Copulas. Modeling dependencies in Financial Risk Management. BMI Master Thesis

Transcription

1 Copulas Modelng dependences n Fnancal Rsk Management BMI Master Thess

2 Modelng dependences n fnancal rsk management

3 Modelng dependences n fnancal rsk management 3 Preface Ths paper has been wrtten as part of my study Busness Mathematcs and Informatcs (BMI) at the Vrje Unverstet n Amsterdam. The BMI-paper s one of the fnal mandatory subjects. The objectve s to nvestgate the avalable lterature n reference to a topc related to at least two out of the three felds ntegrated n the study. For the purpose of etendng my knowledge on Fnancal Rsk Management and technques for rsk measurement, I decded to dedcate my BMI paper to ths area. The subject for ths paper was set n consultaton wth dr. S. Bhula of the OBP research group at the Faculty of Scences. Although several people have supported me durng the realzaton of ths paper, I would especally lke to epress my grattude to my supervsor, dr. S. Bhula. Despte hs busy schedule he took tme to gude me. I am thankful for hs advce, comments, and motvatonal speeches when I needed t. Purmerend, December 007 Farza Habboellah

4 Modelng dependences n fnancal rsk management 4 Summary The goal of Fnancal Rsk Management (FRM) s to measure and manage rsks across a dverse range of actvtes used n fnancal sectors. Rsk can be defned as a hazard, a chance of bad consequences, loss or eposure to mschance. There are dfferent types of rsks, but we wll focus on the three most mportant: market, credt, and operatonal rsk. There are dfferent ways to measure rsk. Two rsks measures that are often used, based on loss dstrbutons, are Value at Rsk (VaR) and Epected Shortfall. Wthn FRM dependences between random varables play an mportant role. A popular and often used dependence measure s correlaton, also called the correlaton coeffcent, whch ndcates the strength and drecton of a lnear relatonshp between two random varables. It s a reasonable measure when the random varables are ellptcally dstrbuted and a good measure when the random varables are multvarate normally dstrbuted. But research shows that the multvarate normal dstrbuton s nadequate because t underestmates both the thckness of the tals of the margnals of the rsks and ther dependence structure. Apart from that correlaton has more dsadvantages. Correlaton s not nvarant under strctly ncreasng transformatons of the rsks. It s a scalar measure of dependence and therefore cannot tell us everythng we would lke to know about the dependence structure of rsks. Correlaton s only defned when the varances of the rsks are fnte. It s not an approprate dependence measure for very heavy-taled rsks where varances tend to nfnty. Ths nadequacy of correlaton requres an approprate dependence measure: the copula. The man objectve of ths thess s as follows: What are the advantages of usng a copula model to model dependences between varables n fnancal rsk models over a more tradtonal method such as correlaton? A copula functon couples n unvarate margnal dstrbutons together to form a multvarate dstrbuton resultng n a jont dstrbuton functon of n standard unform random varables. Assume that you have two random varables (X,Y). Then the standard formulaton s: H (,y) = C(F (), G (y)), where C(u,v) s the copula, F and G are margnal dstrbuton functons, and H s the jont cumulatve dstrbuton functon. Important copula functons are the Fréchet-Hoeffdng upper and lower bound gven by M ( u, v) = mn( u, v) and W ( u, v) = ma( u + v,0 ), respectvely, and the product copula ( u, v) = uv. Copulas are nvarant under strctly ncreasng transformatons of the rsks.

5 Modelng dependences n fnancal rsk management 5 In 959 Abe Sklar was the frst who used the term copula n a mathematcal sense. The theorem, that was named after hm, states that any jont cumulatve dstrbuton functon F can be wrtten n terms of a copula and margnal cumulatve dstrbuton functons. If the margnals are contnuous then the copula s unque for F. The copula s n comparson to correlaton nvarant under transformatons of the rsks. Correlaton s a scalar measure of dependence; t does not tell us everythng we would lke to know about the dependence structure of rsks. A copula determnes the dependence relatonshp by jonng the margnal dstrbutons together to form a jont dstrbuton. The scalng and the shape are entrely determned by the margnals. In contrast to correlaton the copula functon can be appled when rsks are heavly taled. A copula model that has become a standard market model for valuatng collateralzed debt oblgatons (CDOs) s the Gaussan copula model. The rsk of a CDO s dstrbuted over several tranches, where each tranche represents a group of nvestors wth dfferent rsk degrees. To determne the prce of the tranches of a CDO, the followng are needed: the default probablty, the default severty (or recovery), and the default correlaton. The default correlaton s the lkelhood that the default of one asset causes the default of another and s much hgher between credts wthn the same ndustral sector. Default correlatons can be modeled through the use of the one-factor copula model. Suppose a CDO contans assets from n companes, then the default tme of the th company s denoted by T. Its correspondng cumulatve probablty dstrbuton that company wll default before tme t s denoted by Q ( t). It s assumed that T s related to a random varable X, so that for any gven t, there s a correspondng value such that: P X = P T t. The default of the n companes can then be modeled by the ( ) ( ) followng copula model: X percentle to percentle bass. = a M + a Z. The model maps X to t on a When the cumulatve dstrbuton of X s denoted by F and the cumulatve dstrbuton of Z (assumng that the Z are dentcally dstrbuted) by H, then we can wrte Z as X am. Gven that the market varable M = m, then ts probablty can be wrtten as: a am P ( Z < = ) = M m H. a Correlaton comes n trouble when the random varables are not ellptcally dstrbuted. The performance of the copula does not depend on the fact f you are dealng wth ellptcal dstrbutons or not. Add the fact that copulas possess handy propertes and the wnner of the two s the copula.

