Modern Valuation Techniques. Stuart Jarvis Frances Southall FIA Elliot Varnell

Transcription

1 Suar Jarvis Frances Souhall FIA Ellio Varnell Presened o he Saple Inn Acuarial Sociey 6 February 2001

2 Abou he Auhors Suar Jarvis Suar received a firs class degree in mahemaics from Cambridge Universiy, and a docorae in mahemaics from Oxford Universiy. Afer some years of posdocoral research he joined he acuarial profession, and has worked in Bacon and Woodrow's Birmingham office, in is employee benefis deparmen since Frances Souhall She has a firs class honours degree in mahemaics from Oxford Universiy. She joined Bacon & Woodrow in 1995 and qualified as an acuary in Her acuarial work has included: pensions research, pensions consulancy and research and developmen. Ellio Varnell He has a firs class honours degree in Physics from Noingham and Masers in Microwave Sysems from Porsmouh. He worked as a scienis before joining Bacon & Woodrow in He is currenly sudying o become an acuary. His acuarial work has included Prophe developmen, sochasic model developmen and sofware design.

3 Conens Conens 1 Inroducion Arbirage Sae prices Inroducion o sae-price deflaors Uiliy Risk-neural valuaion Exending he deflaor model Valuaion using deflaors Inroducion o he examples General insurance example Pensions example Life insurance example Conclusions References Appendix A Muliple-period model Appendix B Marginal uiliy and deflaors Appendix C Black-Scholes' model deflaors Appendix D Risk-neural measure Appendix E Complee and incomplee markes

4 Inroducion 1 Inroducion 1.1 Purpose of he paper This paper describes echniques for valuing cashflows generaed wihin a sochasic projecion model, on a marke consisen basis. There has been a recen rend in accouning sandards owards disclosures ha are marked o marke. For example, his is a saed inenion of he ASB in he UK. We herefore believe hese echniques provide he acuarial profession wih a valuable ool. The key ool we inroduce is he sae-price deflaor (or, more simply, deflaor). Saeprice deflaors form a bedrock for modern finance, bu are more ofen applied hrough he beer-known mehod of risk-neural valuaion. Alhough risk-neural valuaion is successfully applied in a wide range of marke conexs, i can be confusing, and we believe ha i has no been well received by he acuarial communiy. Tradiionally acuaries value cashflows using deerminisic calculaions or projecion of he cashflows sochasically and discouning a a risk-adjused rae. We advocae using deflaors o value cashflows. Deflaors are consisen wih economic principles and, where he model is calibraed o he marke, will produce marke-consisen valuaions. 1.2 Acuarial valuaion mehodology Acuarial lieraure (e.g. Wilkie (1995); Ranne (1998); Duval e al (1999)) conains many sochasic models ha projec indices and asse prices ino he fuure, bu which do no place a marke value on oher cashflows. This is because he valuaion firs requires a suiable risk-adjused rae a which o discoun any se of cashflows. This means ha he job of a sochasic model is only half-done. For example, a model's resuls may indicae a high probabiliy ha one invesmen ouperforms anoher. Bu before a value judgemen can be made, risk-discoun raes have o be chosen. There is no good way of choosing hese such ha, where hese invesmens are markeable, he model's values are consisen wih hose of he marke. The difficuly in placing marke values on cashflows generaed sochasically has led o some acuaries concluding ha projecion models are necessarily inconsisen wih valuaion models. We disagree wih his view and aim o show how he wo model ypes can be consisen. We illusrae how a projecion model should be consruced in order ha any cashflows can be easily valued. 1.3 Modern finance valuaion mehodology Modern financial heory is able o reconcile projecion and valuaion hrough he use of he deflaor. The risk associaed wih an asse used for calibraion is refleced in he model in such a way ha he marke value of his risk auomaically allows for differences in reurn. We can consider a deflaor as a sochasic discoun funcion. Consider C as a sochasic cashflow a ime. The deflaor echnique calculaes is presen value using he following equaion, where D is he sochasic deflaor for he model: Value = E[D C ] This is similar in principle o (bu much more powerful han) he acuarial mehodology of muliplying he expeced value of he cashflow by a deerminisic discoun funcion, v : Value = v E[C ] 2

