WHERE HAVE WE COME FROM?...

Transcription

1 A SOCHASIC ASSE MODEL & CALIBRAION FOR LONG-ERM FINANCIAL PLANNING PURPOSES John Hibbe, Philip Mowbay & Caig unbull June 00 Copyigh 00 Baie & Hibbe Limied Baie & Hibbe Limied is a membe of he Secuiies and Fuues Auhoiy he infomaion in his epo is believed o be coec bu canno be guaaneed. All opinions and esimaes included in his epo consiue ou judgemen as of he dae indicaed and ae subjec o change wihou noice. his epo is inended fo infomaion puposes only and is no inended as an offe o buy o sell secuiies. he company, is cliens and offices may have a posiion o engage in ansacions in any of he secuiies menioned. his noice does no esic you ighs unde he Financial Sevices Ac 986 o he egulaoy sysem. his documen is no inended fo he use of Pivae Cusomes.

2

3 ABLE OF CONENS. INRODUCION WHY BUILD MODELS? WHA LESSONS FROM HE PAS? WHA CURREN PROBLEMS DO ACUARIES & FINANCIAL PLANNERS FACE? WHA FUURE PROBLEMS? WHERE HAVE WE COME FROM? BACKGROUND HE WILKIE MODEL SOME PROPERIES OF GOOD MODELS REPRESENAIVENESS ECONOMIC INERPREAION PARSIMONY RANSPARENCY EVOLUION IMPLEMENAION OOLS AN ALERNAIVE ASSE MODEL HE ERM SRUCURE MODEL REAL INERES RAES / -FACOR HULL-WHIE HE INFLAION MODEL INFLAION EXPECAIONS NOMINAL ERM SRUCURE NEGAIVE NOMINAL INERES RAES HE EQUIY MODEL HE MARKOV REGIME-SWICHING MODEL EQUIY DIVIDEND YIELDS SOME ALERNAIVE APPROACHES O CALIBRAION INRODUCION EMPIRICAL DAA A LONG-ERM PERSPECIVE H CENURY INERES RAES HISORIC INERES RAE VOLAILIY HISORIC CURVE SHAPES INFLAION EQUIY REURNS MARKE DAA UNDERSANDING HE ERM SRUCURE SWAPION IMPLIED VOLAILIY EQUIY IMPLIED VOLAILIY EXPER OPINION A CALIBRAION A PLAUSIBLE PARAMEER CHOICE SUMMARY SAISICS : SAMPLE MEAN REURNS & SANDARD DEVIAIONS EQUIIES SHOR-ERM INERES RAES & CASH REURNS YEAR CONVENIONAL BOND YIELDS INDEX-LINKED BOND YIELDS NOMINAL INERES RAE ERM-SRUCURE INFLAION INER-RELAIONSHIPS BEWEEN INFLAION, BOND YIELDS & EQUIY REURNS A COMPARISON WIH HE WILKIE MODEL SOME GENERAL OBSERVAIONS REPRESENAION MEAN REVERSION EQUIY MODEL... 64

4 8.5 ERM SRUCURE MODEL INFLAION DERIVAIVE PRICING EXENSIONS O HE BASIC MODEL FOREIGN EQUIY & PROPERY CREDI RISK & CORPORAE BONDS FOREIGN EXCHANGE & FOREIGN ERM SRUCURES EQUIY MEAN REVERSION MORALIY DERIVAIVE PRICING / CONINGEN CLAIMS VALUAION CONCLUSION... 7 APPENDIX A INCREMENING HE ERM SRUCURE... 7 APPENDIX B CALCULAING COVARIANCE ERM IN NOMINAL ERM SRUCURE APPENDIX C WILKIE PARAMEERS USED IN SECION ABSRAC he epo povides a specificaion fo a sochasic model fo equiy euns, inflaion and he em sucues of eal and nominal inees aes ogehe wih a discussion of he possible appoaches o paamee selecion. We conas he model s oupu wih a ypical calibaion of he Wilkie invesmen model.

5 . INRODUCION Wha is i ha makes he ole of financial inemediaies so special? Suely, some pa of he answe o his quesion is he financial inemediay s objecive of pooling and managing isks on behalf of lage goups of individuals. In common wih ohe financial inemediaies, life insuance companies ae in he isk managemen business. he isk exposues accumulaed by he shaeholdes and policyholdes of oday s life companies come in many diffeen foms. In he pas, unceainy in financial plans has be assessed eihe by expe inuiion o alenaively by developing a small numbe of handcafed wha if? scenaios. Such scenaio analysis can be exemely valuable in siuaions whee hee ae only a small numbe of key souces of isk. Unfounaely, in a siuaion whee hee ae many souces of isk o whee he dimension of he poblem is lage, building he scenaios by hand becomes impacical. he isk manage of 5 yeas ago would be asonished by he ange of ools available o his moden counepas. I is now inceasingly common pacice o use a compue o geneae he scenaios using a sochasic model. Mone-Calo MC simulaion echniques can be used o geneae vey lage numbes of scenaios in ode o undesand he poenial behaviou of financial poducs and ohe eniies in a wold of unceainy and unde vaious saegy opions available o he planne. Insead of esing ou a handful of possible oucomes, he planne can es ou lieally housands of possible fuues. In his noe we will focus on one specific model ou of many ineesing possible candidaes. We shall ackle he difficul poblem of how o simulae consisen fuue pahs fo equiy euns, dividend yields, inflaion and complee eal and nominal em sucues. I is woh emphasising ha hee ae pleny of models aound ha deal adequaely wih he nominal em sucue. Hee we shall aim o econcile he behaviou of he inflaion ae wih boh eal and nominal inees aes. he model pesened has some aacive feaues fo he puposes of analysing ceain classes of poblem, bu we do no claim ha i is pefec. We will exploe he issue of how o calibae his model and illusae wo calibaions. he illusaions povided have been judged o be useful by some people. You may pefe a diffeen calibaion. As we shall see, as always, i is necessay o sike a balance beween ou ambiion o make he model as ealisic as possible and a need o keep he model simple. Whee we sike he balance will depend upon a numbe of consideaions: he specific applicaion of he model as well as he needs and sophisicaion of model uses. 5

6 his epo is ahe longe han we had planned. I would have been saighfowad o wie down ou fancy equaions and leave he eade o apply he model. Insead, we have se ou o explain why we believe modelling is useful a all and wha so of poblems ae being analysed wih sochasic models secion. In secion 3, backgound is povided by highlighing some exaodinay changes ha have aken place ove he pas wo decades. In he following secion we lis some popeies of good models. Secion 5 ses ou he specificaion fo he model and he following wo secions povide a discussion of calibaion saegies and wo specific calibaions. Secion 8 conass he model wih he Wilkie invesmen model. Finally, in secions 9 and 0, we biefly discuss possible exensions and se ou some bief conclusions. Much of he pesenaion ha follows is infomal. Ou objecive is o give he eade insighs ino he geneal poblem of sochasic model building as well as a descipion of a pacical ool. We do no peend ha he model pesened in his epo is complee no ha he calibaion could no be beeed in some especs. Rahe his is ou saing poin fo some sochasic invesigaions. he ideas pesened have been developed ove seveal yeas and wih many misakes and blind alleys along he way. Le us begin by addessing a basic quesion: why build sochasic models a all? 6

7 . WHY BUILD MODELS?. WHA LESSONS FROM HE PAS? Life insuance companies ae in he business of managing isk on behalf of lage goups of individuals ove vey long planning hoizons. Some of hese isks can be divesified away whils ohes mus be caied by eihe policyholdes o shaeholdes. he financial poducs sold by life assues conain guaanees of numeous vaieies. Life insuance is a isk managemen business. Life company manages have expeienced a aumaic decade failing o adequaely undesand and manage isk acoss a ange of diffeen poducs. he lis is dismal and familia: Some wih-pofis pomises made o policyholdes in he lae 980s and ealy 990s look hopelessly opimisic in an envionmen of nomal bu supising low inflaion. Annuiy opions offeed in he 970s and 980s have poven cosly as a consequence of unanicipaed falls in nominal inees aes and supising impovemens in moaliy. he basic financial posiion and funding saegy of pension funds wo vey disinc popeies emain confused amid a hee-way debae among acuaies, accounans and egulaos. I is pobably fai o say ha fo many life companies hese poblems emain unesolved. he qualiy of undesanding of he poblems and saegies seleced o manage hem is vaiable.. WHA CURREN PROBLEMS DO ACUARIES & FINANCIAL PLANNERS FACE? Is his he end of ou lis? Sadly no. he indusy needs o face up o a numbe of ongoing and new poblems ove he coming yeas: Wih-pofis poducs emain unde scuiny mos ecenly fom egulaos. Of couse, he cach-all wih-pofis coves a wide ange of poducs wih vey diffeen guaanees and diffeen saegies fo smoohing euns acoss successive geneaions of mauing policies. Poduc povides have suggled unsuccessfully so fa as hei end-cusome is concened o communicae he naue of he poduc. hee exiss a leas wo impoan challenges: fis, o make he poduc o is eplacemen moe anspaen; second, o manage and pice he isks caied by policyholdes and shaeholdes in an appopiae way. he ecen changes o egulaions mean ha he capial equied o suppo uniised wih-pofi business is now moe sensiive o he paen of deliveed asse euns. Alhough he move fom defined benefi owads defined conibuion pensions aangemens including Sakeholde shifs he buden of isk-beaing fom he sponso o he individual save, he isk managemen challenge emains. he cuen geneaion of pension saves will bea fa geae isk han hei paens. 7

8 A pesen hey ae pooly equipped o negoiae he complicaed ade-off beween eiemen benefis, conibuion levels and he age a which hey can finally affod o sop woking. Pension dawdown poducs and ohe pos-eiemen poducs can expose saves o a poen mixue of invesmen isk and moaliy isk ha few ae well equipped o undesand. So-called income poducs come in all shapes and sizes. Many of hese poducs can be expeced in a saisical sense o educe a save s capial someimes by a maeial amoun. As in he pas, i seems likely hese isk exposues will only be popely appeciaed by saves and issues and egulaos afe a poduc failue. he common elemen in all of hese siuaions is isk. Savings poduc povides ae in he isk managemen business..3 WHA FUURE PROBLEMS? We can only guess a he isk managemen challenges o be faced by financial inemediaies in he fuue. In he ligh of pas expeience, i seems likely ha many of he poblems oulined above will ake many yeas o bing unde effecive conol. So long as he life indusy fails o embace fully he new isk managemen echnologies i is likely ha some unanicipaed combinaion of economic, make and demogaphic change will igge anohe ound of poduc failues. As in he pas, he esul will be damage o povide and egulao epuaions and losses of shaeholde and policyholde capial. 8

9 3. WHERE HAVE WE COME FROM? 3. BACKGROUND I is insucive o highligh wo impoan ends ove he pas 0 yeas: i. Fis, exaodinay innovaion in compue echnology has aken place. Slide ules, log ables and punched compue cads have been eplaced wih unimaginably poweful deskop compues and sofwae. hese ools mean ha calculaions which wee unhinkable 0 yeas ago ae now poenially ouine. Pehaps even moe impoanly, he means of displaying infomaion is now eally only limied by he analys s imaginaion. I is woh poining ou ha some financial insiuions paiculaly he invesmen banks have made subsanial invesmens in his new echnology in ode o enhance hei isk managemen capabiliies. ii. Second, a huge volume of eseach has been geneaed by financial academic eseaches and financial paciiones including acuaies. I is impoan o undesand ha his eseach effo has been moivaed by some quie diffeen needs: ades looking fo impoved echniques fo picing, ading and hedging a ange of new financial insumens. Alhough opions conacs have been aound fo many cenuies, he publicaion of he Black-Scholes model in 973 povided a spu owads apid innovaion in deivaive makes. Fo inees aes in paicula, hee now exiss a vas lieaue of models of vaying degees of complexiy. Economiss have developed models lagely fo he pupose of foecasing and policy-making. Long-em financial plannes including acuaies mus combine muliple souces of unceainy geneally ove vey long hoizons compaed o ohe uses of financial models. I is pobably fai o say ha mos academic wok ends o deal only wih pas of he poblem he acuay is ineesed in. hee is much deailed wok on equiy pice behaviou, on inees ae modelling and on inflaion modelling. Howeve, hee is vey lile which pus all of he componens ogehe wihin a consisen famewok. he fundamenal ask of he long-em financial planne is o undesand he join behaviou of hese vaiables and ohes on he poduc o business unde scuiny. 9

10 3. HE WILKIE MODEL he Wilkie model was oiginally developed in he lae 970s agains a backgound of high and volaile inflaion and he excepional UK equiy make volailiy of 974/5. I was exended in 995. Unlike he wok of mainseam academics, Pofesso Wilkie s model did ackle he difficul poblem of how o pu ogehe a model fo inflaion, eal and nominal euns on equiies and bonds and hei yields. Equiy yields which coninue o play an impoan pa in acuaial analysis ae pominen in he model. hey ae only of passing inees o he economis who ends o focus on pice and eun no how i happens o be packaged. he Wilkie model is elaively saighfowad o implemen. As a consequence, he model has been widely used by UK-based acuaies ove he pas wo decades and has se a benchmak agains which any ohe poposed appoach needs o be judged. Howeve, we believe ha hee ae some seious poblems wih he Wilkie model. Wha is moe, he huge developmen in hinking in mainseam academia means ha hee ae now some almos eady-made ools available o fix he shocomings of he Wilkie model. Ou aim is o show how his can be achieved whils avoiding he mind-boggling complexiy ha seems o be a chaaceisic of many of he models poposed in his aea. We do aim o eain one of he pimay aacions of he Wilkie model is ease of implemenaion. o ha end a woking vesion of he model pesened in his epo will be made available on he Baie & Hibbe web sie his model is disibued unde he GNU Public licence. Exended vesions of he model ae available on a commecial basis. Repo of he Mauiy Guaanees Woking Pay, A.D. Wilkie, Jounal of he Insiue of Acuaies, 07, Pa II, No. 435; A Sochasic Invesmen Model fo Acuaial Use, A.D. Wilkie, ansacions of he Faculy Of Acuaies No. 68, Vol 39 Pa 3; Moe on A Sochasic Model fo Acuaial Use, A.D. Wilkie, Biish Acuaial Jounal, 0

11 4. SOME PROPERIES OF GOOD MODELS he boad objecive of he isk analys is o povide insigh acoss a wide ange of poblems faced by business manages, poduc designes, egulaos, cusomes and hei advises ino he impac of a ange of candidae financial saegy choices. Fo example, when he analys eviews business wien wih aaching annuiy opions, business manages will wan o eview a ange of policy opions spanning bonus policy, invesmen policy as well as hedging and einsuance soluions. In ode o undesand and communicae analysis, a model can be vey valuable. he model is inended o be a cu-down, simplified vesion of ealiy ha capues he essenial feaues of he poblem and aids undesanding. I simply is no plausible o ague ha hee is a single model ha can mee he equiemens of he isk analys acoss all possible poblems. Rahe he analys should aim o build a libay of models ha enable him o ackle he diffeen ypes of poblem ha he is faced wih. So, models mus be seleced, bu how? Wha cieia should he analys use o pick a paicula model? We have lised below some of he aibues ha we hink ae impoan in good models. he lis is no inended o be complee. Some of he cieia on he lis pobably ovelap wih each ohe. As you migh have guessed, i uns ou o be vey difficul o mee all of he cieia simulaneously. We aely find models which pass all of he ess. 4. REPRESENAIVENESS he model should aim o povide a good epesenaion of he financial asses conained in he model. he model should mimic he behaviou of eal-wold financial asses by capuing hei mos impoan chaaceisics. If he model is used o geneae Mone- Calo scenaios, we migh expec ha an expe who scuinises he model oupu o be able o say: Yes each of you scenaios looks plausible and he fequencies assigned o paicula oucomes look easonable. his es coves numeous chaaceisics of asse behaviou he shape of disibuions a diffeen ime hoizons as well as he elaionships beween he vaiables in he model. 4. ECONOMIC INERPREAION he behaviou of asses wihin he model should be consisen wih geneally-acceped economic pinciples. he mos fequen demand is ha a model should be abiagefee. Since we do no expec o obseve sysemaic oppouniies fo abiage in he eal wold, i seems sensible o exclude hem fom models. he no fee lunch ule seems like a good one fo he puposes of modelling. Howeve, i is impoan o appeciae ha implemening he ule ofen comes a a pice. hee may be imes when he modelle will be pepaed o allow some limied abiage ino a model in exchange fo some ohe benefi.