6 Modelng dependences n fnancal rsk management 6 Contents Preface... 3 Summary... 4 Contents... 6 Chapter Introducton Background Object and Scope Thess structure... 8 Chapter Measurng dependences Dependences n FRM Correlaton and regresson, what s the dfference? The classc approach: correlaton Correlaton and the normal dstrbuton Ptfalls of correlaton... 5 Chapter 3 Fnancal Rsk Management Introducton Rsk Rsk measures Value at Rsk (VaR) Epected Shortfall... Chapter 4 The basc stuff Introducton What s a copula? Sklar s Theorem Copula famles Correlaton vs. Copula... Chapter 5 Applcaton Introducton Credt Dervatves & CDO s The one-factor copula model... 4 Concluson... 6 Reference... 7

7 Modelng dependences n fnancal rsk management 7 Chapter Introducton. Background Fnancal Rsk Management s currently a hot topc n the fnancal world. The goal of Fnancal Rsk Management (FRM) s to measure and manage rsks across a dverse range of actvtes used n, e.g., bankng, securtes and nsurance sectors. Wthn FRM dependences between random varables play an mportant role. Take for eample the rsk of a portfolo. A portfolo that contans a varety of stock types wll be less rsky than one holdng only a sngle type of stock. A popular and often used dependence measure s the correlaton coeffcent based on a, whch s a reasonable measure when the random varables are ellptcally dstrbuted and a good measure when the random varables are multvarate normal dstrbuted. But what f the margnals are not normal dstrbuted and a model can make the dfference between thousands or even mllons of Euros loss or revenue for an nsttute? Under these crcumstances reasonable s not good enough anymore. The market rsk portfolo value dstrbutons, whch typcally occur n bankng, are often appromated by a normal dstrbuton. It s a standard assumpton n many rsk management applcatons, because ths dstrbuton s easy to mplement. However, research shows that the multvarate normal dstrbuton s nadequate because t underestmates both the thckness of the tals of the margnals and ther dependence structure []. The credt and especally the operatonal rsks are appromated wth more skewed dstrbutons because of occasonal, etreme losses. So t seems that the margnals of rsks are often not ellptcally dstrbuted. However, when you step outsde the ellptcal world, the margnal dstrbutons and correlatons do not suffce to determne the jont multvarate dstrbuton. It no longer tells us everythng we need to know about dependence, partcularly n the tals. In those cases correlaton s no longer a sutable dependence measure. Ths nadequacy of the use of correlaton outsde the ellptcal world requres an approprate dependence measure: the copula.. Object and Scope The focus n ths thess les on modelng the dependence structure between rsks, an essental actvty whch s needed n the assessment of rsk. In ths thess we wll restrct ourselves to the feld of bankng. Note that the multvarate normal s a specal member n the famly of ellptcal dstrbutons.

8 Modelng dependences n fnancal rsk management 8 Due to the fact that the copula maneuvers around the ptfalls of correlaton, t has become qute popular n fnancal rsk management to model dependences between rsks. Areas of applcaton nclude credt rsk modelng, portfolo Value at Rsk calculatons, default and credt rsk dependence, and tal dependence. In ths thess we wll dscuss the ptfalls of correlaton and how the copula functon deals wth them. The man objectve of ths thess s as follows: What are the advantages of usng a copula model to model dependences between varables n fnancal rsk models over a more tradtonal method such as correlaton? Based on the objectve we have derved the followng questons that wll be answered n ths thess: Chapter 3 - When and where does correlaton come short n rsk modelng? Chapter 4 - What s a copula functon and what are ts propertes? Chapter 5 - How can copulas be appled n fnancal rsk management?.3 Thess structure A graphcal representaton of the thess structure can be seen n Fgure.. In Chapter we wll go deeper nto rsks and rsk measures. It s necessary to cover these subjects to gan a better understandng of Fnancal Rsk Management n order to dscuss the man subject: dependence modelng n FRM.. Introducton. FRM 5. Applcaton 3. Measurng Dependences 4. Copula s Concluson Fgure.: Graphcal representaton of the thess structure.