5 Inroducion 1.4 Benefis of he modern finance approach The deflaor is associaed only wih he model and does no depend on he cashflow being valued. This makes valuaions of sochasic cashflows considerably simpler. This is illusraed in he life insurance example in secion 12. The deflaor echnique produces marke consisen valuaions, and so reflecs he fac ha changing an insiuional invesmen sraegy canno (o firs order) creae or desroy economic value. However, changing he sraegy may change he disribuion of fuure cashflows, so ransferring value beween he sakeholders. The deflaor echnique enables acuaries o analyse hese ransfers. This is illusraed in he pensions example in secion 11. Similarly he marke consisency of he deflaor echnique is useful when considering he capial srucure of a firm. Changing he capial srucure of a firm canno (o firs order) change he marke value of he firm. The capial srucure can however affec he marke value hrough second order effecs such as ax and he impac of credi risk upon business dealings. The deflaor echnique enables acuaries o focus on hese so-called fricional coss which can influence he marke value of he firm. This is illusraed in our general insurance example in secion Srucure of he paper In secions 2 o 6 we inroduce and illusrae he basic principles of modern finance heory and valuaion using a simple model. Secions 7 o 8 exend he previous secions ino more complicaed examples and consider he differences beween he deflaor echnique and radiional valuaion mehods. Secions 9 o 12 inroduce and illusrae he use of he deflaor echnique wih examples from differen acuarial work areas. The appendices conain more mahemaical and economic ideas, proofs and examples. These are no necessary for he undersanding of he main paper, bu are included for compleeness and may be of ineres o readers. 1.6 Originaliy This paper is almos enirely an exposiion of sandard maerial from elsewhere. The basic echnique has even appeared under various guises in papers presened o he Insiue and Faculy (and heir suden bodies) in recen years. However we believe here is a need o clarify he ideas and have sough o presen he maerial again here as simply as possible. A lis of references is provided for he reader who wishes o sudy he source maerial. 1.7 Acknowledgemens We are indebed o Paul Coulhard, Shirleen Sibbe, and especially Andrew Smih for heir invaluable conribuions. We are also graeful o Alam Arbaney, Savros Chrisofides, Gareh Collard, Tim Gordon, Cliff Speed, and Brian Wilson for heir commens and o Ralph Maciejewski for his echnical suppor. Finally we would like o hank Richard Chapman for his coninued suppor for our work. 3

6 Arbirage 2 Arbirage 2.1 Inroducion In his chaper we shall be considering a powerful pricing mechanism known as arbirage-free pricing ha forms he bedrock of valuaion wihin modern finance. In his chaper we sae he definiion of arbirage and illusrae he arbirage concep using a well-known forward sock price example. We coninue his wih a discussion abou arbirage in real markes and finish wih he imporance of using an arbirage-free model for valuaion and projecion. 2.2 Definiion of arbirage Colloquially, an arbirage is a free lunch" for invesors; somehing for nohing. More formally, an arbirage is a porfolio of asses, wih zero iniial value, ha provides a non-negaive value in all possible fuure saes and posiive value in a leas one sae. An arbirage opporuniy exiss when an invesor can consruc wo differen porfolios of differing price, which provide he same cash flows. Selling he more expensive porfolio and buying he cheaper porfolio wih he proceeds produces an unlimied reurn wihou capial expendiure on he par of he invesor. 2.3 Example sock forward price The classic place o sar in any exposiion of financial mahemaics is ofen wih he quesion: "Wha is he forward price of a sock?" The forward price of a sock is he price (agreed oday) a which wo invesors will rade he sock a some fuure ime (specified in he conrac). Suppose we have he following informaion for a share: Curren share price 1.20 Expeced growh in share price 10% pa Guaraneed reurn on cash over 1 year 5% We assume ha bid-offer spreads (including dealing coss) are negligible and ha he sock pays no dividend. All invesors are able o lend and borrow cash on he same erms, 5% over one year. Given he above parameers, wha value should we pu on he forward or equivalenly wha value should we agree o rade he share in one year from now? Firs aemp A firs guess a he forward price migh be he expeced price of he share in one year = This seems a reasonable answer a firs sigh. To see he problem wih his price, consider an invesor offering o buy or sell a forward a his price. A second invesor may hen choose o adop he following sraegy: Second invesor s acions now: Sells he forward hereby agreeing o sell he share for 1.32 in one year from now. 4

