CRO Forum Best Practice Paper - Extrapolation of Market Data

Transcription

1 CRO Forum Bes Pracice Paper - Exrapolaion of Marke Daa Augus 00

2 able of conens able of conens 3. Execuive summary 4. Inroducion 6 3. Principles of Exrapolaion 9 4. Exrapolaion of ineres rae curve 6 5. Exrapolaion of equiy implied volailiy Exrapolaion of ineres rae implied volailiy Valuaion of opions on insrumens wih no opions raded in he marke Appendices 50

3 . Execuive summary he CRO Forum has idenified 4 principles for he exrapolaion of marke daa for marke consisen valuaion purposes:. Use all relevan observed marke daa where available. Exrapolaed marke daa should be arbirage-free 3. he exrapolaion mehod should be heoreically sound 4. he exrapolaion should follow a smooh pah from he enry poin o he uncondiional ulimae long-erm level. his bes pracice paper deails consideraions and bes pracices on how hese principles can be applied o he exrapolaion of ineres raes, equiy and ineres rae implied volailiy and in siuaion in which an opion has been wrien on a securiy for which no liquidly raded opions exis a all. We keep he exrapolaion mehod as simple and pragmaic as possible while remaining wihin he principles. For he exrapolaion of ineres rae curves 5 addiional principles have been developed by a Solvency II askforce on he illiquidiy premium. hese principles are suppored by he CRO Forum and are also discussed in his paper. Alhough we include ineres rae implied volailiy, we only make some high level remarks o clarify why ineres rae implied volailiy is no so much a maer of exrapolaion, bu a maer of calibraion and his paper does no inend o provide bes pracice on calibraion o marke daa. he Solvency II askforce on he illiquidiy premium consiss of CEIOPS, EC, CRO Forum, CFO Forum, CEA, Group Consulaiv and academic represenaion. his group is asked o develop principles on illiquidiy premium, risk-free ineres rae curve and ineres rae exrapolaion. 3

4 . Inroducion his paper is par of he Bes Risk Managemen Pracices series of he CRO Forum. he paper oulines principles and consideraions ha should be aken ino accoun for a bes pracice mehod on exrapolaion of marke daa. We hope his paper will faciliae he implemenaion of bes risk managemen pracices and will help in achieving consisency for regulaory purposes and exernal disclosures. Exrapolaion of marke daa is ofen necessary for he marke consisen valuaion of insurance liabiliies. We refer o he CRO Forum publicaion on valuaion of insurance liabiliies 3 for an inroducion of he basic conceps of a marke consisen valuaion. he graph below summarizes he main componens of an insurance company s balance shee and capial requiremens under Solvency II. Figure Main componens of he balance shee of an insurance company. Why is exrapolaion imporan? he marke consisen valuaion of insurance liabiliies requires he use of marke daa like ineres raes and opion prices. Insurance liabiliies ofen have characerisics ha canno be found exacly in financial insrumens wih observable marke prices. Examples are: An insurance produc wih a guaraneed benefi paymen up o 40 years from he valuaion dae, while he longes observable fixed rae risk-free bond maures in 5 years. An equiy linked insurance produc wih a reurn of premium guaranee in year 0, while he longes observable equiy opion has a erm of only 5 years. Exrapolaion of marke daa is required in boh examples o be able o calculae he marke consisen value of he insurance liabiliy. 3 Marke Value of Liabiliies for Insurance Firms, CRO Forum, 8 July 008 4

5 he choice of exrapolaion mehodology will impac he insurer in numerous ways: I impacs he Marke Value of he Liabiliy MVL and hence he solvency posiion of he company. he exrapolaion mehodology will impac he sensiiviy of he MVL o changes in he observable marke daa. his will have consequences for he amoun of Solvency Capial Requiremen SCR and risk managemen pracices like hedge programs. From a financial sabiliy perspecive exrapolaion mehodology is imporan as i can exuberae or dampen volailiy in he financial markes ino he enire indusry. he impac on MVL and SCR can be very significan, paricularly if he financial marke has limied deph and he insurance liabiliy includes long erm guaranees. he examples on he following page show an exreme ye realisic - case of he impac on MVL for wo simplified exrapolaion mehods... How is his paper organized? Firs we will develop a se of principles ha apply o he exrapolaion of all sors of marke daa in any currency. From hese principles we will derive choices and consequences for he pracical implemenaion of a bes pracice exrapolaion mehod. Afer ha we will give implemenaion guidance on how he principles can be applied o marke daa ha is mos relevan for he insurance indusry. ypes of marke daa we will explicily cover are: Ineres Raes 4 Equiy implied volailiies Ineres rae implied volailiies Opions on securiies for which no liquid raded marke exis 4 In his paper we will no discuss wha he mos appropriae discoun rae for insurance liabiliies is. he exrapolaion covers he basis risk free ineres rae erm-srucure and leaves he liquidiy premium and is exrapolaion ouside scope. he paper also does no cover any adjusmens for credi risk in he marke daa. 5

6 EXAMPLE OF IMPAC OF DIFFEREN EXRAPOLAION MEHODS Produc We will consider wo simplified producs. Produc A is produc wih premium paymens in he firs 0 years and benefi paymens in he remaining 40 years. he second produc is a single premium produc wih benefi paymens hroughou he coverage period of 50 years Cashflow produc A Cashflow produc B Ineres raes he risk-free rae is observable for fixed ineres bonds wih mauriy up o 0 years. Beyond 0 years we have applied wo differen exrapolaion mehods.,50% Forward raes Exrapolaion Exrapolaion,00%,50%,00% 0,50% 0,00% Impac on MVL he impac on MVL is summarized in he able below. MVL mehod MVL mehod % change Produc A % Produc B % he relaive ineres rae sensiiviy of he regular premium produc is much higher han ha of he single premium produc. Even for he single premium produc he impac on MVL is 0%. his illusraes he poenial imporance of he choice of exrapolaion mehod. 6

7 3. Principles of Exrapolaion he following four principles apply o he exrapolaion of all ypes of marke daa in scope. 3.. Principle : Use all relevan observed marke daa where available his principle is fundamenal o any marke consisen valuaion. If relevan observed marke daa would be ignored hen objeciviy of he valuaion would be compromised and he cos of he replicaing porfolio would differ from he calculaed MVL. Marke daa is considered relevan when a. he marke insrumen has characerisics ha are similar o a componen of he liabiliy ha is being valued; and b. he daa is from a liquid marke; i.e. marke paricipans can rapidly execue large-volume ransacions wih lile impac on he prices of he financial insrumens used in he replicaion; or based on exper judgmen allow o include marke daa ha does no mee he sric liquidiy requiremen, bu is sill considered o be a good reflecion of he cos of he replicaing porfolio; and c. he marke daa is reliable; rade and quoe informaion of hese prices is supplied by hird paries and accessible and verifiable e.g. muliple providers wih similar price o marke paricipans Consideraions when deermining he las observed liquid daa poin he liquidiy and reliabiliy of marke daa is ypically lowes for marke daa poins wih he longes enor. Hence i is criical o deermine he las observed liquid daa poin. A number of consideraions should be included in deciding where o sop using marke observed daa and sar he exrapolaion. Consider he acual invesmens of insurance companies Especially in less developed markes here is ofen no deep and liquid marke by European sandard. However, many insurance companies have he available longer enor local governmen bonds in heir asse porfolio. hese bonds, while ypically no very liquid, can give a good reflecion of he cos of he replicaing porfolio and should herefore be considered in building he risk free curve. Consider quaniaive measures Quaniaive measures, like bid-ask spread, rading volume or frequency can be used o gain insighs in he liquidiy of a marke daa poin. However i is no sraighforward o arrive a a consisen rule-based approach ha works for all economies. In he appendix we show a able wih bid-ask spread of swaps. I is no feasible o say ha e.g. only poins wihin a spread of 5bp should be included as his would exclude all swap poins in some currencies. he oher way around one could consider developing crieria like: all swap poin wih a spread of 5bp or less should be included, which would no necessarily exclude poins ha do no mee his crieria. Daa can be paricularly helpful o deec rends in liquidiy for a daa series in a specific economy. Daa can also be used as a warning limi. For example, review would be warraned if rades in a specific daa poin occur wih less han daily frequency. 7

8 Anoher example is shown below. he observed spread beween he 50 year and 30 year swap in he EUR marke became exremely volaile during Q4 008 and Q 009 which could poin o decreased liquidiy in he EUR markes, which for example was no observed in he USD marke. Spread in bps beween 50yr and 30yr swap poins EUR USD sep-04 mr-05 sep-05 mr-06 sep-06 mr-07 sep-07 mr-08 sep-08 mr-09 sep-09 Figure Spread in bps beween 50 year and 30 year swap raes Consider he circumsance under which he marke price was arrived a Marke daa ha is biased because of a forced liquidaion or disress sale should no be direcly used. We refer o and suppor he work 5 by he IASB Exper Advisory Panel in his regard. Below we include a direc copy of some of heir advice. Even when a marke has become inacive, i is no appropriae o conclude ha all marke aciviy represens forced ransacions. However, as noed above, an eniy does no conclude auomaically ha any ransacion price is deerminaive of fair value. An eniy considers all available informaion, bu does no use a ransacion price when here is evidence ha he ransacion was forced. Deermining fair value in a marke ha has become inacive depends on he facs and circumsances and may require he use of significan judgemen abou wheher individual ransacions are forced. Any ransacion deermined o be forced does no form par of a fair value measuremen. An imbalance beween supply and demand for example, fewer buyers han sellers is no always a deerminan of a forced ransacion. A seller migh be under financial pressure o sell, bu i is sill able o sell a a marke price if here is more han one poenial buyer in he marke and a reasonable amoun of ime is available o marke he insrumen. Indicaors of a forced ransacion migh include, for example: a. a legal requiremen o ransac, for example a regulaory mandae. b. a necessiy o dispose of an asse immediaely and here is insufficien ime o marke he asse o be sold. c. he exisence of a single poenial buyer as a resul of he legal or ime resricions imposed. 5 See secion 7 o 5 of Measuring and disclosing he fair value of financial insrumens in markes ha are no longer acive, IASB Exper Advisory Panel, Ocober

9 However, if an eniy sells asses o marke paricipans o mee regulaory requiremens, he regulaor does no esablish he ransacion price and he eniy has a reasonable amoun of ime o marke he asses, he ransacion price provides evidence of fair value. Similarly, ransacions iniiaed during bankrupcy should no auomaically be assumed o be forced. he deerminaion of wheher a ransacion is forced requires a horough undersanding of he facs and circumsances of he ransacion. Consider marke bes pracices he CRO Forum has conduced a survey among is members o idenify marke bes pracices in deermining he las marke daa poin o be used. he resuls, based on year end 009 daa, show ha member companies have made raher consisen choices on where exrapolaion should sar. We do also poin ou ha liquidiy was significanly lower in a number of currencies during he crisis period. Indusry consensus of liquid marke daa hroughou he crisis shows e.g. ha marke daa in EUR and USD was only liquid up o 30 years. he graph below summarizes he findings for ineres raes in some key currencies. A summary for more currencies can be found in appendix, which also includes an overview of indusry consensus on liquid enors hroughou he crisis period based on QIS5 submission o European Commission EUR GBP USD 30 year 40 Year 50 year 60 year Figure 3 Number of CRO Forum members using a specific mauriy as he las observed liquid marke daa of he ineres rae curve per 3 December Consider he marke daa beyond he las observed liquid marke daa poin Once a preliminary choice has been made on wha he las liquid marke daa poin is, and he exrapolaion has been performed, i is advised o compare he exrapolaed daa poins o he observed daa poins ha did no ge included. For example, if a daa poin was no included because i was deemed o be biased because of a forced liquidaion or disress sale hen i would be expeced ha he exrapolaed daa poin would reflec a higher price. If his is no he case hen one should reasses he decision o exclude he daa poin or review he exrapolaion parameerizaion. Anoher example would be if a daa poin is excluded because he bid-ask spread is oo wide. If he exrapolaed daa poin would no fall in beween he bid-ask spread one should ry o undersand why his is he case and poenially revise he decision o exclude he daa poin or review he exrapolaion parameerizaion. 6 One CRO Forum member has lowered he las liquid ineres rae daa poin for EUR, GBP and USD o 30 years afer he survey was conduced. 9

10 Consider exrapolaion using muliple daa sources Anoher issue relaed o he relevance of observed daa is when he siuaion occurs in which wo differen insrumens are considered relevan. In paricular for ineres rae exrapolaion one could be in a siuaion in which he preferred insrumen e.g. swap is no longer liquid for a cerain erm, bu ha anoher insrumen e.g. governmen is sill considered relevan. We argue ha in hese siuaions muliple insrumens could be used o develop he observable par of he marke daa. For he case of ineres raes we will give pracical consideraion in secion 4.3 on how his could be done. Consider sressed marke condiions he consideraions o idenify he las liquid marke daa poin would no change under sressed marke condiions. However, he number of daa poins ha would mee he condiions se ou above would ypically be smaller. Consider OC markes Exper judgmen is especially relevan for opion markes where opion prices and implied volailiies are ofen based on OC Over-he-Couner prices. OC markes can be very liquid, bu i is difficul o objecively deermine o wha enor such markes are considered liquid. 3.. Principle : Exrapolaed marke daa should be arbirage-free Arbirage opporuniies should be avoided o ensure ha he bes-esimae is appropriae. For example, when he ineres rae curve is exrapolaed based on he las observed liquid spo rae his implies in mos cases ha you observe an abrup change in slope of he curve, which implies a jump in he insananeous forward ineres rae. Any arbirage free exrapolaion herefore should have a coninuous curve plus a coninuous forward curve. Exrapolaing based on forward rae achieves his by consrucion and is herefore a good saring poin for any exrapolaion mehod. In his conex he inerpolaion mehod used in boosrapping he observed par of he swap curve is also very relevan as his can have significan impac on he value of he las observed liquid forward rae. herefore, i is imporan o have consisency beween inerpolaion and exrapolaion when moving from observed marke daa ino he exrapolaed par of he curve. When combining muliple sources of marke daa o consruc a risk free curve i is also imporan o ensure his is done in an arbirage-free manner. When here is a basis spread beween such sources of marke daa for example governmen swap spread hen he exrapolaion mehod should also clarify how his basis spread is exrapolaed. Arbirage-free exrapolaion is also very relevan for volailiies. A simple mehod ha exrapolaes linearly from he las observed liquid spo marke implied volailiy owards a long-erm volailiy level can resul in illogical paerns of implied forward volailiies and even negaive forward volailiies in some circumsances. For example combining a 5-year spo volailiy of 0% wih a 0- year spo volailiy of 0% would resul in a negaive forward volailiy beween year 5 and 0. 0