9 Modelng dependences n fnancal rsk management 9 As mentoned before, Chapter 3 wll eplan the concept of correlaton and the shortcomngs of correlaton when t comes to rsk modelng. In Chapter 4 we wll eplan what a copula s, what ts propertes are, and how a copula functon can be constructed. Now that the copula methodology s covered, some possble applcatons of copula functons n fnancal rsk management are presented n Chapter 5. Fnally, the last chapter summarzes the conclusons of our study, and furthermore some ponts for further research are stated.

10 Modelng dependences n fnancal rsk management 0 Chapter Fnancal Rsk Management. Introducton The concept of rsk s based on the uncertanty about future outcomes. In the ntroducton we already mentoned three dfferent types of ndvdual rsks: the market, the credt, and the operatonal rsk. In ths chapter we wll go deeper nto these rsks, especally the ones last mentoned and we wll dscuss two generally used rsk measures: the varance and the Value at Rsk (VaR).. Rsk Accordng to the Concse Oford Englsh Dctonary rsk can be defned as: A hazard, a chance of bad consequences, loss or eposure to mschance. In other words, rsk ndcates any uncertanty that mght trgger losses. However, uncertanty s hardly vsble n contrast to revenues or costs. Consequently, rsks reman ntangble untl they have materalzed nto loss. Ths makes the quantfcaton of rsks more dffcult. Fnancal rsks can be splt nto two parts: the ndvdual rsks and the dependence structure between them. Ths requres an approach for combnng dfferent rsk types and, hence, rsk dstrbutons. There are many types of rsks, but for now we wll focus on the most mportant three: the market, the credt, and the operatonal rsk. Market rsk The market rsk s the rsk that the value of an nvestment wll decrease due to adverse movements n market factors. These movements cause volatlty n Proft & Loss. Credt rsk The credt rsk s the rsk that a company or ndvdual wll be unable to pay the contractual nterest or prncpal on ts debt oblgatons. Operatonal rsk The operatonal rsk s the rsk of loss resultng from nadequate or faled nternal processes, people and systems, or from eternal events. The dstrbutonal shapes of each rsk type vary consderably; some are better characterzed and measured than others. For eample, the market rsk has portfolo value

11 Modelng dependences n fnancal rsk management dstrbutons that are often appromated by a normal dstrbuton. Credt and operatonal rsk on the other hand are appromated wth more skewed dstrbutons because of occasonal, etreme losses. These mght be due to large lendng eposures n the case of credt rsk, or large catastrophes such as 9/, n the case of operatonal rsk..3 Rsk measures In practce, rsk measures are used for a varety of purposes. One of the prncpal functons of fnancal rsk measurement s to determne the amount of captal a fnancal nsttuton needs to hold. Ths s mportant because an nsttuton needs a buffer aganst unepected future losses on ts portfolo n order to keep the solvency of the nsttuton healthy. Another purpose s that rsk measures are often used by management as a tool for lmtng the amount of rsk a unt wthn a frm may take. There are dfferent ways to measure rsk. In ths secton we wll dscuss two rsks measures that are based on loss dstrbutons, namely Value at Rsk (VaR) and Epected Shortfall. Losses are the central object of nterest n FRM and so t s natural to base a measure of rsk on ther dstrbuton..3. Value at Rsk (VaR) Value at Rsk (VaR) s defned as the mamum epected loss (measured n monetary unts) of an asset value (or a portfolo) over a gven tme perod and at a gven level of confdence (or wth a gven level of probablty), under normal market condtons [Coronado, 000]. Consder a portfolo of rsky assets, wth a fed tme horzon Δ, and denote by l = P L l the dstrbuton functon of the correspondng loss dstrbuton. It F L () ( ) measures the severty of the rsk of holdng our portfolo over the tme perod Δ. DEFINITION Gven some confdence level α ( 0,). The VaR of the portfolo at confdence level α s gven by the smallest number l such that the probablty that the loss L eceeds l s no larger than ( α). α { l R : P( L > l) α} = nf{ l R : F ( l α} VaR = nf L ). () In other words, the VaR of our portfolo s the loss L that s epected to be eceeded wth a probablty of ( α). In probablstc terms, the VaR s thus smply the quantle of the loss dstrbuton. For eample f a bank s 0-day 99% VaR s 3 mllon Euros, there s consdered to be only a % probablty that losses wll eceed 3 mllon Euro n 0 days. Typcal values for α are 0.95 or 0.99; n market rsk management the tme horzon Δ s usually or 0 days, n credt rsk management and operatonal rsk management Δ s usually one year.