12 A fuhe consideaion elaes o he join behaviou of model vaiables. Again, we expec o see some consisency beween economic pinciples and model behaviou. I is woh noing ha hee ae some impoan popeies of financial asse behaviou on which hee is no clea consensus among economiss. he pickly opic of equiy make mean evesion is a good example ha we will eun o. 4.3 PARSIMONY Keep i simple. Models should be as simple as we can make whils eaining he mos impoan feaues of he poblem. I is clealy ofen difficul o judge when complexiy is eally needed someimes some faco can have a big impac on esuls whils in ohe siuaions i only has a minimal effec. Complexiy mus be balanced. hee is no sense in modelling one aspec of a poblem in mind-boggling deail, and hen making boad bush assumpions in ohe aeas. he model will sand on is weakes assumpion. hee is anohe eason why complexiy should be avoided. A complex model, which ies o mimic as much eal-wold complexiy as he modelle can capue, can ceae he illusion ha we eally can model eveyhing. Someimes his illusion fools he modelle as well as his unlucky audience. 4.4 RANSPARENCY Unless we can explain how he basic model woks in a few minues i will be difficul o gain he confidence of non-expes. Success hee will depend heavily on he qualiy of communicaion. he esuls poduced fom he model should be displayed in clea gaphical fomas wheeve possible. 4.5 EVOLUION Nohing complex can be designed and buil in a single life. Anyhing complicaed mus be allowed o evolve ove a numbe of lifeimes. his applies whehe you se ou o build a Boeing 747 aeoplane o a financial model of a wih-pofis savings conac. 4.6 IMPLEMENAION OOLS Financial models geneally combine a se of ules fo descibing how some payoff o popey of inees is deemined wih a descipion of he behaviou a se of sochasic vaiables ha deemine he payoffs. he models can be implemened in diffeen ways. Implemenaion ools fall ino a numbe of classes: i. Analyic calculaions whee i is possible o find a mahemaical funcion o descibe he vaiables of inees. his is geneally only possible fo a vey limied se of poblems. ii. iii. Hisoical back-ess ae pefomed by using pas daa on euns, yields ec.. wih he model sucue and implicily assuming ha he fuue will be like he pas peiod seleced. Scenaio analysis deeminisic simulaion / sensiiviy analysis whee he modelle maps ou by hand a seies of scenaios of inees.

13 iv. ee-building echniques whee he scenaios ae buil up in he fom of a ee. his is geneally only feasible whee a vey small numbe of sochasic facos influence he poblem. v. Mone-Calo sochasic simulaion can be used o geneae vey lage numbes of plausible scenaios using a compue. In siuaions whee hee is pah dependency, Mone Calo echniques can be paiculaly valuable 3. hee ae advanages and disadvanages associaed wih each of hese appoaches. Clealy, if a model can be implemened in diffeen ways i povides inceased flexibiliy o he modelle. In pacice, models may be a mixue of Mone-Calo, analyic and eebased componens. Fo he eal-wold poblems faced by acuaies, he flexibiliy and inuiive pesenaion offeed by Mone-Calo echniques mean ha i will emain he focus of ou aenion in he emainde of his epo. Mone-Calo simulaion can be used in siuaions whee we believe ha we can say somehing sensible abou he facos ha affec a poblem, bu we don know wha will happen when we pu hese facos ogehe. Road affic enginees have a pey good idea of how cas and dives behave how quickly hey bake and acceleae ec... hey know how oads ae laid ou and he sequences of affic lighs. Bu i is impossible o capue all of his in a mahemaical equaion ha will pedic how affic will behave. he mahs is oo complicaed. Does his mean ha oad affic enginees can pedic how affic will behave when hey fiddle wih a affic ligh sequence? No a all. hey use compue simulaions. he simulaions how up mos of he feaues of eal affic - bolenecks, queues, sudden empy oads ec.. hey allow he enginee o see how changes in some pa of he sysem will affec is oveall behaviou. 3 Whee esuls depend on he pah aken by financial vaiables ove he planning hoizon no simply by whee hey end up we would say ha a esul is pah-dependen. A good example is a wih-pofis savings conac whee he pah of equiy euns can affec he final payoff. 3

14 5. AN ALERNAIVE ASSE MODEL In his secion we descibe a sochasic asse model ha can be used fo long-em financial pojecions. We believe ha he model has a numbe of aacive popeies: he model deals explicily wih he economic elaionship beween inflaion, inflaion expecaions, eal inees aes and nominal inees aes. I poduces a complee and consisen em sucue fo eal and nominal inees aes wih ich vaiaion in boh he level and shape of he yield cuves geneaed. he model fo equiy euns can be used o geneae he negaive skewness and kuosis fa ails ha ae a chaaceisic of eal-wold equiy eun disibuions. Equiy yields and dividends ae geneaed in a naual way. he basic se-up of he equiy model does no incopoae mean evesion. We believe ha his is a puden saing poin fo long-em financial planning puposes. he model is easy o implemen wih analyic expessions available fo discoun bond pices. I is possible o exend he model elaively easily beyond he basic applicaion pesened hee. We biefly discuss exensions in secion 9. he model is compised of a numbe of componen pas ha ae diven by a se of sochasic dives. Le us begin by eviewing he model fo he em sucue of inees aes. 5. HE ERM SRUCURE MODEL he behaviou of he eal-wold em sucue is eviewed in deail in secion 6. A his poin i is wohwhile highlighing he challenge ahead. hee ae many models ha allow he analys o diecly mimic he behaviou of he nominal em sucue. Hee, we aim o deal wih boh eal and nominal inees aes in a consisen way by explicily modelling he link beween inflaion expecaions and nominal yields. EXHIBI 5.: INFLAION & HE SHOR-ERM INERES RAE 30% 5% 0% Sho-em Inees Rae Inflaion Rae 5% 0% 5% 0% Yea 4

15 Exhibi 5. illusaes he song linkage beween inflaion and nominal inees aes obseved ove he pas fou decades. In he model we popose, he basic idea is o build a em sucue fo nominal inees aes fom wo sepaae componens: A em sucue fo eal inees aes. You can hink of his as he index-linked yield cuve. In he model we deal wih he yield on eal inflaion-poeced discoun bonds 4. A model fo inflaion ha allows us also o model he inflaion expecaions of invesos ove diffeen hoizons. I is implicily assumed ha invesos undesand he pocess geneaing inflaion and adjus hei expecaions fo fuue inflaion in a manne ha is consisen wih he expeienced inflaion paen. he wo em sucues ae combined o fom he nominal em sucue wih allowances fo any assumed coelaion beween inflaion and eal inees ae changes and fo any isk pemium associaed wih bond em and inflaion isk. Le us now eview each of hese componens in un. 5.. REAL INERES RAES / -FACOR HULL-WHIE A huge lieaue exiss on abiage-fee em sucue models. Alhough much of his wok has been moivaed by a need o pice inees ae deivaives, many of he conibuions ae diecly applicable o long-em planning and acuaial wok. Some of he lieaue is highly mahemaical and fa fom anspaen. We have made use of an exension o one of he fis sochasic abiage-fee em sucue models he Vasicek model 5. he Vasicek model specifies a coninuous-ime mean-eveing sochasic pocess fo he sho-em inees ae, and hen infes fowad aes and spo aes fom he expeced fuue pah of he sho ae wih an allowance fo any specified isk pemium invesos may demand fo holding longemauiy bonds elaive o cash. he model can be exended wih he addiion of a second sochasic faco. his faco allows he mean evesion level fo he sho ae also o follow a mean-eveing sochasic pocess. Fo hose who ae familia wih he fis-ode auoegessive pocess used by David Wilkie o model inflaion, ou model uses a simila pocess o model he sho-em eal ae, bu addiionally wih a mean evesion level which is also auoegessive 6. 4 We will use he ems discoun bond and zeo-coupon bond o mean he same hing. he -yea spo ae of inees will mean he annualised coninuously compounded inees ae on a -yea discoun bond i.e. logp/, whee P is he discoun bond pice and he em in yeas. 5 See An Equilibium Chaaceizaion of he em Sucue, O.A. Vasicek, Jounal of Financial Economics, We ae gaeful o Andew Cains of Heio Wa Univesiy who povided assisance in he mahemaical developmen of he model. 5

16 his model we have seleced can be shown o be a special case of anohe published em sucue model a -faco model descibed by Hull & Whie he equaions govening he changes in he eal sho ae ae shown below: d = d = d + σ dz µ d + σ dz whee: = he eal sho ae a ime. = he mean evesion level fo he eal sho ae a ime. = he auoegessive paamee fo he eal sho ae pocess. = he auoegessive paamee fo he eal sho ae mean evesion pocess. σ = he annualised volailiy sandad deviaion of he eal sho ae. σ = he annualised volailiy sandad deviaion of he eal sho ae mean evesion level. µ = he mean evesion level fo. g = a paamee o conol he em pemium in eal bond pices. dz = he shock o he eal sho ae pocess which is disibued Ng d, d dz = he shock o he eal sho ae mean evesion pocess which is disibued Ng d, d b = lowe bound fo he eal sho ae,. b = lowe bound fo he eal sho ae mean evesion level,. hese equaions can appea mahemaically ahe dauning, bu hey ae eally jus he coninuous-ime equivalens of wo fis-ode auoegessive ime-seies pocesses and, as we will see, can be implemened in a simila way. Noe ha hey imply ha eal inees aes have a nomal disibuion allowing he possibiliy of negaive eal aes. he poenial fo an inees ae model o geneae negaive aes is nomally viewed as an inconvenience. Howeve, since eal inees aes ae he subjec of he model, i is ofen agued ha he model should be capable of geneaing some negaive aes. I is impoan o undesand ha he enie em sucue fo eal inees aes is implied by hese equaions. hey allow us o calculae an expeced pah fo fuue sho aes. his pah is naually elaed o cuen fowad aes wih an adjusmen fo any assumed isk pemium. Exhibi 5. gives an appoximae idea of how he model woks. In he example shown in he cha, he iniial sochasic mean-evesion level ploed in geen begins a 0% and is pojeced o be pulled ove ime owads is equilibium level. he eal sho ae,, is pojeced o be pulled owads he ime-vaying evesion level and in his case aces ou a humped pah wih a peak of nealy 8% afe 3 yeas. Cudely, and fo he ime being ignoing isk avesion, you can hink of his pah as he se of insananeous fowad inees aes i.e. he fowad inees ae available fo a vey sho ime 7 See Numeical Pocedues Fo Implemening em Sucue Models II: wo-faco Models, Jounal of Deivaives, Wine 994, and B&H echnical Noe 000/04 F Vasicek as a Special Case of Hull & Whie. 6

17 hoizon. You can hen infe spo aes zeo-coupon aes fo any mauiy by combining ogehe he appopiae fowad aes. In he case illusaed below you aive a a humped em sucue wih a peak a aound 0 yeas. EXHIBI 5.: HE BASIC IDEA ~ -FACOR VASICEK MODEL Rae of Inees ~ Fowad 0.07 Spo Rae 0.06 µ em Yeas Memo: FVasicek_PojecRaePah.xls he exhibi gives some idea of how changes in and can affec he shape of he cuve. he σ and σ paamees will influence vaiabiliy of aes. he auoegessive paamees can be seen o have an impac on he cuvaue poduced by he model. In pacice hings ae a bi moe complicaed han suggesed by exhibi 5.. Alhough his is a coninuous-ime model, i can sill be applied in discee ime wihou he need fo any appoximaions fom he above equaions i is possible o calculae he expeced value and vaiance of and ove any ime incemen. We can hen sample fom hese disibuions o incemen he model in discee ime in a manne exacly consisen wih he coninuous-ime model. An addiional paamee, g, is inoduced o deemine he degee o which long-em eal bond euns exceed he sho eal ae i.e. i is possible o inoduce a em pemium ino he model. We have o make a small adjusmen o how we incemen he inees aes when g is non-zeo. Appendix A descibes he equaions ha ae used fo incemening he em sucue. 7

18 8 he equaions se ou above can be used o deive analyic expessions fo he eal spo ae and eal fowad ae a any em. he equaions deemine he enie eal em sucue. he pice of a zeo-coupon bond a ime ha pays one uni in eal ems i.e. poeced fom inflaion a ime is given by he picing equaion: P eal, = exp [ A - - B - - B - ] whee : = S e B s = S S e e B s = + σ σ σ µ µ 4 S S S e e e B B B B B s s s s s s s A s We ecognise ha his is quie a big fomula, bu you only need o ype i ino he compue once. Of couse, once we have obained pices fo eal discoun bonds, i is hen possible o pice any eal coupon bond which can be piced as if i wee a package of discoun bonds each elemen of he package coesponding o one of he coupon o edempion paymens of he bond. We can also calculae he coninuously compounded yield a ime fo mauiy, R, =- log{ P eal, } / -

