese wo markes are never open a he same ime, socks 1 and canno rade concurrenly. Consequenly, heir respecive daily closing prices are recorded a differen imes and he reurn series for asses 1 and, which are calculaed from daily close-o-close prices, are also nonsynchronous. 8 Char 8.5 illusraes he nonsynchronous rading hours of he NYSE and TSE. Char 8.5 Time char NY and Tokyo sock markes NY open 9:30 am 8.5 hrs 6.5 hrs NY close 4:00 pm 3 hrs TKO close 1:00 am 6 hrs TKO open 7:00 pm Day -1 Day 8.5 hours 6.5 3 6 8.5 6.5 TSE close NYSE open NYSE close TSE open TSE close NYSE open NYSE close TSE close-o-close Informaion overlap 30% NYSE close-o-close 8 This erminology began in he nonsynchronous rading lieraure. See, Fisher, L. (1966) and Sholes, M. and Williams (1977). Nonsynchronous rading is ofen associaed wih he siuaion when some asses rade more frequenly han ohers [see, Perry, P. (1985)]. Lo and MacKinlay (1990) noe ha he nonsynchroniciy problem resuls from he assumpion ha muliple ime series are sampled simulaneously when in fac he sampling is nonsynchronous. For a recen discussion of he nonsynchronous rading issue see Boudoukh, e. al (1994). RiskMerics Technical Documen Fourh Ediion

Sec. 8.5 Nonsynchronous daa collecion 189 We see ha he Tokyo exchange opens hree hours afer he New York close and he New York exchange reopens 81/ hours afer he Tokyo close. Because a new calendar day arrives in Tokyo before New York, he Tokyo ime is said o precede New York ime by 14 hours (EST). RiskMerics compues reurns from New York and Tokyo sock markes using daily close-o-close prices. The black orbs in Char 8.5 mark imes when hese prices are recorded. Noe ha he orbs would line up wih each oher if reurns in boh markes were recorded a he same ime. The following secions will: 1. Idenify he problem and verify wheher RiskMerics really does underesimae cerain correlaions.. Presen an algorihm o adjus he correlaion esimaes. 3. Tes he resuls agains acual daa. 8.5.1.1 Idenifying he problem: correlaion and nonsynchronous reurns Wheher differen reurn series are recorded a he same ime or no becomes an issue when hese daa are used o esimae correlaions because he absolue magniude of correlaion (covariance) esimaes may be underesimaed when calculaed from nonsynchronous raher han synchronous daa. Therefore, when compuing correlaions using nonsynchronous daa, we would expec he value of observed correlaion o be below he rue correlaion esimae. In he following analysis we firs esablish he effec ha nonsynchronous reurns have on correlaion esimaes and hen offer a mehod for adjusing correlaion esimaes o accoun for he nonsynchroniciy problem. The firs sep in checking for downward bias is esimaing wha he rue correlaion should be. This is no rivial since hese asses do no rade in he same ime zone and i is ofen no possible o obain synchronous daa. For cerain insrumens, however, i is possible o find limied daases which can provide a glimpse of he rue level of correlaion; his daa would hen become he benchmark agains which he mehodology for adjusing nonsynchronous reurns would be esed. One of hese insrumens is he US Treasury which has he advanage of being raded 4 hours a day. While we generally use nonsynchronous close-o-close prices o esimae RiskMerics correlaions, we obained price daa for boh he US and Ausralian markes quoed in he Asian ime zone (Augus 1994 o une 1995). We compared he correlaion based on synchronous daa wih correlaion esimaes ha are produced under he sandard RiskMerics daa (using he nonsynchronous US and Ausralian marke close). Plos of he wo correlaion series are shown in Char 8.6. Par IV: RiskMerics Daa Ses

190 Chaper 8. Daa and relaed saisical issues Char 8.6 10-year Ausralia/US governmen bond zero correlaion based on daily RiskMerics close/close daa and 0:00 GMT daa Correlaion 1.0 0.8 Synchronous 0.6 0.4 0. 0 RiskMerics -0. February March April May une 1995 While he changes in correlaion esimaes follow similar paerns over ime (already an ineresing resul in iself), he correlaion esimaes obained from price daa aken a he opening of he markes in Asia are subsanially higher. One hing worh noing however, is ha while he synchronous esimae appears o be a beer represenaion of he rue level of correlaion, i is no necessarily equal o he rue correlaion. While we have adjused for he iming issue, we may have inroduced oher problems in he process, such as he fac ha while US Treasuries rade in he Asian ime zone, he marke is no as liquid as during Norh American rading hours and he prices may herefore be less represenaive of normal rading volumes. Marke segmenaion may also affec he resuls. Mos invesors, even hose based in Asia pu on posiions in he US marke during Norh American rading hours. U.S. Treasury rading in Asia is ofen he resul of hedging. Neverheless, from a risk managemen perspecive, his is an imporan resul. Marke paricipans holding posiions in various markes including Ausralia (and possibly oher Asian markes) would be disoring heir risk esimaes by using correlaion esimaes generaed from close of business prices. 8.5.1. An algorihm for adjusing correlaions Correlaion is simply he covariance divided by he produc of wo sandard errors. Since he sandard deviaions are unaffeced by nonsynchronous daa, correlaion is adversely affeced by nonsynchronous daa hrough is covariance. This fac simplifies he analysis because under he curren RiskMerics assumpions, long horizon covariance forecass are simply he 1-day covariance forecass muliplied by he forecas horizon. Le us now invesigae he effec ha nonsynchronous rading has on correlaion esimaes for hisorical rae series from he Unied Saes (USD), Ausralian (AUD) and Canadian (CAD) governmen bond markes. In paricular, we focus on 10-year governmen bond zero raes. Table 8.8 presens he ime ha RiskMerics records hese raes (closing prices). RiskMerics Technical Documen Fourh Ediion

Sec. 8.5 Nonsynchronous daa collecion 191 Table 8.8 RiskMerics closing prices 10-year zero bonds Counry EST London USD 3:30 p.m. 8:00 p.m. CAD 3:30 p.m. 8:00 p.m. AUD :00 a.m. 7:00 a.m. Noe ha he USD and CAD raes are synchronous while he USD and AUD, and CAD and AUD raes are nonsynchronous. We chose o analyze raes in hese hree markes o gain insigh as o how covariances (correlaions) compued from synchronous and nonsynchronous reurn series obs compare wih each oher. For example, a any ime, he observed reurn series, r and obs obs obs USD, r AUD, are nonsynchronous, whereas r USD, and r CAD, are synchronous. We are ineresed in measuring he covariance and auocovariance of hese reurn series. Table 8.9 provides summary saisics on 1-day covariance and auocovariance forecass for he period May 1993 o May 1995. The numbers in he able are inerpreed as follows: over he sample period, he average covariance beween USD and AUD 10-year zero reurns, obs obs cov r USD,, r AUD, is 0.16335 while he average covariance beween curren USD 10-year zero reurns and lagged CAD 10-year zero reurns (auocovariance) is 0.0039. Table 8.9 Sample saisics on RiskMerics daily covariance forecass 10-year zero raes; May 1993 May 1995 Daily forecass Mean Median Sd. dev. Max Min obs cov r USD,, obs cov r USD, 1 obs cov r USD,, obs r AUD, obs r AUD, obs r AUD, 1 0.1633* 0.0995 0.1973 0.8194 0.3396 0.5685 0.4635 0.3559 1.7053 0.1065 0.0085 0.0014 0.1806 0.5667 0.6056 obs cov r USD,, obs cov r USD, 1, obs cov r USD,, obs r CAD, obs r CAD, obs r CAD, 1 0.608 0.491 0.3764 1.9534 0.1356 0.044 0.059 0.1474 0.9768 0.374 0.0039 0.0003 0.1814 0.3333 0.790 * All numbers are muliplied by 10,000. The resuls show ha when reurns are recorded nonsynchronously, he covariaion beween lagged 1-day USD reurns and curren AUD reurns (0.5685) is larger, on average, han he covariance (0.1633) ha would ypically be repored. Conversely, for he USD and CAD reurns, he auocovariance esimaes are negligible relaive o he covariance esimaes. This evidence poins o a ypical finding: firs order auocovariances of reurns for asses ha rade a differen imes are larger han auocovariances for reurns on asses ha rade synchronously. 9 9 One possible explanaion for he large auocovariances has o do wih informaion flows beween markes. The lieraure on informaion flows beween markes include sudies analyzing apanese and US equiy markes (affe and Weserfield (1985), Becker, e.al, (199), Lau and Dilz, (1994)). Papers ha focus on many markes include Eun and Shim, (1989). Par IV: RiskMerics Daa Ses

19 Chaper 8. Daa and relaed saisical issues As a check of he resuls above and o undersand how RiskMerics correlaion forecass are affeced by nonsynchronous reurns, we now focus on covariance forecass for a specific day. We coninue o use USD, CAD and AUD 10-year zero raes. Consider he 1-day forecas period May 1 o May 13, 1995. In RiskMerics, hese 1-day forecass are available a 10 a.m. EST on May 1. The mos recen USD (CAD) reurn is calculaed over he period 3:30 pm EST on 5/10 o 3:30 pm EST on 5/11 whereas he mos recen AUD reurn is calculaed over he period 1:00 am EST on 5/10 o 1:00 am EST on 5/11. Table 8.10 presens covariance forecass for May 1 along wih heir sandard errors. Table 8.10 RiskMerics daily covariance forecass 10-year zero raes; May 1, 1995 Reurn series Covariance T-saisic obs obs r USD, 5 1 r AUD, 5 1 0.305 - obs obs r USD, 5 11 r AUD, 5 1 obs obs r USD, 5 1 r AUD, 5 11 obs obs r USD, 5 11 r CAD, 5 1 obs obs r USD, 5 1 r CAD, 5 1 0.69 (0.074)* 8.5 0.440 (0.074) 5.9 0.530-0.106 (0.058) 1.8 obs obs r USD, 5 1 r CAD, 5 11 0.16 (0.059).13 * Asympoic sandard errors are repored in parenheses. For a discussion on he use of he -saisic for he auocovariances see Shanken (1987). In agreemen wih previous resuls, we find ha while here is srong covariaion beween lagged obs obs USD reurns r USD, 5 11 and curren AUD reurns r USD, 5 1 (as shown by large -saisics), he covariaion beween lagged USD and CAD reurns is no nearly as srong. The resuls also show evidence of covariaion beween lagged AUD reurns and curren USD reurns. The preceding analysis describes a siuaion where he sandard covariances calculaed from nonsynchronous daa do no capure all he covariaion beween reurns. By esimaing auocovariances, i is possible o measure he 1-day lead and lag effecs across reurn series. Wih nonsynchronous daa, hese lead and lag effecs appear quie large. In oher words, curren and pas informaion in one reurn series is correlaed wih curren and pas informaion in anoher series. If we represen informaion by reurns, hen following Cohen, Hawawini, Maier, Schwarz and Whicomb, (CHMSW 1983) we can wrie observed reurns as a funcion of weighed unobserved curren and lag rue reurns. The weighs simply represen how much informaion in a specific rue reurn appears in he reurn ha is observed. Given his, we can wrie observed (nonsynchronous) reurns for he USD and AUD 10-year zero reurns as follows: [8.43] obs r USD, obs r AUD, = = θ USD, R USD, + θ USD, 1 r USD, 1 θ AUD, R USD, + θ AUD, 1 r AUD, 1 θ j The, i s are random variables ha represen he proporion of he rue reurn of asse j generaed in period -i ha is acually incorporaed in observed reurns in period. In oher words, he θ j, s are weighs ha capure how he rue reurn generaed in one period impacs on he observed reurns in he same period and he nex. I is also assumed ha: RiskMerics Technical Documen Fourh Ediion

Sec. 8.5 Nonsynchronous daa collecion 193 θ AUD, and θ USD, τ are independen for all and τ [8.44] θ AUD, and θ USD, τ are independen of R AUD, and R USD, τ E ( θ AUD, ) = E ( θ USD, ) for all and τ E ( θ j, + θ j, 1 ) = 1 for j = AUD, USD and for all and τ Table 8.11 shows, for he example given in he preceding secion, he relaionship beween he dae when he rue reurn is calculaed and he weigh assigned o he rue reurn. Table 8.11 Relaionship beween lagged reurns and applied weighs observed USD and AUD reurns for May 1, 1995 Dae 5/9 5/10 5/9 5/10 5/10 5/11 5/10 5/11 Weigh θ AUD, 1 θ USD, 1 θ AUD, θ USD, obs obs Earlier we compued he covariance based on observed reurns, cov r USD,, r AUD, However, we can use Eq. [8.43] o compue he covariance of he rue reurns cov ( r USD,, r AUD, ), i.e., [8.45] obs cov ( r USD,, r AUD, ) = cov r USD,, obs +cov r USD,, obs r AUD, 1 obs r AUD, obs + cov r USD, 1, obs r AUD, We refer o his esimaor as he adjused covariance. Having esablished he form of he adjused covariance esimaor, he adjused correlaion esimaor for any wo reurn series j and k is: [8.46] ρ jk obs obs obs obs obs obs cov r j,, rk, 1 + cov r j,, rk, + cov r j, 1, r k,, = ------------------------------------------------------------------------------------------------------------------------------------ obs obs sd r j, sd rk, Table 8.1 shows he original and adjused correlaion esimaes for USD-AUD and USD-CAD 10-year zero rae reurns. Table 8.1 Original and adjused correlaion forecass USD-AUD 10-year zero raes; May 1, 1995 Daily forecass Original Adjused % change cov ( r USD, 5 1, r AUD, 5 1 ) cov ( r USD, 5 1, r CAD, 5 1 ) 0.305 0.560 84% 0.530 0.573 8% Noe ha he USD-AUD adjused covariance increases he original covariance esimae by 84%. Earlier (see Table 8.10) we found he lead-lag covariaion for he USD-AUD series o be saisically significan. Applying he adjused covariance esimaor o he synchronous series USD-CAD, we find only an 8% increase over he original covariance esimae. However, he evidence from Table 8.10 would sugges ha his increase is negligible. Par IV: RiskMerics Daa Ses

194 Chaper 8. Daa and relaed saisical issues 8.5.1.3 Checking he resuls How does he adjusmen algorihm perform in pracice? Char 8.7 compares hree daily correlaion esimaes for 10-year zero coupon raes in Ausralia and he Unied Saes: (1) Sandard RiskMerics using nonsynchronous daa, () esimae correlaion using synchronous daa colleced in Asian rading hours and, (3) RiskMerics Adjused using he esimaor in Eq. [8.46]. Char 8.7 Adjusing 10-year USD/AUD bond zero correlaion using daily RiskMerics close/close daa and 0:00 GMT daa 1.0 **RiskMerics Adjused** 0.8 0.6 0.4 Synchronous 0. 0.0 RiskMerics -0. an-95 Mar-95 Apr-95 un-95 The resuls show ha he adjusmen facor capures he effecs of he iming differences ha affec he sandard RiskMerics esimaes which use nonsynchronous daa. A poenial drawback of using his esimaor, however, is ha he adjused series displays more volailiy han eiher he unadjused or he synchronous series. This means ha in pracice, choices may have o be made as o when o apply he mehodology. In he Ausralian/US case, i is clear ha he benefis of he adjusmen in erms of increasing he correlaion o a level consisen wih he one obained when using synchronous daa ouweighs he increased volailiy. The choice, however, may no always be ha clear cu as shown by Char 8.8 which compares adjused and unadjused correlaions for he US and apanese 10-year zero raes. In periods when he underlying correlaion beween he wo markes is significan (an-feb 1995, he algorihm correcly adjuss he esimae). In periods of lower correlaion, he algorihm only increases he volailiy of he esimae. RiskMerics Technical Documen Fourh Ediion

Sec. 8.5 Nonsynchronous daa collecion 195 Char 8.8 10-year apan/us governmen bond zero correlaion using daily RiskMerics close/close daa and 0:00 GMT daa Correlaion 0.8 0.6 0.4 RiskMerics adjused 0. 0-0. -0.4 RiskMerics -0.6 February March April May une 1995 Also, in pracice, esimaion of he adjused correlaion is no necessarily sraighforward because we mus ake ino accoun he chance of geing adjused correlaion esimaes above 1. This poenial problem arises because he numeraor in Eq. [8.46] is being adjused wihou due consideraion of he denominaor. An algorihm ha allows us o esimae he adjused correlaion wihou obaining correlaions greaer han 1 in absolue value is given in Secion 8.5.. Table 8.13 on page 196 repors sample saisics for 1-day correlaion forecass esimaed over various sample periods for boh he original RiskMerics and adjused correlaion esimaors. Correlaions beween Unied Saes and Asia-Pacific are based on non-synchronous daa. 8.5. Using he algorihm in a mulivariae framework Finally, we explain how o compue he adjused correlaion marix. 1. Calculae he unadjused (sandard) RiskMerics covariance marix, Σ. (Σ is an N x N, posiive semi-definie marix).. Compue he nonsynchronous daa adjusmen marix K where he elemens of K are [8.47] cov ( r k,, r ) + cov ( r j, 1 k,, r ) for k j 1 j,, = 0 for k = j k k j 3. The adjused covariance marix M, is given by M = Σ+ fk where 0 f 1. The parameer f ha is used in pracice is he larges possible f such ha M is posiive semi-definie. Par IV: RiskMerics Daa Ses

196 Chaper 8. Daa and relaed saisical issues Table 8.13 Correlaions beween US and foreign insrumens Correlaions beween USD 10-year zero raes and PY, AUD, and NZD 10-year zero raes.* Sample period: May 1991 May 1995. Original Adjused PY AUD NZD PY AUD NZD mean 0.06 0.166 0.047 0.193 0.458 0.319 median 0.040 0.155 0.036 0.1 0.469 0.367 sd dev 0.151 0.151 0.171 0.308 0.1 0.41 max 0.517 0.56 0.613 0.987 0.937 0.91 min 0.491 0.17 0.389 0.76 0.164 0.405 Correlaions beween USD -year swap raes and PY, AUD, NZD, HKD -year swap raes.* Sample period: May 1993 May 1995. Original Adjused PY AUD NZD HKD PY AUD NZD HKD mean 0.018 0.33 0.04 0.139 0.054 0.493 0.49 0.57 median 0.05 0.00 0.00 0.103 0.065 0.50 0.47 0.598 sd dev 0.147 0.183 0.179 0.17 0.196 0.181 0.03 0.33 max 0.319 0.647 0.559 0.696 0.558 0.90 0.745 0.945 min 0.358 0.148 0.350 0.504 0.456 0.096 0.356 0.411 Correlaions beween USD equiy index and PY, AUD, NZD, HKD, SGD equiy indices.* Sample period: May 1993 May 1995. Original Adjused PY AUD NZD HKD SGD PY AUD NZD HKD SGD mean 0.051 0.099-0.03 0.006 0.038 0.14 0.330 0.055 0.013 0.014 median 0.067 0.119-0.01-0.001 0.08 0.140 0.348 0.053 0.056 0.04 sd dev 0.166 0.176 0.18 0.119 0.145 0.199 0.06 0.187 0.6 0.37 max 0.444 0.504 0.83 0.71 0.484 0.653 0.810 0.349 0.645 0.641 min 0.335 0.345 0.455 0.98 0.384 0.395 0.13 0.54 0.57 0.589 * PY = apanese yen, AUD = Ausralian dollar, NZD = New Zealand dollar, HKD = Hong Kong dollar, SGD = Singapore dollar RiskMerics Technical Documen Fourh Ediion

197 Chaper 9. Time series sources 9.1 Foreign exchange 199 9. Money marke raes 199 9.3 Governmen bond zero raes 00 9.4 Swap raes 0 9.5 Equiy indices 03 9.6 Commodiies 05 Par IV: RiskMerics Daa Ses