12 Modelng dependences n fnancal rsk management.3. Epected Shortfall The Epected Shortfall (ES) s closely related to Value at Rsk. Mathematcally ths rsk measure can be defned as: DEFINITION For a loss L wth E( L ) < and dstrbuton functon F L the Epected Shortfall at α 0, s defned as: confdence level ( ) ES α = α α q ( F )du, () u L where q u ( F L ) s the quantle functon of F L. The ES s therefore related to VaR by: ( L) du = E( L L VaR ) ESα = VaRu α α α. (3) So, n other words, the ES s defned as the epected sze of a loss that eceeds VaR. If you take for eample a alpha of 99% over a tme horzon Δ of ten days, the ES s the average amount that s lost over a 0-day perod, assumng that the loss s greater than the 99 th percentle of the loss dstrbuton. In paragraph.3. we dscussed an eample of a bank wth a 0-day 99% VaR of 3 mllon. However there s no dfferentaton between small and very large volatons of the 3 mllon Euro lmt. The eventual loss can be 3 mllon Euro as well as 30 mllon Euro. That s why ES gves a bgger pcture of the rsk, because you look at the average VaR over all levels u α as can be seen n Equaton (3).

13 Modelng dependences n fnancal rsk management 3 Chapter 3 Measurng dependences 3. Introducton In the prevous chapter we generally dscussed Fnancal Rsk Management. In ths chapter we wll focus on the dependences n FRM and how they can be measured wth the correlaton coeffcent (correlaton). In Secton 3.5 we wll dscuss a way to approach the relaton between correlaton and the Normal dstrbuton. The ptfalls of correlaton wll be dealt wth n the last secton of ths chapter. 3. Dependences n FRM Dependences can be found everywhere, so also n the fnancal world. Take for eample market and credt rsk. Both rsk types are related to the nterest rate. But there can also be relatons between rsks. Assume an nvestor has a portfolo wth two loans to two dfferent companes. He can reduce the portfolo rsk by holdng assets that are not related wth each other. For eample, f the nvestor loans money to two Dutch farmers and there s contnung bad weather, whch destroys the harvest of all Dutch farmers. Changes are hgh that both formers wll default. The farmers are related to each other, because they are eposed to many of the same nfluences. That s why t s smart to hold a rsk dversfed portfolo. If the nvestor would loan to a farmer and to a umbrella factory, he balances the rsk of bad weather. Because bad weather s a bad nfluence for the farmer, but t means bg busness for the umbrella factory. So they can reduce ther eposure to asset rsk by holdng a dversfed portfolo of assets. 3.3 Correlaton and regresson, what s the dfference? Correlaton and regresson are two concepts that are often confused wth each other. Before we wll go deeper nto correlaton, we wll shortly dscuss the dfference between both technques by lookng at ther goals. The goal of lnear regresson s to fnd the equaton of the lne that best fts the data. Ths lne s then used to represent the relatonshp between the varables, or for estmatng unknown values of one varable when gven the value of the other. The goal of correlaton s to see whether two varables co-vary, and to measure the strength of any dependency between the varables. The results of correlaton are epressed as a P-value (for the hypothess test) and an r-value (correlaton coeffcent) or r value (coeffcent of determnaton).

14 Modelng dependences n fnancal rsk management The classc approach: correlaton A concept that s often used n rsk management, but whch s often msunderstood, s correlaton. Part of the msunderstandng can be the result of the usage of the word n lterature. Correlaton s a dependence measure, but s often used for almost every meanng of the word dependence. Even when t s used as a rsk measure, there s a tendency to use t as f t were an all-purpose dependence measure. Ths results n correlaton beng msused and appled to problems for whch t s not sutable. DEFINITION The correlaton coeffcent, ndcates the strength and drecton of a lnear relatonshp between two random varables. The best known correlaton measure s the Pearson product-moment correlaton coeffcent (r) ( r ( X Y ) ( X, Y ) ( X ) var( Y ) cov, =, (4) var Wth ( X, Y ) = E( XY ) E( X ) E( Y ) cov, provded that var(x) and var(y) are greater than 0. The correlaton coeffcent s a scalar measure of dependence wth the property that r X, Y. ( ) In Sectons 3. we dscussed that the default of one company can be related to the default of another. Ths s called the default correlaton. Default correlaton s the lkelhood that the default of one asset causes the default of another and s much hgher between credts wthn the same ndustral sector. In Chapter 5 we wll go deeper nto ths matter. 3.5 Correlaton and the normal dstrbuton Although correlaton plays a central role n fnance, t s mportant to realze that the concept s only a natural one n the contet of multvarate normal or more generally, ellptcal models. The vsual relaton between the normal dstrbuton and correlaton s demonstrated n steps one to three n Fgure., whch shows the gradual transformaton from the bvarate normal dstrbuton to a correlaton lne (Fgure.). Fgure.: The unvarate () and bvarate ( & 3) normal dstrbuton.