19 5.. HE INFLAION MODEL Le us now un ou aenion o inflaion. he idea is o use a model fo he behaviou of he inflaion ae o geneae a em sucue fo inflaion expecaions, which can hen be combined wih he eal inees ae em sucue o build a em sucue fo nominal inees aes. Exacly he same model sucue is used o model he behaviou of he sho-em inflaion ae as was used in he pevious secion fo eal sho-em inees aes. Of couse, we may choose o use diffeen model paamees when he wo models ae used in pacice. So, he equaions govening he pah of he sho-em inflaion ae ae: dq dq = = q q q q d + σ dz µ q d + σ dz q q q q q whee: q = he insananeous ae of inflaion a ime. q = he mean evesion level fo he insananeous inflaion ae a ime. q = he auoegessive paamee fo he inflaion ae pocess. q = he auoegessive paamee fo he inflaion ae mean evesion pocess. σ q = he annualised volailiy sandad deviaion of he insananeous inflaion ae. σ q = he annualised volailiy sandad deviaion of he inflaion ae mean evesion level. µ q = he mean evesion level fo q. g q = a paamee o conol he inflaion isk pemium in nominal bonds elaive o index-linked bonds. dz q = he shock o he inflaion ae pocess which is disibued Ng q d, d dz q = he shock o he inflaion ae mean evesion pocess which is disibued Ng q d, d b q = lowe bound fo he inflaion ae, q. b q = lowe bound fo long-em inflaion expecaions, q. As fo he sho-em eal inees ae, he inflaion ae is assumed o be mean-eveing and nomally disibued. Using wo facos o dive he inflaion pocess, ahe han a single faco as in he Wilkie model has some majo advanages. Fisly, i means ha changes in inflaion ae expecaions a diffeen ems can have a coelaion less han so sho-em inflaion aes and long-un expecaions do no always have o move in lock-sep. Secondly, i allows geae conol ove how he volailiy of inflaion decays. In a single faco model, i may no be possible o obain a sensible disibuion fo longem inflaion wihou unfeasibly high sho-em volailiy and vice vesa. he inflaion pocess can be incemened in he same way as fo eal aes and is descibed in Appendix A. 9

20 5..3 INFLAION EXPECAIONS In exacly he same way ha we use an analyic expession o deive he eal inees ae em sucue a any ime given he values of and, a em sucue fo inflaion expecaions can be infeed fom he cuen insananeous inflaion ae, q, and he value of q. We can heefoe use he picing equaion of secion 5.. o calculae he value of a discoun bond a ime ha pays one uni a ime and is discouned only wih espec o inflaion expecaions. We will efe o his quaniy as P inf,. he equivalen yield is R q, =- log{ P inf, }/ - Fuhe, in he same way ha i is possible o embed a em pemium ino he eal inees ae em sucue, he inflaion expecaions em sucue can also incopoae a isk pemium eflecing he addiional eun which may be equied by invesos o induce hem o inves in nominal, ahe han index-linked bonds. his is someimes called he inflaion isk pemium NOMINAL ERM SRUCURE Amed wih he eal inees ae and inflaion expecaions em sucues, i is now possible o combine hem ogehe o obain a nominal inees ae em sucue. Whee movemens in insananeous eal inees aes and inflaion aes ae independen, his sage is ivial he nominal spo ae is simply equal o he sum of he eal spo ae and he annualised inflaion expecaion ove he elevan peiod. Alenaively, he pice of a nominal discoun bond is obained as he poduc of he eal and inflaion discoun bond pices: P nom, = P eal, P inf, whee P eal, and P inf, ae as descibed in secions 5.. and In pacice, innovaions in sho eal aes and inflaion may no be independen. Fo example, we migh believe ha ises in spo inflaion ae ypically associaed wih inceases in eal aes as policymakes aemp o squeeze inflaionay gowh. he model can accommodae an assumpion fo he coelaion beween sho eal aes and inflaion hough a small addiional covaiance em in he zeo-coupon bond pice equaion: P nom, = P eal, P inf,+ρ.sq[vaexp{-r,}.vaexp{-r q,}] whee: ρ = he coelaion beween he shock o he eal sho ae and he insananeous inflaion ae, dz and dz q 8. R, = s ds R, = q s ds q 8 echnically, his coelaion mus also be applied o dz and dz q. 0

21 Analyic expessions fo he vaiances in he covaiance em ae given in Appendix B. Noe ha he nominal em sucue has wo sepaae isk pemiums eihe o boh of which can be se o zeo a em pemium and an inflaion isk pemium. I is possible o se up he model so ha a long-daed index-linked bond will have a highe expeced eun han a shoe-daed one, and a long-daed nominal bond will have a highe expeced eun han long eal bonds and sho-em nominal bonds NEGAIVE NOMINAL INERES RAES I mus be ecognised ha he way he model is specified does no guaanee ha nominal inees aes will always be posiive. Since inflaion, inflaion expecaions and eal inees aes can ake negaive values, negaive nominal aes of some magniude will feaue in he model. he fequency of hese negaive aes will depend on he paamees seleced. Faced wih his poenial poblem, he modelle can adop a numbe of alenaive appoaches: i. Use he negaive aes. If he inenal mahemaical consisency of he model is paiculaly valued, his may be appopiae. ii. iii. iv. If he main pupose of he model is o geneae plausible scenaios, an alenaive appoach is o discad he scenaios in which negaive aes occu. his appeas saighfowad, bu he discaded scenaios will have an impac on some global chaaceisics of he model ha mus be undesood and possibly adjused fo. A simila appoach is o consain he model in a way ha guaanees posiive aes o ensues ha negaive aes appea wih only a vey low fequency. Fo he model specified above his migh be achieved by seing lowe bounds on he value of he sochasic facos. Find a bee model. We have no, ye.

22 5. HE EQUIY MODEL hee is a huge aay of models of vaying degees of complexiy designed o descibe he equiy euns pocess. he model ha is mos widely used by financial economiss is he lognomal model fo equiy euns. I uns ou ha his povides a easonable, bu impefec descipion of he equiy euns pocess. EXHIBI 5.3: END-YEAR & MID-YEAR UK EQUIY ANNUAL EXCESS REURNS Numbe of Yeas 8 6 Dec-o-Dec 4 June-o-June % -60% -40% -0% +0% +0% +40% +60% +80% +00% Log Excess Reun memo: ChaAnnualUKEquiyDisibuions.xls he cha above shows he disibuion of log excess euns 9 fo UK equiies since 900 and a nomal disibuion wih a mean and sandad deviaion esimaed using he daa shown excluding 974/5. he picue suggess ha he nomal disibuion does a easonable job mos of he ime, bu occasionally fails badly. In common wih mos financial make vaiables, he disibuion fo equiy pice changes shows fa ails saisicians measue his chaaceisic of he disibuion wih he kuosis saisic. EXHIBI 5.4: DISRIBUION OF MONHLY EXCESS REURNS COMPARED O NORMAL Numbe of Monhs % -0% -5% -0% -5% 0% 5% 0% 5% 0% 5% Monhly Excess Reun 9 he ae of eun in excess of he sho-em inees ae.

23 Exhibi 5.4 illusaes he vey maked fa ails of he monhly excess euns disibuion. he kuosis of he euns disibuion ends o incease as he measuemen peiod fo euns is shoened. his is no a new conclusion. his feaue of financial makes is familia o mos analyss. he widely-used nomal only povides an appoximaion o he eal-wold behaviou of equiy euns. Fo some puposes his is fine, bu fo ohes paiculaly when we cae abou he ails of he disibuion i simply is no up o he job. EXHIBI 5.5: EXRAC FROM DISRIBUION OF MONHLY UK EQUIY PRICE CHANGES Sep-97 Dec-93 Sep-9 Ap-9 Dec-87 Feb-87 Jan-87 Ma-86 Jan-00 Feb-86 Sep-90 Aug-84 Aug-90 Jan-80 Oc-89 Sep-77 Ma-80 Jan-77 May-74 Nov-76 Dec-73 Jan-76 Jan-73 Ap-74 Aug-98 Jul-69 Nov-7 Nov-87 May-6 Oc-67 May-84 Jun-6 Aug-6 Feb-9 Oc-76 Sep-57 Aug-59 May-90 Jun-75 Nov-5 Ap-56 Nov-8 Jun-74 Jun-49 Ap-5 Jun-80 Sep-7 Feb-48 Jul-40 Ma-79 Ap-70 Aug-47 May-39 Aug-77 Feb-69 Sep-38 Oc-38 Ap-7 Sep-39 May-38 May-33 Ma-68 Dec-3 May-3 Aug-3 Oc-39 Jan-89 Nov-73 May-3 Ap-3 Nov-4 Oc-3 Aug-75 Ma-74 Sep-8 Jul-66 Nov-9 Oc- Ap- Jun-3 Oc-59 Dec-76 Oc-87 Jun-40 Aug-74 Nov-74 May-40 Nov-0 Feb- Oc-9 Jan-0 Jul-3 Ap-75 Feb-75 Jan % o -5.0% -5.0% o -.5% -.5% o -0.0% -0.0% o -7.5% -7.5% o -5.0% -5.0% o -.5% -.5% o -0.0% -0.0% o -7.5% 7.5% o 0.0% Exhibi 5.5 shows he disibuion of equiy pice changes fo monhs when he absolue size of he pice change exceeded 7.5%. Wha is ineesing abou his cha is ha he same yeas seem o appea seveal imes acoss he picue. Lage absolue changes in pice do no seem o be geneaed evenly ove ime. hey appea o be bunched ogehe duing peiods of make volailiy - such as 93-, 940, and 987. hee ae long gaps wihou lage absolue euns. One widely-acceped explanaion fo hese esuls is he noion ha he volailiy of euns changes ove ime. 0.0% o.5%.5% o 5.0% 5.0% o 7.5% 7.5% o 0.0% 0.0% o.5%.5% o 5.0% 5.0% and above he eally ineesing quesion is how we espond o he challenge posed by he exeme values ploed in he chas above. Should he equiy model be capable of geneaing equiy cashes wih he same fequency as his hisoic daa se? Alenaively, should we be pepaed o live wih an impefec model fo he sake of pasimony? As usual, he answe will depend on wha we wan o do wih he model. If he poblem unde invesigaion is vey sensiive o he pesence of exeme values, i may be wohwhile ensuing ha hey ae popely epesened by he model so long as we can agee on wha popely means. If he analysis is no especially sensiive o oulies, he lognomal disibuion may be pefecly adequae fo he job. 3

24 5.. HE MARKOV REGIME-SWICHING MODEL In ode o mimic hese impoan chaaceisics of equiy euns, a Makov egimeswiching model will be implemened. his ype of model is able o geneae euns disibuions ha ae consisen wih he popeies of he empiical daa. he basic idea is ha euns ae no dawn fom a single nomal disibuion; ahe hee ae wo disibuions a wok geneaing he euns obseved. he equiy euns disibuion is assumed o jump beween wo possible saes ove ime. hese saes ae ofen efeed o as egimes. A ansiion maix conols he pobabiliy of moving beween saes. We use his egime swiching appoach o model log equiy euns in excess of he log eun on a iskless asse. In he cuen implemenaion, we have used a defaul-fee sho-daed discoun bond o epesen he iskless asse. Fo he sake of convenience, we se he em of his bond o equal he ime incemen a which he undelying sochasic vaiables ae updaed fo he esuls pesened below his is one monh. his also means ha he iskless asse will behave vey much like cash, and we will efe o he 'cash eun' o 'isk-fee eun' synonymously. Of couse, he choice of iskless asse is abiay - alhough we have chosen a discoun bond wih a em of one-monh, i would be quie possible fo us o consuc he model using some ohe numeaie. hus he oal eun on equiies in a given peiod of lengh, E, is he sum of he eun on he sho em discoun bond, plus he excess eun on equiies, X: E = ln P nom, + X he excess eun on equiies, X has a Nomal disibuion, whee: X has mean µ Ε, and vaiance σ Ε, if he egime swiching model is in Sae X has mean µ Ε, and vaiance σ Ε, if he egime swiching model is in Sae he maix of ansiion pobabiliies which deemines how he equiy eun model swiches beween he wo saes can be denoed: P P = P whee: P P P = Pob{Model in Sae in peiod, + Model in Sae in peiod -, } P = Pob{Model in Sae in peiod, + Model in Sae in peiod -, } 4

25 Fo a ypical calibaion, in one egime equiy euns have a posiive mean and low vaiabiliy. In he second egime equiy euns ae on aveage lowe and exhibi much highe volailiy. Reuns end o be geneaed mos ofen in he benign sae, bu hee ae bous of high vaiabiliy geneaed by he second sae. Exhibi 5.6 gives an idea of wha he wo egimes migh look like if he model is calibaed o he daa shown above in exhibi 5.4. You can see ha jus ove half of euns will be geneaed fom he naow disibuion ploed wih a solid line. Negaive euns of 0% pe monh and below will be geneaed fom he second high-volailiy egime ploed wih whie cicles. One challenge fo he analys who calibaes his model is o decide how long he pocess will sick in he volaile sae: does he sysem visi he volaile egime fequenly bu quickly jump ou again o does i visi only occasionally and end o sick hee? he answe o his quesion will have implicaions fo he disibuion of euns a ohe hoizons. EXHIBI 5.6: REGIME SWICHING ~ HE BASIC IDEA Pobabiliy Densiy Low Vol / High ReunRegime High Vol / Low Reun Regime -5% -0% -5% -0% -5% 0% 5% 0% 5% 0% 5% Equiy Excess Reun In secion 7 a calibaion will be pesened fo a monhly model fo equiy euns in excess of sho-em inees aes. I is possible o calibae and un he egimeswiching model a any ime hoizon, alhough i mus be appeciaed ha i is no possible o adjus paamees developed fo example fo monhly euns o a - monh -sae model. Ove monhs hee ae acually 3 possible saes fo he - monh eun disibuion. 5

26 5.. EQUIY DIVIDEND YIELDS he log of he equiy yield is assumed o follow he coninuous-ime equivalen o a fis-ode auoegessive pocess AR wih long em mean µ y and dif paamee y, so ha: dlog {y} = y µ y - log{y} d + σ y.dz y whee: y = he equiy dividend yield a ime. y = he auoegessive paamee fo he log equiy dividend yield pocess. σ y = he annualised volailiy sandad deviaion of he log equiy dividend yield dz y = is a andom shock disibued N0,d. he model says ha if log{y- } is below he long em aveage µ y, hen log{y} will incease by appoximaely y imes he diffeence plus a andom shock disibued wih zeo mean and sandad deviaion of σ y. he sandad deviaion fo he log yield, σ y equals he equiy eun volailiy, i.e. σ y σ y = σ Ε, = σ Ε, if he equiy eun egime swiching model is in Sae if he equiy eun egime swiching model is in Sae Suppose we aleady know he oal asse eun, E, fo he ime peiod - o and he dividend yield a ime. If he equiy pice a ime is S, we can wie : S + S. y. / S- S So he dividend o income a ime : D = exp{e} = S-. exp{e} / + y. = S. y. So long as a high negaive coelaion is imposed on he shocks o he yield model and he equiy euns model, his specificaion poduces equiy yields ha move in he sho em wih pice changes. Ove longe peiods song equiy euns end o be followed by above-aveage dividend gowh as he dividend yield eves o mean and equiy declines will be followed by below-aveage dividend gowh o falling dividends. he model is simple and has a naual economic inepeaion. 6