198 RiskMerics Technical Documen Fourh Ediion

199 Chaper 9. Time series sources Sco Howard Morgan Guarany Trus Company Risk Managemen Advisory (1-1) 648-4317 howard_james_s@jpmorgan.com Daa is one of he cornersones of any risk managemen mehodology. We examined a number of daa providers and decided ha he sources deailed in his chaper were he mos appropriae for our purposes. 9.1 Foreign exchange Foreign exchange prices are sourced from WM Company and Reuers. They are mid-spo exchange prices recorded a 4:00 p.m. London ime (11:00 a.m. EST). All foreign exchange daa used for RiskMerics is idenical o he daa used by he.p. Morgan family of governmen bond indices. (See Table 9.1.) Table 9.1 Foreign exchange Currency Codes Americas Asia Pacific Europe and Africa ARS Argenine peso AUD Ausralian dollar ATS Ausrian shilling CAD Canadian dollar HKD Hong Kong dollar BEF Belgian franc MXN Mexican peso IDR Indonesian rupiah CHF Swiss franc USD U.S. dollar PY apanese yen DEM Deusche mark EMB EMBI+* KRW Korean won DKK Danish kroner MYR Malaysian ringgi ESP Spanish pesea NZD New Zealand dollar FIM Finnish mark PHP Philippine peso FRF French franc SGD Singapore dollar GBP Serling THB Thailand bah IEP Irish pound TWD Taiwan dollar ITL Ialian lira NLG Duch guilder NOK Norwegian kroner PTE Poruguese escudo SEK Swedish krona XEU ECU ZAR Souh African rand * EMBI+ sands for he.p. Morgan Emerging Markes Bond Index Plus. 9. Money marke raes Mos 1-, -, 3-, 6-, and 1-monh money marke raes (offered side) are recorded on a daily basis by.p. Morgan in London a 4:00 p.m. (11:00 a.m. EST). Those obained from exernal sources are also shown in Table 9.. Par IV: RiskMerics Daa Ses

00 Chaper 9. Time series sources Table 9. Money marke raes: sources and erm srucures Source Time Term Srucure Marke.P. Morgan Third Pary * U.S. EST 1m 3m 6m 1m Ausralia 11:00 a.m. Hong Kong 10:00 p.m. Indonesia 5:00 a.m. apan 11:00 a.m. Malaysia 5:00 a.m. New Zealand 1:00 a.m. Singapore 4:30 a.m. Thailand 5:00 a.m. Ausria 11:00 a.m. Belgium 11:00 a.m. Denmark 11:00 a.m. Finland 11:00 a.m. France 11:00 a.m. Ireland 11:00 a.m. Ialy 11:00 a.m. Neherlands 11:00 a.m. Norway 11:00 a.m. Porugal 11:00 a.m. Souh Africa 11:00 a.m. Spain 11:00 a.m. Sweden 11:00 a.m. Swizerland 11:00 a.m. U.K. 11:00 a.m. ECU 11:00 a.m. Canada 11:00 a.m. Mexico 1:00 p.m. U.S. 11:00 a.m. * Third pary source daa from Reuers Generic excep for Hong Kong (Reuers HIBO), Singapore (Reuers MASX), and New Zealand (Naional Bank of New Zealand). Money marke raes for Indonesia, Malaysia, and Thailand are calculaed using foreign exchange forwardpoins. Mexican raes represen secondary rading in Cees. 9.3 Governmen bond zero raes Zero coupon raes ranging in mauriy from o 30 years for he governmen bond markes included in he.p. Morgan Governmen Bond Index as well as he Irish, ECU, and New Zealand markes. (See Table 9.3.) RiskMerics Technical Documen Fourh Ediion

9.3 Governmen bond zero raes 01 Table 9.3 Governmen bond zero raes: sources and erm srucures Source Time Term srucure Marke.P. Morgan Third Pary U.S. EST y 3y 4y 5y 7y 9y 10y 15y 0y 30y Ausralia 1:30 a.m. apan 1:00 a.m. New Zealand 1:00 a.m. Belgium 11:00 a.m. Denmark 10:30 a.m. France 10:30 a.m. Germany 11:30 a.m. Ireland 10:30 a.m. Ialy 10:45 a.m. Neherlands 11:00 a.m. Souh Africa 11:00 a.m. Spain 11:00 a.m. Sweden 10:00 a.m. U.K. 11:45 a.m. ECU 11:45 a.m. Canada 3:30 p.m. U.S. 3:30 a.m. Emerging Mk. 3:00 p.m. * Third pary daa sourced from Den Danske Bank (Denmark), NCB Sockbrokers (Ireland), Naional Bank of New Zealand (New Zealand), and SE Banken (Sweden).. P. Morgan Emerging Markes Bond Index Plus (EMBI+). If he objecive is o measure he volailiy of individual cash flows, hen one could ask wheher i is appropriae o use a erm srucure model insead of he underlying zero raes which can be direcly observed from insrumens such as Srips. The selecion of a modeled erm srucure as he basis for calculaing marke volailiies was moivaed by he fac ha here are few markes which have observable zero raes in he form of governmen bond Srips from which o esimae volailiies. In fac, only he U.S. and French markes have reasonably liquid Srips which could form he basis for a saisically solid volailiy analysis. Mos oher markes in he OECD have eiher no Srip marke or a relaively illiquid one. The one possible problem of he erm srucure approach is ha i would no be unreasonable o assume he volailiy of poins along he erm srucure may be lower han he marke s real volailiy because of he smoohing impac of passing a curve hrough a universe of real daa poins. To see wheher here was suppor for his assumpion, we compared he volailiy esimaes obained from erm srucure derived zero raes and acual Srip yields for he U.S. marke across four mauriies (3, 5, 7, and 10 years). The resuls of he comparison are shown in Char 9.1. Par IV: RiskMerics Daa Ses

0 Chaper 9. Time series sources Char 9.1 Volailiy esimaes: daily horizon 1.65 sandard deviaion 6-monh moving average Volailiy.9 Volailiy.5.7.5.3 3-year Srip.0 5-year Srip.1 1.9 1.7 3-year Zero rae 1.5 5-year Zero rae 1.5 199 1993 1994 1.0 199 1993 1994 Volailiy.5 Volailiy.0.0 7-year Srip 1.5 10-year Srip 1.5 1.0 7-year Zero rae 199 1993 10-year Zero rae 1.0 1994 199 1993 1994 The resuls show ha here is no clear bias from using he erm srucure versus underlying Srips daa. The differences beween he wo measures decline as mauriy increases and are parially he resul of he lack of liquidiy of he shor end of he U.S. Srip marke. Marke movemens specific o Srips can also be caused by invesor behavior in cerain hedging sraegies ha cause prices o someimes behave erraically in comparison o he coupon curve from which he erm srucure is derived. 9.4 Swap raes Swap par raes from o 10 years are recorded on a daily basis by.p. Morgan, excep for Ireland (provided by NCB Sockbrokers), Hong Kong (Reuers TFHK) and Indonesia, Malaysia and Thailand (Reuers EXOT). (See Table 9.4.) The par raes are hen convered o zero coupon equivalens raes for he purpose of inclusion wihin he RiskMerics daa se. (Refer o Secion 8.1 for deails). RiskMerics Technical Documen Fourh Ediion

9.5 Equiy indices 03 Table 9.4 Swap zero raes: sources and erm srucures Source Time Term srucure Marke.P. Morgan Third Pary * US EST y 3y 4y 5y 7y 10y Ausralia 1:30 a.m. Hong Kong 4:30 a.m. Indonesia 4:00 a.m. apan 1:00 a.m. Malaysia 4:00 a.m. New Zealand 3:00 p.m. Thailand 4:00 a.m. Belgium 10:00 a.m. Denmark 10:00 a.m. Finland 10:00 a.m France 10:00 a.m. Germany 10:00 p.m. Ireland 11:00 a.m. Ialy 10:00 a.m. Neherlands 10:00 a.m. Spain 10:00 a.m. Sweden 10:00 a.m. Swizerland 10:00 a.m. U.K. 10:00 a.m. ECU 10:00 a.m. Canada 3:30 p.m. U.S. 3:30 a.m. * Third pary source daa from Reuers Generic excep for Ireland (NCBI), Hong Kong (TFHK), and Indonesia, Malaysia, Thailand (EXOT). 9.5 Equiy indices The following lis of equiy indices (Table 9.5) have been seleced as benchmarks for measuring he marke risk inheren in holding equiy posiions in heir respecive markes. The facors ha deermined he selecion of hese indices include he exisence of index fuures ha can be used as hedging insrumens, sufficien marke capializaion in relaion o he oal marke, and low racking error versus a represenaion of he oal capializaion. All he indices lised below measure principal reurn excep for he DAX which is a oal reurn index. Par IV: RiskMerics Daa Ses

04 Chaper 9. Time series sources Table 9.5 Equiy indices: sources* Marke Exchange Index Name Weighing % Mk. cap. Time, U.S. EST Ausralia Ausralian Sock Exchange All Ordinaries MC 96 1:10 a.m. Hong Kong Hong Kong Sock Exchange Hang Seng MC 77 1:30 a.m. Indonesia akara Sock Exchange SE MC 4:00 a.m. Korea Seoul Sock Exchange KOPSI MC 3:30 a.m. apan Tokyo Sock Exchange Nikei 5 MC 46 1:00 a.m. Malaysia Kuala Lumpur Sock Exchange KLSE MC 6:00 a.m. New Zealand New Zealand Sock Exchange Capial 40 MC 10:30 p.m. Philippines Manila Sock Exchange MSE Com l &Inusil Price MC 1:00 a.m. Singapore Sock Exchange of Singapore Sing. All Share MC 4:30 a.m. Taiwan Taipei Sock Exchange TSE MC 1:00 a.m. Thailand Bangkok Sock Exchange SET MC 5:00 a.m. Ausria Vienna Sock Exchange Crediansal MC 7:30 a.m. Belgium Brussels Sock Exchange BEL 0 MC 78 10:00 a.m. Denmark Copenhagen Sock Exchange KFX MC 44 9:30 a.m. Finland Helsinki Sock Exchange Hex General MC 10:00 a.m. France Paris Bourse CAC 40 MC 55 11:00 a.m. Germany Frankfur Sock Exchange DAX MC 57 10:00 a.m. Ireland Irish Sock Exchange Irish SE ISEQ 1:30 p.m. Ialy Milan Sock Exchange MIB 30 MC 65 10:30 a.m. apan Tokyo Sock Exchange Nikei 5 MC 46 1:00 a.m. Neherlands Amserdam Sock Exchange AEX MC 80 10:30 a.m. Norway Oslo Sock Exchange Oslo SE General 9:00 a.m. Porugal Lisbon Sock Exchange Banco Toa SI 11:00 a.m. Souh Africa ohannesburg Sock Exchange SE MC 10:00 a.m. Spain Madrid Sock Exchange IBEX 35 MC 80 11:00 a.m. Sweden Sockholm Sock Exchange OMX MC 61 10:00 a.m. Swizerland Zurich Sock Exchange SMI MC 56 10:00 a.m. U.K. London Sock Exchange FTSE 100 MC 69 10:00 a.m. Argenina Buenos Aires Sock Exchange Merval Vol. 5:00 p.m. Canada Torono Sock Exchange TSE 100 MC 63 4:15 p.m. Mexico Mexico Sock Exchange IPC MC 3:00 p.m. U.S. New York Sock Exchange Sandard and Poor s 100 MC 60 4:15 a.m. * Daa sourced from DRI. RiskMerics Technical Documen Fourh Ediion

9.6 Commodiies 05 9.6 Commodiies The commodiy markes ha have been included in RiskMerics are he same markes as he.p. Morgan Commodiy Index (PMCI). The daa for hese markes are shown in Table 9.6. Table 9.6 Commodiies: sources and erm srucures Time, Term srucure Commodiy Source U.S. EST Spo 1m 3m 6m 1m 15m 7m WTI Ligh Swee Crude NYMEX * 3:10 p.m. Heaing Oil NYMEX 3:10 p.m. NY Harbor # unleaded gas NYMEX 3:10 p.m. Naural gas NYMEX 3:10 p.m. Aluminum LME 11:0 a.m. Copper LME 11:15 a.m. Nickel LME 11:10 a.m. Zinc LME 11:30 a.m. Gold LME 11:00 a.m. Silver LFOE 11:00 a.m. Plainum LPPA 11:00 a.m. * NYMEX (New York Mercanile Exchange) LME (London Meals Exchange) LFOE (London fuures and Opions Meal Exchange) LPPA (London Plainum & Palladium Associaion) The choice beween eiher he rolling nearby or inerpolaion (consan mauriy) approach is influenced by he characerisics of each conrac. We use he inerpolaion mehodology wherever possible, bu in cerain cases his approach canno or should no be implemened. We use inerpolaion (I) for all energy conracs. (See Table 9.7.) Table 9.7 Energy mauriies Mauriies Energy 1m 3m 6m 1m 15m 7m Ligh swee crude I* I I I Heaing Oil I I I I Unleaded Gas I I I Naural Gas I I I I * I = Inerpolaed mehodology. The erm srucures for base meals are based upon rolling nearby conracs wih he excepion of he spo (S) and 3-monh conracs. Daa availabiliy is he issue here. Price daa for conracs raded on he London Meals Exchange is available for consan mauriy 3-monh (A) conracs (prices are quoed on a daily basis for 3 monhs forward) and rolling 15- and 7- monh (N) conracs. Nickel exends ou o only 15 monhs. (See Table 9.8.) Par IV: RiskMerics Daa Ses

06 Chaper 9. Time series sources Table 9.8 Base meal mauriies Mauriies Commodiy Spo 3m 6m 1m 15m 7m Aluminum S* A N N Copper S A N N Nickel S A N Zinc S A N N * S = Spo conrac. A = Consan mauriy conrac. N = Rolling conrac. Spo prices are he driving facor in he precious meals markes. Volailiy curves in he gold, silver, and plainum markes are relaively fla (compared o he energy curves) and spo prices are he main deerminan of he fuure value of insrumens: sorage coss are negligible and convenience yields such as hose associaed wih he energy markes are no a consideraion. RiskMerics Technical Documen Fourh Ediion

07 Chaper 10. RiskMerics volailiy and correlaion files 10.1 Availabiliy 09 10. File names 09 10.3 Daa series naming sandards 09 10.4 Forma of volailiy files 11 10.5 Forma of correlaion files 1 10.6 Daa series order 14 10.7 Underlying price/rae availabiliy 14 Par IV: RiskMerics Daa Ses

08 RiskMerics Technical Documen Fourh Ediion

09 Chaper 10. RiskMerics volailiy and correlaion files Sco Howard Morgan Guarany Trus Company Risk Managemen Advisory (1-1) 648-4317 howard_james_s@jpmorgan.com This secion serves as a guide o undersanding he informaion conained in he RiskMerics daily and monhly volailiy and correlaion files. I defines he naming sandards we have adoped for he RiskMerics files and ime series, he file formas, and he order in which he daa is presened in hese files. 10.1 Availabiliy Volailiy and correlaion files are updaed each U.S. business day and posed on he Inerne by 10:30 a.m. EST. They cover daa hrough close-of-business for he previous U.S. business day. Insrucions on downloading hese files are available in Appendix H. 10. File names To ensure compaibiliy wih MS-DOS, file names use he 8.3 forma: 8-characer name and 3-characer exension (see Table 10.1). Table 10.1 RiskMerics file names ddmmyy indicaes he dae on which he marke daa was colleced File name forma Volailiy Correlaion File descripion DVddmmyy.RM3 DCddmmyy.RM3 1-day esimaes MVddmmyy.RM3 MCddmmyy.RM3 5-day esimaes BVddmmyy.RM3 BCddmmyy.RM3 Regulaory daa ses DVddmmyy.vol DCddmmyy.cor Add-In 1-day esimaes MVddmmyy.vol MCddmmyy.cor Add-In 5-day esimaes BVddmmyy.vol BCddmmyy.cor Add-In regulaory The firs wo characers designae wheher he file is daily (D) or monhly (M), and wheher i conains volailiy (V) or correlaion (C) daa. The nex six characers idenify he collecion dae of he marke daa for which he volailiies and correlaions are compued. The exension idenifies he version of he daa se. 10.3 Daa series naming sandards In boh volailiy and correlaion files, all series names follow he same naming convenion. They sar wih a hree-leer code followed by a period and a suffix, for example, USD.R180. The hree-leer code is eiher a SWIFT 1 currency code or, in he case of commodiies, a commodiy code, as shown in Table 10.. The suffix idenifies he asse class (and he mauriy for ineres-rae and commodiy series). Table 10.3 liss insrumen suffix codes, followed by an example of how currency, commodiy, and suffix codes are used. 1 The excepion is EMB. This represens. P. Morgan s Emerging Markes Bond Index Plus. Par IV: RiskMerics Daa Ses

10 Chaper 10. RiskMerics volailiy and correlaion files Table 10. Currency and commodiy idenifiers Currency Codes Americas Asia Pacific Europe and Africa Commodiy Codes ARS Argenine peso AUD Ausralian dollar ATS Ausrian shilling ALU Aluminum CAD Canadian dollar HKD Hong Kong dollar BEF Belgian franc COP Copper MXN Mexican peso IDR Indonesian rupiah CHF Swiss franc GAS Naural gas USD U.S. dollar PY apanese yen DEM Deusche mark GLD Gold EMB EMBI+ * KRW Korean won DKK Danish kroner HTO NY Harbor # heaing oil MYR Malaysian ringgi ESP Spanish pesea NIC Nickel NZD New Zealand dollar FIM Finnish mark PLA Plainum PHP Philippine peso FRF French franc SLV Silver SGD Singapore dollar GBP Serling UNL Unleaded gas THB Thailand bah IEP Irish pound WTI Ligh Swee Crude TWD Taiwan dollar ITL Ialian lira ZNC Zinc NLG Duch guilder NOK Norwegian kroner PTE Poruguese escudo SEK Swedish krona XEU ECU ZAR Souh African rand * EMBI+ sands for he.p. Morgan Emerging Markes Bond Index Plus. Table 10.3 Mauriy and asse class idenifiers Mauriy Foreign exchange Equiy indices Insrumen Suffix Codes Money marke Swaps Gov bonds Commodiies Spo XS SE C00 1m R030 3m R090 C03 6m R180 C06 1m R360 C1 15m C15 18m C18 4m (y) S0 Z0 C4 7m C7 36m (3y) S03 Z03 C36 4y S04 Z04 5y S05 Z05 7y S07 Z07 9y Z09 10y S10 Z10 15y Z15 0y Z0 30y Z30 RiskMerics Technical Documen Fourh Ediion

Sec. 10.4 Forma of volailiy files 11 For example, we idenify he Singapore dollar foreign exchange rae by SGD.XS, he U.S. dollar 6-monh money marke rae by USD.R180, he CAC 40 index by FRF.SE, he -year serling swap rae by GBP.S0, he 10-year apanese governmen bond (GB) by PY.Z10, and he 3-monh naural gas fuure by GAS.C03. 10.4 Forma of volailiy files Each daily and monhly volailiy file sars wih a se of header lines ha begin wih an aserisk (*) and describe he conens of he file. Following he header lines are a se of record lines (wihou an aserisk) conaining he daily or monhly daa. Table 10.4 Sample volailiy file Table 10.4 shows a porion of a daily volailiy file. Line # 1 3 4 5 6 7 8 9 10 11 1 13 14 Volailiy file *Esimae of volailiies for a one day horizon *COLUMNS=, LINES=418, DATE=11/14/96, VERSION.0 *RiskMerics is based on bu differs significanly from he marke risk managemen sysems *developed by.p. Morgan for is own use..p. Morgan does no warrany any resuls obained *from use of he RiskMerics mehodology, documenaion or any informaion derived from *he daa (collecively he Daa ) and does no guaranee is sequence, imeliness, accuracy or *compleeness..p. Morgan may disconinue generaing he Daa a any ime wihou any prior *noice. The Daa is calculaed on he basis of he hisorical observaions and should no be relied *upon o predic fuure marke movemens. The Daa is mean o be used wih sysems developed *by hird paries..p. Morgan does no guaranee he accuracy or qualiy of such sysems. *SERIES, PRICE/YIELD,DECAYFCTR,PRICEVOL,YIELDVOL ATS.XS.VOLD,0.094150,0.940,0.554647,ND AUD.XS.VOLD, 0.791600,0.940,0.64317,ND BEF.XS.VOLD, 0.0315,0.940,0.546484,ND In his able, each line is inerpreed as follows: Line 1 idenifies wheher he file is a daily or monhly file. Line liss file characerisics in he following order: he number of daa columns, he number of record lines, he file creaion dae, and he version number of he file forma. Lines 3 10 are a disclaimer. Line 11 conains comma-separaed column iles under which he volailiy daa is lised. Lines 1 hrough he las line a he end of file (no shown) represen he record lines, which conain he comma-separaed volailiy daa formaed as shown in Table 10.5. Par IV: RiskMerics Daa Ses