15 Modelng dependences n fnancal rsk management 5 In the frst fgure you can see the unvarate normal dstrbuton and n the last two fgures you can see a bvarate normal dstrbuton from above () and drectly down on the bvarate normal dstrbuton (3). The last fgure (3) s a scatter plot and because of the crcularty of the shape t can be sad that there s no relatonshp (r = 0). In Fgure., t can be seen that as the correlaton between and y ncreases, the crcles narrow untl you see the straght lne of the perfect (r =.0) correlaton. Fgure.: A scatter plot of and y () and one when there s perfect correlaton () Ptfalls of correlaton Correlaton s a reasonable measure of dependence when random varables are dstrbuted as multvarate normal, but when ths s not the case correlaton gets nto trouble. The followng ptfalls occur: Possble values of correlaton depend on the margnal dstrbuton of the rsks. Not all values between and are not necessarly attanable. Perfect postvely dependent rsks do not necessarly have a correlaton of ; perfect negatvely dependent rsks do not necessarly have a correlaton of. A correlaton of zero does not ndcate ndependence of rsks. Correlaton s not nvarant under strctly ncreasng transformatons of the rsks. For eample, log(x) and log(y) generally do not have the same correlaton as X and Y. Hence, transformatons of our data can affect our correlaton estmates. Correlaton s a scalar measure of dependence; t cannot tell us everythng we would lke to know about the dependence structure of rsks. Correlaton s only defned when the varances of the rsks are fnte. In Equaton (4) you can see that the denomnator s determned by the product of the varances, whch (see Equaton (4)). It s not an approprate dependence measure for very heavy-taled rsks where varances tend to nfnty.

16 Modelng dependences n fnancal rsk management 6 Chapter 4 The basc stuff 4. Introducton In the last chapter we dscussed the ptfalls of correlaton as a dependence measure n FRM. In ths chapter we wll dscuss a dependence model that deals wth most of the shortcomngs of correlaton, namely the copula. But before we wll gve a defnton of copula functons, one has to be famlar wth the followng propertes and defntons: - Grounded a s the smallest element of Suppose a functon H from n S S y. - -ncreasng S y S and a y s the smallest element of S nto R s grounded f H (, a ) 0 = H ( a y) y, S y. We say =, for all (,y) A functon H s -ncreasng f the H-volume of B s greater than or equal to 0, where B = [, ] [ y, y ] s a rectangle whose vertces are n the doman of H for every (, ),( y, y ) n [0,] wth and y y. The H-volume of B s computed by: ( B) H (, y ) H (, y ) H (, y ) H ( y ) V H = +. (5) - Dstrbuton functon, A dstrbuton functon s a functon F wth doman R such that: - F s non-decreasng; - F( ) = 0 and ( + ) = - Jont dstrbuton functon F. A jont dstrbuton functon s a functon H wth doman - H s grounded (, ) = H (, y) = 0 H, - H s -ncreasng H-volume s 0. R such that: H has as margnals the functons F and G gven by F( ) = H (, ) and G( y) = H (, y) [4]. - Doman of H = S S y = I.

17 Modelng dependences n fnancal rsk management 7 If S and S y are non-empty subsets that contan all possble values of and y, respectvely, then S S y = I means that all the possble values of and y le wthn the unt square I, where I = [0,]. - Margnals of H Suppose b s the greatest element of S and b y s the greatest element of S y. If the above mentoned propertes hold for H(,y), then H has as margnals the functons F and G gven by: F ( ) = H (, b ) for all n S, G ( y ) = H ( b, y ) for all y n S y. After eplanng how a copula works, we wll cover Sklar s Theorem, how a copula functon can be constructed and two copula famles. 4. What s a copula? The term orgnates from Latn, and means a lnk, te, bond and s used to refer to jonng together or connectng words. In ths thess we wll refer to the mathematcal meanng of the term copula, namely the copula functon. DEFINITION A copula functon lnks n unvarate margnal dstrbutons to a full multvarate dstrbuton resultng n a jont dstrbuton functon of n standard unform random varables. In some sense, the copula stll has a connecton to the Latn meanng of the word, because the copula actually couples the margnal dstrbutons together to form a jont dstrbuton. Assume that you have two random varables (X, Y). Then the standard formulaton s: H (,y) = C(F (), G (y)), (6) where C(u,v) s the copula, F and G are margnal dstrbuton functons, and H s the jont cumulatve dstrbuton functon. The nformaton of the margnal dstrbutons are retaned n F() and G(y), and the dependence nformaton s summarzed by C(u,v). The dependence relatonshp s entrely determned by the copula, whle the scalng and the shape (e.g., the mean, the standard devaton, the skewness, and the kurtoss) are entrely determned by the margnals.