27 6. SOME ALERNAIVE APPROACHES O CALIBRAION 6. INRODUCION In secion 5 a se of models was inoduced fo he puposes of modelling he behaviou of equiies, bonds and inflaion using Mone-Calo mehods. he models ae only half he soy. he scenaios geneaed by a sochasic model depend on boh he model sucue and he model paamees. In his secion we explain why calibaion is so difficul and hen se ou some paamee ses which we believe fom a sensible saing poin fo cuen sochasic invesigaions. Calibaion is somehing of a black a. he analys who faces up o he poblem of how o calibae his fancy new model migh conside a numbe of soluions. Le us conside hee alenaive appoaches. 6. EMPIRICAL DAA he mos obvious appoach o calibaion is o look a he pas behaviou of asses and ohe vaiables and o aemp o mach his wih he model. So, le us begin by eviewing some hisoic daa. 6.. A LONG-ERM PERSPECIVE Exhibi 6. plos UK long-em inees aes since 75. I shows a pofile ha falls wihin a ange beween.5% and 6.5% fo ove 00 yeas befoe he exaodinay expeience of he pas 50 yeas 0. Many invesos and financial analyss ae sill coming o ems wih he cuen low-inflaion envionmen. Headline UK inflaion was unning a an annual ae of less ha % pa a he end of 999. his is a siking conas o he peak aes of inflaion of ove 0% pa eached in 975 and 980 and he accompanying high levels of inees aes. Bu i is clea fom he cha ha his expeience is quie diffeen o any ohe peiod ove he pas 300 yeas. he picue shows us ha duing he nineeenh cenuy, having saed he cenuy a ove 6% owads he end of he Napoleonic Was govenmen bond yields neve exceeded 4% afe 830 and wee neve above 3.5% afe Fo inees, noe ha aes peaked in he eigheenh cenuy a jus below 9% in 7. 7

28 EXHIBI 6.: UK LONG-ERM INERES RAES ANNUAL RANGE OF VALUES Long-em Inees Rae 8% 7% 6% 5% 4% 3% % % 0% 9% 8% 7% 6% 5% 4% 3% % Noes: a Souce is A Hisoy of Inees Raes, Home & Sylla, 3 d Ediion b he annual ange of yields is ploed fo yeas afe 790. his so of picue begs an obvious quesion: If we calibae o hisoy, which peiod should be used he whole peiod, he low-inees ae ea befoe 945 o he 945/95 high inees ae peiod? Le us make one ohe poin. We ae no calibaing he model o plan ove he nex 300 yeas. Even fo vey long-em planning poblems he hoizon aely exends beyond 30 o 40 yeas. 8

29 An alenaive way of viewing he inees ae hisoy is o plo he fequency disibuion of aes. his povides a saighfowad way of measuing he fequency wih which aes have eached diffeen levels ove he 75-yea peiod. You can see ha he bulk of he disibuion falls beween % and 6%. he fequency wih which longem aes have exceeded 6% is.5% one yea in evey eigh. Raes have fallen beween 3% and 4% in ove 40% of he pas peiods analysed. EXHIBI 6.: Hisoic Fequency % 50% 45% 40% 35% 30% 5% 0% 5% 0% 5% 0% EMPIRICAL DISRIBUION OF UK LONG-ERM INERES RAES % % % 3% 4% 5% 6% 7% 8% 9% 0% % % 3% 4% 5% 6% 7% 8% 9% 0% Long-em Inees Rae 00% 90% 80% 70% 60% 50% 40% 30% 0% 0% 0% Cumulaive Fequency H CENURY INERES RAES Someone migh ague ha inees aes in he eigheenh cenuy have lile elevance oday. Indeed, he auho s calibaion of he widely used Wilkie model implicily suggess ha he pas cenuy is an appopiae peiod fo model calibaion. Exhibi 6.3 shows he hisoical disibuion fo end-monh sho aes since 98. I suggess ha he ange of plausible aes lies beween % he mos fequen hisoical obsevaion and 8% - he peak fo sho aes in he 970s. he disibuion looks disincly bimodal suggesing ha he daa was geneaed fom wo quie disinc peiods. Given he exaodinay pos-wa inflaion and he esuling impac on inees aes and economic policy-making, his should be no gea supise. hee ae echniques fo capuing his egime-swiching effec, bu we have no aemped o implemen hem in he inees ae model calibaions pesened in his noe. 9

30 EXHIBI 6.3: EMPIRICAL DISRIBUION SHOR RAES % 00% 90% Fequency 0% 5% 0% 5% 0% 0.0%.0% 4.0% 6.0% 8.0% 0.0%.0% 4.0% 6.0% 8.0% 0.0% 80% 70% 60% 50% 40% 30% 0% 0% Cumulaive Fequency 0% Conside a long-em ae of inees ha is ploed in he cha below. Fo efeence we have ploed he disibuion of end-monh long-em bond yields fom 98 o dae. Like he sho-em inees ae, hee is a suggesion ha he disibuion is bi-modal o i-modal as a esul of he vasly diffeen inflaion expeiences befoe and afe 945. EXHIBI 6.4: EMPIRICAL DISRIBUION LONG BOND YIELDS Fequency 5% 00% 90% 0% 80% 70% 5% 60% 50% 0% 40% 30% 5% 0% 0% 0% 0% 0.0%.0% 4.0% 6.0% 8.0% 0.0%.0% 4.0% 6.0% 8.0% 0.0% Cumulaive Fequency 30

31 6..3 HISORIC INERES RAE VOLAILIY Aside fom he uncondiional disibuion of aes ou esimae of he pobabiliies of aes a a long hoizon, any model should poduce plausible behaviou fo ae changes ove sho hoizons. Exhibi 6.5 abulaes he hisoic volailiy of inees aes ove he pas cenuy. wo volailiy measues ae shown. Fis, he annualised sandad deviaions of absolue ae changes ae shown fo 3-monh aes and fo a long-em bond yield. Sho aes have been moe vaiable han long-em aes ove his peiod wih he annual sandad deviaion aveaging ou a a lile above % pa. he boom of he able shows he sandad deviaion of ae changes measued in popoional ems which is he convenional way ades use o expess inees ae volailiy fo shoem insumens. EXHIBI 6.5: HISORIC ANNUALISED INERES RAE VOLAILIY Sho Rae ime Peiod 3M Long-em Rae Consols Yield / 0Y Gil Absolue Volailiy % 0.95% %.6% Popoional Volailiy % % % 4% I is ineesing o noe ha hese measues of volailiy do vay significanly when hey ae measued ove diffeen ime peiods. Exhibi 6.6 plos he olling 0-yea annualised sandad deviaion of ae changes fo sho aes and long-em bond yields. You can see ha ae volailiy iself vaies makedly ove ime. Again, he peak levels of volailiy occued in he 970s and 980s. EXHIBI 6.6: 3.5% ANNUALISED 0-YEAR SANDARD DEVIAION OF RAE CHANGES 0-Yea Inees Rae Volailiy 3.0%.5%.0%.5%.0% 0.5% 0.0% 3-Monh ae Long-em ae We need o ask whehe ou chosen model calibaion can and should mimic hese episodes. 3

32 6..4 HISORIC CURVE SHAPES Finally, when we eview pas daa on inees ae behaviou, i is ineesing o examine he shape of pas em sucues in addiion o he ohe popeies we have eviewed. As exhibi 6.7 illusaes, he em sucue can assume quie a wide ange of shapes. Ove he 6-yea peiod shown which eally is no vey long fo ou puposes we can obseve upwad and downwad-sloping cuves, humped cuves as well as some slighly sauce-shaped cuves. When a calibaion fo he model is seleced, we should ask whehe i is capable of poducing he ichness we can see in he eal-wold daa. EXHIBI 6.7: HISORIC SPO RAE CURVES END-JUNE & END-DECEMBER 6% 4% Spo Inees Rae % % 0% 8% 6% 4% % 0% em Yeas Souce: Bank of England 3

33 6..5 INFLAION Given he fundamenal link beween inflaion aes and inees aes i is no supising ha we face some vey simila issues when he inflaion daa is examined. Exhibi 6.8 shows he pah of inflaion aes his cenuy spanning wo wold was, he gea depession, exi fom he gold sandad and he pos-945 inflaion. EXHIBI 6.8: 30% 0 H CENURY ANNUAL INFLAION RAES Annual Rae of Inflaion 0% 0% 0% -0% -0% -30% I is ineesing o speculae on how much elevance his expeience eally has fo he analys who is concened wih foecasing he disibuion of inflaion fo he nex 0 yeas. Exhibi 6.9 suggess ha hee is a wide ange of values he analys could choose fo he aveage ae of inflaion, depending on which hisoic peiod is seleced o calibae a model. EXHIBI 6.9: SOME INFLAION AVERAGES Las 700 Yeas h Cenuy Las 50 Yeas Las 30 Yeas Las 0 Yeas Las 0 Yeas Cuen Inflaion Rae Annual Aveage Inflaion Rae % pa Souce : ONS, David Wilkie WhaInflaionCha.xls 33

34 EXHIBI 6.0: UK INFLAION & LONG-ERM INERES RAES PRE & POS 945 0% Long-em Inees Rae 8% 6% 4% % 0% 8% 6% 4% % % -5% -0% -5% -0% -5% 0% 5% 0% 5% 0% 5% Annual Inflaion Rae I is woh emembeing ha we do no simply wan o selec a long-em aveage fo ou model alhough is sucue means ha his is less impoan han fo a simple AR model, i is also desiable o undesand he elaionship beween inflaion and ohe vaiables. In exhibi 6.0 he pas elaionship beween long-em inees aes and inflaion is ploed. You can see ha in he peiod since 945 high aes of inflaion have ended o be accompanied by high long-em inees aes as bond invesos inflaion expecaions ae aised. Aguably, he high long-em bond yields of he 970s and 980s also included an embedded inflaion isk pemium. Inflaion also hus equiy euns. Exhibi 6. ells us ha equiies end o pefom elaively pooly in high-inflaion envionmens and when he ae of inflaion is acceleaing. EXHIBI 6.: UK EQUIY REAL REURNS & INFLAION Real Equiy Reun Real Bond Numbe of Reun Yeas Falling Pices 3.9%.% 9 Less han -% Pice Sabiliy 6.6% 4.5% 0 Beween -% and +% Low Inflaion.4% 9.5% 8 Beween % and 4% Modeae Inflaion 5.3% 4.% 9 Beween 4% and 8% High Inflaion -6.7% -8.4% 5 Geae han 8% Falling Inflaion Rae.0% 6.0% 4 Rising Inflaion Rae 3.% -.0% 40 34

35 EXHIBI 6.: ANNUALISED QUARERLY INFLAION RAE DIFFERENCES 5-YEAR PERIODS Annualised Sd Deviaion of Quaely Inflaion Rae Changes % % 0% 9% 8% 7% 6% 5% 4% 3% % % 0% ChaInflaionVolailiy.xls he vaiaion in inflaion fom peiod o peiod will feaue in he model. he cha above plos he annualised sandad deviaion of inflaion ae changes. I ells us ha he vaiabiliy of he inflaion ae measued in his way is a is lowes level fo ove 50 yeas. I is woh poining ou ha he ecen aveage of % p.a. is well below he oiginal o evised esimaes fo use wih he Wilkie model fo inflaion volailiy he QSD paamee EQUIY REURNS Some impoan popeies of equiy euns ae illusaed in secion 6.. Exhibi 6.3 summaises he saisical chaaceisics of excess euns calculaed a diffeen fequencies. EXHIBI 6.3: SUMMARY SAISICS FOR EXCESS REURNS O UK EQUIIES A VARIOUS FREQUENCIES Monhly Quaely Annual Annual excl Basic Saisics Numbe of Obsevaions Mean Sandad Deviaion Sandad Deviaion Annualised Skewness Relaive Kuosis Auocoelaion euns absolue euns squaed euns

36 he mos noable feaues of he daa ae: he mean log eun is almos 5% pa in excess of sho-em inees aes. he annualised sandad deviaion ises fom 5.5% pa a a monhly fequency o almos 9% pa a an annual fequency. Howeve, if he euns fom 97 o 980 ae dopped ou of he daa his annual volailiy dops o a lile ove 4% pa. he skewness saisic is used o assess he symmey of he disibuions. A sample dawn fom a nomal disibuion would be expeced o show skewness close o zeo. All of he esimaes ae negaive. he yeas 974/5 exe a big influence on he saisics. Noice how he annual auocoelaion esimaes change when we dop he 970s daa. kuosis is shown elaive o he nomal disibuion. All of he kuosis esimaes ae geae han ha expeced fom a nomal disibuion. I is vey clea ha he euns geneaing pocess is vey diffeen o he nomal disibuion. If he UK equiy make eally did confom o a nomal disibuion wih an annualised sandad deviaion of 8% p.a. wha would we expec o see? Saisical heoy ells us ha, ove he pas 00 yeas, we would expec o obseve pehaps one monhly excess euns of less han -5%. We have acually expeienced 7. 36