1 Chaper 10. RiskMerics volailiy and correlaion files Table 10.5 Daa columns and forma in volailiy files Column ile (header line) Daa (record lines) Forma of volailiy daa SERIES Series name See Secion 10.3 for series naming convenions. In addiion, each series name is given an exension, eiher.vold (for daily volailiy esimae), or.volm (for monhly volailiy esimae). PRICE/YIELD Price/Yield level #.###### or NM if he daa canno be published. DECAYFCTR Exponenial moving average decay facor #.### PRICEVOL Price volailiy esimae #.###### (% unis) YIELDVOL Yield volailiy esimae #.###### (% unis) or ND if he series has no yield volailiy (e.g., FX raes). For example, in Table 10.4, he firs value ATS.XS.VOLD in Line 1 corresponds o he SERIES column ile, and idenifies he series o be a USD/ATS daily volailiy series. Similarly, he remaining values are inerpreed as follows: The value 0.094150 was used as he price/yield level in he volailiy calculaion. The value 0.940 was used as he exponenial moving average decay facor. The value 0.554647% is he price volailiy esimae. The value ND indicaes ha he series has no yield volailiy. 10.5 Forma of correlaion files Daily and monhly correlaion files are formaed similar o he volailiy files (see Secion 10.4), and conain analogous header and record lines (see Table 10.6). Each file comprises he lower half of he correlaion marix for he series being correlaed, including he diagonal, which has a value of 1.000. (The upper half is no shown since he daily and monhly correlaion marices are symmerical around he diagonal. For example, 3-monh USD LIBOR o 3-monh DEM LIBOR has he same correlaion as 3-monh DEM LIBOR o 3-monh USD LIBOR.) RiskMerics Technical Documen Fourh Ediion

Sec. 10.5 Forma of correlaion files 13 Table 10.6 Sample correlaion file Line # 1 3 4 5 6 7 8 9 10 11 1 13 14 Correlaion file *Esimae of correlaions for a one day horizon *COLUMNS=, LINES=087571, DATE=11/14/96, VERSION.0 *RiskMerics is based on bu differs significanly from he marke risk managemen sysems *developed by.p. Morgan for is own use..p. Morgan does no warrany any resuls obained *from use of he RiskMerics mehodology, documenaion or any informaion derived from *he daa (collecively he Daa ) and does no guaranee is sequence, imeliness, accuracy or *compleeness..p. Morgan may disconinue generaing he Daa a any ime wihou any prior *noice. The Daa is calculaed on he basis of he hisorical observaions and should no be relied *upon o predic fuure marke movemens. The Daa is mean o be used wih sysems developed *by hird paries..p. Morgan does no guaranee he accuracy or qualiy of such sysems. *SERIES, CORRELATION ATS.XS.ATS.XS.CORD,1.000000 ATS.XS.AUD.XS.CORD, -0.51566 ATS.XS.BEF.XS.CORD, 0.985189 In Table 10.6, each line is inerpreed as follows: Line 1 idenifies wheher he file is a daily or monhly file. Line liss file characerisics in he following order: he number of daa columns, he number of record lines, he file creaion dae, and he version number of he file forma. Lines 3 10 are a disclaimer. Line 11 conains comma-separaed column iles under which he correlaion daa is lised. Lines 1 hrough he las line a he end of he file (no shown) represen he record lines, which conain he comma-separaed correlaion daa formaed as shown in Table 10.7. Table 10.7 Daa columns and forma in correlaion files Column ile (header line) Correlaion daa (record lines) Forma of correlaion daa SERIES Series name See Secion 10.3 for series naming convenions. In addiion, each series name is given an exension, eiher.cord (for daily correlaion), or.corm (for monhly correlaion). CORRELATION Correlaion coefficien #.###### Correlaion coefficiens are compued by using he same exponenial moving average mehod as in he volailiy files (i.e., decay facor of 0.940 for a 1-day horizon, and 0.970 for a 1-monh horizon.) For example, Line 13 in Table 10.6 represens a USD/ATS o USD/AUD daily correlaion esimae of 0.51566 measured using an exponenial moving average decay facor of 0.940 (he defaul value for he 1-day horizon). Par IV: RiskMerics Daa Ses

14 Chaper 10. RiskMerics volailiy and correlaion files 10.6 Daa series order Daa series in he volailiy and correlaion files are sored firs alphabeically by SWIFT code and commodiy class indicaor, and hen by mauriy wihin he following asse class hierarchy: foreign exchange, money markes, swaps, governmen bonds, equiy indices, and commodiies. 10.7 Underlying price/rae availabiliy Due o legal consideraions, no all prices or yields are published in he volailiy files. Wha is published are energy fuure conrac prices and he yields on foreign exchange, swaps, and governmen bonds. The curren level of money marke yields can be approximaed from Eq. [10.1] by using he published price volailiies and yield volailiies as well as he insrumens modified duraions. [10.1] Curren yield = σ Price ( σ Yield Modified Duraion) RiskMerics Technical Documen Fourh Ediion

15 Par V Backesing

16 RiskMerics Technical Documen Fourh Ediion

17 Chaper 11. Performance assessmen 11.1 Sample porfolio 19 11. Assessing he RiskMerics model 0 11.3 Summary 3 Par V: Backesing

18 RiskMerics Technical Documen Fourh Ediion

19 Chaper 11. Performance assessmen Peer Zangari Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-8641 zangari_peer@jpmorgan.com In his chaper we presen a process for assessing he accuracy of he RiskMerics model. We would like o make clear ha he purpose of his secion is no o offer a review of he quaniaive measures for VaR model comparison. There is a growing lieraure on such measures and we refer he reader o Crnkovic and Drachman (1996) for he laes developmens in ha area. Insead, we presen simple calculaions ha may prove useful for deermining he appropriaeness of he RiskMerics model. 11.1 Sample porfolio We describe an approach for assessing he RiskMerics model by analyzing a porfolio consising of 15 cashflows ha include foreign exchange (), money marke deposis (), zero coupon governmen bonds (11), equiies (1) and commodiies (33). Using daily prices for he period April 4, 1990 hrough March 6, 1996 (a oal of 1001 observaions), we consruc 1-day VaR forecass over he mos recen 801 days of he sample period. We hen compare hese forecass o heir respecive realized profi/loss (P/L) which are represened by 1-day reurns. Char 11.1 shows he ypical presenaion of 1-day RiskMerics VaR forecass (90% wo-ail confidence inerval) along wih he daily P/L of he porfolio. Char 11.1 One-day Profi/Loss and VaR esimaes VaR bands are given by +/ 1.65σ P/L 1.0 0.8 0.6 0.4 0. 0-0. -0.4-0.6-0.8 +1.65σ -1.65σ 100 00 300 400 500 600 700 800 Observaions In Char 11.1 he black line represens he porfolio reurn r p, consruced from he 15 individual reurns a ime. The ime porfolio reurn is defined as follows: [11.1] 15 r 1 p, = ri, i= 1 -------- 15 where r i, represens he log reurn of he ih underlying cashflow. The Value-a-Risk bands are based on he porfolio s sandard deviaion. The formula for he porfolio s sandard deviaion, is: σ p, 1 Par V: Backesing

0 Chaper 11. Performance assessmen [11.] 15 1 σ P, 1 = i = 1 -------- 15 σ i, 1 + 15 i = 1 j > i 1 -------- 15 ρij 1, σ σ i, 1 j, 1 where σ i, 1 is he variance of he ih reurn series made for ime and ρ ij, 1 is he correlaion beween he ih and jh reurns for ime. 11. Assessing he RiskMerics model The firs measure of model performance is a simple coun he number of imes ha he VaR esimaes underpredic fuure losses (gains). Recall ha in RiskMerics each day i is assumed ha here is a 5% chance ha he observed loss exceeds he VaR forecas. 1 For he sake of generaliy, le s define a random variable X() on any day such ha X() = 1 if a paricular day s observed loss is greaer han is corresponding VaR forecas and X()=0 oherwise. We can wrie he disribuion of X() as follows [11.3] f ( X() 0.05) = 0.05 X () ( 1 0.05) 1 X () X()=0,1 0 oherwise Now, suppose we observe X() for a oal of T days, = 1,,...,T, and we assume ha he X() s are independen over ime. In oher words, wheher a VaR forecas is violaed on a paricular day is independen of wha happened on oher days. The random variable X() is said o follow a Bernoulli disribuion whose expeced value is 0.05.The oal number of VaR violaions over he ime period T is given by [11.4] X T = T = 1 X() The expeced value of X T, i.e., he expeced number of VaR violaions over T days, is T imes 0.05. For example, if we observe T = 0 days of VaR forecass, hen he expeced number of VaR violaions is 0 x 0.05 = 1; hence one would expec o observe one VaR violaion every 0 days. Wha is convenien abou modelling VaR violaions according o Eq. [11.3] is ha he probabiliy of observing a VaR violaion over T days is same as he probabiliy of observing a VaR violaion a any poin in ime,. Therefore, we are able o use VaR forecass consruced over ime o assess he appropriaeness of he RiskMerics model for his porfolio of 15 cashflows. Table 11.1 repors he observed percen of VaR violaions for he upper and lower ails of our sample porfolio. For each day he lower and upper VaR limis are defined as 1.65σ 1 and 1.65σ 1, respecively. Table 11.1 Realized percenages of VaR violaions True probabiliy of VaR violaions = 5% Prob (Loss < 1.65 σ 1 ) Prob (Profi > 1.65 σ 1 ) 5.74% 5.87% A more sraighforward approach o derive he preceding resuls is o apply he mainained assumpions of he RiskMerics model. Recall ha i is assumed ha he reurn disribuion of simple porfolios (i.e., hose wihou nonlinear risk) is condiionally normal. In oher words, he real- 1 The focus of his secion is on losses. However, he following mehodology can also apply o gains. RiskMerics Technical Documen Fourh Ediion

Sec. 11. Assessing he RiskMerics model 1 ized reurn (P/L) divided by he sandard deviaion forecas used o consruc he VaR esimae is assumed o be normally disribued wih mean 0 and variance 1. Char 11. presens a hisogram of sandardized porfolio reurns. We place arrow bars o signify he area where we expec o observe 5% of he observaions. Char 11. Hisogram of sandardized reurns ( r σ 1 ) Probabiliy ha ( r σ 1 ) < (>) 1.65 (1.65) = 5% Frequency 100 90 80 70 60 50 40 30 0 10 5% 5% 0-4.37-3.1 -.04-0.87 0.30 1.47.64 3.5 Sandarized reurn A priori, he RiskMerics model predics ha 5% of he sandardized reurns fall below (above) 1.65 (1.65). In addiion o his predicion, i is possible o derive he expeced value (average) of a reurn given ha reurn violaes a VaR forecas. For he lower ail, his expeced value is defined as follows: [11.5] where E[ ( r σ ) r σ 1 ( ) < 1.65] 1 φ( 1.65) = ------------------------ Φ( 1.65) =.63 φ( 1.65) = he sandard normal densiy funcion evaluaed a -1.65 Φ ( 1.65) = he sandard normal disribuion funcion evaluaed a -1.65 I follows from he symmery of he normal densiy funcion ha he expeced value for upper-ail reurns is E ( r σ ) 1 ( r σ. ) > 1.65σ =.63 1 1 Table 11. repors hese realized expeced values for our sample porfolio. Table 11. Realized ail reurn averages Condiional mean ail forecass of sandardized reurns E[ r σ 1 ( ( r σ ) < 1.65) ] 1 =.63 E[ r σ 1 ( r σ ) > 1.65] =.63 1 1.741 1.88 Par V: Backesing

Chaper 11. Performance assessmen To ge a beer undersanding of he size of he reurns ha violae he VaR forecass, Chars 11.3 and 11.4 plo he observed sandardized reurns (black circles) ha fall in he lower (< 1.65) and upper (> 1.65) ails of he sandard normal disribuion. The horizonal line in each char represens he average value prediced by he condiional normal disribuion. Char 11.3 Sandardized lower-ail reurns r σ 1 < 1.65 Sandarized reurn 0.0-0.5-1.0-1.5 -.0 -.5-3.0-3.5 -.63% -4.0-4.5 0 5 10 15 0 5 30 35 40 45 50 Char 11.4 Sandardized upper-ail reurns r σ 1 > 1.65 Sandarized reurn 4.0 3.5 3.0.63%.5.0 1.5 1.0 0.5 0 0 5 10 15 0 5 30 35 40 45 50 Boh chars show ha he reurns ha violae he VaR forecass rarely exceed he expeced value prediced by he normal disribuion. In fac, we observe abou 3 violaions ou of (approximaely) 46/47 ail reurns for he upper/lower ails. This is approximaely 6.5% of he observaions ha fall in a paricular ail. Noe ha he normal probabiliy model predicion is 8.5%. We derive his number from Prob (X <.63 X < 1.65) = Prob (X <.63) / Prob ( X < 1.65). RiskMerics Technical Documen Fourh Ediion

Sec. 11.3 Summary 3 11.3 Summary In his chaper we presened a brief process by which risk managers may assess he performance of he RiskMerics model. We applied hese saisics o a sample porfolio ha consiss of 15 cashflows covering foreign exchange, fixed income, commodiies and equiies. Specifically, 1-day VaR forecass were consruced for an 801-day sample period and for each day he forecas was measured agains he porfolio s realized P/L. I was found ha overall he RiskMerics model performs reasonably well. Par V: Backesing

4 Chaper 11. Performance assessmen RiskMerics Technical Documen Fourh Ediion

5 Appendices

6 RiskMerics Technical Documen Fourh Ediion

7 Appendix A. Tess of condiional normaliy Peer Zangari Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-8641 zangari_peer@jpmorgan.com A fundamenal assumpion in RiskMerics is ha he underlying reurns on financial prices are disribued according o he condiional normal disribuion. The main implicaion of his assumpion is ha while he reurn disribuion a each poin in ime is normally disribued, he reurn disribuion aken over he enire sample period is no necessarily normal. Alernaively expressed, he sandardized disribuion raher han he observed reurn is assumed o be normal. Char A.1 shows he nonrivial consequence of he condiional normaliy assumpion. The uncondiional disribuion represens an esimae of he hisogram of USD/DEM log price changes ha are sandardized by he sandard deviaion aken over he enire sample (i.e., hey are sandardized by he uncondiional sandard deviaion). As menioned above, relaive o he normal disribuion wih a consan mean and variance, his series has he ypical hin wais, fa ail feaures. The uncondiional disribuion represens he disribuion of sandardized reurns which are consruced by dividing each hisorical reurn by is corresponding sandard deviaion forecas 1, i.e., divide every reurn, r, by is sandard deviaion forecas, σ 1 (i.e., condiional sandard deviaion). Char A.1 Sandard normal disribuion and hisogram of reurns on USD/DEM PDF 0.6 0.5 uncondiional 0.4 condiional 0.3 0. sandard normal 0.1 0-5 -4-3 - -1 0 1 3 4 5 Reurns The difference beween hese wo lines underscores he imporance of disinguishing beween condiional and uncondiional normaliy. 1 The exac consrucion of his forecas is presened in Chaper 5. Appendices

8 Appendix A. Tess of condiional normaliy A.1 Numerical mehods We now presen some compuaional ools used o es for normaliy. We begin by showing how o obain sample esimaes of he wo parameers ha describe he normal disribuion. For a se of reurns, r, where = 1,,T, we obain esimaes of he uncondiional mean, r, and sandard deviaion, σˆ, via he following esimaors: [A.1] r = T 1 ----- r T = 1 [A.] σˆ = T 1 ----------- ( r T 1 r ) = 1 Table A.1 presens sample esimaes of he mean and sandard deviaion for he change series presened in Table 4.1. Table A.1 Sample mean and sandard deviaion esimaes for USD/DEM FX Parameer esimaes Absolue price change Relaive price change Log price change r, mean (%) 0.060 0.089 0.090 σˆ, sandard deviaion, (%) 0.8 0.4 0.4 Several popular ess for normaliy focus on measuring skewness and kurosis. Skewness characerizes he asymmery of a disribuion around is mean. Posiive skewness indicaes an asymmeric ail exending oward posiive values (righ skewed). Negaive skewness implies asymmery oward negaive values (lef skewed). A simple measure of skewness, he coefficien of skewness, γˆ, is given by [A.3] γˆ= 1 -- T T = 1 r r ----------- σˆ 3 Compued values of skewness away from 0 poin owards non-normaliy. Kurosis characerizes he relaive peakedness or flaness of a given disribuion compared o a normal disribuion. The sandardized measure of kurosis, he coefficien of kurosis, κˆ, is given by [A.4] κˆ = 1 -- T T = 1 r r 4 ----------- σˆ The kurosis for he normal disribuion is 3. Ofen, insead of kurosis, researchers alk abou excess kurosis which is defined as kurosis minus 3 so ha in a normal disribuion excess kurosis is zero. Disribuions wih an excess kurosis value greaer han 0 are frequenly referred o as having fa ails. One popular es for normaliy ha is based on skewness and kurosis is presened in Kiefer and Salmon (1983). Shapiro and Wilk (1965) and Bera and arcque (1980) offer more compuaionally inensive ess. To give some idea abou he values of he mean, sandard deviaion, skewness and kurosis coefficiens ha are observed in pracice, Table A. on page 30 presens esimaes of hese saisics as well as wo oher measures ail probabiliy and ail values, o 48 foreign RiskMerics Technical Documen Fourh Ediion

Appendix A. Tess of condiional normaliy 9 exchange series. For each of he 48 ime series we used 86 hisorical weekly prices for he period uly 1, 1994 hrough March 1, 1996. (Noe ha many of he ime series presened in Table A. are no par of he RiskMerics daa se). Each reurn used in he analysis is sandardized by is corresponding 1-week sandard deviaion forecas. Inerpreaions of each of he esimaed saisics are provided in he able foonoes. When large daa samples are available, specific saisics can be consruced o es wheher a given sample is skewed or has excess kurosis. This allows for formal hypohesis esing. The large sample skewness and kurosis measures and heir disribuions are given below: [A.5] [A.6] 1 --- ( r T r ) 3 = 1 Skewness measure Tγ T ------------------------------------------- N ( 06, ) T T 1 -- ( r T r ) = 1 1 --- ( r T r ) 4 = 1 Kurosis measure Tκ T ------------------------------------------- T 3 1 -- ( r T r ) T = 1 3 -- N ( 04, ) Appendices