18 Modelng dependences n fnancal rsk management 8 A bvarate functon C s a copula functon f t satsfes the followng propertes [4]:. Doman of C = S S =, where Sand S are non-empty subsets of I; I. C s a functon that s grounded and -ncreasng; 3. for every u n Sand every v n Note that for every (u,v) n Dom C, 0 ( u v) S, C ( u, ) = u and C (, v) = v. C,, so that Rang C s a subset of I. EXAMPLE 3. We wll show that the bvarate functon H(,y) = y s n fact a copula, y 0,. on the range [ ] Soluton - Doman of H = S S y = - H ( 0, y) = 0 and (,0) - For B = [, ] [ y, I. H = 0 H s grounded. y ] ( B) 0 Ths has to hold for every, y, y [ 0,] V H H s -ncreasng., wth and y y. Defne = + Δ and y = y + Δ y. Fll ths n n V H ( B). : V V V V H H H H ( B) = H ( + Δ y + Δ ) H ( + Δ, y ) H (, y + Δ ) + H (, y ) 0, y y ( B) = ( + Δ )( y + Δ ) ( + Δ ) y ( y + Δ ) + ( y ) 0 y y ( B) = y + Δ + y Δ + Δ Δ y y Δ y Δ + y 0 y y y ( B) Δ Δ 0 = y And ths holds for all, y, y [ 0,] - H (, ) = and H (, y) = y, wth and y y.,. The functon that s used n Eample 3. s also called the product copula ( u v) = uv Other mportant copula functons are the Fréchet-Hoeffdng upper and lower bound gven M u v mn u, v W u, v = ma u + v,0, respectvely. by (, ) = ( ) and ( ) ( )

19 Modelng dependences n fnancal rsk management 9 THEOREM 3.3 For every copula C ( u,..., ) u n we have the bounds n ma u + n,0 C( u ) mn( u,.., un ). = 4.3 Sklar s Theorem In 959 Abe Sklar was the frst who used the term copula n a mathematcal sense. In Sklar s Theorem he descrbes how the copula functon works and proves that the copula C s unque for a gven dstrbuton F f the margnals are contnuous. The theorem states that any jont cumulatve dstrbuton functon can be wrtten n terms of a copula and margnal cumulatve dstrbuton functons. A copula functon shows that t s possble to separately specfy the dependence between varables and the margnal denstes of each varable. That s why Sklar s Theorem s sad to be the most mportant theorem about copula functons. THEOREM 3. Sklar s Theorem Let H be a jont dstrbuton functon wth margnals F and G. Then there ests a copula functon C such that for all,y n R, H (, y) = C( F( ), G( y) ). (7) If the margnals F and G are contnuous, then C s unque. Otherwse C s unquely determned on Range F Range G. Conversely, f C s a copula and F and G are dstrbuton functons, then the functon H defned by (7) s a jont dstrbuton functon wth margnals F and G [4]. Theorem 3. s presented form the pont of vew of the jont dstrbuton, but what f we were to turn ths around and look at t from the drecton of the copula functon. The theorem can be nverted to epress copulas n terms of a jont dstrbuton functon and the nverses of the margnals, but only f the margnals are strctly ncreasng. PROPOSITION 3.7 ( ) y Let F ( ) and F ( y) be two functons. If F ( ) F = and F ( F( y) ) = F s the nverse of F and vce versa. The notaton for the nverse of F s F. then Equaton (7) can then be wrtten as C ( ) ( u, v) H F ( u), G ( v) =, (8)

20 Modelng dependences n fnancal rsk management 0 because when ( ) u F = and G ( y) = v, the nverse functons are ( u) F = and G () v = y. After substtuton you obtan Equaton (8). A graphcal representaton can be seen n Fgure 3.. Fgure 3.: The margnal dstrbuton functons F X and F Y. So a copula s actually a jont cumulatve dstrbuton functon wth unform margnals. It maps ponts on the unt square (u,v E [0,][0,]) to values between zero and one. PROPOSITION 3.8 Let ( X,..., ) X n be a random vector wth contnuous margnals and copula C and let T,...,T n be strctly ncreasng functons. Then ( T ( X ),..., T n ( X n )) also has copula C. Proposton 3.8 shows the nvarance property of the copula under strctly ncreasng transformatons of the margnals. 4.4 Copula famles Copulas can be dstngushed n the Ellptcal and Archmedean famly. Ellptcal copulas are the copulas wth ellptcal dstrbutons, whch have a ellptcal form and therefore symmetry n the tals. Important copulas n ths famly are the Gaussan and the student s copula. The Gaussan copula s often used because of hs smple form as can be seen n Eample 3.. EXAMPLE 3. Gaussan copula Assume there are two random varables X and Y where both varables are standard normal dstrbuted, X ~ N(0,) and Y ~ N(0,) and let the correlaton between X and Y be denoted by ρ ( X, Y ) = ρ. Then the jont dstrbuton can be wrtten as the followng copula: H ( y) = Φ (, y) = C ( Φ( ), Φ( y) ), ρ ρ, (9)