7 Arbirage Borrows 1.20 a 5% o buy one share now. Buys he share. Second invesor s acions one year from now: Complees he forward by selling he share o he firs invesor for he agreed Repays his crediors, = The second invesor s profi can be deermined by considering his cash flows. Sraegy 1 Conrac Sell a Forward Buy one share (120) Borrow cash in he marke 120 Ne cash flow now 0 Forward price receips 132 Repay borrowings (126) Ne cash flow a expiry 6 The cash flows we have shown in he able are enirely deerminisic. The cash flow able is he same regardless of he final share price. The second invesor would make a cerain profi of 6 pence per conrac, a he expense of he firs invesor. Furhermore he second invesor would no doub sell as many forward conracs as possible o he firs invesor unil he firs invesor realised his misake. This unlimied profi opporuniy is an arbirage because i creaes a profi from nohing. The second invesor has no had o use any of his own capial Second aemp Suppose he firs invesor realises his misake and decides o reduce he price of he forward conrac o This ime he second invesor adops a differen sraegy, buying he forward, as shown in he following cash flow able. Sraegy 2 Conrac Buy a Forward Sell one share 120 Inves cash in he marke (120) Ne cash flow now 0 Forward price paymen (124) Invesmen proceeds 126 Ne cash flow a expiry 2 Now he second invesor is guaraneed a profi of 2p per conrac and will once again make unlimied profi unil he firs invesor realises his misake. 5

8 Arbirage I appears ha invesors are able o find an arbirage even when he forward price is low Final aemp I urns ou ha he only price ha avoids an arbirage is he curren share price wih ineres a he guaraneed reurn on cash, = 126p. The reason is ha a hedge porfolio can replicae he forward conrac. The second invesor consruced he hedge porfolio. In he firs aemp he sold he forward, and his hedge porfolio sraegy was o borrow sufficien money o buy he share. The forward and he hedge porfolio have he same iniial cashflows; zero. A he erminaion of he forward conrac, he forward conrac resuls in he exchange of a share for he price of he forward. The hedge porfolio conains a share and an obligaion o repay he borrowing. The cashflows of he forward conrac and he hedge porfolio will be equal if he price of he forward mees he obligaion o repay he money borrowed. To avoid an arbirage opporuniy, he cashflows from he hedge-porfolio mus be he same as he cashflows from he forward conrac. Below we show he invesor following he above sraegy wih he arbirage-free price and he firs wo sraegies for comparison. Sraegy 3 Sraegy 2 Sraegy 1 Conrac Hedge a Forward Buy a Forward Sell a Forward Sell (buy) one share (120) 120 (120) Borrow (inves) cash in he marke 120 (120) 120 Ne cash flow now Forward price receips (paymen) 126 (124) 132 Invesmen proceeds (repay borrowings) (126) 126 (126) Ne cash flow a expiry Arbirage in real markes Arbirage opporuniies are no available o mos invesors. Specialis arbirage players operae in mos markes, seeking ou arbirage opporuniies and aking advanage unil he opporuniy ceases o exis. Unless an invesor has large amouns of capial, a fas reacion ime, very low dealing coss and few regulaory consrains, he opporuniies will have gone before hey become cos-effecive. The forward sock example is a simple demonsraion of arbirage-free pricing. Arbirage-free pricing can also be demonsraed in several oher financial markes. Foreign exchange spo & forward Equiy markes equiy opions & fuures Deb markes Governmen bonds Many insiuional invesors lack he flexibiliy o be specialis arbirage players and as such are well advised o approach hese markes as if hey were arbirage-free. 6