30 Appendix A. Tess of condiional normaliy Table A. Tesing for univariae condiional normaliy 1 normalized reurn series; 85 oal observaions Tail Probabiliy (%) 8 Tail value 9 Skewness Kurosis 3 Mean 4 Sd. Dev. 5 < 1.65 > 1.65 < 1.65 > 1.65 Normal 0.000 0.000-1.000 5.000 5.000.067.067 OECD Ausralia 0.314 3.397 0.10 0.943.900 5.700.586.306 Ausria 0.369 0.673 0.085 1.037 8.600 5.700 1.975.499 Belgium 0.157.961 0.089 0.866 8.600.900 1.859.493 Denmark 0.650 4.399 0.077 0.903 11.400.900 1.915.576 France 0.068 3.557 0.063 0.969 8.600.900.140.85 Germany 0.096 4.453 0.085 0.87 5.700.900 1.81.703 Greece 0.098.59 0.154 0.943 11.400.900 1.971.658 Holland 0.067 4.567 0.086 0.865 5.700.900 1.834.671 Ialy 0.480 0.019 0.101 0.763 0.900 0 1.853 New Zealand 1.746 7.89 0.068 1.075.900.900.739 3.633 Porugal 1.747 0.533 0.06 0.889 11.400.900 1.909.188 Spain 6.995 1.680 0.044 0.957 8.600.900.93 1.845 Turkey 30.566 118.749 0.761 1.16 11.400 0.944 0 UK 7.035.76 0.137 0.955 8.600.900.516 1.811 Swizerland 0.009 0.001 0.001 0.995.900 5.700.415.110 Lain Amer. Econ. Sysem Brazil 0.880 1.549 0.4 0.8 0 0 0 0 Chile 1.049 0.51 0.91 0.904 8.600 0.057 0 Colombia.010 4.31 0.536 1.89 11.400.900 3.305.958 Cosa Rica 0.093 33.360 0.865 0.45 5.700 0.011 0 Dominican Rep 0.06 41.011 0.050 1.183 5.700 5.700 3.053 3.013 El Salvador.708 49.717 0.014 0.504 0.900 0 1.776 Equador 0.00 50.097 0.085 1.16 5.700 5.700 3.053 3.013 Guaemala 0.06 1.946 0.80 1.036 8.600 5.700.365.37 Honduras 4.40 77.77 0.575 1.415 14.300 0 3.59 0 amaica 81.596 451.1 0.301 1.137.900.900 6.163 1.869 Mexico 13.71 30.37 0.158 0.597.900 0.500 0 Nicaragua 0.051.847 0.508 0.117 0 0 0 0 Peru 1.807 67.453 0.78 1.365 5.700 0 5.069 0 Trinidad 0.813 0.339 0.146 1.063 8.600 11.400.171 1.915 Uruguay 0.74 0.106 0.65 0.371 0 0 0 0 RiskMerics Technical Documen Fourh Ediion

Appendix A. Tess of condiional normaliy 31 Table A. (coninued) Tesing for univariae condiional normaliy 1 normalized reurn series; 85 oal observaions ASEAN Tail Probabiliy (%) 8 Tail value 9 Skewness Kurosis 3 Mean 4 Sd. Dev. 5 < 1.65 > 1.65 < 1.65 > 1.65 Malaysia 1.495 0.65 0.318 0.96 8.600 0.366 0 Philippines 1.654 0.494 0.08 0.393 0 0 0 0 Thailand 0.077 0.069 0.69 0.936 8.600.900.184 1.955 Fiji 4.073 6.471 0.19 0.868.900.900 3.10 1.737 Hong Kong 5.360 9.084 0.03 1.001 5.700 5.700.33.76 Reunion Island 0.068 3.558 0.063 0.969 8.600.900.140.853 Souhern African Dev. Comm. Malawi 0.157 9.454 0.001 0.50 0 0 0 0 Souh Africa 34.464 58.844 0.333 1.555 8.600 0 4.480 0 Zambia.686 39.073 0.007 0.011 0 0 0 0 Zimbabwe 0.831 9.34 0.487 0.76 5.700 0.68 0 Ivory Coas 0.068 3.564 0.064 0.970 8.600.900.144.857 Uganda 40.815 80.115 0.03 1.399 8.600.900 4.09 1.953 Ohers China 80.314 567.01 0.107 1.51.900.900 3.616 8.09 Czech Repub 0.167 1.516 0.108 0.84 5.700.900.088.619 Hungary 1.961 0.006 0.34 0.741 5.700 0.135 0 India 5.633 3.6 0.46 1.336 17.100 5.700.715 1.980 Romania 89.973 45.501 1.49 1.71 14.300 0 4.078 0 Russia 0.48.819 0.10 0.369 0 0 0 0 1 Counries are grouped by major economic groupings as defined in Poliical Handbook of he World: 1995 1996. New York: CSA Publishing, Sae Universiy of New York, 1996. Counries no formally par of an economic group are lised in heir respecive geographic areas. If reurns are condiionally normal, he skewness value is zero. 3 If reurns are condiionally normal, he excess kurosis value is zero. 4 Sample mean of he reurn series. 5 Sample sandard deviaion of he normalized reurn series. 8 Tail probabiliies give he observed probabiliies of normalized reurns falling below 1.65 and above +1.65. Under condiional normaliy, hese values are 5%. 9 Tail values give he observed average value of normalized reurns falling below 1.65 and above +1.65. Under condiional normaliy, hese values are.067 and +.067, respecively. Appendices

3 Appendix A. Tess of condiional normaliy A. Graphical mehods Q-Q (quanile-quanile) chars offer a visual assessmen of he deviaions from normaliy. Recall ha he qh quanile is he number ha exceeds q percen of he observaions. A Q-Q char plos he quaniles of he sandardized disribuion of observed reurns (observed quaniles) agains he quaniles of he sandard normal disribuion (normal quaniles). Consider he sample of observed reurns, r, = 1,, T. Denoe he jh observed quanile by q j so ha for all T observed quaniles we have [A.7] Probabiliy ( r < q j ) p j where j 0.5 p j = --------------- T Denoe he jh sandard normal quanile by z j for j = 1,T. For example, if T = 100, hen z 5 = 1.645. In pracice, he five seps o compue he Q-Q plo are given below: 1. Sandardize he daily reurns by heir corresponding sandard deviaion forecas, i.e., compue r from r for = 1,,T.. Order r and compue heir perceniles q j, j = 1,,T. 3. Calculae he probabiliies p j corresponding o each q j. 4. Calculae he sandard normal quaniles, z j ha correspond o each p j. 5. Plo he pairs ( z 1, q 1 ), ( z, q ), ( z T, q T ). Char A. shows an example of a Q-Q plo for USD/DEM daily sandardized reurns for he period anuary 1988 hrough Sepember 1996. Char A. Quanile-quanile plo of USD/DEM sandardized reurns z q j For a complee descripion of his es see ohnson and Wichern (199, pp. 153-158). RiskMerics Technical Documen Fourh Ediion

Appendix A. Tess of condiional normaliy 33 The sraigher he plo, he closer he disribuion of reurns is o a normal disribuion. If all poins were o lie on a sraigh line, hen he disribuion of reurns would be normal. As he char above shows, here is some deviaion from normaliy in he disribuion of daily reurns of USD/DEM over he las 7 years. A good way o measure how much deviaion from normaliy occurs is o calculae he correlaion coefficien of he Q-Q plo, [A.8] T ( q j q) ( z j z) j ρ = 1 Q = ------------------------------------------------------------------------- T T ( q j q ) ( z j z) j = 1 j = 1 For large sample sizes as in he USD/DEM example, needs o be a leas 0.999 o pass a es of normaliy a he 5% significan. 3 In his example, ρ Q = 0.987. The reurns are no normal according o his es. ρ Q Used across asse classes, can provide useful informaion as o how good he univariae normaliy assumpion approximaes realiy. In he example above, while he reurns on he USD/DEM exchange rae are no normal, heir deviaion is sligh. Deviaions from normaliy can be much more significan among oher ime series, especially money marke raes. This is inuiively easy o undersand. Shor-erm ineres raes move in a discreionary fashion as a resul of acions by cenral banks. Counries wih exchange rae policies ha have deviaed significanly from economic fundamenals for some period ofen show money marke rae disribuions ha are clearly no normal. As a resul hey eiher change very lile when moneary policy remains unchanged (mos of he ime), or more significanly when cenral banks change policy, or he markes force hem o do so. Therefore, he shape of he disribuion resuls from discree jumps in he underlying reurns. A ypical example of his phenomenon can be seen from he Q-Q char of sandardized price reurns on he 3-monh serling over he period 3-an-91 o 1-Sep-94. The ρ Q calculaed for ha paricular series is 0.907. ρ Q 3 See ohnson and Wichern (199, p 158) for a able of criical values required o perform his es. Appendices

34 Appendix A. Tess of condiional normaliy Char A.3 Quanile-quanile plo of 3-monh serling sandardized reurns z q j The Q-Q chars are useful because hey allow he researcher a visual depicion of deparures from normaliy. However, as saed before, here are several oher ess for normaliy. I is imporan o remember ha when applied direcly o financial reurns, convenional ess of normaliy should be used wih cauion. A reason is ha he assumpions ha underlie hese ess (e.g., consan variance, nonauocorrelaed reurns) are ofen violaed. For example, if a es for normaliy assumes ha he daa is no auocorrelaed over he sample period when, in fac, he daa are auocorrelaed, hen he es may incorrecly lead one o rejec normaliy (Heus and Rens, 1986). The ess presened above are ess for univariae normaliy and no mulivariae normaliy. In finance, ess of mulivariae normaliy are ofen mos relevan since he focus is on he reurn disribuion of a porfolio ha consiss of a number of underlying securiies. If each reurn series in a porfolio is found o be univariae normal, hen he se of reurns aken as a whole are sill no necessarily mulivariae normal. Conversely, if any one reurn series is found no o be univariae normal hen mulivariae normaliy can be ruled ou. Recenly, Richardson and Smih (1993) propose a direc es for mulivariae normaliy in sock reurns. Also, Looney (1995) describes es for univariae normaliy ha can be used o deermine o wheher a daa sample is mulivariae normaliy. RiskMerics Technical Documen Fourh Ediion

35 Appendix B. Relaxing he assumpion of condiional normaliy Peer Zangari Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-8641 zangari_peer@jpmorgan.com Since is release in Ocober 1994, RiskMerics has inspired an imporan discussion on VaR mehodologies. A focal poin of his discussion has been he assumpion ha reurns follow a condiional normal disribuion. Since he disribuions of many observed financial reurn series have ails ha are faer han hose implied by condiional normaliy, risk managers may underesimae he risk of heir posiions if hey assume reurns follow a condiional normal disribuion. In oher words, large financial reurns are observed o occur more frequenly han prediced by he condiional normal disribuion. Therefore, i is imporan o be able o modify he curren RiskMerics model o accoun for he possibiliy of such large reurns. The purpose of his appendix is o describe wo probabiliy disribuions ha allow for a more realisic model of financial reurn ail disribuions. I is organized as follows: Secion B.1 reviews he fundamenal assumpions behind he curren RiskMerics calculaions, in paricular, he assumpion ha reurns follow a condiional normal disribuion. Secion B. presens he RiskMerics model of reurns under he assumpion ha he reurns are condiionally normally disribued and wo alernaive models (disribuions) where he probabiliy of observing a reurn far away from he mean is relaively larger han he probabiliy implied by he condiional normal disribuion. Secion B.3 explains how we esimae each of he hree models and hen presens resuls on forecasing he 1s and 99h perceniles of 15 reurn series represening 9 emerging markes. B.1 A review of he implicaions of he condiional normaliy assumpion In a normal marke environmen RiskMerics VaR forecass are given by he bands of a confidence inerval ha is symmeric around zero. These bands represen he maximum change in he value of a porfolio wih a specified level of probabiliy. For example, he VaR bands associaed wih a 90% confidence inerval are given by { 1.65σ p, 1.65σ p} where /+1.65 are he 5h/95h perceniles of he sandardized normal disribuion, and σ p is he porfolio sandard deviaion which may depend on correlaions beween reurns on individual insrumens. The scale facors /+ 1.65 resul from he assumpion ha sandardized reurns (i.e., a mean cenered reurn divided by is sandard deviaion) are normally disribued. When his assumpion is rue we expec 5% of he (sandardized) realized reurns o lie below 1.65 and 5% o lie above +1.65. Ofen, wheher complying wih regulaory requiremens or inernal policy, risk managers compue VaR a differen probabiliy levels such as 95% and 98%. Under he assumpion ha reurns are condiionally normal, he scale facors associaed wih hese confidence inervals are /+1.96 and / +.33, respecively. I is our experience ha while RiskMerics VaR esimaes provide reasonable resuls for he 90% confidence inerval, he mehodology does no do as well a he 95% and 98% confidence levels. 1 Therefore, our goal is o exend he RiskMerics model o provide beer VaR esimaes a hese larger confidence levels. Before we can build on he curren RiskMerics mehodology, i is imporan o undersand exacly wha RiskMerics assumes abou he disribuion of financial reurns. RiskMerics assumes ha reurns follow a condiional normal disribuion. This means ha while reurns hemselves are no normal, reurns divided by heir respecive forecased sandard deviaions are normally disribued wih mean 0 and variance 1. For example, le r, denoe he ime reurn, i.e., he reurn on an asse over a one-day period. Furher, le denoe he forecas of he sandard deviaion of reurns for σ 1 See Darryl Hendricks, Evaluaion of Value-a-Risk Models Using Hisorical Daa, FRBNY Economic Policy Review, April, 1996. Appendices

36 Appendix B. Relaxing he assumpion of condiional normaliy ime based on hisorical daa. I hen follows from our assumpions ha while normal, he sandardized reurn,, is normally disribued. r σ is no necessarily To summarize, RiskMerics assumes ha financial reurns divided by heir respecive volailiy forecass are normally disribued wih mean 0 and variance 1. This assumpion is crucial because i recognizes ha volailiy changes over ime. r B. Three models o produce daily VaR forecass In his secion we presen hree models o forecas he disribuion of one-day reurns from which a VaR esimae will be derived. The firs model ha is discussed is referred o as sandard RiskMerics. This model is he basis for VaR calculaions ha are presened in he curren RiskMerics Technical Documen. The second model ha we analyze was inroduced in he nd quarer 1996 RiskMerics Monior. I is referred o in his appendix as he normal mixure model. The name normal mixure refers o he idea ha reurns are assumed o be generaed from a mixure of wo differen normal disribuions. Each day s reurn is assumed o be a draw from one of he wo normal disribuions wih a paricular probabiliy. The hird, and mos sophisicaed model ha we presen is known as RiskMerics-GED. This model is he same as sandard RiskMerics excep he reurns in his model are assumed o follow a condiional generalized error disribuion (GED). The GED is a very flexible disribuion in ha i can ake on various shapes, including he normal disribuion. B..1 Sandard RiskMerics The sandard RiskMerics model assumes ha reurns are generaed as follows [B.1] r σ = = σ ε λσ 1 + ( 1 λ)r 1 where ε is a normally disribued random variable wih mean 0 and variance 1 σ and σ, respecively, are he ime sandard deviaion and variance of reurns ( ) r λ is a parameer (decay facor) ha regulaes he weighing on pas variances. For oneday variance forecass, RiskMerics ses λ =0.94. In summary, he sandard RiskMerics model assumes ha reurns follow a condiional normal disribuion condiional on he sandard deviaion where he variance of reurns is a funcion of he previous day s variance forecas and squared reurn. RiskMerics Technical Documen Fourh Ediion

Appendix B. Relaxing he assumpion of condiional normaliy 37 B.. Normal mixure In he second quarer 1996 RiskMerics Monior we inroduced he normal mixure model of reurns ha was found o more effecively measure he ails of seleced reurn disribuions. In essence, his model allows for a larger probabiliy of observing very large reurns (posiive or negaive) han he condiional normal disribuion. The normal mixure model assumes ha reurns are generaed as follows [B.] r = σ 1, ε 1, + σ 1, δ ε, where r is he ime coninuously compounded reurn ε 1, is a normally disribued random variable wih mean 0 and variance 1 ε, is a normally disribued random variable wih mean µ, and variance σ, δ is a 0/1 variable ha akes he value 1 wih probabiliy p and 0 wih probabiliy 1 p σ 1, is he sandard deviaion given in he RiskMerics model Alernaively saed, he normal mixure model assumes ha daily reurns sandardized by he RiskMerics volailiy forecass, r, are generaed according o he model [B.3] r = ε 1, + δ ε, Inuiively, we can hink of Eq. [B.3] as represening a model where each day s sandardized reurn is generaed from one of wo disribuions: 1. If δ = 0 hen he sandardized reurn is generaed from a sandard normal disribuion, ha is, a normal disribuion wih mean 0 and variance 1.. If δ = 1 hen he reurn is generaed from a normal disribuion wih mean µ, and variance 1 + σ,. We can hink of δ as a variable ha signifies wheher a reurn ha is inconsisen wih he sandard normal disribuion has occurred. The parameer p is he probabiliy of observing such a reurn. I is imporan o remember ha alhough he assumed mixure disribuion is composed of normal disribuions, he mixure disribuion iself is no normal. Also, noe ha when consrucing a VaR forecas, he normal mixure model applies he sandard RiskMerics volailiy. Char B.1 shows he ails of wo normal mixure models (and he sandard normal disribuion) for differen values of µ,, and σ,. Mixure(1) is he normal mixure model wih parameer values se a µ, =-4, σ, =1, p=%, µ 1, =0 σ 1 =1. Mixure() is he normal mixure model wih he same parameer values as mixure(1) excep now =0, =10., µ, σ, Appendices

38 Appendix B. Relaxing he assumpion of condiional normaliy Char B.1 Tails of normal mixure densiies Mixure(1) µ, =-4, σ, =1, p=%, µ 1, =0 σ 1, =1; Mixure() =0, =10, p=%, =0 =1 µ, σ, µ 1, σ 1, PDF 0.10 0.08 0.06 0.04 0.0 Mixure (1) µ = -4, σ = 1 Mixure () µ = 0, σ = 10 Normal 0-5 -4-3 3 4 5 Sandard deviaion Char B.1 shows ha when here is a large negaive mean for one of he normal disribuions as in mixure(1), his ranslaes ino a larger probabiliy of observing a large negaive reurn relaive o he sandard normal disribuion. Also, as in he case of mixure () we can consruc a probabiliy disribuion wih hicker ails han he sandard normal disribuion by mixing he sandard normal wih a normal disribuion wih a large sandard deviaion. B..3 RiskMerics-GED According o his model, reurns are generaed as follows [B.4] r σ = = σ ξ λσ 1 + ( 1 λ)r 1 where r is he ime coninuously compounded reurn ξ is a random variable disribued according o he GED (generalized error disribuion) wih parameer ν. As will be shown below, ν regulaes he shape of he GED disribuion. σ is he ime variance of reurns ( ) r ξ The random variable ( ) in Eq. [B.4] is assumed o follow a generalized error disribuion (GED). This disribuion is quie popular among researchers in finance because of he variey of shapes he GED can ake. The probabiliy densiy funcion for he GED is RiskMerics Technical Documen Fourh Ediion

Appendix B. Relaxing he assumpion of condiional normaliy 39 [B.5] f ( ξ ) = 1 ν νexp -- ξ λ ------------------------------------------- λ 1 ν ( + 1 ) 1 Γ ν where Γ is he gamma funcion and [B.6] λ = ( ν ) Γ( 1 ν) ( 3 ν) 1 When ν = his produces a normal densiy while ν>(<) is more hin (fla) ailed han a normal. Char B. shows he shape of he GED disribuion for values of ν = 1, 1.5 and. Char B. GED disribuion ν=1, 1.5 and PDF 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 GED (1) GED (1.5) Normal (GED ()) 0-5 -4-3 -1-0 1 4 5 Sandarized reurns Noice ha when he parameer of he GED disribuion is below (normal), he resul is a disribuion wih greaer likelihood of very small reurns (around 0) and a relaively large probabiliy of reurns far away from he mean. To beer undersand he effec ha he parameer ν has on he ails of he GED disribuion, Char B.3 plos he lef (lower) ail of he GED disribuion when ν =1, 1.5 and. Appendices