21 Modelng dependences n fnancal rsk management where Φ denotes the standard unvarate and dstrbuton functon. (9) can be wrtten as: Φ ρ the standard bvarate normal C ρ ( ) ( u, v) = Φ Φ ( u), Φ ( v) ρ = ( ) ( ) Φ u Φ v y π ( ρ) ( + y ρ ) ( ρ ) e ddy. (0) Archmedean copulas are wdely appled, because they are not dffcult to construct. In comparson to Ellptcal copulas, Archmedean copulas have only one dependency parameter (nstead of a dependency matr) and have many dfferent forms. In Table 3. you can see three often used Archmedean copulas. Copula C θ ( u, v) θ ( t) Gumbel θ θ ( ln t) ep ( ln u ) + ( ln v) θ θ θ ( ) Clayton θ θ ( ) u + v ( ) θ θ ln + θ ep( θu) ep( θv) ep( θ ) Frank ( )( ) Table 3.: Summary of three Archmedean copulas. ϕ Parameter θ t θ - ln e θ e t The copulas n Table 3. are of the form: C u, v) = ϕ ( ϕ( u) + ϕ( v) ) θ (, where ϕ s a decreasng functon from [0,] to [0, ], satsfyng ϕ ( 0) =, () Archmedean copulas enjoy ther popularty due to the followng reasons: - Many copula famles belong to ths class, - they can be easly constructed, - they posses nce propertes (.e. C ( u, v) = C( v, u) ). θ R ϕ = Correlaton vs. Copula Now that we have covered the bascs of copula we wll compare ths method wth correlaton. As we have seen n the last chapter correlaton comes n trouble when the random varables are not Normal dstrbuted. In market rsk ths s not so much of a problem, but credt and operatonal rsk are appromated wth more skewed dstrbutons because of occasonal, etreme losses. In those cases copula s preferred over correlaton. Copula s n comparson to correlaton nvarant under transformatons of the rsks. For eample, log(x) and log(y) generally do not have the same correlaton as X and Y. Hence,

22 Modelng dependences n fnancal rsk management transformatons of our data can affect our correlaton estmates. Proposton 3. shows that f you transform a random varable X, whch has a copula functon C, wth a strctly ncreasng functon T, then T(X) also has copula C. Correlaton s a scalar measure of dependence; t cannot tell us everythng we would lke to know about the dependence structure of rsks. Copula on the other hand works by jonng together the margnal dstrbutons to form a jont dstrbuton. In ths way the dependence relatonshp s entrely determned by the copula, whle the scalng and the shape are entrely determned by the margnals. Correlaton s only defned when the varances of the rsks are fnte (see Equaton ()). It s not an approprate dependence measure for very heavy-taled rsks where varances appear nfnte.

23 Modelng dependences n fnancal rsk management 3 Chapter 5 Applcaton 5. Introducton A copula model that has become a standard market model for valuatng collateralzed debt oblgatons (CDOs) s the Gaussan copula model. In ths chapter we wll show how CDO s can be modeled by a copula. In paragraph 4. we wll go nto the bascs of Credt Dervatves and CDO s and n paragraph 4.3 we wll show how an one-factor copula model can be appled to prze CDO tranches. 5. Credt Dervatves & CDO s Before we dscuss the copula model, some knowledge about Credt Dervates and CDO s s requred. The purpose of these nstruments s to allow market partcpants to trade the credt rsk assocated wth certan debt nstruments. Credt dervatves can be seen as fnancal contracts between two partes. One party wants to gan protecton aganst credt rsk eposure (the protecton buyer) and the other party wants to nvest by sellng protecton (protecton seller) n echange for the cash flows of ther nvestments (premum). In other words, credt dervatves allow one party to transfer credt rsk to another n echange for a fee. CDO s are categorzed as credt dervatves and generally secure a portfolo of bonds and loans by transferrng the asset rsk to captal market nvestors n return for the cash flows that are generated from the asset portfolo. Ths s done by dvdng the portfolo nto packages named securtes, whch are sold to nvestors. Not all securtes carry the same rsk. Fgure 4.: Establshment of a (Cash) CDO.