11 Well-known models o fi marke implied volailiies eiher model volailiy as a sochasic process e.g. Heson model or apply parameric funcions o describe he volailiy process e.g. SABR assuming in boh cases coninuous forward volailiies. Exrapolaing forward volailiy resuls by consrucion in arbirage free marke daa and is herefore a good saring poin for any exrapolaion mehod. Arbirage-free should also be inerpolaed in he sense ha here should be consisency in exrapolaion beween highly similar markes. For example, if he observed marke daa for implied volailiy in a paricular index e.g. Nasdaq 00 is consisenly higher han he observed marke daa in anoher index e.g. S&P500 and boh indices are highly correlaed, hen you would also expec ha he exrapolaion of boh indices o ensure his propery is mainained in he exrapolaion Principle 3: he exrapolaion mehod should be heoreically sound I is imporan ha he chosen mehod is heoreically sound. Exrapolaion mehods ha are in violaion of generally acceped economic heory or clearly observed hisorical paerns should be avoided. he exrapolaion mehod should no resul in spurious volailiy. Volailiy arises from genuine changes in expeced levels of long-erm bes-esimaes and changes in risk premia. hus he resuls of he exrapolaion mehod should be consisen wih he observed paerns of marke daa. For example, realized ineres rae volailiy is ypically declining wih he enor of he rae and he resul of he exrapolaion mehod should be consisen wih ha. In general implied volailiies are used o calibrae heoreical models for he underlying marke risk facors. E.g. swapion volailiies are used o calibrae sochasic ineres rae models e.g. Libor marke model or exended Vasicek. hese models are hen used o value embedded insurance guaranees and opions based on hese marke risk facors. Such models ypically ry o capure well described properies of marke daa such as mean revering behavior. herefore such heoreical models provide a solid basis for any exrapolaion mehod. Moreover, by consrucion such models are arbirage-free. heoreical models also imply cerain feaures for long-erm levels. here is for example a lo of lieraure showing ha impac of convexiy in long-erm forward ineres raes 7. Anoher example is ha for long-erm equiy opions i can be shown ha ineres rae volailiy can have a significan impac on he pricing and herefore he implied volailiy which is ypically used based in combinaion wih he assumpion of deerminisic ineres raes. 7 A well known aricle in his conex is Ineres rae volailiy and he shape of he erm srucure [and Discussion] by Brown, Schaefer, Rogers, Meha and Pezier, 994 he Royal Sociey.

12 3.4. Principle 4: he exrapolaion should follow a smooh pah from he enry poin o he uncondiional ulimae long-erm level he view on he long-erm level should be forward looking, reflecing marke analysis, fuure expecaions, economic heory and hisoric experience. his view should be documened appropriaely and regularly reviewed. wo exrapolaion mehods can be disinguished:. Grading from he las observed liquid marke forward daa poin o a long-erm ulimae level;. Keeping he las observed liquid marke forward daa poin consan for he enire exrapolaion. In he firs mehod a long-erm ulimae level of he marke forward daa is se and an appropriae grading mehod is chosen for he area in beween he las observed liquid marke forward daa poin and he ulimae level. his mehod requires hree key assumpions: a. Wha is he ulimae forward level? he view on long-erm level should be forward looking, and based on marke analysis, fuure expecaions, economic heory and hisoric experience. his view should be documened appropriaely and regularly reviewed. b. When is he ulimae forward level reached? his should be based on an analysis of when he predicive power of he currenly observed marke daa on he fuure level of he marke daa becomes negligible. he forward level is kep consan once he ulimae level has been reached. c. Wha is he grading process beween he las observed liquid forward daa and he ulimae level? he shape and speed of he grading process o he long-erm ulimae level should comply wih principle and 3; i.e. i should be arbirage-free and based on sound economic heory and relevan hisoric daa, e.g. from oher currencies wih a deeper marke. In he second mehod he simplifying assumpion is made ha he ulimae forward level is equal o he curren level. In general his mehod can work fine when he exrapolaion is limied in lengh. his mehod may no be appropriae in cases where he liabiliies are long and he observable marke is much shorer. Principle #4 requires exrapolaion o an ulimae long-erm level. he observed marke daa which forms he basis for he exrapolaion is regarded as including an implici risk margin. his is paricularly he case where here is a naural mismach in supply-demand as for he longer enors in he swap curve. Consequenly, he exrapolaed par of he curve ha resuls from i will also include an implici risk margin. When he ulimae long-erm level is based on marke daa and economic heory as se ou in his paper, he exrapolaion is consisen wih he observed daa. If a risk margin were o be excluded from he ulimae long-erm level, hen he exrapolaion would have only a parial risk margin embedded in he exrapolaion. Excluding he risk margin compleely from he exrapolaion is no possible as i would violae principle 4 smooh pah. Adding an explici risk margin i.e. firs calculae presen value of cashflows, nex calculae a risk margin would represen double couning of risk and be highly impracical. In he box on page 5 he relaionship beween exrapolaion and risk margin is described. We will describe hese issues and implemenaion of he principles in more deail for each of he ypes of daa in he nex chapers.

13 RELAIONSHIP BEWEEN EXRAPOLAION AND RISK MARGINS One can disinguish hree ypes of exrapolaion: A simple direc exrapolaion of he marke price; i is assumed ha risk margins embedded in he marke price say consan in he exrapolaion. An advanced direc exrapolaion of he marke price; in his mehod he componens of he marke price are exrapolaed based on a macro-economic view of he long-erm marke price i.e. no necessarily he same as in he las observed liquid marke daa. 3 Sar wih an exrapolaion o a bes-esimae long-erm level; i.e. expeced real world realizaion of ineres raes or volailiies. his does no include compensaion for all risk; a risk margin needs o be calculaed separaely and included in he MVL. he firs mehod can be comforably applied as long as he exrapolaion is of limied lengh. Say, he marke is liquid up o 40 years, bu some of he liabiliy cashflows go ou o 45 years. Major deviaions beween he forward rae in year 40 and 45 are highly unlikely and would have a negligible impac on he MVL anyway. his is also evidenced by he resuls in Figure 4. Invesmen banks would ypically apply such an approach o make a marke. However, unlike insurance companies, hey end no o ake maerial amouns of non-hedgeable financial risk and hence his mehod suffices for hem. Even if one would ry o build in a risk margin, his would urn ou o be negligible since one could hedge he posiion quie accuraely wih he 40 year insrumen and he lile remaining risk would be drasically reduced due o he discouning effec over 40+ years. Now consider he same produc in a siuaion in which we have observable ineres rae forwards for only he firs 5 years. his creaes a number of complicaions: Yield curves end o be seep on he shorer end of he curve. his could resul in exrapolaion o exraordinary levels Yield curves end o be more volaile in he shorer end of he curve. his could resul in exraordinary volailiy in he exrapolaed curve. he fac ha he liquid curve ends a 5 years indicaes ha here is no naural supply of long daed fixed income. Any raional player who would offer a 45 year guaranee would be exposed o a very large amoun of risk and would wan o be compensaed for his in excess of he bes esimae level. For his ype of siuaions i is desirable o use he second mehod, as i allows o ake ino accoun he long erm naure of he produc. he hird mehod is exremely difficul, if no impossible, o apply accuraely. Since he las observed liquid daa poin includes a risk margin - bu he long-erm level does no - he exrapolaion will consis of a parial risk margin. he remaining risk margin will need o be capured separaely. o make his mehod work would require a significan amoun of assumpions ha will be difficul o verify. Where should one draw he line where he firs mehod can sill be applied and where he second mehod above should be applied? In pracice, his quesion may no be so relevan. Once a more sophisicaed mehod i.e. mehod is developed i can jus as well be applied o liabiliies ha have limied exposure o exrapolaed marke daa. 3

14 4. Exrapolaion of ineres rae curve 4.. Solvency II askforce For he exrapolaion of ineres rae curve 5 addiional principles have been developed by he Solvency II askforce on he illiquidiy premium. hese principles are suppored by he CRO Forum: a. he exrapolaed par of he basis risk free ineres rae curve should be calculaed and published by a cenral EU insiuion, based on ransparen procedures and mehodologies, wih he same frequency and according o he same procedures as he non exrapolaed par. b. Exrapolaion should be based on forward raes converging from one or a se of las observed liquid marke daa poins o an uncondiional ulimae long-erm forward rae o be deermined for each currency by macro-economic mehods. Mehods can ake differences beween currencies ino accoun. he principles used o deermine he macro-economic long-erm forward rae should be explicily communicaed. c. Crieria should be developed o deermine he las observed liquid marke daa poins which serve as enry poin ino he exrapolaed par of he ineres curve and for he pace of convergence of exrapolaion wih he uncondiional ulimae long-erm forward rae. d. echniques should be developed regarding he consideraion o be given o observed marke daa poins siuaed in he exrapolaed par of he ineres curve. e. he calibraion of he shock o he risk free ineres rae erm srucure used for he calculaion of he SCR should be reviewed in order o be compaible wih he relaive invariance of he uncondiional ulimae long-erm forward rae. he benefi of principle A is consisency and comparabiliy beween companies; see secion Guidance on principle B can be found in secion 4.5. Consideraions on C and D are included in secion 3... We believe ha i is difficul o develop easy o apply crieria C and echniques D and raher refer o consideraions ha need o be aken ino accoun for deermining he las observed liquid daa poin. Some level of exper judgmen will ulimaely be required o deermine he las liquid marke daa poin based on such consideraions. Key is ha he marke daa used should be a good reflecion of he cos of he replicaing porfolio. Principle E is discussed in secion Exrapolaion mehods From he principles i is firsly derived ha any exrapolaion mehod should be based on exrapolaing forward raes. BES PRACICE I is bes pracice o apply he exrapolaion o coninuously-compounded forward ineres raes. From he las observed liquid forward rae, he exrapolaion should follow a pah owards a robus long-erm level. We consider wo mehods for seing his long-erm bes-esimae level. More precisely he second mehod is a special case of he firs mehod and can be appropriae in cerain circumsances. he wo mehods for seing he long-erm forward rae level are: 4

15 Macro-economic mehod: A long-erm marke forward rae is se based on marke analysis, fuure expecaions, economic heory and hisorical experience. Below he componens of his long-erm bes esimae forward rae are discussed in deail. Consan forward rae: he las observed liquid forward rae derived from marke daa is assumed o be kep consan beyond he las observed liquid marke daa poin. Below i is discussed how his las observed liquid forward rae should be deermined. his mehod is a special case of he firs mehod wih he long-erm marke forward rae se equal o he las observed liquid forward rae. he second mehod is only appropriae when he available marke daa is sufficienly long compared o he cash flows ha need o be discouned. We performed ess for a number of currencies and acual liabiliy porfolios. o illusrae ha his ulimae level is especially relevan for markes where limied marke daa is available we show he absolue valuaion difference in he Expeced Presen Value of Liabiliy Cash Flows for acual porfolios of CRO Forum companies based on he above wo mehods for a number of currencies. I can be clearly seen ha he range in valuaion sar o deviae significanly when he available marke daa becomes shorer. For mos markes, i seems ha 30 years of marke daa is sufficien for using he second mehod. 6% Difference in Expeced Presen Value of Cashflow 4% % 0% 8% 6% 4% % 0% % CNY KRW HKD EUR GBP JPY USD Number of years of available marke daa Figure 4 8 Absolue difference in Expeced PV of Liabiliy Cash Flows beween a simple consan forward rae and advance macro-economic exrapolaion. Informaion provided by CRO Forum working group members We also noe ha here is a specrum of feasible approaches in beween he wo mehods described above. A moving average of long-erm forward raes could for insance be used as a pracical basis for esimaing he ulimae rae in he macroeconomic mehod. Boh mehods sar heir exrapolaion based on he las observed liquid marke forward rae and herefore he implicaion for risk managemen is ha ineres rae risk can be neuralized hrough posiions mimicking his las observed liquid forward rae. However, we would like o sress ha here remains exposure o he forward raes beyond he las observed liquid marke daa poin ha canno be fully hedged. 8 Poins depiced in diagram depend on chosen porfolio 5