40 Appendix B. Relaxing he assumpion of condiional normaliy Char B.3 Lef ail of GED (ν) disribuion ν = 1, 1.5, and PDF 0.5% 0.0% GED (1) 0.15% 0.10% GED (1.5) 0.05% Normal (GED ()) 0.00% -5-4.9-4.8-4.7-4.6-4.5-4.4-4.3-4. -4.1-4 Sandarized reurns Char B.3 shows ha as ν becomes smaller, away from (normal), here is more probabiliy placed on relaively large negaive reurns. B.3 Applying he models o emerging marke currencies and equiy indices We applied he hree models described above o 15 ime series represening 9 emerging marke counries o deermine how well each model performs a esimaing he 1s and 99h perceniles of he reurn disribuions. The ime series cover foreign exchange and equiy indices. In order o faciliae our exposiion of he process by which we fi each of he models and abulae he resuls on forecasing he perceniles, we focus on one specific ime series, he Souh African rand. B.3.1 Model esimaion and assessmen We firs fi each model o 115 reurns on each of he 15 ime series for he period May 5, 199 hrough Ocober 3, 1996. Table B.1 shows he parameer esimaes from each of he hree models for he Souh African rand. Table B.1 Parameer esimaes for he Souh African rand Normal Mixure Sandard RiskMerics RiskMerics-GED Parameer Esimae Parameer Esimae Parameer Esimae µ, 5.086 λ 0.94 ν 0.97 σ, 9.087 p 0.010 σ 1, 1.88 Table B.1 poins o some ineresing resuls: In he RiskMerics-GED model, he esimae of ν implies ha he disribuion of reurns on he rand are much hicker han he normal disribuion (recall ha ν= is a normal dis- RiskMerics Technical Documen Fourh Ediion

Appendix B. Relaxing he assumpion of condiional normaliy 41 ribuion). In oher words, we are much more likely o observe a reurn ha is far away from he mean reurn han is implied by he normal disribuion. In he normal mixure model here is a 1% chance of observing a normally disribued reurn wih a mean 5 and sandard deviaion 9 and a 99% chance of observing a normally disribued reurn wih mean 0 and sandard deviaion 1.88. The RiskMerics opimal decay facor for he Souh African rand is 0.940. This decay facor was found by minimizing he roo mean squared error of volailiy forecass. Coincidenally, his happens o be he same decay facor applied o all imes series in RiskMerics when esimaing one-day volailiy. If a volailiy model such as RiskMerics fis he daa well is sandardized reurns (i.e., he reurns divided by heir volailiy forecas) should have a volailiy of 1. Table B. presens four sample saisics mean, sandard deviaion, skewness and kurosis for he sandard RiskMerics model and esimaes of ν for he RiskMerics-GED model. Recall ha skewness is a measure of a disribuion s symmery. A value of 0 implies ha he disribuion is symmeric. Kurosis measures a disribuion s ail hickness. For example, since he kurosis for a normal disribuion is 3, values of kurosis greaer han 3 indicae ha here is a greaer likelihood of observing reurns ha are far away from he mean reurn han implied by he normal disribuion. Table B. Sample saisics on sandardized reurns Sandard RiskMerics model Insrumen ype Source Mean Sd dev Skewness Kurosis GED parameer, ν Foreign exchange Mexico 0.033 3.50 1.744 553.035 0.749 Philippines 0.061 1.75 13.865 37.377 0.368 Taiwan 0.069 1.70 8.00 16.34 0.49 Argenina 0.08 1.177 5.67 11.30 0.19 Indonesia 0.013 1.081 1.410 1.314 0.460 Korea 0.013 1.106 1.14 10.188 0.778 Malaysia 0.09 1.10 0.589 1.488 0.908 Souh Africa 0.040 1.91 6.514 116.45 0.97 Thailand 0.004 1.003 0.168 4.865 1.101 Equiy Argenina 0.043 1.007 0.376 3.817 1.1 Indonesia 0.00 1.085 1.069 1.436 0.868 Malaysia 0.00 1.130 0.346 5.966 1.03 Mexico 0.007 1.046 0.04 4.389 0.798 Souh Africa 0.07 1.03 0.081 5.41 1.136 Thailand 0.019 1.056 0.008 5.014 0.999 Under he mainained assumpion of he RiskMerics model he saisics of he sandardized reurns should be as follows; mean = 0, sandard deviaion = 1, skewness = 0, kurosis = 3. Table B. shows ha excep for Mexico, Philippines and Taiwan foreign exchange, sandard RiskMerics does a good job a recovering he sandard deviaion. The fac ha kurosis for many of he ime series are well above hree signifies ha he ails of hese reurn disribuions are much larger han he normal disribuion. Also, noe he esimaes of ν produced from RiskMerics-GED. Remember ha if he disribuion of he sandardized reurns is normal, ν= and values of ν< signify ha he disribuion has hicker ails han ha implied by he normal disribuion. The fac ha all of he esimaes of ν are well below indicae ha hese series conain a relaively large number of reurns (negaive and posiive). Appendices

4 Appendix B. Relaxing he assumpion of condiional normaliy B.3. VaR analysis In his secion we repor he resuls of an experimen o deermine how well each of he models described above can predic he 1s and 99h perceniles of he 15 reurn disribuion. These resuls are provided in Table B.3. Our analysis consised of he following seps: Firs, we esimae he parameers in each of he hree models using price daa from May, 5, 199 hrough Ocober 3, 1996. This sample consiss of 115 hisorical reurns on each of he 15 ime series. Second, we consruc one-day volailiy esimaes for each of he hree models using he mos recen 95 reurns. Third, we use he 95 volailiy esimaes and he hree probabiliy disribuions (normal, mixure normal and GED) evaluaed a he parameer esimaes o consruc VaR forecass a he 1s and 99h perceniles. Fourh, we coun he number of imes he nex day realized reurn exceeds each of he VaR forecass. This number is hen convered o a percenage by dividing i by he oal number of rials 95 in his experimen. The ideal model would yield percenages of 1%. Table B.3 presens hese percenages for he hree models. Table B.3 VaR saisics (in %) for he 1s and 99h perceniles RGD = RiskMerics-GED; RM = RiskMerics; MX = Normal mixure 1s percenile (1%) 99h percenile (99%) Insrumen ype Source RGD RM MX RGD RM MX Foreign exchange Mexico 1.477.346 0.434 1.043 1.998 0.434 Philippines 1.390.50 1.043 1.16 1.998 1.043 Taiwan 0.956 1.651 0.78 1.043 1.911 0.78 Argenina 1.998 1.998 1.303.17.17 1.39 Indonesia 1.651 3.56 1.651 1.19 1.998 1.411 Korea 1.16.433 1.303 0.51 1.303 0.956 Malaysia 1.564.433 1.477 1.911 3.15 1.911 Souh Africa 1.390 1.998 1.16 1.19 1.998 1.303 Thailand 0.695 1.19 1.043 1.651.17 1.07 Equiy Argenina 1.85.50 1.646 0.869 1.19 1.19 Indonesia 0.608 1.998 1.564 1.564.50 1.564 Malaysia 1.651.433 1.698 1.651.693 1.738 Mexico 0.956.59 1.738 1.043 1.651 1.303 Souh Africa 1.16.17 1.651 1.477.085 1.738 Thailand 1.19 1.564 1.190 0.869.346 1.611 Column average 1.315.01 1.396 1.86.060 1.70 Table B.3 shows ha for he RiskMerics-GED model he VaR forecass a he 1s percenile are exceeded 1.315 percen of he ime whereas he VaR forecass a he 99h percenile are exceeded 1.86% of he ime. Similarly, he VaR forecass produced from he mixure model are exceeded a he 1s and 99h perceniles by 1.396% and 1.70% of he realized reurns, respecively. Boh models are marked improvemens over he sandard RiskMerics model ha assumes condiional normaliy. RiskMerics Technical Documen Fourh Ediion

43 Appendix C. Mehods for deermining he opimal decay facor Peer Zangari Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-8641 zangari_peer@jpmorgan.com Chrisopher C. Finger Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-4657 finger_chrisopher@jpmorgan.com In his appendix we presen alernaive measures o assess forecas accuracy of volailiy and correlaion forecass. C.1 Normal likelihood (LKHD) crierion Under he assumpion ha reurns are condiionally normal, he objecive here is o specify he join probabiliy densiy of reurns given a value of he decay facor. For he reurn on day his can be wrien as: [C.1] f ( r λ) 1 ----------------------------------- 1 = exp -- --------------------------- ( ) ( ) πσ 1 λ r σ 1 λ Combining he condiional disribuions from all he days in hisory for which we have daa, we ge: [C.] f ( r 1,, r T λ) = T = 1 1 ----------------------------------- exp ( ) πσ 1 λ 1 -- --------------------------- ( ) r σ 1 λ Equaion [C.] is known as he normal likelihood funcion. Is value depends on λ. In pracice, i is ofen easier o work wih he log-likelihood funcion which is simply he naural logarihm of he likelihood funcion. The maximum likelihood (ML) principle sipulaes ha he opimal value of he decay facor λ is one which maximizes he likelihood funcion Eq. [C.]. Wih some algebra, i can be shown ha his is equivalen o finding he value of λ ha minimizes he following funcion: [C.3] T 1 LKLHD v = ln [ σ 1 ( λ) ] + -- --------------------------- ( ) = 1 r σ 1 λ Noice ha he crierion Eq. [C.3] imposes he assumpion ha reurns are disribued condiionally normal when deermining he opimal value of λ. The RMSE crierion, on he oher hand, does no impose any probabiliy assumpions in he deerminaion of he opimal value of λ. C. Oher measures In addiion o he RMSE and Normal likelihood measures alernaive measures could also be applied such as he mean absolue error measure for he variance [C.4] 1 MAE v = ---- r T + 1 σˆ T = 1 + 1 For individual cashflows, RiskMerics VaR forecass are based on sandard deviaions. Therefore, we may wish o measure he error in he sandard deviaion forecas raher han he variance forecas. If we ake as a proxy for he one period ahead sandard deviaion, r, hen we can define he RMSE of he sandard deviaion forecas as Appendices

44 Appendix C. Mehods for deermining he opimal decay facor [C.5] T 1 RMSE σ = ---- ( r T + 1 σˆ + 1 ) = 1 Noice in Eq. [C.5] ha E r + 1 σ + 1. In fac for he normal disribuion, he following equaion holds: E r + 1 = ( π) 1 σ + 1. Oher ways of choosing opimal λ include he Q-saisic described by Crnkovic and Drachman (RISK, Sepember, 1996) and, under he assumpion ha reurns are normally disribued, a likelihood raio es ha is based on he normal probabiliy densiy likelihood funcion. C.3 Measures for choosing an opimal decay facor for muliple ime series. In Chaper 5, we explained how an opimal decay facor for he 480 RiskMerics ime series was chosen. This mehod involved finding opimal decay facors for each series, and hen aking a weighed average of hese facors, wih hose facors which provided superior performance in forecasing volailiy receiving he greaes weigh. In his secion, we briefly describe some alernaive mehods which accoun for he performance of he correlaion forecass as well. The firs such mehod is an exension of he likelihood crierion o a mulivariae seing. If we consider a collecion of n asses whose reurns on day are represened by he vecor r, hen he join probabiliy densiy for hese reurns is 1 1 [C.6] f ( r λ) ----------------------------------------------, n 1 -- r T Σ -- -- 1 ( λ) 1 = exp r ( π) Σ 1 ( λ) where Σ 1 ( λ) is he marix represening he forecased covariance of reurns on day using decay facor λ. The likelihood for he reurns for all of he days in our daa se may be consruced analogously o Eq. [C.]. Using he same reasoning as above, i can be shown ha he value of λ which maximizes his likelihood is he one which also maximizes [C.7] LKLHD v = { ln [ Σ 1 ( λ) ] + r T Σ. ( λ) 1 r 1 } = 1 As noed before, choosing he decay facor according o his crierion imposes he assumpion of condiional normaliy. In addiion, o evaluae he likelihood funcion in Eq. [C.7], i is necessary a each ime o inver he esimaed covariance marix Σ 1 ( λ). In heory, his marix will always be inverible, alhough in pracice, due o limied precision calculaions, here will likely be cases where he inversion is impossible, and he likelihood funcion canno be compued. A second approach is a generalizaion of he RMSE crierion for he covariance forecass. Recall from Chaper 5 ha he covariance forecas error on day for he ih and jh reurns is [C.8] ε ij, 1 ( λ) = r i, r j, Σ ij, 1 ( λ). T (Recall also ha under he RiskMerics assumpions, E [ 1 ε ] ij, 1 = 0.) The oal squared error for day is hen obained by summing he above over all pairs ( i, j), and he mean oal squared error (MTSE) for he enire daa se is hen RiskMerics Technical Documen Fourh Ediion

Appendix C. Mehods for deermining he opimal decay facor 45 1 [C.9] MTSE = -- ε. T ij, 1 ( λ) T = 1 i, j The value of λ which minimizes he MTSE above can be hough of as he decay facor which hisorically has given he bes covariance forecass across all of he daa series. The above descripion presens a myriad of choices faced by he researcher when deermining opimal λ. The simple answer is ha here is no clear-cu, simple way of choosing he opimal predicion crierion. There has been an exensive discussion among academics and praciioners on wha error measure o use when assessing pos-sample predicion. 1 Ulimaely, he forecasing crierion should be moivaed by he modeler s objecive. For example, Wes, Edison and Cho (1993) noe an appropriae measure of performance depends on he use o which one pus he esimaes of volailiy. Recenly, Diebold and Mariano (1995) remind us, of grea imporance and almos always ignored, is he fac ha he economic loss associaed wih a forecas may be poorly assessed by he usual saisical measures. Tha is, forecass are used o guide decisions, and he loss associaed wih a forecas error of a paricular sign and size induced direcly by he naure of he decision problem a hand. In fac, Leich and Tanner (1991) use profiabiliy raher han size of he forecas error or is squared value as a es of forecas accuracy. 1 For a comprehensive discussion on various saisical error measures (including he RMSE) o assess forecasing mehods, see he following: Ahlburg, D. Armsrong,. S., and Collopy, F. Fildes, R. in he Inernaional ournal of Forecasing, 8, 199, pp. 69 111. Appendices

46 Appendix C. Mehods for deermining he opimal decay facor RiskMerics Technical Documen Fourh Ediion

47 Appendix D. Assessing he accuracy of he dela-gamma approach Peer Zangari Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-8641 zangari_peer@jpmorgan.comf In his appendix we compare he VaR forecass of he dela-gamma approach o hose produced by full simulaion. Before doing so, however, we invesigae briefly when he dela-gamma approach is expeced o perform poorly in relaion o full simulaion. The accuracy of he dela-gamma approach depends on he accuracy of he approximaion used o derive he reurn on he opion. The expression for he opion s reurn is derived using wha is known as a Taylor series expansion. We now presen he derivaion. [D.1] + V + n V δ P + n P ( ) + 0.5 Γ ( P + P ) θ τ n + ( + τ n ) This expression can be rewrien as follows: [D.] V + n V δ P + n P ( ) + 0.5 Γ ( P + P ) θ τ n + ( + τ n ) We now express he changes in he value of he opion and he underlying in relaive erms: [D.3] + V V V n V ----------------------- P n = δ P P + P ---------------------- + n P + 0.5 Γ P ---------------------- P P + θ ( τ + n τ ) Dividing Eq. [D.3] by P, we ge [D.4] V ---- V n ----------------------- P + V V P + n P P = δ ---------------------- + n P + 0.5 Γ P ---------------------- P P θ + ---- ( τ + n τ ) P and define he following erms: + V V + P P V n P n R V = -----------------------, R, n =, and P = ---------------------- ( τ + n τ ) η We can now wrie he reurn on he opion as follows: = P ---- V [D.5] R V = ηδr P 0.5 ( αγp ) ( R ) θ + + P ---- V n = δ R P + 0.5Γ ( R P ) + θ ( τ + n τ ) This expansion is a reasonable approximaion when he greeks δ and Γare sable as he underlying price changes. In our example, he underlying price is he US dollar/deuschemark exchange rae. If changes in he underlying price causes large changes in hese parameers hen we should no expec he dela-gamma approach o perform well. Char D.1 shows he changes in he value of dela ( δ) when he underlying price and he ime o he opion s expiry boh change. This example assumes ha he opion has a srike price of 5. Appendices

48 Appendix D. Assessing he accuracy of he dela-gamma approach Char D.1 Dela vs. ime o expiraion and underlying price Dela (δ) 1.00 0.75 0.50 0.5 0.96 0.86 0.76 0.66 0.56 0.46 0.36 Time o expiraion 0.6 0.16 0.06.8 4.8 6.8 8.8 Underlying price (srike = 5) 0 10.8 Noice ha large changes in dela occur when he curren price in he underlying insrumen is near he srike. In oher words, we should expec o see large changes in dela for small changes in he underlying price when he opion is exacly, or close o being, an a-he-money opion. Since he dela and gamma componens of an opion are closely relaed, we should expec a similar relaionship beween he curren underlying price and he gamma of he opion. For he same opion, Char D. presens values of gamma as he underlying price and he ime o expiry boh change. The char shows ha gamma changes abruply when he opion is near o being an a-he-money opion and he ime o expiry is close o zero. RiskMerics Technical Documen Fourh Ediion

Appendix D. Assessing he accuracy of he dela-gamma approach 49 Char D. Gamma vs. ime o expiraion and underlying price Gamma (Γ) 3.5 3.0.5.0 1.5 1.0 0.5 0.96 0.86 0.76 0.66 0.56 0.46 Time o expiraion 0.36 0.6 0.16 0.06.0 4.0 6.0 8.0 Underlying price (srike = 5) 0 10.0 Togeher, Chars D.1 and D. demonsrae ha we should expec he dela-gamma mehod o do mos poorly when porfolios conain opions ha are close o being a-he-money and he ime o expiry is shor (abou one week or less). D.1 Comparing full simulaion and dela-gamma: A Mone Carlo sudy In his secion we describe an experimen underaken o deermine he difference in VaR forecass produced by he full simulaion and dela-gamma mehodologies. The sudy focuses on one call opion. (For more complee resuls, see he hird quarer 1996 RiskMerics Monior.) VaR forecass, defined as he 5h percenile of he disribuion of fuure changes in he value of he opion, were made over horizons of one day. The Black-Scholes formula was used o boh revalue he opion and o derive he greeks. We se he parameers used o value he opion, deermine he greeks, and generae fuure prices (for full simulaion) as shown in Table D.1. Table D.1 Parameers used in opion valuaion Parameer Value Srike price (K) 5.0 Sandard deviaion (annualized) 3.0% Risk-free ineres rae 8.0% Given hese parameer seings we generae a series of underlying spo prices,, wih values 4.5, 4.6, 4.7,..., 5.6. Here he ime subscrip denoes he ime he VaR forecas is made. These spo prices imply a se of raios of spo-o-srike price, P K, ha define he moneyness of he opion. The values of P K are 0.90, 0.9,0.94,...,1.1. In addiion, we generae a se of ime o expiraions, τ, (expressed in years) for he opion. Values of τ range from 1 day (0.004) o 1 year (1.0). P Appendices

50 Appendix D. Assessing he accuracy of he dela-gamma approach In full simulaion, we are required o simulae fuure prices of he underlying insrumen. Denoe he fuure price of he underlying insrumen by P + n where n denoes he VaR forecas horizon (i.e., n = 1 day, 1 week, 1 monh and 3 monhs). We simulae underlying prices a ime +n, P + n, according o he densiy for a lognormal random variable [D.6] ( ( r σ ) + zσ n) P + n = P e where z is a sandard uni normal random variable. In full simulaion, VaR is defined as he difference beween he value of he opion a ime + n (he forecas horizon) and oday, ime. This means ha all insrumens are revalued. [D.7] Exac = BS ( P + n, + n) BS ( P, ), where BS() sands for he Black-Scholes formula. We use he erm Exac o represen he fac ha he opion is being revalued using is exac opion pricing formula. In he dela-gamma approach, VaR is approximaed in erms of he Taylor series expansion discussed earlier: [D.8] Approx = δ ( P + n P ) + 0.5 Γ ( P + P n ) + θ n Here, he erm Approx denoes he approximaion involved in using only he dela, gamma and hea componens of he opion. To compare VaR forecass we define he saisics VaR E and VaR A as follows: VaR E VaR A = he 5h percenile of he Exac disribuion which represens full simulaion. = he 5h percenile of he Approx disribuion which represens dela-gamma. P For a given spo price,, ime o expiraion, τ, and VaR forecas horizon, n, we generae 5,000 fuure prices, P + n, and calculae VaR E and VaR A. This experimen is hen repeaed 50 imes o produce 50 VaR E s and VaR A s. We hen measure he difference in hese VaR forecass by compuing wo merics: [D.9] MAPE = 1 ----- 50 50 i = 1 i i VaR A VaR E ---------------------------------------- (Mean Absolue Percenage Error) i VaR E [D.10] ME = 50 1 50 ----- VaR i A i = 1 i VaR E (Mean Error) Tables D. and D.3 repor he resuls of his experimen. Specifically, Tables D. and D.3 show, respecively, he mean absolue percenage error (MAPE) and he mean error (ME) for a call opion for a one-day forecas horizon. Each row of a able corresponds o a differen ime o expiraion (mauriy). Time o expiraion is measured as a fracion of a year (e.g., 1 day = 1/50 or 0.004) and canno be less han he VaR forecas horizon which is one day. Each column represens a raio of he price of he underlying when he VaR forecas was made (spo) o he opion s srike price, P K. This raio represens he opion s moneyness a he ime VaR was compued. All enries greaer han or equal o 10 percen are repored wihou decimal places. RiskMerics Technical Documen Fourh Ediion