37 6.3 MARKE DAA hee is a second appoach ha an analys migh adop in ode o calibae a model. I is o use make infomaion o help find appopiae paamees. A poponen of he make-based appoach would ague ha make asse and deivaive pices conain infomaion ha can be used o guide he calibaion choice. I uns ou ha make pices mus be inepeed caefully when hey ae used in he calibaion pocess. Le us look a some examples of how make daa can be used UNDERSANDING HE ERM SRUCURE Befoe we discuss make-implied daa fo inees aes, i is wohwhile eviewing some of he heoies ha have been poposed o explain he shape of he em sucue. You view of hese heoies will bea on how you choose o inepe some of he mixed signals poduced by his appoach o calibaion. hee ae a numbe of classical heoies of he yield cuve. hese heoies aim o ell us abou he deeminans of he level and shape of he em sucue. he heoies ean he classical label because hey have been aound fo a long ime. he economis s classical appoach povides hee boad explanaions and los of vaiaions, as follows: he expecaions heoy can be inepeed in a numbe of ways ha ae no quie equivalen. One inepeaion says ha fowad inees aes measue make paicipans aggegae expecaion fo he coesponding fuue sho-em ae of inees. he heoy suggess ha an upwad-sloping cuve means ha sho-em inees aes ae expeced o ise. Convesely a downwad-sloping cuve implies an expecaion of falling sho-em aes. he basic idea behind he expecaions heoy is ha invesos selec bonds puely on he basis of hei expeced euns hey don demand a pemium fo acceping isk. Alenaively, he liquidiy pemium heoy suggess ha, if invesos have a pefeence fo sable pofolios compaed o volaile pofolios, hey will also exhibi a pefeence fo sho-daed bonds compaed o volaile long-daed bonds. In ode o induce invesos o hold bonds wih long mauiies, invesos mus expec o eceive a highe ae of eun han on sho mauiy insumens. his inceased expeced eun is called a liquidiy pemium he exa expeced eun fo beaing he inees ae isk of long-daed insumens. he heoy ells us ha long-em bonds should, on aveage, offe highe euns han cash invesmens. As a esul, he naual shape of he yield cuve should be upwad sloping. Noice ha, if he liquidiy pemium heoy eally does hold ue, we should expec long-em fowad aes o be a biased expecaion fo fuue sho aes. In ohe wods, if we use fowad aes o foecas he sho ae we would oveesimae he sho ae on aveage by he liquidiy pemium. he liquidiy pemium heoy suggess ha invesos ae avese o vaiabiliy in pofolio values. Bu suppose ha invesos do no jus look a he value of hei asses, bu a he oal value of asses and liabiliies. A pension fund may have 37

38 long-em liabiliies fo which he closes mach is a long-em bond. he asse ha poduces he lowes vaiabiliy in asses plus liabiliies may un ou o be a long-em bond ahe han cash. You can see ha he pefeed o naual mauiy of any inveso who is avese o oal pofolio vaiabiliy will mach he mauiy of he inveso s liabiliies whee he oal pofolio includes asses and liabiliies. Now, in jus he same way ha he liquidiy pemium heoy ewads long-em insumens wih a isk pemium ha ises in line wih em, he pefeed habia heoy ewads bonds which fall ouside he pefeed habia of invesos wih a pemium. Invesos mus be compensaed fo moving away fom hei pefeed mauiy. Noice ha his could mean ha he isk pemium acually deceases o becomes negaive wih he em of bonds, if bond issuance of longdaed pape fails o mach a naual appeie among invesos. he hee heoies oulined above should no be viewed as muually exclusive. Mos commenaos and eseaches believe ha all of he facos behind he diffeen heoies play a pa in he deeminaion of inees aes fom ime o ime. Now conside exhibi 6.4. I shows a em sucue fo Seling govenmen bond yields. he aes ploed wih whie cicles ae coninuously compounded log zeocoupon yields consisen wih obseved gil pices. he sho ae is a lile ove 6.5%. he 0-yea spo ae is aound 4.5% wih he 30-yea ae a 4%. Noe ha, alhough hese aes ae expessed in ems of noional discoun bonds, hey ae he mos convenien way of measuing he em sucue. We can hink of any coupon govenmen bond as a package of discoun bonds. he noional schedule shown below can be used wih he coupon and pincipal bond cash flows o eplicae faily closely he obseved pices of gils las Ocobe. he solid line above he zeo-coupon cuve shows he pa yield cuve. his is he schedule of yields fo noional coupon bonds ading a pa. Ou analysis shows ha a 4.7% 0-yea gil would ade a pa unde he zeo-coupon em sucue. EXHIBI 6.4: GOVERNMEN BOND INERES RAES END-OCOBER 000 7% 6% Inees Rae 5% 4% 3% % % Zeo-Coupon Yield Pa Yield Fowad Rae Fowad 5-Yea Bond Yield 0% Oc-00 em Souce: Baie & Hibbe 38

39 he cha also plos fowad aes of inees fo he govenmen yield cuve 6-monh aes. You can see ha, fo vey sho-em aes hese ae close o spo aes, bu fo longe mauiies hey ae lowe han spo aes. he fowad ae a 0 yeas was 4.5% and a 5 yeas he fowad ae is only a lile above 3%. A believe in he expecaions hypohesis would say ha sho aes wee anicipaed on aveage o fall o 3% in 5 yeas ime. Noice ha because he em sucue of spo aes is downwad sloping fowad aes fall moe apidly wih em han spo aes. Finally, fowad 5-yea bond yields have been ploed. his is he noional yield on a 5-yea bond ha we could buy fowad a vaious ems. Again, a believe in he expecaions hypohesis would view hese as unbiased expecaions of 5-yea bond yields a he ems shown. Fo example, he migh say: he expeced yield on a 5-yea gil a he end of 0 yeas is 4%. he equivalen figue fo 0 yeas is 3%. We should add ha a believe in he liquidiy pemium heoy would view hese as ove-esimaes fo fuue gil yields biased upwads by an impossible-o-measue isk pemium. On he ohe hand, he poponen of he pefeed habia heoy would ell us ha he fowad 5-yea bond yields do no ell us anyhing useful abou he expeced level of long-em yields. Rahe hey simply eflec supply and demand fo gils. he low levels of fowad aes migh only eflec an imbalance beween he demand fo long-daed fixed income insumens and supply fom he Biish govenmen and ohe high-qualiy bond issues. Anyone looking o he make fo clues o how hey should calibae a model of inees aes should no necessaily aim o mach hese fowad aes in a simulaion execise. Anohe eason why you migh be suspicious of long-em fowad aes as a measue of long-em fowad ae expecaions is he maked diffeence acoss diffeen cuencies. Long-em fowad aes have been consisenly lowe in he Seling bond seco han fo Euo- o dolla-denominaed asses. I is difficul o explain away hese diffeences wih aionale economic agumens. Rahe hey suppo he view ha Seling aes ae cuenly biased as a measue of expeced fuue sho aes by song demand fo longdaed pape. EXHIBI 6.5: ALERNAIVE ASSUMPIONS FOR HE FUURE AVERAGE LONG-ERM INERES RAES Full Peiod h Cenuy 9h Cenuy 0h Cenuy Las 50 Yeas Las 5 Yeas Las 0 Yeas 0-Yea Bond Yield 0-Yea Fowad Bond Yield 0% % % 3% 4% 5% 6% 7% 8% 9% 0% % Aveage Long-em Inees Rae WhaGilYieldCha.xls 39

40 6.3. SWAPION IMPLIED VOLAILIY he analys seeking infomaion o calibae a model looks o he yield cuve fo invesos expecaions fo inees aes. By conas, he opions makes offe he enicing pospec of infomaion on he disibuion of fuue inees aes. Some analyss believe ha i is possible o back ou he pobabiliy disibuion of he undelying asse implied by opions by consideing how he implied volailiy of opions vaies wih he opions sike pices. Fo example, by consideing swapion pices on he 5-yea swap ae wih a ange of sike aes and a common expiy dae, i is possible o calculae a pobabiliy disibuion of 5-yea swap aes implied fo he expiy dae. his appoach can povide insigh ino make expecaions, bu we also need o be way of he assumpions implici in using opion pices in his way in paicula, he appoach assumes ha opions can be dela-hedged wihou cos o isk, and so opion pices ae assumed o be solely a funcion of he make s isk-neual expecaions of he fuue behaviou of he undelying asse. Exhibi 6.6 below shows he disibuion of 5-yea swap aes on he 8 h Febuay 00 as implied by swapion pices on 7 h Novembe 000. EXHIBI 6.6: OPION-IMPLIED PROBABILIY DISRIBUION FOR 5-YEAR SWAP RAES Pobabiliy Densiy % 90% 80% 70% 60% 50% 40% 30% 0% 0% Cumulaive Pobabiliy 0 0% % 3% 4% 5% 6% 7% 8% 9% 0% % % 5-yea Swap Rae on 8 Feb 00 Souce: SwapRaesNov_000_Y he fowad swap ae was 5.79% and, by consucion, his is equal o he mean of he implied-disibuion. Howeve, he cha suggess ha swapion pices imply significan poenial deviaion fom his expeced swap ae he sandad deviaion of he disibuion is.75%. Swapions ae no exchange-aded, and so OC pices have o be used. We ae gaeful o GenRe Financial Poducs fo supplying us wih epesenaive swapion pices. 40

41 6.3.3 EQUIY IMPLIED VOLAILIY We can also apply his mehodology o equiy pices, by using aded opion pices. Below is an opion-implied pobabiliy densiy fo he s Mach 00 as implied by LIFFE aded opion pices on he 0 h Mach 00. Sicly speaking, his is he iskneual pobabiliy densiy i assumes he expeced eun on equiies is equal o he isk-fee ae. Howeve, we can ansfom his densiy ino a eal-wold disibuion wih a given equiy isk pemium by making some assumpions egading invesos uiliy funcions. Such a ansfom would shif he disibuion o he igh, and would change he shape of he disibuion, hough no usually vey significanly. EXHIBI 6.7: OPION-IMPLIED PROBABILIY DISRIBUION FOR FSE 00 Pobabiliy Densiy Souce: UKEquiies030_030 FSE 00 a Mach % Noe ha, like hisoical equiy make daa, he opion-implied disibuion exhibis negaive skew unlike he lognomal disibuion. he lef-hand ail is significanly fae han would be implied by a lognomal disibuion, and suggess o he poponen of his appoach ha hee is a pobabiliy of aound 0% ha of he FSE 00 index will fall below 4000 a s Mach % 80% 70% 60% 50% 40% 30% 0% 0% 0% Cumulaive Pobabiliy 4

42 6.4 EXPER OPINION A final poenial souce of infomaion ha can be used o judge he easonableness of he disibuions geneaed by a model is he judgemen of expes. his infomaion can be accessed in diffeen ways: Published independen foecass. his infomaion comes in he fom of expeced asse euns and, on occasion, disibuional infomaion. A good example of his ype of daa is he inflaion foecas disibuions published by he Bank of England. he bank esimaes a 90% confidence ange of fo he inflaion ae in yeas ime beween 0.9% and 3.7% wih a median value in line wih he bank s age of.5%. Cudely, his ange implies an annual sandad deviaion fo he inflaion ae of 0.6% fo he nex yeas. In-house expes. Mos lage financial insiuions employ economiss and foecases. hey ae capable of commening on he locaion and shape of disibuions poduced by models. Alhough hee ae no clea ules fo how expe opinion is incopoaed ino he calibaion pocess, i is impoan o undesand ha i is a poenially hugely valuable souce of insigh. As we have aleady seen, naïve calibaion o hisoic daa o make infomaion is likely o poduce poo model oupu. he expe s opinion povides a useful check agains his isk. Le us now focus on wo specific calibaions fo he model. 4

43 7. A CALIBRAION 7. A PLAUSIBLE PARAMEER CHOICE hee is a wide ange of possible paamee choices fo he model specified in Secion 5. Hee we eview wo possible choices ha we judge o be a easonable saing poin in he ligh of he analysis pesened above. he se-up of he model is shown in exhibis 7.A and 7.B below. Exhibi 7.A ses ou paamee values fo a base case and a modified se-up fo he model whee posiive inees aes ae poduced a all imes. EXHIBI 7.A: BASE CASE CALIBRAION & WIH REFLECION Calibaion Case Paamee A. Base Case Yield Cuve Reflecion OFF B. Posiive Inees Yield Cuve Reflecion ON Real Raes Yields Equiy Reuns Inflaion Expecaions σ σ µ γ b No b No 0 Pe Py q q σ q σ q µ q γ q q q b q No b q No 0 Pe Py µ E, σ E, µ E, σ E, p, p, π π y σ y, σ y, µ y log.035 log.035 y0 log.05 log.05 Foce +ve Raes No Yes Coelaion A A 43

44 Exhibi 7.B povides one example of a se of coelaion coefficiens which will conol he elaionships beween he sochasic innovaions he dz's: EXHIBI 7.B : CORRELAION MARIX Z Z Z q Z µ Z E Z Y Z Z Z q Z µ Z E Z Y Calibaion A epesens a vey simple base case. We have se boh g and g q o zeo, implying a zeo em pemium on index-linked bonds and a zeo inflaion isk pemium. In ohe wods, his fis calibaion should geneae scenaios whee expeced euns on boh nominal and index-linked bonds of all ems ae he same. Impoanly, his fis calibaion will also poduce some negaive nominal inees aes. he advanage of saing wih his simplisic calibaion is ha much of he complexiy wihin he model elaing o he implemenaion of isk pemia falls away. Using his simple calibaion allows us o ensue ha he model is poducing he esuls expeced. In calibaion B, we have incopoaed isk pemia fo boh inees aes and inflaion, by seing g and g q o equal hese isk pemia have he effec of inoducing a em pemium o he em-sucues fo boh eal inees aes and inflaion expecaions. In Exhibi 7.A, we expess hese em pemia in ems of he coninuously compounded log ae of eun, Pe, and he zeo-coupon yield, Py. Fuhemoe, wihin calibaion B we have used sepaae devices o conol he value of eal inees aes and inflaion expecaions, and o foce nominal yields o emain posiive: We have imposed minimum baies on he sochasic vaiables descibing he em-sucues of eal aes and inflaion expecaions: b, b, q b q, q b q. In his calibaion, we have used b and b q o ensue ha ou model scenaios exclude boh vey lage negaive eal aes, and vey lage negaive inflaion aes. b and b q ensue ha he vey long-em eal inees ae and inflaion expecaion do no fall below zeo. We have also 'efleced' he nominal yield cuve off zeo: if + q < 0, hen se q = if + q < 0, hen se q = We have ouched on he limiaions of hese adjusmens in secion hey have he obvious benefi of emoving implausible oucomes fom he model oupu. he disadvanage of his appoach is he inoducion of inconsisency ino he model beween is solid heoeical foundaion and he pacical implemenaion. We believe ha - fo many sochasic simulaion applicaions - hese limiaions ae ouweighed by he benefis in ems of geneaing plausible individual scenaios. 44