24 Modelng dependences n fnancal rsk management 4 The rsk on the portfolo s dstrbuted over several tranches, where each tranche represents a group of nvestors wth dfferent rsk degrees. The assets that have the hghest rsk of default are classfed n the lowest tranche and vce versa. So the hgher the tranche the lower the rsk that the correspondng assets wll default. To determne the prce of the tranches of a CDO the followng are needed: default probablty, default severty (or recovery) and default correlaton. The default correlaton s the lkelhood that the default of one asset causes the default of another and s much hgher between credts wthn the same ndustral sector. For eample, the default of an US arlne wll have a hgh default correlaton assocated wth other US arlnes. There could even be a correlaton assocated wth arlnes n Europe, because they are eposed to many of the same nfluences. On the other hand the default correlaton of the US arlne assocated wth a telecom company n Europe would be low or even zero. The structure of a general CDO can be seen n Fgure 4.. Under the CDO a specal purpose vehcle (SPV) s created by a sponsorng organzaton (a bank or other fnancal nsttuton). The SPV s a legal entty, wth ts own assets, labltes and management, whose operatons are lmted to the acquston and fnancng of specfc assets. Often, the sponsorng organzaton acts as the manager of the SPV. The assets, whch are generally bought from the sponsorng organzaton, serve as collateral for the securtes that the SPV ssued. The SPV funds these assets wth the cash proceeds of the securtes, whch are sold n the captal market to nvestors. 5.3 The one-factor copula model In the last paragraph we brefly dscussed default correlatons. In ths secton we wll show how they can be modeled through the use of the one-factor copula model, see Equaton (). Suppose a CDO contans assets from n companes, then the default tme of the th company s denoted by T. Its correspondng cumulatve probablty dstrbuton that company wll default before tme t s denoted by Q ( t). It s assumed that the tme to default correspondng value such that: P T s related to a random varable ( X ) = P( T t) < X, so that for any gven t, there s a <, =,,n. () X can be seen as a varable that ndcates when an oblgator wll default: the lower the value of the varable, the earler a default s lkely to occur.

25 Modelng dependences n fnancal rsk management 5 The default of the n companes can then be modeled by the followng copula model: X = a M + a Z where, () X the default ndctor varable for the th company; a correlaton of the th company wth the market, where a satsfes a ; M a market factor, whch s the same for all X ; Z ndvdual component affectng only X. The two stochastc components of X, M and Z ' s, have ndependent probablty dstrbutons. In the case of the Gaussan copula model both components are Normal dstrbuted. The correlaton of the th company wth the market s denoted by a, where a satsfes a. Because the correlaton s wth respect to the common market factor, the correlaton between two trgger levels for names and j s gven by a a. j The one factor copula model maps X to t on a percentle to percentle bass, whch means that the 5% pont on the X dstrbuton s mapped to the 5% pont on the t dstrbuton. In other words: when X s small, the tme t before default s also small. When the cumulatve dstrbuton of X s denoted by F and the cumulatve dstrbuton of Z (assumng that the Z are dentcally dstrbuted) by H, then, n general the pont X = s transformed to t F Q t t = Q F. We can rewrte () nto Z and gven that the market varable t = where = ( ( )) or ( ( )) Z X a M =. Snce H s the cumulatve dstrbuton of a M = m, t follows from Equaton () that: am P ( Z < = ) = M m H, (3) a and the condtonal default probablty s ( Q () t ) F am P ( t < = ) = t M m H. (4) a The dea behnd the copula model s that we do not defne the correlaton structure between the varables of nterest drectly ( t ), but we map the varables of nterest nto other more manageable varables ( X ) and defne a correlaton structure between those varables.

26 Modelng dependences n fnancal rsk management 6 Concluson The man object of ths thess was what the advantages of usng a copula over correlaton were to model dependences between varables n fnancal rsk models. Par wse correlaton and the margnal dstrbutons of a random vector s then not enough to determne ts jont dstrbuton. The copula works by jonng together the margnal dstrbutons to form a jont dstrbuton. In ths way the dependence relatonshp s entrely determned by the copula, whle the scalng and the shape are entrely determned by the margnals. When we look at the dstrbutons of fnancal rsk we see that market rsk s often appromated wth a Normal dstrbuton and therefore does not cause much of a problem. Credt and operatonal rsk on the other hand are appromated wth more skewed dstrbutons because of occasonal, etreme losses. Correlaton comes n trouble when the random varables are not ellptcally dstrbuted. The performance of the copula does not depend on the fact f you are dealng wth ellptcal dstrbutons or not. Add the fact that copulas possess handy propertes and the wnner of the race s copula.

27 Modelng dependences n fnancal rsk management 7 Reference [] Dowel, K., An nformal ntroducton to copulas, Fnancal Engneerng News, Aprl 004, pp [] Romano, C., Applyng copula functon to rsk management, Rome. [3] Embrechts, P., A. McNel and D. Straumann, Correlaton: Ptfalls and Alternatves, March 999, ETH Zurch. [4] Nelsen, R. B., An Introducton to copulas, Sprnger, New York. [5] Embrechts, P., A. McNel and R. Frey, Quanttatve Rsk Management, 005.