Appendix D. Assessing he accuracy of he dela-gamma approach 51 Table D. MAPE (%) for call, 1-day forecas horizon Time o mauriy, (years) Spo/Srike 0.90 0.9 0.94 0.96 0.98 1.00 1.0 1.04 1.06 1.08 1.10 1.1 0.004 3838 455 1350 633 187 8 1 1.7463 0.004 0.004 0.004 0.0036 0.054 11 7.1 3.960 1.561 0.04 0.76 0.946 0.733 0.37 0.08 0.05 0.069 0.104 3.6.061 1.180 0.486 0.01 0.71 0.395 0.399 0.330 0.9 0.133 0.061 0.154 1.59 1.039 0.615 0.7 0.08 0.130 0.16 0.45 0.3 0.195 0.148 0.10 0.04 0.983 0.655 0.400 0.190 0.036 0.069 0.133 0.163 0.168 0.155 0.131 0.104 0.54 0.685 0.465 0.93 0.148 0.040 0.037 0.087 0.115 0.15 0.1 0.111 0.095 0.304 0.515 0.355 0.30 0.13 0.041 0.018 0.059 0.084 0.096 0.098 0.093 0.084 0.354 0.407 0.85 0.189 0.106 0.04 0.007 0.041 0.063 0.075 0.079 0.078 0.073 0.404 0.333 0.37 0.160 0.094 0.041 0.003 0.08 0.047 0.059 0.065 0.066 0.063 0.454 0.8 0.0 0.139 0.084 0.041 0.007 0.019 0.036 0.048 0.054 0.056 0.055 0.504 0.41 0.175 0.13 0.077 0.040 0.010 0.01 0.08 0.038 0.045 0.048 0.048 0.554 0.11 0.155 0.111 0.071 0.039 0.013 0.007 0.01 0.031 0.038 0.041 0.04 0.604 0.187 0.139 0.101 0.066 0.038 0.015 0.003 0.016 0.05 0.03 0.035 0.037 0.654 0.167 0.15 0.09 0.06 0.037 0.016 0.001 0.01 0.01 0.07 0.031 0.033 0.704 0.151 0.114 0.085 0.058 0.035 0.017 0.003 0.008 0.017 0.03 0.06 0.09 0.754 0.138 0.105 0.079 0.055 0.034 0.018 0.005 0.006 0.013 0.019 0.03 0.05 0.804 0.16 0.097 0.074 0.05 0.033 0.018 0.006 0.003 0.011 0.016 0.00 0.0 0.854 0.117 0.09 0.069 0.049 0.033 0.019 0.007 0.00 0.008 0.014 0.017 0.00 0.904 0.108 0.084 0.065 0.047 0.03 0.019 0.008 0.001 0.006 0.011 0.015 0.018 Table D.3 ME (%) for call, 1-day forecas horizons Time o mauriy, (years) Spo/Srike 0.90 0.9 0.94 0.96 0.98 1.00 1.0 1.04 1.06 1.08 1.10 1.1 0.004 0.000 0.000 0.000 0.003 0.186 0.569 0.180 0.010 0.000 0.000 0.000 0.000 0.054 0.004 0.006 0.008 0.005 0.000 0.005 0.007 0.005 0.003 0.001 0.000 0.000 0.104 0.003 0.004 0.003 0.00 0.000 0.001 0.00 0.003 0.00 0.00 0.001 0.000 0.154 0.00 0.00 0.00 0.001 0.000 0.001 0.001 0.00 0.00 0.001 0.001 0.001 0.04 0.00 0.00 0.001 0.001 0.000 0.000 0.001 0.001 0.001 0.001 0.001 0.001 0.54 0.001 0.001 0.001 0.001 0.000 0.000 0.000 0.001 0.001 0.001 0.001 0.001 0.304 0.001 0.001 0.001 0.001 0.000 0.000 0.000 0.000 0.001 0.001 0.001 0.001 0.354 0.001 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.404 0.001 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.454 0.001 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.504 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.554 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.604 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.654 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.704 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.754 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.804 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.854 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.904 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 D. Conclusions The resuls repored in his appendix show ha he relaive error beween dela-gamma and full simulaion is reasonably low, bu becomes large as he opion nears expiraion and is a-he-money. Noe ha he exremely large errors in he case where he opion is ou-of-he-money reflecs he fac ha he opion is valueless. Refer o Tables C.10 and C.19 in he RiskMerics Monior (hird quarer, 1996) o see he value of he opion a various spo prices and ime o expiraions. Therefore, Appendices

5 Appendix D. Assessing he accuracy of he dela-gamma approach aside from he case where he opion is near expiraion and a-he-money, he dela-gamma mehodology seems o perform well in comparison o full simulaion. Overall, he usefulness of he dela-gamma mehod depends on how users view he rade-off beween compuaional speed and accuracy. For risk managers seeking a quick, efficien means of compuing VaR ha measures gamma risk, dela-gamma offers an aracive mehod for doing so. RiskMerics Technical Documen Fourh Ediion

53 Appendix E. Rouines o simulae correlaed normal random variables Peer Zangari Morgan Guarany Trus Company Risk Managemen Research (1-1) 648-8641 zangari_peer@jpmorgan.com In Secion E.1 of his appendix we briefly inroduce hree algorihms for simulaing correlaed normal random variables from a specified covariance marix Σ (Σ is square and symmeric). In Secion E. we presen he deails of he Cholesky decomposiion. E.1 Three algorihms o simulae correlaed normal random variables This secion describes he Cholesky decomposiion (CD), eigenvalue decomposiion (ED) and he singular value decomposiion (SVD). CD is efficien when Σ is posiive definie. However, CD is no applicable for posiive semi-definie marices. ED and SVD, while compuaionally more inensive, are useful when Σ is posiive semi-definie. Cholesky decomposiion We begin by decomposing he covariance marix as follows: [E.1] Σ = P T P where P is an upper riangular marix. To simulae random variables from a mulivariae normal disribuion wih covariance marix Σ we would perform he following seps: 1. Find he upper riangular marix P.. Compue a vecor of sandard normal random variables denoed ε. In oher words, ε has a covariance marix I (ideniy marix). 3. Compue he vecor y = P T ε. The random vecor y has a mulivariae normal disribuion wih a covariance marix Σ. Sep 3 follows from he fac ha [E.] V ( y) P T E εε T T T = P = P IP = P P = Σ where V( ) and E( ) represen he variance and mahemaical expecaion, respecively. Eigenvalue decomposiion Applying specral decomposiion o Σ yields [E.3] Σ = C C T = Q T Q where C is an NxN orhogonal marix of eigenvecors, i.e., C T C = I is an NxN marix wih he N-eigenvalues of X along is diagonal and zeros elsewhere [E.4] Q 1 C T = Appendices

54 Appendix E. Rouines o simulae correlaed normal random variables To simulae random variables from a mulivariae normal disribuion wih covariance marix Σ we would perform he following seps: 1. Find he eigenvecors and eigenvalues of Σ.. Compue a vecor of sandard normal random variables denoed ε. In oher words, ε has a covariance marix I (ideniy marix). 3. Compue he vecor y = Q T ε. The random vecor y has a mulivariae normal disribuion wih a covariance marix Σ. Sep 3 follows from he fac ha [E.5] V ( y) Q T E εε T T T 1 = Q = Q IQ = Q Q = C 1 C T = C C T = Σ The final algorihm ha is proposed is known as he singular value decomposiion. Singular Value decomposiion We begin wih he following represenaion of he covariance marix [E.6] Σ = UDV T where U and V are NxN orhogonal marices, i.e., V T V = U T U = I, and D is an NxN marix wih he N singular values of Σ along is diagonal and zeros elsewhere. I follows direcly from Takagi s decomposiion ha for any square, symmeric marix, Σ = VDV T. Therefore, o simulae random variables from a mulivariae normal disribuion wih covariance marix Σ we would perform he following seps: 1. Apply he singular value decomposiion o Σ o ge V and D.. Compue a vecor of sandard normal random variables denoed ε. In oher words, ε has a covariance marix I (ideniy marix). 3. Compue he vecor y = Q T ε where Q= D 1 V T. The random vecor y has a mulivariae normal disribuion wih a covariance marix Σ. Sep 3 follows from he fac ha [E.7] V ( y) Q T E εε T T T 1 = Q = Q IQ = Q Q = VD D 1 V T = VDV T = Σ E. Applying he Cholesky decomposiion In his secion we explain exacly how o creae he A marix which is necessary for simulaing mulivariae normal random variables from he covariance marix Σ. In paricular, Σ can be decomposed as: [E.8] Σ = A T A If we simulae a vecor of independen normal random variables X hen we can creae a vecor of normal random variables wih covariance marix Σ by using he ransformaion Y=A X. To show how o obain he elemens of he marix A, we describe he Cholesky decomposiion when he dimension of he covariance marix is 3 x 3. Afer, we give he general recursive equaions used o derive he elemens of A from Σ. RiskMerics Technical Documen Fourh Ediion

Appendix E. Rouines o simulae correlaed normal random variables 55 Consider he following definiions: [E.9] Σ s 11 s 1 s 13 a 11 0 0 = s 1 s s 3 A T = a 1 a 0 A = a 11 a 1 a 31 0 a a 3 s 31 s 3 s 33 a 31 a 3 a 33 0 0 a 33 Then we have [E.10] s 11 s 1 s 13 s 1 s s 3 = a 11 0 0 a 1 a 0 a 11 a 1 a 31 0 a a 3 s 31 s 3 s 33 a 31 a 3 a 33 0 0 a 33 equivalen o s 11 s 1 s 13 a 11 a 11 a 1 a 11 a 31 [E.11] s 1 s s 3 = a 11 a 1 a 1 a + a 1 a 31 + a 3 a s 31 s 3 s 33 a 11 a 31 a 1 a 31 + a 3 a a 31 + a 3 + a 33 s he elemens of A. This is done recur- Now we can use he elemens of Σ o solve for he sively as follows: s 11 = a 11 a 11 = s 11 s s 1 = a 11 a 1 a 1 1 = ------ a 11 a i, j [E.1] a s = a 1 + a = s a 1 s s 31 = a 11 a 31 a 31 31 = ------ a 11 1 s 3 = a 1 a 31 + a 3 a a 3 = ------ ( s 3 a 1 a 31 ) a a 33 s 33 = a 11 + a + a 33 = a 33 a 11 a Having shown how o solve recursively for he elemens in A we now give a more general resul. Le i and j index he row and column of an N x N marix. Then he elemens of A can be solved for using [E.13] and [E.14] i 1 a ii s ii 1 = a ik k = 1 1 a ij ----- s ij 1 = a ik a jk j = i+ 1, i +,, N a ii i 1 k = 1 Appendices

56 Appendix E. Rouines o simulae correlaed normal random variables RiskMerics Technical Documen Fourh Ediion

57 Appendix F. BIS regulaory requiremens acques Longersaey Morgan Guarany Trus Company Risk Managemen Advisory (1-1) 648-4936 riskmerics@jpmorgan.com The Basel Commiee on Banking Supervision under he auspices of BIS issued in anuary 1996 a final Amendmen o he 1988 Capial Accord ha requires capial charges o cover marke risks in addiion o he exising framework covering credi risk. The framework covers risks of losses in on- and off- balance shee posiions arising from movemens in marke prices. Banks' minimum capial charges will be calculaed as he sum of credi risk requiremens under he 1988 Capial Accord, excluding deb and equiy securiies in he rading book and all posiions in commodiies, bu including he credi counerpary risk on all OTC derivaives, and capial charges for marke risks. The proposal ses forh guidelines for he measuremen of marke risks and he calculaion of a capial charge for marke risks. I. Measuremen of marke risk Marke risk may be measured using banks' inernal models (subjec o approval by he naional supervisor) and incorporaes he following: 1. Marke risk in he rading accoun (i.e., deb and equiy securiies and derivaives): Sandardized mehod uses a building block approach where charges for general risk and issuer specific risk for deb and equiies risks are calculaed separaely. Inernal model mus include a se of risk facors corresponding o ineres raes in each currency in which he bank has ineres sensiive on- and off-balance shee posiions and corresponding o each of he equiy markes in which he bank holds significan posiions.. Foreign exchange risk across he firm (including gold: Sandardized mehod uses he shorhand mehod of calculaing he capial requiremen. Inernal model mus include FX risk facors of he bank s exposures. 3. Commodiies risk across he firm (including precious meals bu excluding gold) Sandardized mehod risk can be measured using he sandardized approach or he simplified approach. Inernal model mus include commodiy risk facors of he bank s exposures. 4. Opions risk across he firm: Sandardized mehod banks using only purchased opions should use a simplified approach and banks using wrien opions should, a a minimum, use one of he inermediae approaches ( dela plus or simulaion mehod). Inernal model mus include risk facors (ineres rae, equiy, FX, commodiy) of he bank s exposures. Appendices

58 Appendix F. BIS regulaory requiremens II. Capial charge for marke risk Sandardized mehod simple sum of measured risk for all facors (i.e., deb/equiy/fx/commodiies/opions) Inernal model Higher of he previous day's VaR (calculaed in accordance wih specific quaniaive sandards) or average of daily VaR on each of he preceding 60 days imes a muliplicaion facor, subjec o a minimum of 3. A separae capial charge o cover he specific risk of raded deb and equiy securiies if no incorporaed in model. A plus will be added ha is direcly relaed o he ex-pos performance of he model (derived from back-esing oucome) Among oher qualiaive facors, sress esing should be in place as a supplemen o he riskanalysis based on he day-o-day oupu of he model. III. Mehods of measuring marke risks A choice beween a Sandardized Mehodology and an Alernaive Mehodology (i.e., use of banks' inernal models) will be permied for he measuremen of marke risks subjec o he approval of he naional supervisor. 1. The sandardized mehodology This mehod uses a building block approach for deb and equiy posiions, where issuer-specific risk and general risk are calculaed separaely. The capial charge under he sandardized mehod will be he arihmeic sum of he measures of each marke risk (i.e., deb/equiy/foreign exchange/ commodiies/opions). Deb securiies Insrumens covered include: deb securiies (and insrumens ha behave like hem including non-converible preferred shares) and ineres rae derivaives in he rading accoun. Mached posiions in idenical issues (e.g., same issuer, coupon raes, liquidiy, call feaures) as well as closely mached swaps, forwards, fuures and FRAs which mee addiional condiions are permied o be offse. The capial charge for deb securiies is he sum of he specific risk charge and general risk charge. Specific risk The specific risk charge is designed o proec agains an adverse movemen in he price of an individual securiy owing o facors relaed o he individual issuer. Deb securiies and derivaives are classified ino broad caegories (governmen, qualified, and oher) wih a varying capial charge applied o gross long posiions in each caegory. Capial charges range from 0% for he governmen caegory o 8% for he Oher caegory. General marke risk The general risk charge is designed o capure he risk of loss arising from changes in marke ineres raes. A general risk charge would be calculaed separaely for each currency in which he bank has a significan posiion. There are wo principal mehods o choose from: RiskMerics Technical Documen Fourh Ediion

Appendix F. BIS regulaory requiremens 59 1. Mauriy mehod long and shor posiions in deb securiies and derivaives are sloed ino a mauriy ladder wih 13 ime bands (15 for deep discoun securiies). The ne posiion in each ime band is risk weighed by a facor designed o reflec he price sensiiviy of he posiions o changes in ineres raes.. Duraion mehod achieves a more accurae measure of general marke risk by calculaing he price sensiiviy of each posiion separaely. The general risk charge is he sum of he risk-weighed ne shor or long posiion in he whole rading book, a small proporion of he mached posiions in each ime-band (verical disallowance 10% for mauriy mehod; 5% for duraion mehod), and a larger proporion of he mached posiions across differen ime bands (horizonal disallowance). Equiies Insrumens covered include: common socks, converible securiies ha behave like equiies, commimens o buy or sell equiies, and equiy derivaives. Mached posiions in each idenical equiy in each marke may be fully offse, resuling in a single ne shor or long posiion o which he specific and general marke risk charges apply. The capial charge for equiies is he sum of he specific risk charge and general risk charge. Specific risk Specific risk is he risk of holding a long or shor posiion in an individual equiy, i.e., he bank's absolue equiy posiions (he sum of all long and shor equiy posiions). The specific risk charge is 8% (or 4% if he porfolio is liquid and diversified). A specific risk charge of % will apply o he ne long or ne shor posiion in an index comprising a diversified porfolio of equiies. General marke risk General marke risk is he risk of holding a long or shor posiion in he marke as a whole, i.e., he difference beween he sum of he longs and he sum of he shors (he overall ne posiion in an equiy marke). The general marke risk charge is 8% and is calculaed on a marke by marke basis. Foreign exchange risk (including gold) The shorhand mehod of calculaing he capial requiremen for foreign exchange risk is performed by measuring he ne posiion in each foreign currency and gold a he spo rae and applying an 8% capial charge o he ne open posiion (i.e., he higher of ne long or ne shor posiions in foreign currency and 8% of he ne posiion in gold). Commodiies risk Commodiies risk including precious meals, bu excluding gold, can be measured using he sandardized approach or he simplified approach for banks which conduc only a limied amoun of commodiies business. Under he sandardized approach, ne long and shor spo and forward posiions in each commodiy will be enered ino a mauriy ladder. The capial charge will be calculaed by applying a 1.5% spread rae o mached posiions (o capure mauriy mismaches) and a capial charge applied o he ne posiion in each bucke. Under he simplified mehod, a 15% capial charge will be applied o he ne posiion in each commodiy. Treamen of opions Banks ha solely use purchased opions are permied o use a simplified approach; however, banks ha also wrie opions will be expeced o use one of he inermediae approaches or a comprehensive risk managemen model. Under he sandardized approach, opions should be carved ou and become subjec o separaely calculaed capial charges on paricular rades o be added o oher capial charges assessed. Inermediae approaches are he dela plus approach and scenario Appendices