16 BES PRACICE We consider he macro-economic mehod bes pracice, as i is appropriae irrespecive of he amoun of marke daa available Deermining he las observed liquid ineres rae daa poin he general consideraions as laid ou in secion 3.. apply for deermining he las observed liquid ineres rae daa poin. Specific for ineres raes we will discuss in more deail he use of muliple daa sources and wha o do wih marke daa poins ha are beyond he las liquid marke daa poin Using muliple daa sources he CRO Forum recommends he use of swap raes. However, if oher relevan daa ypically governmen curve has availabiliy beyond swap hen his should also be used. In mos markes he swap marke is available for he same or longer enors han governmen bonds. E.g. in he EUR and USD marke he swap marke currenly goes up o 50 years where governmen bonds are mosly only available up o 30 years. However, in less developed markes he swap marke is no ha developed ye. For example aiwan swap up o 0 years, governmen bonds up o 0 years, Malaysia swap up o 5 years, governmen bonds up o 0 years and hailand swap up o 5 years, governmen bonds up o 30 years. he quesion is how hese exra marke daa poins can be included in he risk free yield curve while mainaining consisency wih he principles of yield curve exension; In paricular avoiding disconinuiies in he curve ha could resul in arbirage opporuniies. One should also ake ino accoun ha boh sources of marke daa e.g. swap and governmen bonds show a difference spread for he observed par of he curve. he bes pracice would be o add addiional raw poins o he curve before a coninuous compounded yield curve is boosrapped. An example of how such an exra poin could be creaed using aiwan swap curve as an example: WD 0yr swap poin = WD 0yr Gov. Bond yield + 0 yr Swap-Gov Spread. = WD 0yr swap poin + WD 0yr Gov. Bond yield WD 0yr Gov. Bond yield he new swap poin is herefore equal o he las observed liquid swap poin plus he observed slope in he Governmen bond curve. Afer consrucing his addiional swap poin, he coninuously compounded yield curve can be boosrapped in a sandard manner including his addiional consruced swap poin. Given he imporance of he slope of he forward curve as he saring poin for he exrapolaion one has o be careful in fiing he slope beween wo marke daa poins based on a differen source, e.g. swap vs governmen bond see secion

17 As long as he swap and governmen curves hemselves are arbirage free his combinaion will also resul in an arbirage free coninuous yield curve. Noe ha in he example we have chosen o keep he las observed spread beween swap and governmens consan. A more refined mehod could be developed in which he exrapolaed spread increases or decreases wih mauriy. However, ofen here is no sufficien marke daa o suppor a more refined mehod Consideraions for using ineres rae daa poins ha are no fully incorporaed I is imporan o realize ha here is no a very clear line beween marke daa poins ha are liquid and should be fully incorporaed in he observed par of he curve and marke daa poins ha are no liquid and should no be aken ino accoun. Below a few consideraions on how o deal wih daa poins before and afer he las liquid marke daa poin. he slope of he las observed liquid forward rae should be relaively sable he las observed liquid marke daa poins are relaively imporan as hey deermine he shape of he forward curve which is he saring poin for he exrapolaion. A las liquid daa poin ha is rading slighly ou of sync wih is previous marke daa poin can resul in spurious volailiy in he exrapolaed par of he curve. herefore i should be feasible o deviae from he las liquid marke daa poin in fiing he curve o ensure a sable slope of he forward curve. Such deviaions should be ypically limied o he bid-ask spread of such poins o avoid arbirage opporuniies. Marke daa poins no included in he observed par of he curve. here migh be swap poins or governmen bond yields ha are no ye considered o be liquid enough o be included in he observed par of he curve. Such poin should give a sensibiliy check on he slope of he exrapolaed par of he curve. While i is no necessary ha he exrapolaion says wihin he bid-ask spread of such poins, hey should no be subsanially ou of sync wih each oher. Ideally i should be undersood why hey deviae. Lasly, once such poins become more liquid hey should ideally converge such ha including hese poins in he observed par of he curve should no resul in a big change in he curve. Imporance of marke daa cu-off enor in sressed marke condiions he exrapolaion of marke daa has an imporan role in avoiding oo much pro-cyclicaliy in he Solvency II framework. A solvency regime based on marke values, already has build in mechanisms ha in imes of sressed marke condiions here is an exra endency o de-risk and herefore pu exra pressure on financial markes, which could worsen again he solvency on a marke value basis. he naure of in paricular life insurance companies is such ha in general heir liabiliies have longer duraions han he available asses in he markes. As already referred o his resuls in supply-demand pressure on ineres raes for longer enors. In a crisis siuaion where boh insurance companies and pension funds ry o de-risk such pressure can resul in unbalanced markes. A very clear example of his is he period end-008 o mid-009 in he EUR ineres rae marke, where ineres rae forward raes plummeed. Similar issues also arose in he equiy implied volailiy marke wih forward volailiies shooing sky-high. he role of exrapolaion in 7

18 such siuaions is o ignore marke daa poins ha are subjec o significan unbalanced marke and promoe sabiliy in liabiliy valuaion o avoid ha addiional pro-cyclical effecs worsen he solvency posiion of insurance companies. How can his be achieved? Firsly, he fac ha he long-erm uncondiional forward rae is se in a sable manner and is no impaced by economic cycles is a good basis condiion. However, exrapolaion sars from he las observed liquid marke daa forward rae. So any insabiliy in such poin is exended o longer enors. he mechanism ha provides sabiliy is ha in a crisis siuaion he ransiion poin of where marke daa is used and where exrapolaion sars is moved o an earlier poin. his is also in line wih he principles se ou in his paper as long-enor swap poins in such marke condiions and unbalanced demand/supply would be assessed as being illiquid. So while in liquid imes he EUR swap marke would be considered somewha liquid up o a long erm, in a crisis such as end-008 his would reduce o 30 years. he sable long-erm forward rae will hen ensure enough sabiliy in he remaining enors. How can such an unbalance be observed? here are wo elemens ha could indicae an unbalanced marke. Firsly an ineres rae forward rae ha is significanly seeper downward sloping hen pre-crisis and secondly, he fac ha long-erm swap raes dropped significanly below raded AAA governmen bonds in imes of crisis. Special aenion has o be paid in imes of changes in regulaion. In paricular markes have o be wached closely around he inroducion of SII and IFRS phase wheher shor erm hedging aciviies of insurers impac he long end of he curve emporarily and herefore markes have o be regarded as disored Calculaion of he las observed liquid forward rae I is considered bes pracice o base he las observed liquid forward rae on he las wo liquid marke daa poins. I is of crucial imporance how he yield curve is being boosrapped as his impacs he forward rae beween he las wo observed liquid marke daa poins. his is shown wih below simplified ye realisic example. Assume ha he yield curve is based on below five swap poins. hese poins are quoed on an acual/acual basis wih annual paymen frequency % 3.40% 3.80% 4.0% 4.30% We boosrap his swap curve based on wo mehods: Assuming a piece-wise linear zero rae curve: Zero raes are fied for he, 5, 0, 5 and 0 year enors and raes in beween are found wih linear inerpolaion. Assuming a piece-wise linear forward rae curve: year forward raes are fied for he 0, 4, 9, 4 and 9 year poins and forward raes in beween are found wih linear inerpolaion. Boh mehods resul in nearly idenical zero curves wih differences of below bp for all enors. However, boh mehods resul in significan differences in year forward raes as shown from below graph. Moreover, he ulimae forward rae differs by 4 bps. he second mehod is considered bes pracice. 8

19 5.50% 5.00% 4.50% 4.00% 3.50% 3.00%.50% Forward Rae based on piece-wise linear zero raes Forward Rae based on piece-wise linear forward raes Figure 5 wo mehods o inerpolae he yield curve An alernaive mehod for fiing he exising daa poins as seen in pracice is fiing a spline curve which simulaneously fis he daa poins. his mehod is also considered bes pracice, bu is more difficul o implemen and can give weird resuls if daa poins are no fully consisen. We herefore prefer he simpler linear piece-wise forward rae mehod as i can be implemened more easily Exrapolaion owards ulimae forward rae Bes pracice mehod is o se ulimae forward rae based on an esimae of long-erm economic environmen. Forward raes will be inerpolaed beween las observed liquid forward rae and ulimae forward rae. wo principles have been developed on how his ulimae long-erm forward rae should be se: he uncondiional long-erm forward rae should be relaively sable over ime and only change due o fundamenal changes in long-erm expecaions and should no be affeced maerially by shor erm economic changes. he uncondiional long-erm forward rae is he sum of forward looking expecaions of he: i. Long-erm Inflaion Level ii. Long-erm Real Rae iii. Long-erm erm Premium iv. Long-erm Convexiy Adjusmen he esimaion of hese componens should be forward looking, and based on marke analysis, fuure expecaions, economic heory, hisorical experience and should reflec a risk margin already. In case one componen is no refleced in he exrapolaion i should be a leas refleced wihin an addiional risk margin - as his inroduces addiional complexiy in some cases i's no our recommendaion bu feasible as long as he risk margin is se in a consisen manner wih he long-erm forward rae. 9

20 4.5. Componens of ulimae rae hese componens of he uncondiional long-erm forward rae defined by above principle can be esimaed individually or can be esimaed ogeher. he following secions describe bes-pracice mehods o esimae hese componens Long-erm real forward ineres rae We consider 3 mehods for deermining he real forward ineres raes: Hisorical average Derived from nominal forward raes and inflaion swaps 3 Direcly observed in he marke Hisorical Average his mehod for deermining real ineres rae levels is o look a hisorical averages for real cash reurns consruced as he yield on a shor erm ineres rae less CPI inflaion. Work by B&H 9 shows ha his resuls in a range of oucomes for various developed counries e.g. 0.6% for Japan versus.4% for US and.% for Germany and he median esimae is se a.8%. hese esimaes are based on a reference period. Main drawback of his mehod is ha is i backward looking raher han forward looking and assumes ha hisorical observaions are represenaive for he far fuure. Derived from nominal forward raes and inflaion swaps his mehod looks a real forward ineres raes implied by nominal forward ineres raes and inflaion swap prices. Since inflaion swaps are available up o 30 years, i is feasible o esimae he forward real rae a he 30 year poin. Looking a he reference period his mehod would have resuled in real raes around.5% for boh EUR and USD. During he 009 crisis he implied real raes for boh currencies dropped significanly o around.0% and since hen recovered o around.0%. he advanage is ha his mehod is forward looking. However, he cos of his is more insabiliy. his could be solved by applying some averaging over ime. Direcly observed in he marke A hird alernaive is available for some counries only. In many Lain-American counries, boh nominal and real ineres rae curves are direcly quoed in he marke. So he real forward rae can be direcly observed in he marke for significanly long enors. Examples include Mexico UDI curve up o 30 years, Chile CLF curve up o 0 years and Colombia UVR curve up o 0 years. he real raes in hese markes are ypically somewha higher han hose seen for EUR and USD, a around 3.0% o 4.0% as invesors demand a higher real reurn in hese markes o compensae for more inflaion uncerainy. 9 Ineres rae calibraion, how o se long-erm ineres raes in he absence of marke prices, Seffen Sorensen, B&H, Financial Economic Research, Version, Sepember 008, Pg 6. 0

21 he choice for one of hese hree mehods or a combinaion will depend on he availabiliy of daa. If sufficien daa is available hen all hree mehods should give a roughly similar exrapolaion of he forward rae. Wheher his will be he case in pracice depends on he speed of convergence o he long-erm expeced forward rae. BES PRACICE Boh marke implied mehod & 3 and hisorical mehod should be evaluaed in esimaing he real forward rae. hese mehods are no muually exclusive since hisorical daa of marke implied real raes is available. Key is ha enough hisorical daa a leas one economic cycle is used o ensure sabiliy in he long-erm forward real rae and avoiding ha he shor-erm economic cycle impacs he esimaion process. I should also be avoided ha oo much hisorical daa is used o avoid ha srucural changes in he real raes are no refleced in he esimaion Long-erm expeced Inflaion We consider 3 mehods for deermining he long-erm expeced inflaion: Weighed average of hisorical inflaion and inflaion arge Weighed average of economic forecas esimaion per currency 3 Derived from inflaion swaps/bonds. Weighed average of hisorical inflaion and inflaion arge Hisorical analysis as shown by B&H 0 highlighs a large discrepancy in average inflaion beween differen counries. o comba he high level of inflaion in he 970s and 980s, a number of cenral banks now operae wih an explici arge for inflaion. he level of such inflaion arges ends o be around %. Hisorical inflaion averages are raher high compared o his arge level, while more recen hisory is quie close o his level. he quesion is herefore wheher we believe ha cenral banks would credibly succeed in keeping inflaion a low and sable levels in he very long-erm. B&H explicily summarizes his dilemma in hree ways o se he long-erm expeced inflaion: Mehod Descripion No prior on inflaion arge credibiliy Compue an esimae for long-erm inflaion expecaions using he exponenially weighed average of CPI inflaion Srong prior on credible inflaion Aach a weigh of x o an inflaion arge of % for all economies and -x o he exponenially weighed average of hisorical CPI inflaion. he closer x is o one, he higher a prior on he inflaion arge. 3 Credible inflaion arge Impose long run inflaion expecaions of % a any poin in ime across each of he economies. 0 Ineres rae calibraion, how o se long-erm ineres raes in he absence of marke prices, Seffen Sorensen, B&H, Financial Economic Research, Version, Sepember 008, Pg 8.