45 We have used he model o simulae 000 scenaios ove a hoizon of 30 yeas. he simulaion ials wee buil up in ime incemens of monh i.e. d = /, alhough he esuls wee ecoded on an annual basis. he choice of ime incemen and oupu fequency is eniely flexible. he enie simulaion execise ook a few minues o un on a deskop PC, and poduced simulaed esuls fo evey asse, in evey oupu ime peiod, fo each of he simulaion ials. As we have aleady menioned, geneaing oupu in his foma means hee is huge flexibiliy in he way he esuls can be pesened. 7. SUMMARY SAISICS : SAMPLE MEAN REURNS & SANDARD DEVIAIONS Fis, i is wohwhile consideing some summay saisics ha descibe he disibuions of asse euns geneaed fom he model. hese summay saisics ae povided fo he wo alenaive calibaions: EXHIBI 7.A: SAMPLE MEANS & SANDARD DEVIAIONS REFLECION OFF Real Odinay Log Log Expeced Sd. Dev. Asse Reun Reun Reun % pa. % pa. % pa. % pa. 0Y Hisoic Sd. Dev. Equiies Cash Consan Mauiy Coupon Bond Consan Mauiy Coupon IL Bond Undelying Inflaion na EXHIBI 7.B: SAMPLE MEANS & SANDARD DEVIAIONS REFLECION ON Asse Log Reun % pa. Real Log Reun % pa. Odinay Expeced Reun % pa. Sandad Deviaion 0Y Hisoic Sd. Dev. Equiies Cash Consan Mauiy Coupon Bond Consan Mauiy Coupon IL Bond Undelying Inflaion na Fisly noe ha in calibaion A, he expeced odinay euns on all he fixed inees asses is equal o 5%, he expeced ae of eun on cash. Calibaion B poduces a em pemium on 0-yea bonds of abou %, and on 0-yea index-linked bonds of abou 60 basis poins i.e. fo 0-yea bonds hee is an inflaion isk pemium of 40 basis poins. hese esuls show ha he effec of excluding negaive nominal aes, as descibed above, is o incease he expeced euns on he vaious asses whils poducing simila sandad deviaions. I is quie possible o offse his effec by educing he µ and µ q paamees. 45

46 7.3 EQUIIES Exhibi 7.3 shows one way of ploing esuls fo log coninuously compounded annual equiy euns and compaing hem o he hisoic and make-implied disibuions. Noe ha he solid ed band in he cene of he cha shows he spead fom 5 h o 75 h pecenile fo annual log euns. he oue pink bands plo he 5 h /5 h and 75 h /95 h anges. Noice ha he alenaive model calibaions A & B poduce vey simila disibuions fo equiy euns, and his disibuion appeas o lie somewhee beween hisoic expeience and he disibuion implied by cuen make pices as of end- Mach 00. EXHIBI 7.3: DISRIBUION OF -YEAR EQUIY REURNS 50% Annual Equiy Reun Log 40% 30% 0% 0% 0% -0% -0% -30% -40% -50% Reflec OFF Reflec ON Hisoic Make Implied Noe: Hisoic peiod=900/000; Make-implied daa fo FSE 3/03/00 EXHIBI 7.4: 00 Cumulaive Pobabiliy % UNCONDIIONAL DISRIBUION OF UK EQUIY REURNS Reflec OFF Reflec ON Hisoic Make Implied 0-60% -40% -0% 0% 0% 40% 60% Annual Equiy Reun Log Exhibi 7.4 illusaes he enie cumulaive pobabiliy disibuion, ahe han jus five seleced pecenile poins. Again, you can see ha he disibuions poduced by he model appea o fall somewhee beween he hisoic disibuion and he cuen make 46

47 implied disibuion. An impoan feaue highlighed by his cha, which was no eviden in exhibi 7.3, is he vey heavy downside ail in he cuen make implied disibuion of equiy euns fo he peiod 3/3/00 o 3/3/00. Accoding o opion pices a end-mach, he make appeaed o be assigning a pobabiliy of oughly 5% o a eun of -40% ove he following monhs. Alhough, he cuen calibaion of he egime-swiching model fo equiy euns does no capue he size of his ail, i does vey much bee han a simple lognomal assumpion. Fuhemoe, you migh ge much close o he make implied disibuion by adjusing he paamees of he egimeswiching model if judged appopiae. 7.4 SHOR-ERM INERES RAES & CASH REURNS In exhibi 7.5 we conside he disibuion of sho-em nominal inees aes simulaed by he model, and compae his disibuion wih he disibuion of inees aes ove wo hisoic peiods. he yellow disibuion is posiioned well o he igh of he geen disibuion, eflecing he fac ha pos-wa inees aes have been vey high elaive o pe-wa aes and he aes ha have been obseved moe ecenly. he disibuions of sho aes poduced by boh calibaions geneally fall beween he wo hisoic disibuions, wih he excepion ha calibaion A poduces sho aes below zeo in jus unde 0% of he simulaions. hese negaive aes ae emoved in calibaion B, whee we foce nominal aes o be posiive. EXHIBI 7.5: 00 Cumulaive Pobabiliy % UNCONDIIONAL DISRIBUION FOR SHOR RAES Reflec OFF Reflec ON Hisoic ~ Pas 00 yeas Hisoic ~ Pas 50 yeas 0-5.0% 0.0% 5.0% 0.0% 5.0% Sho-em Inees Rae Exhibis 7.6A and 7.6B illusae he disibuion of he pah of sho aes ove he couse of he 30-yea simulaion hoizon. hese chas illusae he peceniles of a disibuion in a manne simila o he ba chas in exhibi 7.3, he only diffeence being ha we ae now ineesed in he change in he disibuion as we pogess hough he 30-yea simulaion hoizon, ahe han a single disibuion acoss all ime peiods an uncondiional disibuion. In calibaion A, we have iniialised he model fo eal and nominal inees aes in is equilibium posiion, meaning ha 0 and 0 equal µ, and q 0 and q 0 equal µ q. One consequence of his is ha he cene of he sho ae disibuion he median 47

48 emains a 5% houghou he couse of he 30-yea hoizon. In he wo chas, we can see how unceainy in movemens in he sho ae ceaes a disibuion aound his cenal value - a funnel of doub. he spead of he disibuion aound is median inceases ove he fis few yeas, bu he effec of mean evesion is such ha he disibuion sabilizes afe abou 0 yeas o so. Impoanly, in his basic fis calibaion, sho aes become negaive in appoximaely 0% of he 000 simulaions. his feaue is likely o be viewed as an undesiable popey of his calibaion. EXHIBI 7.6A: DISRIBUION OF PAH OF SHOR RAE OVER 30-YEAR HORIZON REFLECION OFF 5% 3% 0% Sho Rae 8% 5% 3% 0% -3% -5% ime yeas In exhibi 7.6B, you can see ha focing nominal inees aes o be posiive means ha he lowe peceniles of he disibuion emain posiive. An awkwad consequence of he ick used o ensue posiive aes is ha he expeced sho ae ends o dif up ove ime. his effec can be paly offse by educing he value of he µ and µ q paamees. EXHIBI 7.6B: DISRIBUION OF PAH OF SHOR RAE OVER 30-YEAR HORIZON REFLECION ON 5% 3% 0% Sho Rae 8% 5% 3% 0% -3% -5% ime yeas 48

49 In Exhibi 7.7 we illusae he disibuion of euns on cash, and compae he disibuions poduced by he wo calibaions wih he hisoic disibuion fom he las cenuy. Ou second calibaion poduces a disibuion which looks simila o he hisoic disibuion, alhough he uppe peceniles of he cash euns geneaed by he model ae somewha less he hisoic values. his is a diec consequence of his paicula calibaion, which assigns lowe likelihood o he vey high inees aes ha have been obseved hisoically. I is sensible o ask wha we believe he chances ae of obseving a sho-em inees ae of 5% ove he couse of he nex 30 yeas. Ou second calibaion assigns a pobabiliy of jus less han %. EXHIBI 7.7: 0% DISRIBUION OF ANNUAL CASH REURNS Annual Cash Reun Log 5% 0% 5% 0% -5% Reflec OFF Reflec ON Hisoic 49

50 7.5 5-YEAR CONVENIONAL BOND YIELDS Exhibi 7.8 shows he esuling cumulaive fequency plo fo long-em bond yields. Fo compaison, we have also ploed cumulaive plos fo hisoic long-em yields and make-implied aes. he cha ells us wha we aleady know ha hee is a ange of plausible disibuions. Expe opinion suggess ha he immediae inflaion and inees ae fuue will look nohing like he las 50 yeas and ha we should assign much lowe pobabiliy o 0% bond yields han he 30% fequency ove he pas 50 yeas. he wo disibuions geneaed by he model look easonable, alhough a efinemen would be o exend he lef-hand o ail o assign geae pobabiliy o Japanese-syle bond yields. EXHIBI 7.8: 00 Cumulaive Pobabiliy % UNCONDIIONAL DISRIBUION OF 0-YEAR COUPON BOND YIELDS Reflec OFF Reflec ON Hisoic ~ Pas 00 yeas Hisoic ~ Pas 50 yeas GBP 5Y swap EUR 5Y swap 0-5% 0% 5% 0% 5% Long-em 0-Yea Bond Yield Exhibis 7.9A and 7.9B ae analogous o he funnels of doub fo cash euns illusaed in exhibis 7.6A and 7.6B. Again, we can see a funnel ha gows ove he fis 0-5 yeas of he simulaion, and hen flaens ou as mean evesion akes effec. Anohe obvious effec of mean evesion is ha he spead of he disibuion of long-em yields is less han ha fo he sho ae. he cenal 98% of he disibuion fo he 5-yea coupon bond yield in exhibi 7.9A coves he ange -.5% o.5%, compaed o -5% o 5% fo he sho ae in exhibi 7.6A. Exhibi 7.9B shows ha he second calibaion ensues ha no negaive bond yields ae geneaed. In fac, bond yields neve appea o fall below abou.5%. Again, his is a feaue of a paicula calibaion. hee ae many ohe calibaions we could use o educe o emove his implici lowe bound on bond yields. hee ae also ohe devices we could use which would ensue nominal aes emained posiive, bu which did no peclude vey low bond yields. 50

51 EXHIBI 7.9A: DISRIBUION OF PAH OF 5-YEAR COUPON BOND YIELDS OVER 30 YEAR HORIZON REFLECION OFF 5Y Coupon Bond Yield 5.0%.5% 0.0% 7.5% 5.0%.5% 0.0% -.5% -5.0% ime yeas In exhibi 7.0, we show he disibuion fo yields on 5-yea coupon bonds, and compae he disibuions fom he wo model calibaions wih boh he hisoic and make implied disibuions. Noice ha he disibuion poduced by he second calibaion is somewha naowe han he hisoic expeience. As fo sho aes, in using his paicula se of paamees, we have assigned lowe pobabiliies o vey high long-em nominal inees aes han he fequencies been expeienced ove he las cenuy. Noice ha he second calibaion poduces a disibuion ha is ahe simila o he disibuion fo long-em bond yields implied by cuen make pices fo longdaed swap conacs. EXHIBI 7.9B: DISRIBUION OF PAH OF 5-YEAR COUPON BOND YIELDS OVER 30 YEAR HORIZON REFLECION ON 5Y Coupon Bond Yield 5.0%.5% 0.0% 7.5% 5.0%.5% 0.0% -.5% -5.0% ime yeas 5

52 EXHIBI 7.0: DISRIBUION OF COUPON BOND YIELDS 6% 4% % Long Bond Yield 0% 8% 6% 4% % 0% -% Reflec OFF Reflec ON Hisoic Make Implied 7.6 INDEX-LINKED BOND YIELDS Now le us conside yields on index-linked bonds. In he fis calibaion, we can see fom exhibi 7. ha negaive index-linked bond yields ae geneaed wih a pobabiliy of jus ove 0% of he simulaion ials. When he minimum baies ae applied o he values of,, q, q, and he nominal yield cuve eflecion is acivaed in he second calibaion, he lowes simulaed index-linked bond yield is jus ove %. he median of he disibuion fo index-linked yields also inceases fom.5% o ove 3%. his effec is eviden when we compae he disibuion of he pah fo index-linked yields unde he wo calibaions in exhibis 7.A and 7.B. In he second calibaion, he em pemium in eal inees aes will mean ha he expeced eun on indexlinked bonds will incease wih em. EXHIBI 7.: UNCONDIIONAL DISRIBUION OF 0-YEAR INDEX-LINKED BOND YIELDS 00 Cumulaive Pobabiliy % Reflec OFF Reflec ON 0 -% 0% % 4% 6% 8% Index-Linked Bond Yield 5

53 EXHIBI 7.A: DISRIBUION OF PAH OF 0-YEAR INDEX-LINKED YIELDS REFLECION OFF 8% 0Y Index-Linked Bond Yield 6% 4% % 0% -% -4% ime yeas EXHIBI 7.B: DISRIBUION OF PAH OF 0-YEAR INDEX-LINKED YIELDS REFLECION ON 8% 0Y Index-Linked Bond Yield 6% 4% % 0% -% -4% ime yeas 7.7 NOMINAL INERES RAE ERM-SRUCURE Rahe han consideing he disibuion of nominal and index-linked yields on bonds of paicula mauiies, i is impoan ha he model capues movemens in he enie em-sucue in a ealisic way. he value of a pofolio of asses is affeced by movemens in he level and shape of he yield cuve, and so i can be useful if he model can poduce a epesenaive ange of yield cuve shapes and changes in shape. he following wo chas illusae a couple of ahe diffeen scenaios fo changes in he yield cuve ove he couse of individual 30-yea simulaion ials, fom is 'fla' saing posiion of 5%. Exhibi 7.3A illusaes a scenaio whee, fo much of he 30 yeas, he nominal yield cuve lies below is saing posiion, including wo o hee yeas whee zeo coupon yields fo many mauiies fall below %. Exhibi 7.3B illusaes a 53

54 scenaio whee yields have geneally inceased ove he couse of he 30-yea simulaion. EXHIBI 7.3A: SIMULAED PAH OF NOMINAL YIELD CURVE SIMULAION #5 5.0% Nominal Zeo-Coupon Yield.5% 0.0% 7.5% 5.0%.5% 0.0% em yeas EXHIBI 7.3B: SIMULAED PAH OF NOMINAL YIELD CURVE SIMULAION #47 5.0% Nominal Zeo-Coupon Yield.5% 0.0% 7.5% 5.0%.5% 0.0% em yeas As well as looking a he behaviou of he yield cuve wihin individual simulaion ials, i is useful o undesand he way in which zeo-coupon yields ae disibued as he model is un ou ove longe ems he uncondiional disibuion of he nominal em sucue. Needless o say, he disibuions illusaed in exhibis 7.4A and 7.4B look ahe diffeen. Fo he fis calibaion, he disibuion of nominal zeocoupon yields acoss all ems ae cened a slighly below 5% 3. his cha ceainly 3 Alhough he expeced insananeous ae of eun a all ems is exacly equal o 5%, he expeced yields fall slighly wih inceasing em due o Jensen's inequaliy [Bullein of he Ausalian Mahemaical Sociey, 997, 55, ] 54

55 highlighs he poblem of negaive nominal inees aes inheen in he fis calibaion: ove 5% of nominal zeo-coupon yields fall below zeo. When we use he eflecion adjusmen o he model o foce posiive nominal aes in he second calibaion exhibi 7.4B, he enie disibuion is shifed up, and we expeience no vey low long-em nominal aes. hese issues have been menioned peviously and i is impoan o eieae ha such feaues ae a consequence of his paicula calibaion. Uses who believe ha such a disibuion fo long-em nominal aes was implausible should invesigae ohe paamee choices. EXHIBI 7.4A: UNCONDIIONAL DISRIBUION OF ERM-SRUCURE OF NOMINAL INERES RAES REFLECION OFF 0% Zeo Coupon Yield 5% 0% 5% 0% -5% -0% em yeas EXHIBI 7.4B: UNCONDIIONAL DISRIBUION OF ERM-SRUCURE OF NOMINAL INERES RAES REFLECION ON 0% Zeo Coupon Yield 5% 0% 5% 0% -5% -0% em yeas 55