60 Appendix F. BIS regulaory requiremens analysis. Under he dela plus approach, dela-weighed opions would be included in he sandardized mehodology for each risk ype.. Alernaive mehodology: inernal models This mehod allows banks o use risk measures derived from heir own inernal risk managemen models, subjec o a general se of sandards and condiions. Approval by he supervisory auhoriy will only be graned if here are sufficien numbers of saff (including rading, risk conrol, audi and back office areas) skilled in using he models, he model has a proven rack record of accuracy in predicing losses, and he bank regularly conducs sress ess. Calculaion of capial charge under he inernal model approach Each bank mus mee on a daily basis a capial requiremen expressed as he higher of is previous day's value a risk number measured according o he parameers specified or an average of he daily value a risk measures on each of he preceding sixy business days, muliplied by a muliplicaion facor. The muliplicaion facor will be se by supervisors on he basis of heir assessmen of he qualiy of he bank's risk managemen sysem, subjec o a minimum of 3. The plus facor will range from 0 o 1 based on backesing resuls and ha banks ha mee all of he qualiaive sandards wih saisfacory backesing resuls will have a plus facor of zero.the exen o which banks mee he qualiaive crieria may influence he level a which supervisors will se he muliplicaion facor. Banks using models will be subjec o a separae capial charge o cover he specific risk of raded deb and equiy securiies o he exen ha his risk is no incorporaed ino heir models. However, for banks using models, he specific risk charge applied o deb securiies or equiies should no be less han half he specific risk charge calculaed under he sandardized mehodology. Any elemens of marke risk no capured by he inernal model will remain subjec o he sandardized measuremen framework. Capial charges assessed under he sandardized approach and he inernal model approach will be aggregaed according o he simple sum mehod. Requiremens for he use of inernal models: Qualiaive sandards Exisence of an independen risk conrol uni wih acive involvemen of senior managemen Model mus be closely inegraed ino day-o-day risk managemen and should be used in conjuncion wih inernal rading and exposure limis Rouine and rigorous programs of sress esing and back-esing should be in place A rouine for ensuring compliance and an independen review of boh risk managemen and risk measuremen should be carried ou a regular inervals Procedures are prescribed for inernal and exernal validaion of he risk measuremen process RiskMerics Technical Documen Fourh Ediion

Appendix F. BIS regulaory requiremens 61 Specificaion of marke risk facors The risk facors conained in a risk measuremen sysem should be sufficien o capure he risk inheren in he bank's porfolio, i.e., ineres raes, exchange raes, equiy prices, commodiy prices. Quaniaive sandards Value a risk should be compued daily using a 99h percenile, one-ailed confidence inerval and a minimum holding period of 10 rading days. Banks are allowed o scale up heir 1-day VaR measure for opions by he square roo of 10 for a cerain period of ime afer he inernal models approach akes effec a he end of 1997. Hisorical observaion period will be subjec o a minimum lengh of one year. For banks ha use a weighing scheme or oher mehods for he hisorical observaion period, he effecive observaion period mus be a leas one year. Banks will have discreion o recognize empirical correlaions wihin broad risk caegories. Use of correlaion esimaes across broad risk caegories is subjec o regulaory approval of he esimaion mehodology used. Banks should updae heir daa ses no less frequenly han once every hree monhs and should reassess hem whenever marke prices are subjec o maerial change Models mus accuraely capure he unique risks associaed wih opions wihin he broad risk caegories (using dela/gamma facors if analyical approach is chosen) IV. Calculaion of he capial raio The minimum capial raio represening capial available o mee credi and marke risks is 8%. The denominaor of he raio is calculaed by muliplying he measure of marke risk by 1.5 (reciprocal of he 8% raio) and adding he resuls o credi risk-weighed asses. The numeraor is eligible capial, i.e., sum of he bank's Tier 1 capial, Tier capial under he limis permied by he 1988 Accord, and Tier 3 capial, consising of shor-erm subordinaed deb. Tier 3 capial is permied o be used for he sole purpose of meeing capial requiremens for marke risks and is subjec o cerain quaniaive limiaions. Alhough regular reporing will in principle ake place only a inervals (in mos counries quarerly), banks are expeced o manage he marke risk in heir rading porfolios in such a way ha he capial requiremens are being me on a coninuous basis, i.e., a he close of each business day. Appendices

6 Appendix F. BIS regulaory requiremens V. Supervisory framework for he use of backesing Backesing represens he comparison of daily profis and losses wih model-generaed risk measures o gauge he qualiy and accuracy of banks risk measuremen sysems. The backess o be applied compare wheher he observed percenage of oucomes covered by he risk measure is consisen wih a 99% level of confidence. The backesing framework should use risk measures calibraed o a 1-day holding period. The Commiee urges banks o develop he capabiliy o perform backess using boh hypoheical (based on he changes in porfolio value ha would occur were end-of-day posiions o remain unchanged) and acual rading oucomes. The framework adoped by he Commiee calculaes he number of imes ha he rading oucomes are no covered by he risk measures (excepions) on a quarerly basis using he mos recen 1 monhs of daa. The framework encompasses a range of possible responses which are classified ino 3 zones. The boundaries are based on a sample of 50 observaions. Green zone he backesing resuls do no sugges a problem wih he qualiy or accuracy of a bank s model (only four excepions are allowed here). Yellow zone he backesing resuls do raise quesions, bu such a conclusion is no definiive (only 9 excepions are allowed here). Oucomes in his range are plausible for boh accurae and inaccurae models. The number of excepions will guide he size of poenial supervisory increases in a firm s capial requiremen. The purpose of he increase in he muliplicaion facor is o reurn he model o a 99h percenile sandard. Backesing resuls in he yellow zone will generally be presumed o imply an increase in he muliplicaion facor unless he bank can demonsrae ha such an increase is no warraned. The burden of proof in hese siuaions should no be on he supervisor o prove ha a problem exiss, bu raher should be on he bank o prove ha heir model is fundamenally sound. Red zone he backesing resuls almos cerainly indicae a problem wih a bank s risk model (10 or more excepions). If a bank s model falls here, he supervisor will auomaically increase he muliplicaion facor by one and begin invesigaion. RiskMerics Technical Documen Fourh Ediion

63 Appendix G. Using he RiskMerics examples diskee Sco Howard Morgan Guarany Trus Company Risk Managemen Advisory (1-1) 648-4317 howard_james_s@jpmorgan.com A number of he examples in his Technical Documen, are included on he enclosed examples diskee. This diskee conains a Microsof Excel workbook file conaining six spreadshees and one macro file. The workbook can be used under Excel Version 4.0 or higher. Some of he spreadshees allow he user o modify inpus in order o invesigae differen scenarios. Oher spreadshees are non-ineracive. In he laer case, he objecive is o provide he user wih a deailed illusraion of he calculaions. This workbook and user guide is presened o he experienced user of Microsof Excel, alhough we hope he maerial is meaningful o less experienced users. Please make a duplicae of he Examples.XLW workbook and save a leas one copy on your hard drive and a leas one copy on a floppy disk. This will allow you o manipulae he enclosed spreadshees wihou sacrificing heir original forma. Opening he Examples.XLW workbook will show he following file srucure: The files lised above are described as follows: File Secion, page Descripion CFMapTD.xls Secion 6.4, page 134 Decomposiion of he 10-year benchmark OAT ino RiskMerics verices CFMap.xls Secion 6.4, page 135 Generic Excel cash flow mapping spreadshee (users are given flexibiliy o map sandard bulle bonds) FRA.xls Secion 6.4, page 136 Mapping and VaR calculaion of a 6x1 French franc FRA FX_Fwd.xls Secion 6.4, page 143 Mapping and VaR calculaion of a DEM/USD 1-year forward Sr_noe.xls Secion 6.4, page 139 Mapping and VaR calculaion of a 1-year Noe indexed o -year DEM swap raes FXBase.xls Secion 8.4, page 183 Generic calculaor o conver U.S. dollar based volailiies and correlaions o anoher base currency Examples.XLM Macro shee ha links o buons on he various spreadshees CFMapTD.xls & CFMap.xls These wo spreadshees are similar, alhough CFMap.xls allows he user o change more of he inpus in order o invesigae differen scenarios, or o perform sensiiviy analysis. CFMap.xls allows provides more verices o which o map he cash flows. Noe ha only daa in red is changeable on all spreadshees. Appendices

64 Appendix G. Using he RiskMerics examples diskee In CFMapTD.xls, Example Par 1 illusraes he mapping of a single cash flow, while Example Par II illusraes he mapping of he enire bond. To begin mapping on eiher spreadshee, ener your chosen daa in all cells ha display red fon. Then click he Creae cash flows buon. Wai for he macro o execue, hen click he Map he cash flows o verices buon o iniiae he second macro, which execues for he final oupu of Diversified Value a Risk, Marke Value, and Percenage of marke value. If you wish o prin he cash flow mapping oupu, simply click he Prin Mapping macro buon. FRA.xls This Forward Rae Agreemen example is for illusraive purposes only. We encourage he user, however, o manipulae he spreadshee is such a way as o increase i s funcionaliy. Changing any spreadshee, of course, should be done afer creaing a duplicae workbook. In his spreadshee, cells are named so ha formulae show he inpus o heir respecive calculaions. This naming convenion, we hope, increases user friendliness. For example, looking a cell C1 shows he calculaion for he FRA rae uilizing he daa in 1. Basic Conrac Daa and daa under he Mauriy column under. FRA Mapping and VaR on 6-an-95. Cells are named according o he heading under which hey fall, or he cell o heir lef ha bes describes he daa. For example, cell B30 is named Mauriy_1, while cell K3 is named Divers_VaR_1. Also noe ha he RiskMerics Correlaions are named in wo-dimensional arrays: cells L30:M31 are named Corr_Marx_1, while cells L40:N4 are named Corr_Marx_. If you have any confusion abou he naming convenion, simply go o he Formula Define name command. The Define Name dialogue box will appear, where he cell names are lised in alphabeical order along wih heir respecive cell references. The cells conaining he individual VaR calculaions (K30, K31, K40, K41, K4) conain he absolue value of he value a risk. In order o calculae he Diversified VaR, however, in cells K3 and K43, we have placed he acual VaR values o he righ of he correlaion marices. If you go o cell K3, you will see ha he formula makes use of VaR_Array_1, which refers o cells O31:O3. This VaR array conains he acual values of VaR_1 and VaR_, which are essenial o calculaing he Divers_VaR_1. Cells O31 and O3 are formaed in whie fon in order o mainain he clariy of he spreadshee. Similarly, he calculaion in cell K43 uilizes VaR_Array_, found in cells O40:O4. FX_Fwd.xls This spreadshee offers some ineracion whereby he user can ener daa in all red cells. Before examining his spreadshee, please review he names of he cells in he 1. Basic conrac daa secion in order o beer undersand he essenial calculaions. If you have any confusion abou he naming convenion, simply go o he Formula Define name command. The Define Name dialogue box will appear, where he cell names are lised in alphabeical order along wih heir respecive cell references. Please noe ha he Diversified Value a Risk calculaion uilizes he var_array inpu, which refers o cells I33:I35. These cells are formaed in whie fon in order o mainain he clariy of he workshee. RiskMerics Technical Documen Fourh Ediion

Appendix G. Using he RiskMerics examples diskee 65 Sr_noe.xls This spreadshee is for illusraive purposes only. Again, we encourage he user o forma he spreadshee for cusom use. Please noice ha he Diversified VaR calculaions make use of VaR_Array1 and VaR_Array. VaR_Array1 references cells N6:N8, while VaR_Array references cells O37:O40. These wo arrays are formaed in whie fon in order o mainain he clariy of he workshee. If you have any confusion abou he naming convenion, simply go o he Formula Define name command. The Define Name dialogue box will appear, where he cell names are lised in alphabeical order along wih heir respecive cell references. Appendices

66 Appendix G. Using he RiskMerics examples diskee RiskMerics Technical Documen Fourh Ediion

67 Appendix H. RiskMerics on he Inerne Sco Howard Morgan Guarany Trus Company Risk Managemen Advisory (1-1) 648-4317 howard_james_s@jpmorgan.com RiskMerics home pages on he Inerne are currenly locaed a hp://www.jpmorgan.com/riskmanagemen/riskmerics/riskmerics.hml and hp://www.riskmerics.reuers.com The RiskMerics home page on he Reuers Web is locaed a: hp://riskmerics.session.rservices.com The Inerne can be accessed hrough such services as CompuServe, Prodigy, or America Online, or hrough service providers by using browsers such as Nescape Navigaor, Microsof Inerne Explorer, Mosaic or heir equivalens. The Reuers Web is available wih he Reuers 3000 series. RiskMerics daa ses can be downloaded from he Inerne and from he Reuers Web. RiskMerics documenaion and a lising of hird paries, boh consulans and sofware developers who incorporae RiskMerics mehodology and/or daa ses, are also freely available from hese sies or from local Reuers offices. Users can receive e-mail noificaion of new publicaions or oher informaion relevan o RiskMerics by regisering a he following address: hp://www.jpmorgan.com/riskmanagemen/riskmerics/rmform.hml Noe ha URL addresses are subjec o change. H.1 Daa ses RiskMerics daa ses are updaed daily on he Inerne a hp://www.riskmerics.reuers.com and on he Reuers Web a hp://riskmerics.session.rservices.com. The daa ses are available by 10:30 a.m. U. S. Easern Sandard Time. They are based on he previous day s daa hrough close of business, and provide he laes esimaes of volailiies and correlaions for daily and monhly horizons, as well as for regulaory requiremens. The daa ses are no updaed on official U.S. holidays. For hese holidays, foreign marke daa is included in he following business day s daa ses; U.S. marke daa is adjused according o he Expecaion Maximizaion (EM) algorihm described in Secion 8.5. EM is also used in a consisen fashion for filling in missing daa in oher markes. The daa ses are supplied in compressed file forma for DOS, Macinosh, and UNIX plaforms. The DOS and Macinosh files are auo-exracing, i.e., he decompression sofware is enclosed in he file. Appendices

68 Appendix H. RiskMerics on he Inerne On he same page as he daa ses is he Excel add-in, which enables users o perform DEaR/VaR calculaions on oher han a US dollar currency basis. The add-in allows users full access o he daa ses when building cusomized spreadshees. Curren rae, price volailiy, and correlaion of specified pairs can be reurned. The add-in was compiled in 16 bi and runs under Excel 4.0 and 5.0. I does no run under Windows NT. The name of he add-in is PMVAR for he Mac and PMVAR.XLL for he PC. I has an expiraion dae of November 1, 1997. H. Publicaions hp://www.jpmorgan.com/riskmanagemen/riskmerics/pubs.hml The annual RiskMerics Technical Documen, he quarerly Monior and all oher RiskMerics documens are available for downloading in Adobe Acroba pdf file forma. Adobe Acroba Reader is required o view hese files. I can be downloaded from hp://www.adobe.com. RiskMerics documens are also available from your local Reuers office. H.3 Third paries hp://www.jpmorgan.com/riskmanagemen/riskmerics/third_pary_direcory.hml Seing up a risk managemen framework wihin an organizaion requires more han a quaniaive mehodology. A lising of several consuling firms who have capial advisory pracices o help ensure he implemenaion of effecive risk managemen and sysem developers who have inegraed RiskMerics mehodology and/or daa ses is available for viewing or saving as a file. Users should be able o choose from a number of applicaions ha will achieve differen goals, offer various levels of performance, and run on a number of differen plaforms. Cliens should review he capabiliies of hese sysems horoughly before commiing o heir implemenaion.. P. Morgan and Reuers do no endorse he producs of hese hird paries nor do hey warran heir accuracy in he applicaion of he RiskMerics mehodology and in he use of he underlying daa accompanying i. RiskMerics Technical Documen Fourh Ediion

69 Reference

70 RiskMerics Technical Documen Fourh Ediion

71 Glossary of erms absolue marke risk. Risk associaed wih he change in value of a posiion or a porfolio resuling from changes in marke condiions i.e., yield levels or prices. adverse move X. Defined in RiskMerics as 1.65 imes he sandard error of reurns. I is a measure of he mos he reurn will move over a specified ime period. ARCH. Auoregressive Condiional Heeroskedasciciy. A ime series process which models volailiy as dependen on pas reurns. GARCH Generalized ARCH, models volailiy as a funcion of pas reurns and pas values of volailiy. EGARCH Exponenial GARCH, IGARCH Inegraed GARCH. SWARCH Swiching Regime ARCH. auocorrelaion (serial correlaion). When observaions are correlaed over ime. In oher words, he covariance beween daa recorded on he same series sequenially in ime is non-zero. bea. A volailiy measure relaing he rae of reurn on a securiy wih ha of is marke over ime. I is defined as he covariance beween a securiy s reurn and he reurn on he marke porfolio divided by he variance of he reurn of he marke porfolio. boosrapping. A mehod o generae random samples from he observed daa s underlying, possibly unknown, disribuion by randomly resampling he observed daa. The generaed samples can be used o compue summary saisics such as he median. In his documen, boosrapping is used o show monhly reurns can be generaed from daa which are sampled daily. CAPM. Capial Asse Pricing Model. A model which relaes he expeced reurn on an asse o he expeced reurn on he marke porfolio. Cholesky facorizaion/decomposiion. A mehod o simulaion of mulivariae normal reurns based on he assumpion ha he covariance marix is symmeric and posiive-definie. consan mauriy. The process of inducing fixed mauriies on a ime series of bonds. This is done o accoun for bonds rolling down he yield curve. decision horizon. The ime period beween enering and unwinding or revaluing a posiion. Currenly, RiskMerics offers saisics for 1-day and 1-monh horizons. decay facor. See lambda. dela equivalen cash flow. In siuaions when he underlying cash flows are uncerain (e.g. opion), he dela equivalen cash flow is defined as he change in an insrumen's fair marke value when is respecive discoun facor changes. These cash flows are used o find he ne presen value of an insrumen. dela neural cash flows. These are cash flows ha exacly replicae a callable bond s sensiiviy o shifs in he yield curve. A single dela neural cash flow is he change in he price of he callable bond divided by he change in he value of he discoun facor. duraion (Macaulay). The weighed average erm of a securiy s cash flow. EM algorihm. A saisical algorihm ha can esimae parameers of a funcion in he presence of incomplee daa (e.g. missing daa). EM sands for Expecaion Maximizaion. Simply pu, he missing values are replaced by heir expeced values given he observed daa. Reference

7 Glossary of erms exponenial moving average. Applying weighs o a se of daa poins wih he weighs declining exponenially over ime. In a ime series conex, his resuls in weighing recen daa more han he disan pas. GAAP. Generally Acceped Accouning Principles. hisorical simulaion. A non-parameric mehod of using pas daa o make inferences abou he fuure. One applicaion of his echnique is o ake oday s porfolio and revalue i using pas hisorical price and raes daa. kurosis. Characerizes relaive peakedness or flaness of a given disribuion compared o a normal disribuion. 1 N N + 3 K x = ------------------------------------------------------------ ( N 1) ( N ) ( N 3) i = 1 X i σ x Since he uncondiional normal disribuion has a kurosis of 3, excess kurosis is defined as K x -3. λ lambda (decay facor). The weigh applied in he exponenial moving average. I akes a value beween 0 and 1. In he RiskMerics lambda is 0.94 in he calculaion of volailiies and correlaions for a 1-day horizons and 0.97 for 1-monh horizon. lepokurosis (fa ails). The siuaion where here are more occurrences far away from he mean han prediced by a sandard normal disribuion. linear risk (nonlinear). For a given porfolio, when he underlying prices/raes change, he incremenal change in he payoff of he porfolio remains consan for all values of he underlying prices/raes. When his does no occur, he risk is said o be nonlinear. log vs. change reurns. For any price or rae P, log reurn is defined as ln (P /P 1 ) whereas he change reurn is defined by (P P 1 )/P 1. For small values of (P P 1 ), hese wo ypes of reurns give very similar resuls. Also, boh expressions can be convered o percenage reurns/changes by simply muliplying hem by 100. mapping. The process of ranslaing he cash flow of acual posiions ino sandardized posiion (verices). Duraion, Principal, and cash flow. N 1 x = --- X N i i = 1 mean. A Measure of cenral endency. Sum of daily rae changes divided by coun mean reversion. When shor raes will end over ime reurn o a long-run value. modified duraion. An indicaion of price sensiiviy. I is equal o a securiy s Macaulay duraion divided by one plus he yield. ouliers. Sudden, unexpecedly large rae or price reurns. N x 4 ------------- 3-------------------------------------------- ( N 1) ( N 3) N ( N ) ( N 3) 1 We would like o hank Seven Hellinger of he New York Sae Banking Deparmen for poining his formula ou for us. RiskMerics Technical Documen Fourh Ediion