22 B&H op for he second mehod and apply an 80% weigh on he % inflaion arge and a 0% weigh on he exponenially weighed hisorical average. his resuls in a.4% long-erm expeced inflaion for all counries. We believe i makes sense o differeniae he inflaion arge by currency. For example he Norwegian cenral bank has an inflaion arge of.5% while mos oher cenral banks have a lower arge. In seing he inflaion arge one should also consider he credibiliy of such a arge if he acual inflaion level consisenly over or undershoos he arge se by he cenral bank. Weighed average of economic forecas esimaion per currency he disadvanage of above esimae is ha i assumes ha inflaion expecaions will be he same for all counries. Hisorically here are significan differences over longer periods in inflaion levels in differen counries. herefore i can be argued o use a differen inflaion forecas for differen blocks of counries. One good source of inflaion expecaions is a semi-annual survey of Consensus Economics which includes a wide range of counries and sources boh banks and economic insiuions and predics inflaion forward up o 0 years. he quesion is how represenaive his inflaion forecas is for he very far fuure. One could use his forecas o derive he expeced inflaion levels going forward. Below able shows for some example currencies he average 0 year inflaion forecas over he las years average las four surveys. he.0% inflaion arge is credible for hard currencies such as EUR, SEK and CAD. Some counries wih less explici inflaion arges have higher inflaion and for counries like Japan and aiwan he forecas is persisenly below.0%. We noe ha forecass are broadly consisen wih inflaion arges se by cenral banks e.g. EUR.0% and NOK.5%. Europe Americas Asia Pacific EUR.0 USD.3 JPY. GBP.3 CAD.0 KRW.5 NOK.4 MXN 3.8 WD.8 SEK.0 BRL 4. AUD.6 In summary, his mehod applies a separae inflaion forecas for each currency based on a moving average of Consensus Economics forecass. One excepion migh be for he currencies in Cenral Europe. Since hese currencies are expeced o sooner or laer join he EUR, i makes sense o apply he EUR inflaion forecas o se he long-erm forward rae. Compared o he previous mehod, which assumes one inflaion arge for all currencies based on hisorical daa, his mehod akes he oher exreme by seing a counry specific inflaion arge based on relaively shor-erm projecions. he ideal se up migh be in he middle aking a weighed average of a generic long-erm arge inflaion and a counry specific projecion. Derived from inflaion swaps/bonds. Inflaion forecas can also be derived from raded insrumens. Mos specifically one can use inflaion swaps which exchange he acual observed inflaion agains a fixed coupon. Based on such insrumens one can predic he inflaion up o 30 years. Drawback is ha such inflaion swaps are only raded of a limied number of currencies such as EUR, USD, GBP and JPY. An alernaive is o derive inflaion from inflaion linked bonds or he difference beween nominal and real ineres raes such as quoed in many Lain-American counries e.g. MXN versus UDI raes.

23 he curren levels of he 30-year inflaion swap quoes for EUR and USD are respecively.4% and.8%. here is evidence ha inflaion swaps are rading a a premium of abou 40 bps over acual inflaion. his is consisen wih he difference versus he Economic Consensus forecass. his mehod can be used in combinaion wih he previous mehod and can be used insead of a Consensus Economics forecas. E.g. a weigh of 50% o he inflaion arge and 50% o an average of years of survey daa would resuls in below long-erm inflaion levels for some example currencies. Inflaion argeavg Survey Weigh L inflaion EUR.0%.0% 50.0%.0% GBP.0%.3% 50.0%.% JPY.0%.% 50.0%.6% HKD.0%.6% 50.0%.3% BES PRACICE We believe i is imporan o keep a balance beween on one hand a generic inflaion level for all counries and on he oher hand he curren inflaion credibiliy of a counry. herefore we propose o calculae for each counry a weighed average beween he currency specific inflaion arge weigh X% and a hisorical average of long-erm forecass based on eiher survey daa or marke raded inflaion weigh -X%. he weigh can differ by currency erm premium he componen of he long-erm forward rae ha is perhaps he mos difficul o esimae is he long-erm erm premium. he basic heory of he erm srucure of ineres raes is he expecaions hypohesis. According o his hypohesis, he expeced reurn from holding a long bond unil mauriy is he same as he expeced reurn from rolling over a series of shor bonds wih a oal mauriy equal o ha of he long bond. ha is, he long bond yield is he average of he expeced shor-erm raes. hough he expecaions hypohesis provides a simple and inuiive appealing inerpreaion of he yield curve, i ignores ineres rae risk and invesors may require compensaion for his risk. he erm premium refers o such compensaion. here are also oher facors influencing erm premia such as liquidiy consideraions and invesor habias. One example is he fligh o qualiy effec in some major governmen securiies markes a imes of exreme volailiy. Demand for governmen securiies from large insiuion such as pension funds and insurance companies in cerain marke segmens is an example of invesor habias. 3

24 Even hough various research has been done here is lile consensus in he lieraure on he empirical properies of he erm premium. he discussion is furher complicaed by he exisence of muliple definiions of he erm premium. hree commonly used definiions are: he expeced reurn of holding a muli-period zero coupon bond for one period minus he one-period yield shor rae. he forward rae minus he expeced fuure spo rae. 3 he yield on a zero coupon bond minus he average expeced shor raes from he presen o he mauriy of he bond. Mos sudies are based on 0-5 year bond daa mosly governmen bonds and indeed show a significan posiive erm premium in he order of % o %, alhough wih a big sandard deviaion. Work by B&H esimaes his erm premium o be.7% for swap raes. Some caveas need o be made. Firsly, ha erm premiums show a declining rend over ime as shown by Orphanides. Secondly, he quesion arises on how o exend he erm premium beyond he 0-5 years as used in hose sudies. here is also lieraure ha clearly shows ha forward ineres rae curve consisenly decline beyond 5-0 years as shown by Brown and Schaefer 000. We indeed observe his behaviour also consisenly for all major currencies over he las 6 years wih 30 year forward raes in he order of % lower han 0 year forward raes. he demand supply mismach in longer enor bonds plays definiely a role in his observed decline in forward raes and herefore erm premium for hose enors. In summary, lieraure does no give us conclusive informaion on wha he long-erm erm premium should be. When he real ineres rae level is based on he observed cash real rae hen a erm premium will be required o esimae he ulimae forward rae. When real raes are derived from forward looking long-erm nominal raes and inflaion swaps hen his is no applicable. he erm premium can be hough of as a risk premium. If here would be a naural balance beween demand and supply hen he erm premium could be close o zero. If here is no naural balance hen he erm premium gives he addiional incenive on op of he bes-esimae o eiher buyer or seller o reach a muually accepable price. In observed bond prices his is ypically hough of he addiional premium required o compensae an invesor for locking in he money for a long ime and aking he associaed ineres rae risk. he same argumen would hold for a policyholder buying a guaraneed insurance produc wih erm beyond wha is available in he fixed income marke 3. However, if one would look a i from he perspecive from he insurance company hen i is he oher way around. he insurance company would wan o be compensaed for offering a longerm guaranee i canno hedge. his would argue for a negaive adjusmen o he bes-esimae. Basically, one can see he bid-ask spread widening. In his framework he size and sign of he erm premium depends on he perspecive one akes: bid, ask or mid. While using mid prices has he advanage ha asses and liabiliies wih exacly he same cashflow profile have he same value, i does creae he problem ha he insurance company would no be able o earn a reurn on he capial i would need o hold for he non-hedgeable ineres rae mismach risk; i.e. i would value he produc a a price below wha i would cos o manufacure i. In his ex we use muliple reference from he BIS paper of Orphanides, he bond marke erm premium: wha is i, and how can we measure i? Ineres rae calibraion, how o se long-erm ineres raes in he absence of marke prices, Seffen Sorensen, B&H, Financial Economic Research, Version, Sepember 008, Pg 3 3 his would also explain why he invesor is willing o pay exra disribuion cos, profi margin on op of he cos o manufacure an insurance produc. 4

25 he impac on he risk margin calculaion is anoher imporan consequence of he approach aken in he exrapolaion of he erm premium. One would be double couning if a bid price is modeled and in addiion a risk margin would be calculaed. Vice versa, if mid or ask price is modeled hen an addiional risk margin would be required. he criical poin here is ha he risk margin for non-hedgeable marke risk should be se in a consisen manner wih he erm premium in he ulimae long-erm forward rae. Below we illusrae his based on wo examples. Consider wo nearly idenical exrapolaions based on he principles and proposals in he bes pracice paper. he only difference is he assumpion in he erm premium for which we applied eiher.7% or 0.0%. he goal of hese examples is no o claim ha eiher value is righ or wrong, bu o show he need for a consisen risk margin o arrive a marke consisen value of insurance liabiliies. We look a hree currencies: WD, PLN and EUR. he firs wo have limied marke daa we use respecively 0 and 0 years of marke daa, while we use 50 years of marke daa for EUR. he graphs below show he average forward curve over he pas five years excluding Q4 008 o Q for WD and PLN versus EUR based on above wo exrapolaions. he firs graph shows PLN, which is a candidae currency o ener he EUR a some poin in he relaively nearby fuure. 6,0% 5,5% 5,0% 4,5% 4,0% 3,5% 3,0%,5% EUR PLN P = 0.0% PLN P =.7% Figure 6 Average forward rae curves for EUR and PLN implied by swap curves over he las 5 years excl Q4 008 Q 009 based on quarerly observaion poins. Key poin: Assuming a higher erm premium in he exrapolaion as implied from observed marke daa i.e. EUR in above example implies ha forward raes in relaed markes wih less marke daa no necessary have convergence in forward raes. o ge a consisen MVL he risk margin would need o be se consisen o ensure such convergence afer applying he risk margin for un-hedgeable marke risk. 4 During his period we noed ha EUR forwards beyond 0 years deviaed significanly from he period before and afer hese daes due o he credi crisis and reduced liquidiy in such poins. 5

26 he second graph shows WD where forward raes have been consisenly below he level seen in EUR over he las five years. 5,5% 5,0% 4,5% 4,0% 3,5% 3,0%,5%,0%,5%,0% EUR WD P = 0.0% WD P =.7% Figure 7 Average forward rae curves for EUR and WD implied by swap curves over he las 5 years excl Q4 008 Q 009 based on quarerly observaion poins. Key Poin: One would expec o see some level of consisency beween he difference and shape of he observed par he curve and he exrapolaed par. Assuming a higher risk premium in he exrapolaion as implied from observed marke daa i.e. EUR in above example implies ha forward raes can rise above he levels seen in observed markes even hough he observed par of he curve has consisenly been lower. o ge a consisen MVL he risk margin would need o be se consisenly o ensure ha i is NO more aracive o wrie long-erm guaranees in currencies where we CANNO hedge han in currencies where we CAN hedge our exposure. he CRO Forum prefers o embed he risk margin for non-hedgeable marke risk in he yield curve exrapolaion. he size of his embedded risk margin depends on he currency, and how effecive he exposure in he exrapolaed par of he curve can be hedged wih available insrumens. his also o achieve consisency wih marke daa poin ha lie in he exrapolaed par of he curve, bu are sill relevan for risk managemen purposes. Refer o secion for a discussion on risk managemen consideraions. In he conex of Solvency II direcive, i is imporan o noe ha as he marke erm premium already includes an allowance for he risk margin for non-hedgeable marke risk i is no necessary o separae he calibraion beween a bes-esimae and charge a separae risk margin for non-hedgeable marke risk. his is equally relevan for implied volailiies, which also include a risk margin a any enor. BES PRACICE he CRO Forum prefers a erm premium derived from observed marke daa and economic heory ha herefore embeds he risk margin implicily in he exrapolaion. his allows for a naural exension from observed marke daa and would ensure a consisen MVL calculaion across companies. 6

27 Convexiy From exrapolaion principles # and #3 one can derive he exisence of convexiy of ineres raes. Convexiy arises when here is uncerainy abou wha ineres raes will be in he fuure. he reurn o he invesor in a fixed income securiy is higher when raes go down wih a cerain amoun compared o he siuaion in which raes go up wih he same amoun. his phenomenon is called convexiy. A simplified example can help undersand he convexiy adjusmen. Consider an invesor ha knows he curren shor-erm ineres is 5%. he invesor knows raes in he fuure are uncerain, bu expecs raes on average o remain 5%. Furhermore, for illusraion purpose we assume he invesor is indifferen beween an invesmen wih ineres rae risk fixed income and wihou ineres rae risk cash. he invesor has a choice beween invesing in a 5% shor-erm asse and rolling i forward every ime, or o inves in a fixed long-erm rae. he long-erm rae a which he invesor is indifferen beween he shor-erm sraegy and he long-erm sraegy mus be lower han he shor-erm rae, even hough one expecs ineres raes o say on average he same. he volailiy in he raes ensures ha he expeced reurn o he invesor is he same. In his example he convexiy adjusmen is he difference in he shor-erm and long-erm rae. Of course, in realiy risk aversion and expecaions of changes in he average level of raes have an impac on he shape of he curve, bu he concep of convexiy adjusmen remains in ac under hose circumsances 5. Depending on he selecion of he shor rae model, he convexiy adjusmen is somewha differen. ypically he size of he adjusmen depends on he volailiy and enor of he forward rae. he higher he volailiy, he higher he adjusmen; he longer he enor, he higher he adjusmen. he esimaion based on he real ineres rae, he inflaion and he erm premium is sill an expeced rae. We need o apply he convexiy adjusmen o ge he forward rae. he observed swap curves in major markes such as EUR, GBP, JPY and USD suppor he exisence of he convexiy adjusmen oo. hey poin o consisenly downward sloping forward ineres rae curves for enors beyond 0 years. Mehods o esimae he convexiy adjusmens Derived from he variance of a 0 year Governmen bond Derived based on shor rae model 3 Observed from long-erm swap raes. 5 For empirical evidence see Brown and Schaefer, Why Long erm Forward Ineres Raes Almos Always Slope Downwards, May 00. 7