56 7.8 INFLAION In his secion we will illusae he behaviou of inflaion aes and he em-sucue fo inflaion expecaions as we have done fo bond yields and he nominal yield cuve. Exhibi 7.5 shows he enie cumulaive pobabiliy disibuion fo he insananeous ae of inflaion unde he wo calibaions, and compaes hese disibuions wih hisoic inflaion aes fom he las 00 yeas. he median value geneaed by boh model calibaions is appoximaely.5% pe annum. he mos siking feaue of his cha is ha he hisoic disibuion has much geae spead han he disibuion geneaed by he model. We can look back o exhibi 6.8 o confim ha he UK inflaion ae has indeed exceeded 5% fo oughly 0 of he las 00 yeas. We have chosen he model paamees o eflec a view ha he likelihood of he inflaion ae eaching 5% a some poin ove he nex 30 yeas is vey much smalle han obseved ove he couse of he 0 h cenuy. Exhibi 7.5 shows ha ou boh calibaions assign a 0% pobabiliy o he inflaion ae exceeding abou 6%, ahe han 5%! EXHIBI 7.5: UNCONDIIONAL DISRIBUION OF INFLAION RAE Cumulaive Pobabiliy % Reflec OFF Reflec ON Hisoic ~ Las 00 Yeas 0-5.0% -.5% 0.0%.5% 5.0% 7.5% 0.0%.5% 5.0% Inflaion Rae Exhibis 7.6A and 7.6B show he disibuion of he pah of inflaion ove he couse of he 30-yea simulaion hoizon. Vey like he disibuions fo bond yields and inees aes, he mean evesion wihin he inflaion model means ha hese funnels of doub spead ou ove he fis few yeas of he simulaion befoe sabilising afe abou 5 yeas. Noice ha, even when we apply he minimum baies o q and q, and foce nominal aes o be posiive, deflaionay scenaios ae sill geneaed in abou 5% of he simulaions. 56

57 EXHIBI 7.6A:DISRIBUION FOR PAH OF INFLAION RAE OVER 30 YEARS REFLECION OFF 0.0% 7.5% Inflaion Rae 5.0%.5% 0.0% -.5% -5.0% ime yeas EXHIBI 7.6B:DISRIBUION FOR PAH OF INFLAION RAE OVER 30 YEARS REFLECION ON 0.0% 7.5% Inflaion Rae 5.0%.5% 0.0% -.5% -5.0% ime yeas 7.9 INER-RELAIONSHIPS BEWEEN INFLAION, BOND YIELDS & EQUIY REURNS Alhough ou model may poduce individual asse scenaios and disibuions ha appea quie sensible when compaed wih empiical daa, when we ae consideing enie pofolios consising of a ange of asses and liabiliies, i is impoan o ensue ha he model geneaes plausible ine-elaionships beween diffeen asse classes. Fo insance, geneaing a significan numbe of scenaios whee high inees aes coincide wih sable, low inflaion aes would seem ahe uneasonable. Fisly we look a he elaionship beween he simulaed inflaion ae and 5-yea coupon bond yield a a paicula poin in ime yea 5, in each of he 000 simulaion ials. Exhibis 7.7A and 7.7B demonsae ha he model geneaes quie song 57

58 coelaion beween he simulaed inflaion expeience and nominal bond yields. Lowe inflaion scenaios end o coincide wih lowe bond yields, and vice-vesa. EXHIBI 7.7A : INFLAION VS BOND YIELD REFLECION OFF 0% 5-Yea Bond Yield 5% 0% 5% 0% -5% -8% -6% -4% -% 0% % 4% 6% 8% 0% % Simulaion Clock=5 Inflaion Rae EXHIBI 7.7B : 0% INFLAION VS BOND YIELD REFLECION ON 5-Yea Bond Yield 5% 0% 5% 0% -5% -4% -% 0% % 4% 6% 8% 0% % Simulaion Clock=5 Inflaion Rae 58

59 In exhibis 7.8A and 7.8B we look a he elaionship beween simulaed scenaios fo olled-up equiy euns and he coesponding level of he inflaion index. Geneally speaking, high inflaion has ended o hinde equiy pefomance, paiculaly in eal ems. his effec can be seen modeaely wihin he scenaios geneaed by his model. High inflaion scenaios ae ofen associaed wih lowe olled-up equiy euns. his is paiculaly ue when we conside equiy euns in eal, ahe han nominal, ems. EXHIBI 7.7A:EQUIY ROLL-UP VS INFLAION REFLECION OFF 0.0 Inflaion Index.0 UKE.hisRollUp Real Equiy RollUp Simulaion Clock=5 Equiy Rollup EXHIBI 7.7B: 0.0 EQUIY ROLL-UP VS INFLAION REFLECION ON Inflaion Index.0 UKE.hisRollUp Real Equiy RollUp Simulaion Clock=5 Equiy Rollup 59

60 8. A COMPARISON WIH HE WILKIE MODEL Someone migh quie easonably ask: Who needs anohe model? Suely he Wilkie model is good enough fo my puposes? As we have aleady obseved, any model will be fi fo some pupose bu no ohes. I would be difficul o ague ha he Wilkie model is no fi o analyse ceain poblems poviding he modelle holds a paicula se of beliefs abou equiy make behaviou. On he ohe hand, ou expeience suggess some seious failings. We now compae ou poposed model and calibaion which we will efe o as he B&H model wih he Wilkie model. Of couse, hee could be as many calibaions of he Wilkie model as hee ae model uses. Pofesso Wilkie has ceainly encouaged uses o y ou diffeen paamees. We will pesen a Wilkie calibaion ha we believe is faily epesenaive of cuen paamee selecions see Appendix C fo he paamees used in his secion. he compaison beween he wo models is undeaken boh in ems of compaing model oupus and making some geneal obsevaions wih egad o diffeences beween he wo models. 8. SOME GENERAL OBSERVAIONS he wo models adop vey diffeen appoaches o he challenge of modelling he longem behaviou of financial vaiables. In he Wilkie model, saisical ime-seies elaionships ae developed fo a numbe of obsevable vaiables inflaion, sho-em inees aes, consol yields, dividend income and he dividend yield. Make pices ae hen deived fom hese pocesses. Fo example, in he Wilkie model, equiy pices ae infeed fom he aio of dividend income o dividend yield, wheeas in he B&H model, he oal eun on equiies is modelled sepaaely fom he dividend yield, and he pocess fo he equiy pice is heefoe anspaen and pasimonious. Whils he sucue used by Wilkie may seem simple o hose wih an avesion o mahs, i has some majo dawbacks. he sucue of he model is ahe convolued and lacks anspaency. In a saisical analysis of he popeies of he Wilkie model, Hube 4 found ha he model did no povide a good epesenaion of hisoical daa and was ove-paameeised, wih a numbe of paamees being effecively edundan: he agued ha he complexiy of he model sucue does no add o he effeciveness of he model. Whils calibaion is a challenging poblem in almos any financial modelling, he poo saisical fi may seem slighly moe supising in he case of he Wilkie model given he complex sucue of he model seems o have been diven moe by consideaion of fiing obseved paens in he daa ahe han by any efeence o building sucues consisen wih economic heoy. We believe ha he sucue of he model pesened in secion 5 is simple and moe economically logical. We model eal inees aes, inflaion aes and he equiy eun in excess of he nominal inees ae. his sucue ensues a consisency in he join asse behaviou geneaed a any given ime ha is lacking in he Wilkie model. 4 A Review of Wilkie s Sochasic Invesmen Model, P. Hube

61 he following secions eview some oupu of he Wilkie model. We highligh some of he poenial poblems wih he oupu and make compaisons wih oupu geneaed by he B&H model. 8. REPRESENAION As has been menioned above, he Wilkie model does a poo job of epesening ceain feaues of he eal wold. Fo example: he elaionship beween simulaed inflaion ouuns and long-em bond yields simply is no plausible. he model using a ypical paamee se geneaes fequen scenaios of vey low aveage inflaion accompanied by high bond yields and high inflaion coupled wih low bond yields. wo examples of ypical - bu ahe supising - join pahs fo inflaion and bond yields ae ploed below. EXHIBI 8.A: A WILKIE SCENARIO 5% Log Inflaion Rae Consols Yield Bank Rae 0% 5% 0% -5% Hoizon Yeas EXHIBI 8.B: ANOHER WILKIE SCENARIO 5% Log Inflaion Rae Consols Yield Bank Rae 0% 5% 0% -5% Hoizon Yeas 6

62 EXHIBI 8.: WILKIE INFLAION RAES & CONSOLS YIELD 0-YEAR HORIZON 8% Consols Yield a 0-Yea Hoizon 6% 4% % 0% 8% 6% 4% % 0% -0% -5% 0% 5% 0% 5% Inflaion Rae a 0-Yea Hoizon Exhibi 8. illusaes he elaionship beween he inflaion ae a a 0-yea hoizon and he esuling Consols yield. hee is no appaen coelaion beween hese wo quaniies. Fo an analys who uses he model o invesigae he popeies of convenional bonds in diffeen sos of inflaion envionmen, he model fails o capue he fundamenal link beween inflaion and nominal yields. he model using a ypical paamee se geneaes plausible vaiabiliy in equiy euns ove sho hoizons, bu vey naow disibuions a long-em hoizons. he pobabiliies assigned o equiy make declines ove long-em hoizons look implausibly low given 0 h cenuy expeience. Since one of he pimay puposes of he model is ofen o show uses he poenial impac of low-pobabiliy oucomes, his is paiculaly supising. Exhibi 8.3 illusaes hee 0 h cenuy episodes of 0-yea yea equiy index declines one fo each of he UK, US & Japanese makes. hee ae ohe examples fo hese makes. Indeed, equiy claims wee wiped ou alogehe in Japan in 945. Noe ha he scale of he UK decline is inceased damaically if he illusoy impac of inflaion is emoved by ploing he eal pice of equiies. EXHIBI 8.3: 0 H CENURY BEAR MARKES.00 Equiy Pice US June 99 UK Dec 97 Real~UK Dec 97 Japan Dec ime Yeas

63 So, vey cudely, you migh guess ha since each of hese makes has expeienced a 0-yea 50% pice decline somewha less in ems of oal eun and appoximaely he same in ems of excess euns a leas once in he pas 00 yeas, he pobabiliy assigned o his so of scenaio would be somehing like 0%. he mean evesion which is an impoan pa of he Wilkie model means ha he pobabiliy assigned o long-em declines of his magniude is much lowe. he juy is sill ou in he complicaed mean evesion debae. Whaeve you migh hink, i seems easonable ha he modelle should begin by excluding mean evesion fom models a puden assumpion? unless hee is compelling evidence o suppo is exisence 5. I is possible o incease long-em vaiaion in Wilkie equiy euns by aising sho-em vaiabiliy. Unfounaely, his has he effec of poducing implausible vaiabiliy in sho-em equiy euns. he mean evesion feaue means ha sho-em euns geneaed fom he model can be sensiive o he way he model is iniialised. his equies exeme cae fom he model use. Even wih a faily damaic educion in he paamee descibing he vaiabiliy of inflaion QSD, sho-em inflaion vaiaion looks oo high by compaison wih almos any economic foecas and he disibuion a long hoizons looks oo naow. 8.3 MEAN REVERSION he sucue of he Wilkie model means ha wih ypical paamee choices he model geneaes mean-evesion in equiy euns. Whils he exen o which equiy makes acually mean-eve is he subjec of much debae, we would ague ha assuming mean-evesion o he exen geneaed by he Wilkie model is pehaps unwise. Such mean-evesion means ha simple e-balancing ules can incease euns whils a he same ime educing isk. I could be agued ha aking cedi fo he on-going exisence of his supposed fee lunch going fowad is a ahe impuden saing poin fo making long-em equiy pojecions. hough hee is some saisical evidence fo mean-evesion in equiy makes, is saisical significance is dubious, paiculaly when we conside he excess eun i.e. he eun in excess of he pevailing cash ae. Given he sucue of he Wilkie model, wheeby a sochasic pocess fo he dividend yield is used o deive equiy pices, mean-evesion seems difficul o emove fom he model. I could be agued ha deiving pices fom dividend yields is eally a case of he ail wagging he dog. In he B&H model, equiy pices and dividend yields ae modelled as wo sepaae bu highly negaively coelaed pocesses. As in he Wilkie model, he dividend yield is modelled as a mean-eveing pocess. Howeve, he B&H model has 5 hee is lage lieaue on mean evesion. Fo a ecen example see Mean Revesion in Sock Reuns: Evidence & Implicaions, L Summes & J Poeba Financial Makes Goup discussion pape, LSE

64 quie diffeen implicaions fo he behaviou of dividend income. Fo example, he B&H model implies ha when makes cash, yields will immediaely ise and hen gadually fall back o hei long-em mean, wihou any sysemaic mean-evesion in equiy euns. his implies ha following a make cash, he ae of expeced dividend gowh is lowe ha is, he make has fallen due o a educion in expeced dividend gowh. Pu simply, he expeced eun in excess of cash is assumed o be consan, so any change in dividend yield implies a coesponding change in he opposie diecion fo dividend gowh. Noe ha we ae consideing he excess eun hee he expeced nominal eun on equiies in he B&H model will be highe when sho-em inees aes ae highe. his is fundamenally diffeen fom he Wilkie model hee he expeced eun on equiies depends on he level of he dividend yield elaive o is assumed mean, as expeced fuue dividend gowh is no a funcion of he cuen dividend yield excep insofa as hey boh ae a funcion of inflaion. So when yields ae high, expeced euns ae high ha is, expeced dividend gowh has no fallen o he exen ha i offses he highe yield. In his case, he model is implying ha equiies have fallen by moe han is implied by he educion in expeced dividend income, so equiies hen mean-eve back o hei fai value. So he B&H model is consisen wih a wold whee equiy pices ae based on aional expecaions of fuue dividend gowh, whils he Wilkie model assumes ha dividend income is fa moe sable han would be suggesed by he volailiy of he dividend yield, and equiy makes go hough peiods whee he expeced eun on equiies can vay vey significanly and noe ha a change in inflaion o inees aes is no necessay fo such a change in expeced eun o occu. We now un ou aenion o how hese diffeences in he model affec he disibuions of equiy euns geneaed. 8.4 EQUIY MODEL he diffeences in he sucue of he wo models pehaps have he geaes impac on he simulaed disibuions of equiies, paiculaly a longe hoizons. he meanevesion inheen in he Wilkie model implies fa lowe equiy isk a longe ime hoizons han he B&H model he mean-evesion esuls in annualised equiy volailiy decaying much fase han i ohewise would. In a model wih no mean evesion, we would expec he annualised volailiy o decay a ae of /sq. Le us call he annualised volailiy muliplied by he squae oo of ime he sandadised annualised volailiy. Exhibi 8. plos his saisic as simulaed fo he wo models ove a en-yea hoizon. 64