Glossary of erms 73 overlapping daa. Consecuive reurns ha share common daa poins. An example would be monhly reurns (5-day horizon) compued on a daily basis. In his insance adjacen reurns share 4 daa poins. nonparameric. Poenial marke movemens are described by assumed scenarios, no saisical parameers. parameric. When a funcional form for he disribuion a se of daa poins is assumed. For example, when he normal disribuion is used o characerize a se of reurns. principle of expeced reurn. The expeced oal change in marke value of he porfolio over he evaluaion period. relaive marke risk. Risk measured relaive o an index or benchmark residual risk. The risk in a posiion ha is issue specific. skewness. Characerizes he degree of asymmery of he disribuion around is mean. Posiive skews indicae asymmeric ail exending oward posiive values (righ-hand side). Negaive skewness implies asymmery oward negaive values (lef-hand side). S x = N ---------------------------------------- ( N 1) ( N ) N i = 1 X i x ---------------- σ x speed of adjusmen. A parameer used in modelling forward raes. I is esimaed from pas daa on shor raes. A fas speed of adjusmen will resul in a forward curve ha approaches he longrun rae a a relaively shor mauriy. sochasic volailiy. Applied in ime series models ha ake volailiy as an unobservable random process. Volailiy is ofen modeled as a firs order auoregressive process. sandard deviaion. Indicaion of he widh of he disribuion of changes around he mean. 1 σ x = ------------ ( N 1 X x ) i N i = 1 Srucured Mone Carlo. Using he RiskMerics covariance marix o generae random normal variaes o simulae fuure price scenarios. oal variance. The variance of he marke porfolio plus he variance of he reurn on an individual asse. zero mean. When compuing sample saisics such as a variance or covariance, seing he mean o a prior value of zero. This is ofen done because i is difficul o ge a good esimae of he rue mean. Reference

74 Glossary of erms RiskMerics Technical Documen Fourh Ediion

75 Bibliography Ahlburg, Dennis A. A commenary on error measures, Inernaional ournal of Forecasing, 8, (199), pp. 99 111. Armsrong,. Sco and Fred Collopy. Error measures for generalizing abou forecasing mehods: Empirical comparisons, Inernaional ournal of Forecasing, 8, (199), pp. 69 80. Bachelier, L. Theorie de la Speculaion (Paris: Gauhier-Villars, 1900). Becker, K., Finnery,., and A. Tucker. The inraday inerdependence srucure beween U.S. and apanese Equiy markes, ournal of Financial Research, 1, (199), pp. 7 37. Belsley, D.A. Condiioning Diagnosics: Collineariy and Weak Daa in Regression (New York: ohn Wiley & Sons, 1980). Bera, A.K. and C.M. arque. Efficien ess for Normaliy, Heeroscedasiciy, and Serial Independence in Regression Residuals, Economics Leers, 6, (1980), pp. 55 59. Blaberg, R. and N. Gonedes. A comparison of Sable and Suden Disribuions as Saisical Models for Sock Prices, ournal of Business, 47, (1974), pp. 44 80. Bollersev, T. Generalized Auoregressive Condiional Heeroscedasiciy, ournal of Economerics, 31, (1986), pp. 307 37. Bollersev, T. A Condiional Heeroskedasic Model for Speculaive Prices and Raes of Reurn, Review of Economics and Saisics, 69, (1987), pp. 54 547. Boudoukh,., Richardson, M., and R. F. Whielaw. A Tale of Three Schools: Insighs on Auocorrelaions of Shor-Horizon Sock Reurns, The Review of Financial Sudies, 3, (1994), pp. 539 573. Braun, P.A., Nelson, D.B., and Alain M. Sunier. Good News, Bad News, Volailiy and Beas, Working Paper 90-93, Graduae School of Business, Universiy of Chicago, (Augus 1991). Campbell, B. and. Dufour. Exac NonParameric Orhogonaliy and Random Walk Tess, Review of Economics and Saisics, 1, (February 1995), pp. 1 15. Campbell,.Y., Lo, A.W., and A.C. MacKinley. The Economerics of Financial Markes, manuscrip, une 1995. Cohen, K., e al. Fricion in he rading process and he esimaion of sysemaic risk, ournal of Financial Economics, (1983), pp. 63 78. Crnkovic, Cedomir and ordan Drachman. Model Risk Qualiy Conrol, RISK, 9, (Sepember 1996), pp. 138 143 Davidian, M. and R.. Carroll. Variance Funcion Esimaion, ournal of he American Saisical Associaion, 400, (1987), pp. 1079 1091. DeGroo, M. Probabiliy and Saisics, nd ediion, (Reading, MA: Addison-Wesley Publishing Company, 1989), Chaper 8. Reference

76 Bibliography Dempser, A.P., Laird, N.M., and D. B. Rubin. Maximum likelihood from incomplee daa via he EM algorihm, ournal of he Royal Saisical Sociey, 39, (1977), pp. 1 38. Diebold, F.X. and R.S. Mariano. Comparing predicive accuracy, ournal of Business and Economic Saisics, 13, (1995), pp. 53 63. Engle, R.F. Auoregressive condiional heeroscedasiciy wih esimaes of he variance of UK inflaion, Economerica, 50, (198), pp. 987 1007. Engle, R.F. and T. Bollerslev. Modelling he Persisence of Condiional Variances, Economeric Reviews, 5, (1986), pp. 1 50. Engle, R.F. and Ken Kroner. Mulivariae Simulaneous Generalized ARCH, Economeric Theory, (1995), pp. 1 150. Eun, C.S. and S. Shim. Inernaional Transmission of Sock Marke Movemens, ournal of Financial and Quaniaive Analysis, 4, (1989), pp. 41 56. Fabozzi, Frank (edior), The Handbook of Fixed Income Opions, (Chicago, IL: Probus Publishing, 1989), Chaper 3. Fama, E. The Behavior of Sock Marke Prices, ournal of Business, 38, (1965), pp. 34 105. Fama, E. and K. French. Permanen and Temporary Componens of Sock Prices, ournal of Poliical Economy, 96, (1988), pp. 46 73. Figlewski, Sephen. Forecasing Volailiy using Hisorical Daa, New York Universiy Working Paper, S-94-13, (1994). Fildes, Rober. The evaluaion of exrapolaive forecasing mehods, Inernaional ournal of Forecasing, 8, (199), pp. 81 98. Finger, Chrisopher C. Accouning for he pull o par and roll down for RiskMerics cashflows, RiskMerics Monior, (Sepember 16, 1996). Fisher, L. Some New Sock-Marke Indexes, ournal of Business, 39, (1966), pp. 191 5. Ghose, Dev and Ken Kroner. The Relaionship beween GARCH and Sable Processes: Finding he source of fa-ails in financial daa, Universiy of Arizona Working Paper, 93-1, (une 1993). Governmen Bond Oulines, 8h ediion, C. Dulligan, 1995. Greene, William. Economeric Analysis, nd ediion. (New York: Macmillan Publishing Company, 1993). Harvey, A.C. Time Series Models, nd ediion. (Cambridge, MA: MIT Press, 1993). Harvey, A.C., E. Ruiz and N.G. Shepard. Mulivariae Sochasic Variance Models, Review of Economic Sudies, 61, (1994), pp. 47 64. Hendricks, Darryl. Evaluaion of Value-a-Risk Models Using Hisorical Daa, Federal Reserve Bank of New York Economic Policy Review, (April 1996). Heus, R.M.. and S. Rens. Tesing Normaliy When Observaions Saisfy a Cerain Low Order ARMA-Scheme, Compuaional Saisics Quarerly, 1, (1986), pp. 49 60. RiskMerics Technical Documen Fourh Ediion

Bibliography 77 Heynen, R. and H. Ka. Volailiy predicion: A comparison of GARCH(1,1), EGARCH(1,1) and Sochasic Volailiy model. Mimeograph, (1993). Hill, I.D., Hill, R. and R.L. Holder. Fiing ohnson Curves by Momens (Algorihm AS 99), Applied Saisics, (1976), pp. 180 189. Inroducing he Emerging Markes Bond Index Plus,.P. Morgan publicaion, uly 1, 1995. affe and Weserfield. Paerns in apanese Common Sock Reurns: Day of he Week and Turn of he Year Effecs, ournal of Financial and Quaniaive Analysis, 0, (1985), pp. 61 7. arrow, Rober and Donald van Devener. Disease or Cure? RISK, 9, (February 1996), pp. 54 57. ohnson, N.L. Sysems of frequency curves generaed by mehods of ranslaion, Biomerika, (1949), pp. 149 175. ohnson, R.A. and D.W. Wichern. Applied Mulivariae Saisical Analysis, 3rd ediion (Englewood Cliffs, N: Prenice-Hall, Inc., 199). ohnson,. Economeric Mehods. (New York: McGraw-Hill, Inc., 1984). orion, Phillipe. Predicing Volailiy in he Foreign Exchange Marke, ournal of Finance,, (une 1995), pp. 507 58. orion, P. On ump Processes in he Foreign Exchange and Sock Markes, Review of Financial Sudies, 6, (1988), pp. 81 300. Karolyi, G.A. A Mulivariae GARCH Model of Inernaional Transmissions of Sock Reurns and Volailiy: The Case of he Unied Saes and Canada, ournal of Business and Economic Saisics, (anuary 1995), pp. 11 5. Kemphorne,. and M. Vyas. Risk Measuremen in Global Financial Markes wih Asynchronous, Parially Missing Price Daa. IFSRC Working Paper No. 81 94, (1994). Kiefer, N. and M. Salmon. Tesing Normaliy in Economeric Models, Economic Leers, 11, (1983), pp. 13 17. King, M., Senena, E., and S. Wadhwani. Volailiy and links beween naional sock markes, Economerica, (uly 1994), pp. 901 933. Kon, S.. Models of Sock Reurns: A comparison, ournal of Finance, 39, (1988), pp. 147 165. Kroner, K., Kneafsey, K.P., and S. Claessens. Forecasing volailiy in commodiy markes, forhcoming, Inernaional ournal of Forecasing, (1995). Lau, S.T. and.d. Dilz. Sock reurns and he ransfer of informaion beween he New York and Tokyo sock exchanges, ournal of Inernaional Money and Finance, 13, (1994), pp. 11. LeBaron, B. Chaos and Nonlinear Forecasabiliy in Economics and Finance. Working Paper Universiy of Wisconsin - Madison, (February 1994). Leich, G. and.e. Tanner. Economic Forecas Evaluaion: Profi Versus The Convenional Error Measures, American Economic Review, 3, (1991), pp. 580 590. Reference

78 Bibliography Li, W.K. and T.K. Mak. On he squared residual auocorrelaions in non-linear ime series wih condiional heeroskedasiciy, ournal of Time Series Analysis, 6, (1994), pp. 67 635. Lo and MacKinlay. An economeric analysis of nonsynchronous rading, ournal of Economerics, 45, (1990), pp. 181. Looney, Sephen. How o use es for univariae normaliy o assess mulivariae normaliy, The American Saisician, 49, (1995), pp. 64 70. Lumsdaine, R.L. Finie-Sample properies of he maximum likelihood esimaor in GARCH(1,1) and IGARCH(1,1) Models: A Mone Carlo Invesigaion, ournal of Business and Economic Saisics, 1, (anuary 1995), pp. 1 9. Maomoni, Hakim. Valuing and Using FRAs,.P. Morgan publicaion, Ocober 1994. Mandlebro, B. The Variaions of Cerain Speculaive Prices, ournal of Business, 36, (1963), pp. 394 419. McLeod, A.I. and W.K. Li. Diagnosic checking ARMA ime series models using squared-residual auocorrelaions, ournal of Time Series Analysis, 4, (1983), pp. 69 73. Meese, Richard A. and Kenneh S. Rogoff. Empirical exchange rae models of he sevenies: Do hey fi ou of sample? ournal of Inernaional Economics, 14, (1983), pp. 3 4..P. Morgan & Co., Inc., Arhur Andersen & Co. SC, and Financial Engineering Limied, The.P. Morgan/Arhur Andersen Guide o Corporae Exposure Managemen, published by RISK magazine, (Augus 1994). Nelson, D. B. Saionariy and persisence in he GARCH (1,1) model, Economeric Theory, 6, (1990), pp. 318 334. Pagan, A. and G. W. Schwer. Alernaive Models for Condiional Sock Volailiy, ournal of Economerics, 45, (1990), pp. 67 90. Parkinson, M. The Exreme Value Mehod for Esimaing he Variance of he Rae of Reurn, ournal of Business, 1, (1980), pp. 61 65. Perry, P. Porfolio Serial Correlaion and Nonsynchronous Trading, ournal of Financial and Quaniaive Analysis, 4, (1985), pp. 517 53. Poliical Handbook of he World: 1995 1996. (New York: CSA Publishing, Sae Universiy of New York, 1996). Richardson, M. and T. Smih. A Tes for Mulivariae Normaliy in Sock Reurns, ournal of Business,, (1993), pp. 95 31. Ruiz, Esher. Sochasic Volailiy verses Auoregressive Condiional Heeroskedasiciy, Working Paper 93-44, Universidad Carlos III de Madrid (1993). Ruiz, Esher. Quasi-maximum likelihood esimaion of sochasic volailiy models, ournal of Economerics, 63, (1994), pp. 89 306. Schwer, W.G. Why does Sock Marke Volailiy Change over Time? ournal of Finance, 44, (1989), pp. 1115 1153. RiskMerics Technical Documen Fourh Ediion

Bibliography 79 Shanken,. Nonsynchronous daa and covariance-facor srucure of reurns, ournal of Finance, 4, (1987), pp. 1 31. Shapiro, S.S. and M.B. Wilk. An Analysis of Variance Tes for Normaliy, Biomerika, 5, (1965), pp. 591 611. Sholes, M. and. Williams. Esimaing bea from non-synchronous daa, ournal of Financial Economics, 5, (1977), pp. 309 37. Silverman, B.W. Densiy Esimaion for Saisics and Daa Analysis, (New York: Chapman and Hall, 1986). Taylor, S.. Modeling Financial Time Series (Chicheser, UK: ohn Wiley and Sons, 1986). Tucker, A. L. A Reexaminaion of Finie- and Infinie-Variance Disribuions as Models of Daily Sock Reurns, ournal of Business and Economic Saisics, 1, (199), pp. 73 81. Warga, Arhur. Bond Reurns, Liquidiy, and Missing Daa, ournal of Financial and Quaniaive Analysis, 7, (199), pp. 605 617. Wes, Ken and D. Cho. The predicive abiliy of several models of exchange rae volailiy, ournal of Economerics, 69, (1995), pp. 367 391. Wes, K.D., Edison, H.., and D. Cho. A uiliy-based comparison of some models of exchange rae volailiy, ournal of Inernaional Economics, 35, (1993), pp. 3 45. Whie, H. Maximum likelihood esimaion of unspecified models, Economerica, (198), pp. 1 16. Xu, X. and S. Taylor. Condiional volailiy and he informaion efficiency of he PHLX currency opions marke, ournal of Banking and Finance, 19, (1995), pp. 803 81. Reference

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8 RiskMerics Technical Documen Fourh Ediion

83 Look o he.p. Morgan sie on he Inerne for updaes of hese examples or useful new ools. If here is no a diskee in he pocke below, he examples i conains are available from he Inerne web pages. hp://www.jpmorgan.com/riskmanagemen/risk- Merics/pubs.hml The enclosed Excel workbook is inended as a demonraion of he RiskMerics marke risk managemen mehodology and volailiy and correlaion daases. The workshees have been designed as an educaional ool and should no be used for he risk esimaion of acuual porfolios. Cliens should conac firms specialized in he design of risk managemen sofware for he implemenaion of a marke risk esimaion sysem. If you have any quesions abou he use of his workbook conac your local.p.morgan represenaive or: Norh America New York Sco Howard (1-1) 648-4317 howard_james_s@jpmorgan.com Europe London Guy Coughlan(44-171) 35-5384 coughlan_g@jpmorgan.com Asia Singapore Michael Wilson (65) 36-9901

RiskMerics Technical Documen Fourh Ediion, 1996 page 84 RiskMerics producs Inroducion o RiskMerics : An eigh-page documen ha broadly describes he RiskMerics mehodology for measuring marke risks. RiskMerics Direcory: Available exclusively on-line, a lis of consuling pracices and sofware producs ha incorporae he RiskMerics mehodology and daa ses. RiskMerics Technical Documen: A manual describing he RiskMerics mehodology for esimaing marke risks. I specifies how financial insrumens should be mapped and describes how volailiies and correlaions are esimaed in order o compue marke risks for rading and invesmen horizons. The manual also describes he forma of he volailiy and correlaion daa and he sources from which daily updaes can be downloaded. RiskMerics Monior: A quarerly publicaion ha discusses broad marke risk managemen issues and saisical quesions as well as new sofware producs buil by hird-pary vendors o suppor RiskMerics. RiskMerics daa ses: Two ses of daily esimaes of fuure volailiies and correlaions of approximaely 480 raes and prices, wih each daa se oaling 115,000+ daa poins. One se is for compuing shor-erm rading risks, he oher for mediumerm invesmen risks. The daa ses currenly cover foreign exchange, governmen bond, swap, and equiy markes in up o 31 currencies. Eleven commodiies are also included. A RiskMerics Regulaory daa se, which incorporaes he laes recommendaions from he Basel Commiee on he use of inernal models o measure marke risk, is also available. Worldwide RiskMerics conacs For more informaion abou RiskMerics, please conac he auhors or any oher person lised below. Norh America New York acques Longersaey (1-1) 648-4936 longersaey_j@jpmorgan.com Chicago Michael Moore (1-31) 541-3511 moore_mike@jpmorgan.com Mexico Bearice Sibblies (5-5) 540-9554 sibblies_bearice@jpmorgan.com San Francisco Paul Schoffelen (1-415) 954-340 schoffelen_paul@jpmorgan.com Torono Dawn Desjardins (1-416) 981-964 desjardins_dawn@jpmorgan.com Europe London Guy Coughlan (44-71) 35-5384 coughlan_g@jpmorgan.com Brussels Lauren Fransole (3-) 508-8517 fransole_l@jpmorgan.com Paris Ciaran O Hagan (33-1) 4015-4058 ohagan_c@jpmorgan.com Frankfur Rober Bierich (49-69) 71-4331 bierich_r@jpmorgan.com Milan Robero Fumagalli (39-) 774-430 fumagalli_r@jpmorgan.com Madrid ose Anonio Carreero (34-1) 577-199 carreero@jpmorgan.com Zurich Vikor Tschirky (41-1) 06-8686 schirky_v@jpmorgan.com Asia Singapore Michael Wilson (65) 36-9901 wilson_mike@jpmorgan.com Tokyo Yuri Nagai (81-3) 5573-1168 nagai_y@jpmorgan.com Hong Kong Marin Masui (85-) 973-5480 masui_marin@jpmorgan.com Ausralia Debra Roberson (61-) 551-600 roberson_d@jpmorgan.com RiskMerics is based on, bu differs significanly from, he marke risk managemen sysems developed by.p. Morgan for is own use..p. Morgan does no warran any resuls obained from use of he RiskMerics daa, mehodology, documenaion or any informaion derived from he daa (collecively he Daa ) and does no guaranee is sequence, imeliness, accuracy, compleeness or coninued availabiliy. The Daa is calculaed on he basis of hisorical observaions and should no be relied upon o predic fuure marke movemens. The Daa is mean o be used wih sysems developed by hird paries..p. Morgan does no guaranee he accuracy or qualiy of such sysems. Addiional informaion is available upon reques. Informaion herein is believed o be reliable, bu.p. Morgan does no warran is compleeness or accuracy. Opinions and esimaes consiue our judgemen and are subjec o change wihou noice. Pas performance is no indicaive of fuure resuls. This maerial is no inended as an offer or soliciaion for he purchase or sale of any financial insrumen..p. Morgan may hold a posiion or ac as marke maker in he financial insrumens of any issuer discussed herein or ac as advisor or lender o such issuer. Morgan Guarany Trus Company is a member of FDIC and SFA. Copyrigh 1996.P. Morgan & Co. Incorporaed. Cliens should conac analyss a and execue ransacions hrough a.p. Morgan eniy in heir home jurisdicion unless governing law permis oherwise.