28 Derived from he variance of a 0 year Governmen bond From finance heory i can be found as shown by B&H 6 ha he expeced reurn of holding a long-erm bond, which equals he uncondiional nominal forward rae is equal o he one period rae, he one period expeced inflaion, a holding period risk premium and a convexiy adjusmen. his convexiy adjusmen is equal o minus half he variance of he excess reurn of he bond. B&H use a 0-year nominal Governmen bond o esimae his convexiy adjusmen. heir analysis esimaes his adjusmen as -40 bps. Derived from Shor rae models Based on he Ho Lee model, he convexiy adjusmen is one half of volailiy squared imes he beginning and ending enor. I does no converge as he enor goes o infiniy. Based on he Hull Whie model, he convexiy adjusmen is more complicaed. Depending on he parameer, i does converge o a specific level; see figure 8 for he convexiy adjusmen a differen mauriies. If he esimaion of he real rae is based on he shor-erm rae he ulimae convexiy adjusmen would be slighly more han -30 bps. If he esimaion of he real rae is based on he 30 year daa poin, hen hose raes would already include a convexiy adjusmen of around -30 bps. Based on he chosen parameers of he Hull-Whie model here is hardly any addiional increase in convexiy adjusmen ha should be included in he exrapolaion i.e. - bps a 00 years % 0.05% % 0.5% 0.0% 0.5% 0.30% 0.35% convexiy adjusmen Addiional adjusmen afer 30 year Figure 8 Convexiy adjusmen using Hull Whie ineres rae model. 6 Ineres rae calibraion, how o se long-erm ineres raes in he absence of marke prices, Seffen Sorensen, B&H, Financial Economic Research, Version, Sepember 008, Appendix. 7 We have assumed a volailiy of.% and speed of he mean reversion facor of 5%. 8

29 Observed from long-erm swap raes One can also observe convexiy direcly in swap raes in various markes. Coninuously compounded forward raes derived from swap raes ypically are downward sloping in all markes. Moreover his convexiy adjusmen also seems o increase for 40 and 50 year enors in EUR and USD markes. However, his marke implied convexiy adjusmen combines wo elemens: he heoreically jusified convexiy adjusmen as shown above and a risk margin as here are no 40 and 50 year bonds available o hedge he exposure ha raders in such ransacions ake. herefore a risk premium for such non-hedgeable risk is included. he increase in he observed spreads beween Q3 008 and Q 009 could be inerpreed as an increase in he risk premium added by marke paricipans bu could be misleading due o a drop in liquidiy in hose enors. EUR Spreads beween nominal forward raes - -0,50 -,00 -,50 -,00 -,50-3, Figure 9 Observed differences beween he 50, 40, 30 year yr forward raes versus he 0yr yr forward rae for EUR during he las 5 years. he downward sloping forward curves can be inerpreed as he combined effec of he erm premium and he convexiy effec. he drop in he 30, 40 and 50 year forwards versus he 0yr forward beween Q4 008 and Q 009 could be hough of as a combined effecs from increased volailiy and herefore higher convexiy and a emporary increased risk premia i.e. lower erm premium due o hedging aciviies by various marke players resuling in more supply-demand mismach. BES PRACICE he CRO Forum prefers an esimaion of he convexiy adjusmen using a model approach mehod and. Boh models have some limiaions, so i is advised o calculae he convexiy adjusmen based on boh models. Exper judgmen will be required o ake ino accoun he limiaions of hese mehods in seing he convexiy adjusmen. Mehod 3 has significan limiaion and should only be used as a reasonableness check. 9

30 4.5.. When is he ulimae rae reached? he quesion ha auomaically arises when seing a long-erm uncondiional forward rae is when his rae migh be reached. Is i reasonable o assume such a limiing rae is reached afer 50, 00 or even more years? Evidence suggess ha expecaions for real shor raes and inflaion change only very slowly for horizons beyond 0 years 8. his is a leas for developed markes wih high credibiliy in argeing low inflaion. For counries wih limied credibiliy in seing arge inflaion such expecaions can change more rapidly. E.g. forecased 0 year inflaion for Argenina jumped from 3.8% o 7.3% in 008 when acual inflaion increased rapidly. I seems ha variabiliy in required risk premia accouns for a major source of asse price volailiy. he quesion B&H 9 pose is: how far ino he fuure do hese risk premiums variaions exend and, how far i is reasonable o adjus he 50-year, 00- year and 000-year forward rae? As long as he ulimae rae is se sufficienly far beyond he las observed liquid marke daa his choice will have negligible impac on he valuaion of insurance liabiliies. he more criical decision is he speed of he grading process, which will be discussed in he nex secion. Ulimaely, he poin a which he ulimae rae is reached, he speed of convergence in he grading process and marke daa lying in he exrapolaed par of he curve have o be evaluaed in combinaion o achieve he bes resul from a risk managemen perspecive. We also refer o secion for a discussion on risk managemen consideraions. BES PRACICE he CRO Forum recommends ha he ulimae rae is reached sufficienly far beyond he las observed liquid marke daa Consisency in seing all componens of he uncondiional long-erm forward rae We would like o emphasize ha he four componens of he uncondiional long-erm forward rae need o be se in a consisen manner. Overall a good balance needs o be found beween hisorical observed informaion and marke observaions o ensure ha on one hand he mehod resuls in a sable anchor poin for exrapolaion and does no creae spurious ineres rae volailiy and on he oher hand ensures marke consisency. So he esimaion canno look a each componen in isolaion, bu should look a he oal framework. An implemenaion of all componens of he uncondiional long-erm forward rae for end 008 and end 009 is shown in Appendix 4. 8 See for example Kim and Orphanides, 005: erm Srucure Esimaion wih Survey Daa on Ineres Rae Forecass, Finance and Economics Discussion Paper , Board of Governors of he Federal Reserve Sysem. 9 Marke-consisen valuaion of ulra long-erm cash flows, John Hibber, B&H. 30

31 How o grade from las observed liquid forward rae o he ulimae rae? Defining an ulimae rae ranslaes he exrapolaion problem of yield curves ino an inerpolaion grading problem so ha i can be reaed wih known mehods as described in he following. he grading mehodology chosen has a large effec on he variaion in yields beyond he las observed liquid marke daa poin. Spurious variaion in hese yields can feed hrough o give excessive balance shee and valuaion variaion ha should be avoided. Wih his in mind i becomes imporan o develop a robus exrapolaion mehod ha reflecs boh curren marke condiions and empirical views of long rae volailiy, while simulaneously displaying adequae sabiliy. One mehod ha combines hese properies very well is he Nelson-Siegel mehod. Below we exensively leverage work by B&H on he Nelson-Siegel N-S mehod 0 and consider heir work on fiing his mehod bes pracice. his N-S parameric form of he forward curve is described by he following formula: F = β + β + β 3 max exp -λ max We show in he appendix how o derive his represenaion of N-S from oher commonly used represenaions. Here β corresponds o he uncondiional long-erm forward rae, λ is a free parameer which we can use o conrol he speed of convergence and β and β 3 can be specified by maching he value and derivaive of he curve a he las observed liquid forward rae. he mauriy of he las observed liquid ineres rae is denoed max. he N-S mehod has a number of desirable properies. A long mauriies i becomes fla, and as we approach our uncondiional arge is second derivaives is already negaive and increasing if approaching from below and posiive and decreasing if approaching from above reflecing prevailing marke condiions. Because forward raes are described using a very sraighforward form we can analyically mach he level and gradien of he curve a he erminus of he marke daa, and seing he uncondiional level of raes is rivial. Finally, speed of convergence can be direcly and inuiively conrolled using he λ parameer, giving us an easy handle on long rae volailiy and correlaion avoiding spurious variaion in yields. he erm exp-λ max fades he informaion on β and β 3 ou and can be ranslaed in a inuiively more accessible half life of ln/λ i.e. he recommended value of λ=0.06 by B&H ranslaes ino a half life of.6 years fading ou half of he value of β + β 3 max every.6 years please be aware ha max works in he opposie direcion bu wih rising exp-λ max will ouweigh his effec. 0 Fiing he yield curve, Spline inerpolaion and Nelson-Siegel exrapolaion, David Anonio, David Roseburgh, B&H, Version.0, Sepember

32 his approach leads implicily o falling hisoric volailiies wih increasing mauriies. his is clearly preferred over he yield curve consrucion mehods - used by many praciioners and cenral banks - ha inroduce spurious variaion in he long-erm nominal forward raes. he N-S mehod does no inroduce he problem of some oher mehods in which small changes due o measuremen/pricing errors for long mauriy financial insrumens can induce large variaion in he long end of he forward rae curve [see B&H: How o consruc a volailiy erm-srucure of ineres raes in he absence of marke prices ]. he following is a char of he realized swap coupon rae volailiy for major currencies from May 000 o Sepember 008. he numbers are calculaed based on he absolue change of he swap raes. % means ha he one sandard deviaion of he ineres rae movemen over he nex year is eiher up or down %. he swap raes beyond 30 years are calculaed using he N- S exension. his mehod exends he decline of realized volailiy beyond he observed period quie nicely. Realized volailiy of swap raes.4%.%.0% 0.8% 0.6% 0.4% USD EUR GBP 0.% 0.0% Years Figure 0 Realized volailiy of swap raes since 000 including he exrapolaion Noe ha he realized volailiy in he exrapolaed par of he curve is raher sensiive o he las observed liquid marke daa poins especially wih regards o parameer bea3 which migh lead o reducing he impac of bea3 in an implemenaion. Anoher approach would be applying a ypically small adjusmen o his par of he curve o ensure ha he volailiy decay is in line wih he observed par of he curve. he ulimae es for any grading mehod is o back-es he mehod for a range of currencies and over a significan ime horizon e.g. las 5 years o see ha he mehod achieves i goal of a sable curve wih declining forward volailiy over ime and broad consisency wih marke daa lying in he exrapolaed par of he curve e.g. wihin a cerain band. In his area he B&H mehod has proven iself in pracice, alhough alernaives can be evaluaed o come up wih he mos appropriae mehod. If muliple mehods work well hen he mos simple mehod would have he preference. BES PRACICE he CRO Forum suppors he grading mehod based on Nelson-Siegel developed by B&H as being appropriae, bu alernaives can be evaluaed in order o achieve he goal of a curve wih he desired properies wih less echnical effors. 3

33 Risk managemen consideraions he exrapolaion mehodology should adequaely reflec robus and marke consisen valuaion of reserves, as well as being risk manageable. A risk manageable exrapolaion mehodology allows he indusry o neuralize effecively he hedge-able par of he ineres rae exposure, which implies ha he behavior of he curve given he observed par of he curve should be predicable given he observed marke daa and ha here should be no excess volailiy in he exrapolaed par of he curve. Wih predicable we imply ha he shape of he curve canno change overnigh. As he las observed liquid forward rae is he saring poin of he exrapolaion, such forward rae has a high impac on he sar and direcion of he forward curve. herefore, i is key ha in fiing he curve o he observed marke daa ha here is some smoohing mechanism ha avoids ha small inconsisencies in he las observed liquid marke daa poins can resul in big swings in he level and he shape of he las observed liquid forward raes. Furhermore, he exrapolaion mehod should no resul in abrup changes in he shape of he forward curve. his would resul in spurious volailiy in he exrapolaed par of he curve. Any mehod used should be back-esed o show ha he volailiy in he exrapolaed par of he curve shows he same declining paern as observed in he observed par of he curve. Lasly, here are ofen marke insrumens available beyond he las liquid marke daa poin. herefore, such insrumens lie in he exrapolaed par of he curve. Alhough hese insrumens are no exremely liquid, hey can sill be use in effecively managing he risks. herefore, he exrapolaion should be fied such ha he valuaion he liabiliies lie wihin a reasonable boundary of such insrumens o avoid ha using such insrumens for risk managemen purposes does no produce counerinuiive resuls Solvency II: adjusmen for credi risk and illiquidiy premium In he conex of Solvency II he swap curve is proposed o be adjused o reflec he credi risk associaed wih earning he floaing par of a ineres rae swap ransacion he illiquidiy premium associaed wih he predicabiliy of he liabiliy cashflows. Exrapolaion of hese adjusmens is sraighforward. One of he quesions ha arises is how he illiquidiy premium and he exrapolaion of his premium inerferes wih he exrapolaion of he base risk free curve. One of he principles for applying he illiquidiy premium is here should be available asses o lock in he illiquidiy premium a he valuaion dae. Corporae bonds and covered bonds are ypically referred o as he insrumens for locking in he illiquidiy premium. he swap curve can have liquidiy beyond he corporae and covered bond marke as here migh be risk-free asses ha are liquid and hence do no generae an illiquidiy premiums. So he cu-off enor unil which he full illiquidiy premium can be earned is no necessary equal o he cu-off poin of liquidiy for he base risk free curve, alhough in many cases i is. he quesion is wheher here are insances of when he illiquidiy premium can be earned beyond he las observable liquid poin for he base risk free curve. Illiquidiy ask force principle #3 he liquidiy premium applicable o a liabiliy should no exceed he exra reurn which can be earned by he insurer by holding illiquid asses free of credi risk, available in he financial markes and maching he cash flows of he liabiliy. 33