9 Arbirage 2.5 Arbirage and valuaion Alhough he discussion of his chaper has been phrased in erms of marke prices he underlying concep has as much relevance for he acuary calculaing presen values. A model mus provide a unique value for a unique se of fuure cashflows regardless of he porfolio used o produce hem. If his were no he case, he model would no produce consisen valuaions. A valuaion model which claims o be able o value cashflows mus herefore be arbirage-free if i is o make any sense. 2.6 Summary Expeced reurns on asses may be valid, bu are misleading when i comes o valuaion. In markes where a hedge porfolio can be consruced for a conrac, he value of he conrac is he value of he hedge porfolio. This is he arbirage-free pricing mehod. Alhough shor-lived arbirage opporuniies migh exis, acuarial models should respec he fac ha markes are effecively arbirage-free if he models are o produce meaningful valuaions. 7

10 Sae prices 3 Sae prices 3.1 Inroducion In his chaper we will build on he arbirage-free concep by considering how an arbirage-free model can be consruced using he concep of a sae-price securiy. A sae-price securiy (or Arrow-Debreu Securiy) is a conrac ha agrees o pay one uni of currency if a paricular sae occurs a a paricular ime in he fuure. The prices of hese securiies, sae prices, can be used as he building blocks for an arbirage-free model. We shall be using a simple model o illusrae he sae price concep. We sar by calibraing he model and calculaing sae prices, hen we use he sae prices o value cerain asses. We show how he simple model can be exended before finishing he chaper wih a discussion of sae prices. 3.2 Example feas and famine Inroducion In his secion we will use a small example, called he feas and famine example, o demonsrae how valuaion in an arbirage-free model leads naurally o sae prices. The model is a wo-sae example over one ime-period ha enables us o concenrae on how a produc can be valued in a very simple world. I will enable us o gain a good undersanding of he basic conceps Calibraion of sae prices All models ha hope o replicae marke dynamics mus be calibraed o marke daa. In our simple model here are only wo possible fuure saes (feas and famine), so only wo asses are required o calibrae he model. The wo asses shown in he able below herefore deermine he model. Asse A Asse B Marke Price Feas sae payou 3 2 Famine sae payou Each asse is compleely specified by he cashflow i produces in each sae and is curren marke price. We will have fully calibraed he model if we knew he marke value of he sae-price securiies. We need o fill in he unknowns of he following able: Feas sae price securiy Famine sae price securiy Marke price?? Feas sae payou 1 0 Famine sae payou 0 1 8