65 EXHIBI 8.: SANDARDISED ANNUALISED VOLAILIY 0% 9% Sandadised Ann. Vol. 8% 7% 6% 5% 4% 3% % % 0% B&H Wilkie em Yeas Exhibi 8. suggess some siking diffeences in he simulaed long-em equiy behaviou of he wo models. Noe ha he one-yea volailiies ae vey simila. Bu as he ime hoizon is exended, he Wilkie model suggess ha volailiy falls vey significanly, wheeas in he B&H model he volailiy emains consan ignoing sampling eo. Wha does his imply fo he disibuions of equiy euns? Exhibi 8.3 plos he simulaed cumulaive disibuions fo he annualised equiy euns geneaed by he wo models ove en yeas. EXHIBI 8.3: CUMULAIVE DISRIBUIONS OF 0-YEAR ANNUALISED EQUIY REURN 00% Cumulaive Fequency % 90% 80% 70% 60% 50% 40% 30% 0% 0% Wilkie model B&H model 0% -0% -5% 0% 5% 0% 5% 0% 5% 0-Yea Annualised Equiy Reun You can see ha he Wilkie model aibues a vey low pobabiliy o equiies geneaing a negaive eun ove a 0-yea peiod: he 99 h pecenile is -0.7% p.a., whils he coesponding B&H value is 6.0 %p.a. he olled-up equiy values ove he en-yea peiod ae 88 % and 53%. Whils you migh hink he B&H value is vey beaish, as was menioned above hee is pleny of evidence of such poo euns in equiy makes. 65

66 8.5 ERM SRUCURE MODEL he Wilkie model does no geneae a full em sucue of inees aes. his means ha he analys who wishes o model he euns on a bond which is no a consol o cash will need o make some difficul assumpions, e.g. some fom of inepolaion of yields. he lack of a full em sucue makes analysis of some foms of inees ae isk vey difficul. he wo models can be configued o geneae simila disibuions fo sho and longem inees aes. Bu his, o an exen, misses he poin he B&H model geneaes an economically consisen elaionship beween he sho and long-em aes he longem spo ae can be egaded as he expeced pah of he sho-em ae plus any specified isk pemium, wheeas he Wilkie model has no such economically meaningful elaionship beween he aes as hee is no model fo he em sucue. his feaue, and indeed he way he model is buil fom a seies of saisical elaionships ahe han being based on any noion of aional economic expecaions, can lead o some vey odd join pahs fo inees aes and inflaion easily occuing, as we saw above. 8.6 INFLAION he sucue of he inflaion models used in Wilkie and B&H ae acually quie simila in boh models, he ae of inflaion is nomally disibued and is modelled as a meaneveing pocess. Whils he B&H model woks in coninuous ime and he Wilkie model woks in discee ime, hee is no eason why his should have a significan impac on simulaed inflaion aes. Howeve, hee is one vey impoan diffeence: he Wilkie model is a single faco model, whee as he B&H model has wo sochasic vaiables as well as he ae of inflaion being able o vay, so oo can he ae o which i is pulled a any momen in ime. his allows moe vaied shapes of implied fuue inflaion expecaions o be geneaed which can be impoan in geneaing nominal yield cuves in he B&H models. Fuhe, as we shall see below, i also pemis geae flexibiliy in he paen of sho-em and long-em disibuions of he inflaion ae. Exhibi 8.4 illusaes he disibuions of he inflaion ae simulaed fo a 0-yea hoizon by he wo models. EXHIBI 8.4: UNCONDIIONAL DISRIBUION OF HE INFLAION RAE Fequency % 66 0% 8% 6% 4% % 0% 8% 6% 4% % 0% Wilkie B&H Wilkie Cumulaive B&H Cumulaive -6% -5% -4% -3% -% Uncondiional Inflaion Rae -% 0% % % 3% 4% 5% 6% 7% 8% 9% 0% 00% 90% 80% 70% 60% 50% 40% 30% 0% 0% 0% Cumulaive fequ. %

67 I can be seen ha he wo models poduce boadly simila shapes of uncondiional disibuions fo inflaion wih ou selecion of paamees fo he Wilkie model. Alhough he mean paamees of he wo models ae assumed o be equal a.5%, he B&H simulaed mean is acually highe a 3.%. his is because we he simulaions have been un wih eflecion uned on when geneaing he nominal em sucue. his has he effec of poducing a highe mean han when he model is un wihou eflecion. Clealy we can quanify his effec and adjus he paamees of he model o offse he incease in he simulaed mean if we wish. Now ake a look a exhibi 8.5 below, which shows he disibuions of nex yea s inflaion ae. EXHIBI 8.5: DISRIBUION OF NEX YEAR S INFLAION RAE 50% 00% Fequency % 45% 40% 35% 30% 5% 0% 5% 0% 5% 0% -6% -5% -4% -3% -% Wilkie B&H Wilkie Cumulaive B&H Cumulaive -% 0% % % 3% 4% 5% 6% 7% 8% 9% 0% Inflaion Rae afe yea 90% 80% 70% 60% 50% 40% 30% 0% 0% 0% Cumulaive fequ. % Ineesingly, he diffeences beween hese disibuions ae fa moe significan he Wilkie model geneaes unfeasibly high sho-em inflaion volailiy. In he Wilkie model, he sucue of he model makes such high sho-em volailiy necessay in ode o geneae longe-hoizon disibuions wih sufficienly high volailiy. he use of a second sochasic faco in he B&H model means ealisic disibuions can be geneaed fo boh sho and long-em disibuions of he inflaion ae. Advocaes of he Wilkie model migh ague ha he sho-em disibuion of inflaion is no eally impoan he model is designed fo analysing long-em oucomes. Howeve, he diec dependence of asse euns on pevailing and pas inflaion aes suggess he pah aken by inflaion could be vey impoan o he model oupu. 67

68 8.7 DERIVAIVE PRICING he indiec appoach o modelling equiy pices and he lack of an abiage-fee em sucue fo inees aes makes boh equiy opion and inees opion picing and modelling vey difficul wihin he Wilkie model. In picing equiy opions, finding a closed-fom equiy opion picing fomula as implied by he equiy model is vey difficul. We could of couse simulae opion pay-offs unde a isk-neual se of assumpions fo equiy euns. We would find ha opion pices implied by he Wilkie model ae lowe han make pices as make paicipans anicipae highe long-em equiy volailiy han ha implied by he mean-eveing equiy eun behaviou of he Wilkie model. he lack of a full em sucue pus he ask of picing/modelling almos any inees ae deivaive beyond he scope of he Wilkie model. Geneally, he analysis of deivaives boh in ems of picing and in ems of hei behaviou wihin a pofolio will equie a moe sophisicaed sochasic asse model, and one which is moe consisen wih he economic pinciples of efficien, aional makes. 68

69 9. EXENSIONS O HE BASIC MODEL We believe ha he model descibed above is useful in siuaions whee we seek plausible join pahs fo equiies, inflaion and boh eal and convenional bonds. As always, new poblems will aise which equie exensions o he basic model. We have se ou below some vey bief commens on poenial exensions o he model some of which ae elaively saighfowad and some fo which fuhe eseach is equied. 9. FOREIGN EQUIY & PROPERY he addiion of equiy-ype asses and yields is faily saighfowad if as fo he domesic equiy asse class euns ae specified in excess of sho aes and he modelle is pepaed o make all equiy-ype asses swich egimes a he same ime. Coelaion among equiy excess euns is imposed hough he coelaion sucue specified beween he impulses o he vaious componens of he model. 9. CREDI RISK & CORPORAE BONDS his is anohe aea in which academic models can be used o exend he basic model o mimic he behaviou of cedi speads in geneal and he cedi behaviou of individual asses. he Jaow-Lando-unbull cedi model 6 can be used o exend ou model o any ohe model of he defaul-fee em sucue o allow he analysis of cedi-isky bonds wihin a Mone-Calo famewok. 9.3 FOREIGN EXCHANGE & FOREIGN ERM SRUCURES he exension of he basic model o foeign cuency and foeign inflaion adds consideable complexiy o he model o ensue: Exchange aes espec puchasing powe paiy elaionships a long hoizons. Inflaion paens can accommodae a global componen given he appaen synchonisaion of pas inflaion paens. Whils hese feaues can be added o he model, we have come acoss few siuaions o dae whee hey add significanly o he insighs gained fom modelling a educed se of asse ypes. 9.4 EQUIY MEAN REVERSION Ou saing poin has been o ignoe mean evesion because he saisical evidence whils suggesive does no compel us o include i in he model. Of couse, i is possible o impose mean evesion on he model if we choose o. Ceainly, hee ae pleny of people who do believe ha mean evesion is a song feaue of equiy makes and expec o see i epesened in models. 6 Jaow RA, Lando D, unbull SM, "A Makov Model fo he em Sucue of Cedi Risk Speads", he Review of Financial Sudies 997, Vol 0, No, pp

70 9.5 MORALIY Eos in he saisical sense in foecasing moaliy impovemen aes ove he pas 0 yeas have caused poblems o life assues, conibuing o he magniude of annuiy opion losses. Again, i is possible o incopoae a sochasic moaliy feaue ino he model o mimic fuue moaliy unceainy. 9.6 DERIVAIVE PRICING / CONINGEN CLAIMS VALUAION Ou model can be used o esimae he value of ceain ypes of deivaive o coningen claim using he isk-neual picing mehodology ouinely used by invesmen banks and academics. Mean euns on all asses ae se equal o he isk-fee ae of inees and hen a mean esimae fo he expeced cash flow fom he deivaive is calculaed using Mone-Calo mehods. his cash flow is discouned a he isk-fee ae o obain he economic value of he deivaive o coningen claim. he isk-neual picing mehodology makes many assumpions ha ae violaed in he eal wold and in pa by ou model if implemened as descibed above. As a consequence, hese esimaed values mus be inepeed caefully. Neveheless, hey do give a meaningful benchmak and ae ofen makedly diffeen o esimaes fo coningen claims made using convenional acuaial echniques. Fo ha eason alone, hey deseve he aenion of he acuay. 70

71 0. CONCLUSION Life companies ae in he isk managemen business. he isks caied by hem come fom many diffeen souces. Some isks can be divesified and ohes mus be bone by shaeholdes and policyholdes. Ove he pas 0 yeas hee has been an enomous incease in he compuing powe available o he financial planne who ses ou o build financial models. In paallel wih his echnological innovaion hee has also been apid developmen of models by academics and paciiones. he new echnology and echnical know-how offes he oppouniy o addess old poblems in new ways. Ou epo pesens ou ideas on wha consiues a good model and hen ses ou a single example of a sochasic model ou of many ineesing poenial candidaes. We give some backgound infomaion o he messy ask of calibaing he model and povide some sample calibaions. Again, hee ae many ineesing alenaives. he oupu of he model is illusaed wih he specific calibaions pesened. Finally, we conas he model wih he Wilkie model and vey biefly discuss exensions. he model we have pesened is fa fom pefec no model eve is. Howeve, we do believe ha is elaive pasimony, eady economic inepeaion and is abiliy o mimic some impoan feaues of financial makes means ha i deseves he aenion of analyss seeking o model joinly he behaviou of inflaion, inees aes and equiy makes. 7

72 7 APPENDIX A INCREMENING HE ERM SRUCURE Given he cuen values and, we can calculae he expeced values and vaiances of and. hey ae as follows: e e e E µ µ µ + + =, e E µ µ + =..., + + = e e Va σ σ e e e σ e Va = σ As we know and ae nomally disibued, so all we need ae he above momens in ode o sample fom he disibuions and incemen he em sucue. ha is:,, Z Va E + = and Z Va E + = whee Z and Z ae independen sandad nomal deviaes. Howeve, if we have a non-zeo em pemium paamee, g, hese equaions ae adjused as follows: g Z Va E + + =,, and g Z Va E + + = he em pemium paamee has he effec of adjusing he evoluion of he sho-em inees ae pah, so ha he shape of he yield cuve is no an un-biased esimae of he pah of fuue sho-em inees aes, bu insead has a loading which eflecs invesos isk pefeences. Fo example, if invesos equie an addiional eun o inves in longe-em bonds, hen g is negaive, and he expeced value of a olled-up cash accoun will be lowe han he expeced value of an accoun invesed and coninuously-ebalanced in, say, 0-yea bonds. he em pemium can be expessed in

73 ems of expeced zeo-coupon bond yields, o expeced euns. he em pemiums ae as follows whee he em pemium is defined as he diffeence beween he expeced yield/eun on an infinie-mauiy zeo-coupon bond and he expeced yield/eun on a insananeously-mauing zeo-coupon bond: em P emium Yields = g σ σ + σ σ + σ σ em P emiumreuns = g + Noe ha a posiive eun em pemium equies a negaive g. In he inflaion model, he isk pemium paamee eflecs he addiional eun available on a nominal bond elaive o index-linked bond, and woks in exacly he same way as in he eal inees ae model. 73

74 74 APPENDIX B CALCULAING COVARIANCE ERM IN NOMINAL ERM SRUCURE o calculae he covaiance em, we need o evaluae he following vaiance: { } { }, exp.,, exp, exp + = R Va R Va R E R Va whee:., x x R E + + = µ , y y y y y y y y y y y R Va + + = } exp{ x µ = } exp{ } exp{ x µ = + = y σ σ y σ σ = 3 } exp{ y = 4 y σ = 5 } exp{ y = 6 y σ σ + = 7 } exp{ y = 8 y σ = 9 } exp{ y + + = 0 y σ = } exp{ y =

75 An analogous expession exiss fo his vaiance wih egad o he inflaion em sucue. By aking he poduc of hese vaiances and he coelaion, we find he covaiance em ha applies o he zeo-coupon nominal bond pice. APPENDIX C WILKIE PARAMEERS USED IN SECION 8 EXHIBI C.: WILKIE MODEL PARAMEERS Inflaion Dividend Yields qmu.5% yw.35 qa 0.58 ya 0.6 qsd.7% ymu 0.05 i0.5% ysd 0.55 yn0 Ln0.05 Dividends Consols dw 0.58 cw dd 0.3 cd dmu cmu dy ca. dsd 0.07 ca dx 0.4 ca3 0. cy 0.06 csd 0.4 Sho-em inees aes bmu 0.98 ba 0.74 bsd 0.8 bd

76 76