34 While here maybe circumsances in which for example illiquid bonds wih a mauriy beyond he las liquid risk-free insrumen are available, i would be pracically impossible o quanify he illiquidiy premium ha far ou, as one would need he risk free rae o deermine he size of he liquidiy premium. he exrapolaion of he illiquidiy premium can be applied separaely from he exrapolaion of he base risk free curve. he same principle is applied ha his exrapolaion requires a coninuous forward curve. Based on he principle ha he illiquidiy premium can only be applied when asses are available o lock i in, he exrapolaion of he forward liquidiy premium goes o zero. In pracice i is suggesed o grade he forward liquidiy premium o zero in five years. I is no problem when his period overlaps wih he exrapolaed par of he base risk free curve. Liquidiy in he base risk free curve also does no sop immediaely, so he cu-off enor of he base risk free curve should be inerpreed as he maximum enor on which he full illiquidiy premium can be earned Solvency II: exrapolaion by an EU insiuion he CRO Forum is supporive of CEIOPS proposal ha a cenral EU insiuion shall calculae and publish he exrapolaed par of he basis risk free ineres rae curve based on ransparen procedures and mehodologies. However, we noe ha exper judgemen plays a crucial role in seing he uncondiional ulimae long erm forward rae. I is herefore vial ha he indusry has a good undersanding of how his exper judgemen is deermined and used o se he long erm rae. o monior solvency posiion on an ongoing basis, as required by he Solvency II direcive, reinsurers would need o be able o fully undersand and replicae he exrapolaion process inernally. I would defea he purpose of Solvency II if he enire indusry canno monior is solvency posiion because hey can' predic he calculaions from he cenral EU body. One specific concern he indusry has relaes o how he risk free curve is adjused in siuaions where marke liquidiy drops and long-erm observed marke is no longer a good reflecion of he value of risk free asses and exrapolaion should sar a an earlier poin. herefore i would be good o pre-define upfron riggers for an adjusmen in he saring poin of he exrapolaed par of he curve. Insurers may also wan o perform projecions and sress esing on he risk free rae for inernal purposes when making business and sraegic decisions. herefore he abiliy o accuraely and independenly replicae he exrapolaed risk free ineres rae would be imporan for sound risk aking, as promoed by he Solvency II Direcive. In secion 4. we discussed wo ineres rae exrapolaion mehods: macro-economic and consan forward rae. In he ligh of he principles ha he risk free curve should be calculaed and published by a cenral EU insiuion, based on ransparen procedure and mehodology, i seems logical o apply he macro-economic mehod for all currencies. 34

35 Relaionship beween exrapolaion and calculaion of SCR he Solvency Capial Requiremen for ineres rae risk in Solvency II conex is calculaed by applying shocks o he ineres rae environmen and revaluing asses and liabiliies in he shocked environmen. For he observed par of he curve he shocks will be derived from hisorical volailiy. he exrapolaion of he shocked curve should be based he same bes pracice mehods described in his paper. he calibraion of he long-erm forward rae should ake ino accoun he shocked marke environmen. However, since he long-erm forward rae by design is raher sable and he shock only has a one year horizon, a shock if any - o his rae should be minor. he pah owards he long-erm forward rae would change as a resul of he change in he observed par of he curve. 35

36 5. Exrapolaion of equiy implied volailiy 5.. Imporance of Exrapolaion for Equiy Implied Volailiy Many ypes of Insurance conracs, mainly life insurance, have opions and guaranees linked o equiy prices. hese opions ypically have quie long duraions which can be significanly beyond opions raded on exchanges and in OC markes. Moreover, hese opions are no held for rading, bu embedded in our liabiliies and herefore held unil mauriy or when cliens surrender heir policies. Given he long enor of such opions heir marke consisen value is very sensiive o he volailiy used in he calculaion. Marke equiy implied volailiies are very liquid for shorer enors, bu liquidiy ypically declines rapidly beyond 5 years and here is ofen no liquidiy a all beyond 0 years. During a crisis marke liquidiy can even evaporae as we have winessed a he end of 008. I is also very difficul o objecively measure liquidiy as mos equiy opions beyond a -year enor are raded in OC markes. One imporan elemen of equiy implied volailiy ha is relevan for is exrapolaion is mean reversion. Afer a period of high implied volailiy levels, ypically pared wih large drops in equiy levels, he volailiy level reurns o normalized levels. his feaure is also very clearly visible in he erm-srucure of implied volailiy. During periods of high marke volailiy he ermsrucure of declining as longer enor implied volailiies incorporae an expecaion ha volailiy is reurning o a long-erm level and visa versa during periods of low marke volailiy. herefore he exrapolaion of equiy implied volailiy should have a buil in feaure o incorporae mean reversion in volailiy levels. 5.. Exrapolaion mehod Before discussing specific exrapolaion mehods, i is also imporan o realize how equiy implied volailiy is used o value embedded insurance opions and guaranees linked o equiy price levels. ypically opions are priced wih Mone Carlo simulaion based on an underlying sochasic model. his model is ofen a Black-Scholes model which direcly uses he a-hemoney implied volailiy erm-srucure or more advanced models such as he Heson or Baes sochasic volailiy models which are calibraed o he a-he-money implied volailiy erm-srucure and ofen addiionally o he implied volailiy skew. he focus here is o provide a bes-pracice mehodology for he exrapolaion of he a-he-money implied volailiy erm-srucure. he complexiy of he sochasic model used o value embedded opions and guaranees is direcly linked o he complexiy of he underlying insurance conrac and ou of scope in his paper. We also need o clarify how a-he-money is inerpreed. he mos widely followed definiion of a-he-money opion for a ime in he fuure is he opion wih a srike level equal o he forward equiy price level for ime. We follow his definiion. Key elemens of equiy implied exrapolaion: Deermining he las liquid marke daa poin Funcional form of grading from he las observed liquid forward volailiy owards he long-erm forward volailiy 3 Long-erm forward volailiy 4 Adjusmen for sochasic ineres raes 36

37 As noed above he funcional form used o grade owards he long-erm level should incorporae ha implied volailiy follows a mean revering process and how quickly such mean-reversion akes place. he long-erm forward volailiy is capuring boh he long-erm bes esimae volailiy and an adjusmen for non-hedgeable marke risk o capure ha implied volailiy is srucurally higher han hisorical volailiy due o he capial cos involved in hedging opions. Lasly, implied volailiy is ypically quoed in he marke based on he Black & Scholes formula which assumes consan ineres raes. As ineres raes are in fac sochasic, his needs o be aken ino accoun when pricing long-erm opions & guaranees. BES PRACICE I is bes pracice o apply he exrapolaed equiy implied volailiy from he las observed liquid forward volailiy owards he long-erm forward volailiy. he funcional form of grading akes ino accoun ha volailiy is following a mean revering process. 5.. Deermining he las available liquid marke daa poin Deermining he liquidiy of equiy opion markes is very difficul. For mos indices exchange raded opions only have enors up o wo years. Mos liquidiy is coming from OC markes, bu such liquidiy is difficul o observe. Early in 009 a CRO Forum Survey on marke daa indicaed ha companies use beween 5 and 0 years of marke daa before saring o exrapolae. However, liquidiy is no as deep for each marke. Key indices such as DJ EuroSoxx50 and S&P500 have a higher liquidiy han smaller indices such as IBEX or DAX for example. Furhermore, liquidiy can dry up during a crisis as we have winessed end 008 when long-erm equiy implied volailiies where driven by liquidiy raher han by fundamenal expecaions. Quanifying specific crieria for deermining he las liquid marke daa poin is difficul and subjec o exper judgmen. Some elemens o consider include: Marke deph: open ineres for opion conracs raded in exchange markes versus hisorical levels. Such informaion is no easily obainable and does no give informaion abou he OC marke. Observed forward implied volailiy levels and heir volailiy. One should expec ha hese mean rever owards a longerm level and ha he volailiy of implied volailiy declines wih he enor. Analysis can poin our where such rends are broken. E.g. a he end of 008 forward implied volailiies beyond yrs did no show any mean reversion anymore. Marke surveys among large insurance companies ha paricipae in opion hedging of heir long-erm opions. Here we can refer o he above menioned CRO Forum survey and for example a 008 US survey among variable annuiy wriers who are acive in hedging programs. Abnormaliies in he marke: Large shocks o he marke ypically impac he liquidiy of he marke. 37

38 5.3. Funcional form of grading from he las observed liquid forward volailiy owards he long-erm forward volailiy Funcional form of grading from he las observed liquid forward volailiy owards he long-erm forward volailiy has o incorporae ha volailiy follows a mean revering process. A logical saring poin herefore is o look a he Heson model, which is widely used in financial markes o price equiy implied volailiies. he Heson model describes he volailiy as a meanrevering sochasic process, where he process revers back o is long-erm levels wih a cerain speed. he model is very simple as i has very limied number of parameers and gives a good fi o boh he erm-srucure of implied volailiy and he implied volailiy skew. he basic Heson model assumes ha boh S, he price of he asse, and v, he asses insananeous variance, are deermined by sochasic processes. ds S d dv Where v - v v dw S dw and v S dw S v dw are Wiener processes wih correlaion. In his forma one can see clearly ha he variance follows a mean revering process in which is he long-erm volailiy and he rae a which v revers o. From he Heson model i is feasible o derive a simple and inuiive funcional form for he forward implied volailiy or more precisely he forward variance as he Heson model describes he variance process. Forward IV = Mk Forward IV Forward IV = Mk Forward IV w + L Forward IV [ w] for for > Wih w = e -K- and where : Sar poin of exrapolaion, so cu-off poin of marke daa used. L Forward IV : Long-erm forward implied volailiy level K : Mean reversion speed owards long-erm level Mk Forward IV : Forward implied volailiy from observed marke prices a ime he forward implied volailiy is in his funcional form simply a weighed average of he las observed liquid forward variance and he long-erm implied variance level. he mean reversion speed K can easily be implied from observed marke daa. his mehod is in line wih he mehodology applied by B&H. In he Appendix i is shown how K could be esimaed from observed variaion in implied volailiy markes. See hp://en.wikipedia.org/wiki/heson_model for more deails on he Heson model. 38

39 5.4. Long-erm forward implied volailiy Implied volailiy has some differen feaures han jus he equiy price level volailiy observed in he marke. A marke paricipan ha wans o replicae opion prices by a dynamic rebalancing sraegy also known as dela-hedging is exposed o non-linear equiy price movemens gamma exposure. here are capial coss associaed wih such risk. In paricular here is he risk of a jump in equiy markes which can resul in losses for marke paricipans following a dynamic rebalancing sraegy. Furhermore, here is uncerainy in he long-erm implied volailiy level which canno be hedged in he marke. We noe ha such capial coss for non-hedgeable marke risk is no only relaed o he exrapolaed par of he erm-srucure, bu already is incorporaed in he observed par as well. When comparing shor-erm very liquid implied volailiy levels wih shor-erm realized hisorical volailiy hen one can observe a consisen raio beween implied volailiy and hisorical volailiy of 0% o 0% for mos indices. In he appendix an analysis is included for several indices o show his difference. For longer enor implied volailiies here is he addiional uncerainy of he long-erm implied volailiy level. By calibraing his long-erm implied volailiy level from marke daa one can observe ha marke paricipans do no view his long-erm level as fully cerain. herefore he long-erm forward implied volailiy in fac is square, he variance can be described as: Long-erm Forward IV = Long-erm Bes Esimae Volailiy + Capial Cos Non-hedgeable Marke Risk + Uncerainy L Bes Esimae Volailiy In he following secion we will discuss each of hese componens. However, we should sress ha hey canno be deermined fully independenly as hey are relaed Long-erm Bes Esimae Volailiy he firs quesion is wha hisorical volailiy is represenaive for he long-erm bes esimae volailiy. he balance has o be found beween using a long enough hisorical period o warran sabiliy in he esimae and using a relevan enough period as he volailiy is used for projecing forward which implies ha more recen informaion is more relevan. One observaion when looking a curren hisorical volailiies for several enors is ha beyond 7 years he volailiy level sabilizes. 39

40 Hisorical Volailiies 45.0% 40.0% 35.0% 30.0% 5.0% 0.0% 5.0% Duraion.4GSPC.AEX.FSE.HSI.IBEX.KS.N00.N5.RUX.SPX.SOXX50E.CECEEUR.OPX Figure Hisorical volailiies of seleced equiy indices based on differen lengh of hisorical period Anoher observaion is ha he volailiy of he las 0 years has been higher han over he previous 0 years. So when looking a very long hisorical daa one migh no capure ha hisorical volailiy has gone up over ime. Below he 0yr volailiy of differen indices over he hree blocks of 0yr daily reurn daa from //978 o //008. DJ EuroSoxx 50 S&P 500 Nikkei % 7.97%.0% % 3.%.88% % 3.0% he second quesion is wheher one single hisorical volailiy can be used for all indices or wheher i is index specific. Our view is ha here can be significan differences beween indices due o he number of socks in an index, E.g. 50 socks in DJ EuroSoxx50 versus 500 in he S&P500; he correlaion of socks wihin an index, e.g. a world index is more diversified han a counry index; And he ype of socks of which he index is composed of e.g. he Nasdaq 00 was much more volaile han he S&P500 in he 990s during he inerne-bubble. We herefore prefer a separae esimae of he long-erm bes esimae volailiy per index. Lasly, one should consider consisency beween he long-erm forward implied volailiy and he long-erm level implied from he marke. When calibraing he funcional form by esimaing he long-erm level and he mean reversion speed by fiing he funcional form o he marke, one ges an esimae of he long-erm level implied by he marke. he long-erm bes esimae volailiy and he adjusmens discussed in he nex secion should be se such ha hey are broadly consisen wih he marke. 40