11 Sae prices To calculae he prices of hese asses, we make use of he arbirage-free pricing mehod illusraed in secion 2. Tha is, we find a hedge porfolio ha will replicae he payous of he sae price securiies. A lile linear algebra can be used o show ha porfolios conaining he following unis of Asse A and B will hedge he sae price securiies. Unis of asse A Unis of asse B Feas sae price securiy -1 2 Famine sae price securiy 4-6 A negaive number of unis indicaes ha he hedge porfolio should be shor in ha asse, a posiive number indicaes ha i should be long in ha asse. To be long in an asse means o buy, and so hold, he asse. To be shor in an asse means o sell an asse ha you do no own. This is possible as long as he buyer does no require physical delivery of he produc, which is ofen he case in financial markes. If your porfolio is shor in a paricular share you would make a profi if is price fell. You can do his because you sold he share earlier a a high price and can close your posiion by buying he same share a a lower price. The hedge porfolios replicae he cashflows of he securiies. The principle of arbirage-free pricing indicaes ha he securiies mus have he same value as heir hedge porfolio. By valuing he hedge porfolios using curren marke prices, we ge he following prices for he sae price securiies. Feas-sae securiy Famine-sae securiy Marke price now The calculaions are readily checked by calculaing marke prices for Asses A and B using he sae-price securiies. Considering he payous in each sae, we see ha he price of Asse A should be equivalen o 3 feas-sae securiies plus 1 famine-sae securiy. As expeced, he calculaions, shown in he able below, generae he original Asse A price. Asse A payous Sae securiy prices Hedge porfolio price calculaions Feas sae x 0.35 = 1.05 Famine sae x 0.6 = 0.6 Marke price = Valuaion of a new asse using sae prices Sae price securiies grealy simplify he valuaion of arbirary producs. Simple linear algebra is all ha is required. The nex able illusraes how a new risky asse, wih known payous in each sae, can be valued using he sae prices in he feas and famine example. The calculaions are very similar o hose used o check he price of asse A a he end of he calibraion. 9

12 Sae prices New risky asse Marke price now Feas sae payou 1.5 Famine sae payou Risk-free asse A risk-free asse provides a guaraneed reurn. Is cashflows are herefore he same regardless of which sae occurs in he fuure. We calculae he price using he same mehodology as above. The calculaion reduces o he sum of he sae prices. Risk free asse Marke price now 0.95 Feas sae payou 1 Famine sae payou 1 Guaraneed payou 1 Guaraneed reurn 5.26% This calculaion shows ha he marke price of he risk free asse is 0.95 which indicaes a guaraneed reurn of 5.26%. The guaraneed reurn is also known as he riskfree rae. 3.3 Muliple-sae model The wo-sae model over one-period can be exended o a muliple-sae model. The same calibraion and valuaion approach can be used as in he wo-sae model alhough his becomes a edious exercise in marix algebra for a model wih a large number of discree saes. The valuaion is underaken using he following equaion: where: C = ψ () s C() s AllSaes Marke Price of Asse : Payoff in Sae s : Sae Price for Sae s : ψ(s) Sae prices become unwieldy in a muliple-period seing. Forunaely here are beer echniques for dealing wih a muliple-period model. These are inroduced laer in his paper. 3.4 Consrains on Sae Prices There are some consrains on he sae prices which need o be observed if he model is o remain arbirage-free. All sae prices mus be posiive. If his were no he case, hen buying a sae-price securiy would presen an arbirage wihin he model. C C(s) 10

13 Sae prices Since we know ha he risk-free rae canno be negaive, he marke price of he risk-free asse mus always be less han or equal o one, hence he sum of he sae prices should be less han or equal o one. 3.5 Summary A sae-price securiy pays ou 1 if sae s occurs and 0 oherwise. A sae price ψ(s) is he price of he sae-price securiy and is posiive. An n-sae model is arbirage-free if and only if sae prices exis, and in ha case, for any asse C, defined by is payous C(s) in each sae s, Value now of asse C = C() s ψ () s AllSaes The risk-free asse is defined as he asse whose payou a some fuure ime is independen of he sae ha occurs a ha ime. The sum over all saes of he sae prices is less han or equal o 1. AllSaes ψ () s 1 11