41 5.4. Capial Cos of Non-Hedgeable Marke Risk As explained before i can be observed ha implied volailiy is consisenly higher han hisorical volailiy over ime. I can also be heoreically explained why his is observed in he marke. he marke invesor who holds an equiy opion canno fully offse he risk by a saic replicaion wih he underlying sock or index. A common hedging sraegy is o regularly rebalance he underlying posiion such ha he linear risk is neuralized. Such a sraegy is called dela hedging. In such a dela-hedging sraegy he invesor is subjec o he risk of a big overnigh shock in equiy prices resuling in a loss due o he non-lineariy of opions. herefore, he invesor will need o hold capial for his risk. In he appendix i is shown ha a simple model can be consruced in which he capial ha he invesor needs o hold is a direc add-on o he long-erm implied volailiy based on jus a few parameers. Capial cos Non-Hedgeable Marke Risk = where π is he cos-of-capial and J J -- ln, J he jump risk in equiy markes o which he invesor is sensiive J = 70% implies a 30% jump in markes. For example applying a cos-of-capial of 6% and a jump of 30% in markes J=70% and a bes esimae volailiy of 5% resuls in an add-on for implied volailiy of.3%. We also noe ha he mehod implies a consan add-on in variance resuling o a higher add-on in sandard deviaion for when he bes esimae volailiy is lower as can be seen from below able. Bes esimae volailiy 5.0% 0.0% 5.0% 30.0% Bes esimae variance Non-hedgeable Marke Risk oal variance Long-erm implied volailiy 7.%.6% 6.3% 3.% Implied add-on for non-hedgeable Marke Risk.%.6%.3%.% Uncerainy Long-erm Bes Esimae Volailiy here is uncerainy in he long-erm bes esimae volailiy. As his long-erm bes esimae volailiy canno be locked in effecively his implies ha capial is required o capure he uncerainy in his long-erm bes esimae volailiy. his can already be observed o some exen in observed implied volailiies for longer enors, alhough i is hard o measure given oher elemens less liquidiy, demand/supply mismach ha sar o play are role. However, echnically i can be shown as by Manisre 00 3 ha he add-on for he uncerainy in he long-erm bes-esimae volailiy can be wrien as Uncerainy Long-erm Bes Esimae Volailiy = where Δσ α a shock in he bes-esimae volailiy over a year period. a parameer ha governs poenial fuure shocks over ime. 3 Manisre, B.J. A Cos of Capial Approach o Exrapolaing an Implied Volailiy Surface. Proceedings of he 00 ERM Symposium Chicago. hp:// 4

42 he shock is indirecly relaed o he level of he hisorical bes esimae volailiy. When his level is already near an hisorical high hen even a shock in he marke will no resul in a big jump in he long-erm esimae, while he shock is bigger when he hisorical volailiy is a a low level. In fac his becomes a kind of sabilizing componen in he overall long-erm volailiy ha offses acual changes in he bes-esimae volailiy o some degree. Manisre argues an of around 50%. Below is graph showing how he 7yr and 0yr absolue hisorical volailiies behave on a daa se of 30yrs of daily daa of he S&P 500. I shows ha annual movemens in he hisorical volailiy moved by up o 4% for he 0yr bes esimae volailiy and up o 6% in he 7yr bes esimae volailiy. 7.0% 6.0% 5.0% 4.0% 3.0%.0%.0% 0.0% Movemen 0yr his. vol. Movemen 7yr his. vol Figure Absolue change in hisorical volailiy of S&P 500 based on 0 and 7 year period o calculae he hisorical volailiy. Applying he 4% based on shocks of 0yr hisorical enor and an alpha facor of 50% resuls in an add-on of abou 0.6% on he long-erm implied volailiy based on a bes-esimae volailiy of 5%. See below he impac for various levels of he bes esimae volailiy. Bes esimae volailiy 5.0% 0.0% 5.0% 30.0% Bes esimae variance Uncerainy Long-erm bes esimae oal variance Long-erm implied volailiy 6.0% 0.8% 5.6% 30.5% Implied add-on for uncerainy L bes esimae.0% 0.8% 0.6% 0.5% Combining he impac he impac of non-hedgeable marke risk and uncerainy in he long-erm bes esimae one can calculae he long-erm implied volailiy for differen levels of he bes esimae volailiy. Bes esimae volailiy 5.0% 0.0% 5.0% 30.0% Bes esimae variance Non-hedgeable Marke Risk Uncerainy Long-erm bes esimae oal variance Long-erm implied volailiy 8.0%.4% 6.9% 3.6% Absolue add-on vs BE volailiy 3.0%.4%.9%.6% Relaive add-on vs BE volailiy 0.%.8% 07.7% 05.4% 4

43 We noe ha above numbers are o illusrae he mehodology and ha he parameers used are represenaive, bu no specific recommended values by he CRO Forum. We believe ha some more work needs o be done in esimaing he parameers. Moreover, he resuling long-erm implied volailiies need o be benchmarked wih values implied from opion prices in he marke such ha he mehodology does no resul in a consisenly over- or under-pricing of long-erm equiy guaranees Adjusmen for impac of sochasic ineres rae volailiy Indusry pracice is o quoe implied volailiy as he volailiy o plug ino he Black & Scholes formula o ge he marke implied volailiy. One of he assumpions in he B&S formula is ha ineres raes are consan over ime. When pricing shor-erm opions his assumpion is no maerial as forward equiy prices for shorer enor are driven by equiy volailiy raher han ineres rae volailiy. However, for longer enor opions he impac of sochasic ineres rae canno be ignored. herefore here are wo opions in calibraing our sochasic models:. We inerpre he implied volailiy in he exrapolaed par of he implied volailiy erm-srucure as a pure equiy volailiy. When calibraing hese volailiies we assume herefore also deerminisic ineres raes. However, when simulaing scenarios for our opions and guaranees we combine he sochasic process for equiy wih a sochasic ineres rae model. As a resul he oal volailiy of forward equiy prices will be higher han under deerminisic ineres raes. he implied volailiies of he observed par of he curve need o be adjused down for he sochasic process o reproduce he dollar price of he raded opions.. We inerpre he implied volailiy in he exrapolaed par of he implied volailiy erm-srucure as he implied volailiy ha we need o ener ino he B&S formula o price opions. In such case an adjusmen needs o be made o compensae for he fac ha he B&S formula assumes deerminisic ineres raes. his adjusmen needs o be consisen wih he ineres rae model used o simulae ineres rae levels in he valuaion of embedded opions and guaranees in insurance producs. In he Appendix his adjusmen is worked ou for he Hull & Whie ineres rae model. he wo opions will give he same resul. his adjusmen is broadly acknowledged by marke paricipans working wih longerm opions and herefore any long-erm equiy guaranee also needs o be valued in combinaion wih a sochasic ineres rae model Implied volailiy surface In he above we fully focused on he exrapolaion of he a-he-money erm-srucure of implied volailiy. In realiy he implied volailiy level no only depends on he enor, bu also on he moneyness level of he opion raio beween srike and forward equiy level. In pracice his aspec should be capured by he sochasic model used o generae equiy scenarios. he complexiy of he model used should ake ino accoun he complexiy of he liabiliy and is embedded equiy guaranees e.g. complex knock-in guaranees a differen levels are more complex han plain guaranees like reurn of iniial invesmen. he calibraion of any model should a leas consider he a-he-money erm-srucure and for more complex models e.g. Heson should also consider he full volailiy surface of he liquid par of he volailiy marke. his hen auomaically implies exrapolaed volailiies for in- and ou-he-money volailiies in he exrapolaed par of he curve. 43

44 6. Exrapolaion of ineres rae implied volailiy Exrapolaion of ineres rae implied volailiy should follow he principles idenified in his paper. However, ineres rae volailiy canno be exrapolaed in a sraighforward way. Conrary o equiy implied volailiy, ineres rae volailiy is no ypically quoed in erm of he volailiy of he ineres rae erm-srucure iself i.e. zero rae volailiy, bu in erms of swapion volailiies. herefore, ineres rae volailiy exrapolaion and calibraion of sochasic ineres rae models are wo very relaed opics. Alhough calibraion of sochasic models o marke daa is ouside he scope of his paper we would like o illusrae his wih an example. Any sochasic ineres rae model such as he Libor Marke model, Hull & Whie or Black-Karasinski allow us o describe he erm-srucure of ineres rae volailiy in erms of he parameers of he model. herefore, exrapolaing ineres rae volailiy owards a long-erm level would imply puing resricions on he model parameers. Anoher elemen ha needs o be considered is consisency in marke daa used o calibrae he sochasic ineres rae model. When he swap marke is only considered o be liquid up o 30yrs hen i does no make sense o include swapion in he calibraion where he combined opion and swap enor is beyond 30yrs e.g. 0y x 30yr or 0yr x 0yr swapions. Lasly, one imporan observaion is ha for many currencies here is no developed ineres rae opion marke ye. herefore, in line wih he nex chaper, proxies need o be consruced based on hisorical daa. Overall, he calibraion of sochasic ineres raes and he implici exrapolaon of ineres rae volailiy hrough his calibraion is an area ha requires more research o develop bes pracices. 44

45 7. Valuaion of opions on insrumens wih no opions raded in he marke 7.. Implied Volailiies for Unobservable Volailiy Markes Implied volailiies are based on opion prices, and are indicaive of he marke s consensus on he fuure volailiy of he underlying price and he risk aversion in he marke. Where raded opion prices are no accessible or non-exisen, alernaive approaches o consrucing suiable implied volailiy surfaces are needed. wo general approaches can be used o consruc implied volailiy surface proxies: Consrucion from an implied volailiy surface of anoher index accessible in he marke Consrucion from hisorical price volailiy of he arge index, adjused wih a risk margin add-on 7.. Firs Approach his firs approach is based on he assumpion ha hisorical price volailiies are indicaive of implied volailiies. A benchmark index wih accessible implied volailiy daa is firs seleced. he benchmark and arge index prices should ideally be highly correlaed, or should share comparable characerisics. Nex he numerical raio is deermined based on he indices price volailiies. his raio indicaes he exen by which he benchmark s implied volailiies are adjused o consruc he arge s implied volailiies. his mehod may be exended o furher allow for volailiy smiles, hrough applying his raio approach on up- /down-side srikes. In case he relaive raio oversaes implied volailiies in very skewed areas of he surface one soluion is o replace i by an absolue difference. o illusrae, if he raio of hisorical volailiies of he arge index o ha of he benchmark is 5%, hen his approach would esimae he arge s implied volailiies o be in he order of 5% of he implied volailiy benchmark s. I should be noed ha he raio need no necessarily be consan hroughou he enire surface. Raios a differen enors may be deermined via differen periodic lenghs of hisorical volailiy, for example. Consequences of he firs approach: In his approach, he arge volailiy surface mimics he benchmark s characerisics. his has implicaions for volailiy arbirage rades, especially when boh arge and benchmark indices are involved. Diversificaion of such opion porfolios may be undersaed, given he assumed perfec correlaions underlying he consrucion of he arge s implied volailiies. In such cases, he basis risk should be capured wihin he risk module Second Approach In he second approach, he implied volailiy of he arge index is esimaed using he hisorical price volailiies of he underlying, wih a risk margin adjusmen. his mehod is simply a generalizaion of he equiy implied volailiy mehod developed by Manisre described in secion 5.4 in which he implied volailiy is esimaed as he sum of he expeced bes-esimae volailiy plus adjusmens for a shock in he underlying invesmen and o he bes-esimae parameer. Alernaively a combinaion of he firs and second approach can be used in which a marker is firs seleced whereby he implied volailiy will be esimaed from hisorical index price volailiies, adjused for he difference beween implied and hisoric volailiies. Ideally, he enor of his marker should be of shor-erm. he implied volailiy surface is hen consruced by graduaing he esimaed shor-erm volailiy wih he long-erm volailiy referred o in secion Also required by Final Advice of CEIOPS, CP3, on risk miigaion echniques. 45

46 his approach capures he variabiliy of shor-erm implied volailiies and he relaive sabiliy of long-erm ones; where he graduaion of implied volailiy erm srucure can be conrolled by a mean-reversion parameer kappa. o illusrae, he shor 3-Monh marker enors is firs seleced. he 3-monh implied volailiy is esimaed as 0% of he hisorical 3-monh index price volailiy, where 0% is he assumed risk margin adjusmen in his case. he remaining implied volailiy erm srucure can hen be deermined by inerpolaing beween his poin and he long-erm volailiy from secion

47 8. Appendices Appendix. Swap Bid-Ask Spread Overview 0Y 0Y 30Y 40Y 50Y AUD CAD CHF EUR GBP JPY KRW 3 4 MXN 4 4 MYR 0 NOK PLN 7 7 SEK WD 6 USD Average bid-ask spread on swaps raes in basis poins over period based on broker daa mainly ICAP. Noe ha spreads migh be smaller for some currencies when looking a differen brokers. 47

48 Appendix. CRO Forum Survey on Exrapolaion of Marke Daa Below able summarizes he CRO Forum survey resuls for all currencies wih a leas 4 companies submiing resuls. he Mos used column shows he enor ha was mos frequenly used as he las liquid swap enor all companies use he swap curve as heir risk free curve wih he excepion of a few currencies. he able furher shows he longes available swap enor, he number of companies in he survey using his paricular currency and he % of hese companies using he mos used enor abou 65% on average. Daa is based on marke siuaion per December 3, 009. In he las column we also show he indusry submission of he cu-off enor for he QIS5 enor. his column represens he max enor of marke daa poins ha have been liquid during he crisis, which indicaes ha for a number of markes his enor is significanly shorer hen mos companies used per end-009. * In some currencies he swap marke was exended by governmen yields. In wo currencies he bes pracice enor is beyond he las available liquid swap enor. ** In hese currencies a minoriy of companies exended he curve using Governmen yields. 48

49 Appendix 3. Nelson-Siegel: As mos readers migh be more familiar wih a differen represenaion of he Nelson-Siegel formula like his: f exp exp wih being he long erm componen, being he shor erm componen and 3 radiional applicaion of Nelson-Siegel. being he mid-erm componen in a we show shorly how o derive i from he more convenien represenaion we made use of: F max b b b3 max e b b e b b e b max e max b b max b e max max 3 b e b e e e max max max e e max e max Boh represenaions are equivalen as: be b3e max max b e max 3 max 49

50 Appendix 5 Mean Reversion during Some analysis was done o show how well he funcional form capures he mean reversion observed in he equiy implied volailiy erm srucure. We esimaed he annual and monhly sandard deviaion of m, 3m, 6m, y and y a-he-money opions for he DJ EuroSoxx 50. his is he mos liquid segmen of he marke. We hen applied a sandard deviaion shock o he implied volailiy erm srucure assuming a fla 5% erm-srucure o sar wih. In he graphs below he red dos show he shocked implied volailiies afer a year lef graph or monh righ graph sandard deviaion shock. We hen applied he funcional form wih he m opion as he shor-erm volailiy and he iniial 5% as he long-erm implied volailiy level. Forward IV = m shocked IV w + L Forward IV [ w] for > m We hen calibraed he opimal mean reversion parameer k o ge he opimal fi o he shocked marke daa for all enors beyond monh by consrucion he monh poin is perfecly fied. I urned ou ha a K equal o.0 for he annual shocks and o. for he monhly shocks resuled in he mos opimal fi. In below graphs we show he modeled equiy implied volailiies for K=.0. Acual mean reversion vs model for annual shocks based on DJ EuroSoxx50 daa Acual mean reversion vs model for monhly shocks based on DJ EuroSoxx50 daa m 3m 6m Marke y Fied by model y m 3m 6m Marke y Fied by model y Conclusion is ha we indeed see a clear mean reversion implied from he acual movemens in implied volailiy daa and ha proposed funcional form provides a good fi wih his observed behavior. 50

51 Appendix 6 Observed difference beween shor-erm hisorical volailiies and implied volailiies In observed marke daa a clear difference can be observed beween hisorical and implied volailiy. A es was performed on daily observed marke daa for he period 005 and 008. Each day he hisorical 3 monh volailiy and he observed 3 monh implied volailiy have been observed. Below able shows he average value of boh he hisorical and he implied volailiy for a range of indices. I can be observed ha implied volailiy is consisenly higher by up o % poins and on a relaive basis in he range of 0-0%. AEX IBEX FSE 00 DJ EuroSoxx50 S&P 500 Nikkei 5 Hang Seng Average Avg. 3M HV 3.3% 3.5%.8% 4.%.5% 7.3% 6.7% 4.% Avg. 3m IV 4.9% 4.8% 3.7% 5.5% 4.0% 8.% 8.3% 5.6% Difference.6%.3% 0.8%.4%.5% 0.8%.6%.4% Relaive Diff 8% 8% 0% 7% 0% 4% 6% 3% 5

52 Appendix 7 Cos of capial approach for exrapolaing he implied volailiy surface his secion is a shor summary of he 00 paper by Manisre 5 referred o earlier. he basic idea is o sar wih a bes esimae or P measure model of equiy dynamics and hen consider he problem of valuing an opion using only he cos of capial conceps ha have been developed o value oher forms of non-hedgeable risk. Models ha have been calibraed o fi observed marke daa hen grade o his cos of capial model over some reasonable ime frame. Saring wih he sandard lognormal model of equiy price movemen ds S d S dw S he paper assumes insurers hold economic capial for wo very differen risks. A conagion shock or jump S JS where he facor J is chosen o be consisen wih he economic capial model s VaR level. Mos capial models assume a J facor in he 30%-40% range. his is analogous o a life insurer holding capial for a repea of he 98 flu epidemic. If V, S is he value of he opion we valuing hen he model assumes we hold capial equal o V, JS V, S. his ensures ha he enerprise can wihsand a shock even even if i is no compaible wih he bes esimae model.. he second risk aken ino accoun is he fac ha our bes esimae P measure parameers, can be wrong. If ˆ is a plausible shock o he volailiy parameer over a year ime frame hen he insurer needs o hold sufficien capial o cover he increase in value V, S, ˆ V, S,. his is analogous o a life insurer holding capial for a shock o a moraliy or lapse assumpion. Having idenified he key risks he paper hen defines he value of he opion o be he expeced presen value of opion payous plus he cos of holding economic capial for he wo risks described above. Considering he conagion shock firs, he principles above ell us ha he opion value saisfies a differenial equaion of he form V V V S S rv [ V, JS V, S]. S S his equaion says we expec he value of he opion o grow wih ineres while releasing sufficien margin o pay for he cos of capial. he quaniy is he cos of capial rae. An invesor who is willing o pu up he risk capial can expec o earn he risk free rae r plus on heir invesmen. If we wan our model o be marke consisen, in he sense ha i prices he equiy index back o iself, hen he funcion V, S S mus be a soluion of he equaion above. Demanding ha his be he case allows us o solve for he cos of capial. We find ha he cos of capial is he leveraged equiy premium r /. J 5 Manisre, B.J. A Cos of Capial Approach o Exrapolaing an Implied Volailiy Surface. Proceedings of he 00 ERM Symposium Chicago. hp:// 5

53 If we assume a ypical equiy premium of 3% r and J 70% hen 0%. he leverage facor makes sense from he perspecive of a shareholder who pus up he risk capial. Such an invesmen has a bea facor of / J. Noe ha his is a very differen siuaion from he cos of capial for underwriing risk which is usually considered o be a low bea risk. If we subsiue r J ino he valuaion equaion above we can rewrie i in a way ha makes comparison wih he sandard Black-Scholes model easy. We find, afer a simple rearrangemen V V rs S V V S rv [ V, JS V, S J S ]. S S he way o inerpre his form of he equaion is o say ha V is he risk neural presen value of opion cash flows plus he cos of holding capial for he loss ha would occur if a large jump occurred while dela hedging. he cos of capial is posiive as long as he insrumen being valued is convex. We have found a model where hedgers and speculaors can agree on value as long hey each hold capial for heir respecive unhedged risk and hey boh use r / J as he cos of capial. he model derived herefore does no assume dela hedging bu is no inconsisen wih i eiher. One of he paper s more echnical resuls is ha effec of using he above model o value long enor opions is equivalen o using he Black-Scholes model wih an implied volailiy of he form J ln J. his is essenially a law of large imp numbers limiing resul which becomes more accurae as he mauriy dae of he opion increases. A deailed derivaion of his useful formula is in he paper s appendix. o pu a value on he parameer shock ˆ risk we need o add a erm o equaion o ge somehing like V V V S S rv [ V, JS V, S] ˆ[ Vˆ, S V, S] S S he paper argues ha he cos of capial for parameer risk ˆ is no he same as he leveraged equiy premium r / J because i is no a direc marke risk. ˆ should be similar o raes used o value underwriing risk. he shocked value Vˆ is calculaed using a similar equaion bu wih shocked volailiy Vˆ ˆ V ˆ V S ˆ S rv [ Vˆ, JS Vˆ, S] ˆ[ Vˆ, S Vˆ, S] 3 S S Equaion 3 illusraes a conundrum ha arises when valuing parameer shocks. In order o compue V we need o know Vˆ and ha requires a double shocked value Vˆ ec.. his is known as he circulariy problem and arises when compuing marke value margins for insurance risks. he paper explains more fully why his model srucure makes heoreical sense and, a he same ime, finds a very pracical way o solve he conundrum wihou over engineering. 53

54 A pracical model ha capures he essence of he srucure oulined above is o use a ime dependen, bu deerminisic, variance ha evolves, in he valuaion measure, according o d d ˆ ˆ. he valuaion equaion becomes V S V JS V rv S V S S V S V ˆ ˆ ],,,, [. he way o inerpre his equaion is o say ha, in he P measure, is consan so ha in he real world we ge a coninuous margin release equal o V V ˆ ˆ ˆ ˆ. he quaniy ˆ V is clearly he model s required parameer risk capial. he parameer governs he model s simplified approach o he shock hierarchy or circulariy issue. Given he firs shock ˆ we are basically assuming ha he nex shock is ˆ ˆ ˆ wih subsequen shocks similarly scaled by he facor. If his sequence of values will converge o a limiing value of ˆ which we can hink of as an upper bound on any plausible sequence of variance assumpion changes. his is he value o which he models variance is mean revering owards a he rae ˆ. A plausible value of he rae ˆ is somehing like % 3.5 % 6 which means ha i can ake a long ime for he variance o reach is ulimae value. When we pu hese wo pieces ogeher we conclude ha he ulimae long erm implied volailiy ha capures boh ypes of risk for long daed opions is ˆ ln J J J r imp. A final poin o noe is ha we could have sared wih a more sophisicaed P measure model e.g. a Heson ype model and gone hrough a similar risk analysis o arrive a very similar conclusions. 54

55 Appendix 8 Adjusmen o equiy implied volailiy erm-srucure o compensae of sochasic ineres raes In his appendix, we illusrae he impac of sochasic ineres raes on he equiy implied volailiy erm srucure by way of wo popular ineres models he Hull-Whie -facor and Hull-Whie -facor models. As explained in he main ex, exrapolaed equiy implied volailiies should be adjused o allow for ineres rae volailiy. hese adjused implied volailiies can hen be direcly plugged in o opion pricing formulae. his adjusmen is based on he principle ha ineres rae volailiies are no represened in he exrapolaed implied volailiies. his adjusmen may also be applied o he exrapolaed implied volailiies of oher marke variables such as Foreign Exchange if hese exrapolaes do no possess any ineres rae volailiy feaures. We commence wih a brief heoreical consruc of he Hull-Whie models before providing numerical illusraion on how he implied volailiies are adjused. he adjused volailiies can hen be plugged-and-played in rouine opion pricing formulae such as he Black-Scholes Meron. I should be noed ha he implied volailiies may be also adjused via oher ineres rae models, wih he appropriae adjusmen formulae; alhough many of such models are seldom racable. We begin wih he following definiions: S denoes he underlying equiy S adj α and forward implied volailiy of an opion adjused forward implied volailiy of an opion shor rae mean-reversion and sigma parameers in he Hull-Whie -facor model, respecively correlaion beween he equiy and ineres rae processes in he HW- model α i and i shor rae mean-reversion and sigma parameers in he Hull-Whie -facor model, respecively MN correlaion beween he processes M and N in he HW- model; where M,N = sock or ineres rae facors Hull-Whie -facor model In he Hull-Whie -facor model, he adjused volailiy is given by adj. { S S y. B s, y. B s, }. ds s where B s,. e he above equaion can be inerpreed as he oal forward adjused equiy volailiy being composed of he sum of: he equiy volailiy, as represened by he forward implied volailiy he ineres rae volailiy; and he covariance beween he equiy and ineres rae processes Since S is implicily assumed as a fla volailiy forecas, he inegraion of he above formula should arrive a 55

56 y y S S adj ds s B ds s B.,..,... he wo inegrands in he above equaion can hen be simplified as., e ds s B., e e ds s B such ha he adjused volailiy, under he Hull-Whie -facor model, is hus 3.. y S y S adj e e e Hull-Whie -facor model In he Hull-Whie -facor model, he adjused volailiy is similar and given by y y x x y x y x xy y y S ys x x S xs S adj ds s B s B s B s B s B s B.,,,,,,. where he ineres rae dynamics are described by he wo facors, denoed by x and y: Y y X x dw d y dy dw d x dx y x r Noe ha,, and are four model parameers derived from calibraion. he iner-facor correlaions are given by: d dw dw d dw dw d dw dw ys S y xs S x xy y x and,, s y s x e s B e s B Working hrough in similar manner as he Hull-Whie -facor derivaion, he adjused implied volailiy under he Hull-Whie -facor model can be presened as: 56

57 y x xy y S ys x S xs y x S adj e e e e e e e e e he following able illusraes he impac of sochasic ineres raes on implied volailiies. As one would expec, hese adjusmens increase wih he opion lifespan. he following poins are noeworhy for he above illusraion: he cuoff enor denoes he enor a which hisorical equiy-based exrapolaion commences. he higher he cuoff enor, he furher he impac of sochasic ineres raes is delayed in he implied volailiy srucure. Alhough he adjusmens are o be applied o forward volailiies, we have applied he adjusmens o he spo volailiies in his illusraion for purposes of simplificaion. his adjusmen may also be applied for non-complee markes, where implied volailiy surfaces are fabricaed from hisorical equiy prices. In he case where he enire implied volailiy surface is consruced from equiy price movemens, he cuoff enor should be se o 0. 57

58 Recen publicaions of he CRO Forum Calibraion recommendaion for he correlaions in he Solvency II sandard formula December 009 Calibraion Principles May 009 Inernal Model Admissibiliy April 009 Insurance Risk Managemen Response o he Crisis April 009 Addressing he pro-cyclical naure of Solvency II November 008 Members of he CRO Forum he CRO Forum is suppored by a Secrearia ha is run by KPMG Advisory N.V. Laan van Langerhuize, 86 DS Amselveen, or PO Box 74500, 070 DB Amserdam he Neherlands el Fax