14 Inroducion o sae-price deflaors 4 Inroducion o sae-price deflaors 4.1 Inroducion In secion 3, sae prices were defined and consruced for a one-period model wih wo saes: he feas and famine example. Now we exend his example o consider he probabiliy of he wo saes occurring. Inroducing he probabiliy of saes occurring allows us o calculae sae-price deflaors for his example and we illusrae basic resuls using sae-price deflaors. These resuls will be exended o more general cases, for example muliple-period models, in laer secions. 4.2 Sae-price deflaors for he feas and famine example We reurn o he feas and famine example and inroduce he probabiliy of each sae, feas and famine, occurring. We assume he sae-probabiliies are equal. We also inroduce a new concep called he sae-price deflaor, D(s), as he raio of he sae price, ψ(s), and he sae-probabiliy, p(s): () D s () s () ψ = p s In he able below we have calculaed he sae-price deflaor for he feas and famine example. Probabiliy Sae price Deflaor Feas sae Famine sae Muliple-sae model We consider a one-period model wih a finie number of possible saes s a a paricular fuure ime. Then, as has been described in previous secions, he arbirage-free model has sae prices ψ(s) a ime 0 for an asse paying 1 uni if sae s occurs a he fuure ime. The sae-probabiliy a ime 0 is 1 and he value of a sae-price securiy paying 1 a ime 0 is 1. So he deflaor a ime 0 is 1. We can use he sae-price deflaor and sae probabiliy, insead of he sae-price, when puing a value on a cashflow or an asse. The inroducion of probabiliies means ha valuaion formulae are wrien in erms of expecaions. Using he deflaor D in he valuaion of any asse C gives: Value of asse C = ψ () s C() s = p() s D() s C() s = E[ DC] AllSaes AllSaes where C(s) is he cashflow from C in sae s. We can consider he price of he risk-free asse paying 1 uni in all fuure saes. Value of risk free asse = p() s D() s = E[ D] AllSaes So he expeced value of he sae-price deflaor a a fuure ime is less han 1. 12

15 Inroducion o sae-price deflaors There are a couple of imporan poins worh emphasising. We noe ha as sae prices are always posiive, sae-price deflaors are always posiive. The sae prices ψ(s) are known a ime 0. The deflaor, on he oher hand, is a random variable; i akes he value 1 a ime 0, bu afer ime 0 is value depends on he paricular sae generaed by he model. 4.4 Summary In a single-period model, he sae-price deflaor D(s) is dependen on he fuure sae s a ime and is defined as () D s () s () ψ = p s where ψ(s) are he sae prices and p(s) is he probabiliy of sae s occurring. The sae-price deflaor a ime 0, akes he value 1. The sae-price deflaor D(s) is always posiive. E[D] is he value of he risk-free asse paying 1 a ime and so E[D] is less han 1. The presen value of cashflows C(s) in saes s occurring a fuure ime is Value of asse C= E[DC] 13

16 Uiliy 5 Uiliy 5.1 Inroducion In secion 4, we inroduced he concep of sae-price deflaors for he feas and famine example. In his secion we inroduce he uiliy funcion and we reurn o he feas and famine example o illusrae how a uiliy funcion can generae sae-price deflaors. We sae he use of a uiliy funcion for he generaion of deflaors in general erms; he formal mahemaical proof is conained in Appendix B. 5.2 Inroducion o uiliy Some praciioners assume ha each invesor has a uiliy funcion U ha enables hem o rank differen porfolios. U is a funcion of he invesor's wealh. Economic argumens sugges ha U is an increasing funcion (more wealh is always beer). U has a decreasing gradien (he rae of increase in uiliy decreases wih increasing wealh). The gradien of U is he marginal uiliy, so U has decreasing marginal uiliy. For simpliciy, we assume ha U is differeniable. This enables resuls o be saed more cleanly below. An invesor seeks o inves so as o maximise his expeced uiliy by choosing an opimal porfolio. In order for an opimal porfolio o exis, he model mus be free of arbirage. An arbirage would enable he invesor o increase his expeced uiliy a nil cos wihou bound. 5.3 Feas and famine example Uiliy funcion and opimal porfolio Le he uiliy funcion be aken as () x x U =, where x is he level of wealh. This saisfies he above requiremens for uiliy funcions. Using he feas and famine example described in secion 3, an invesor can use his uiliy funcion o esablish wha invesmen mix of asses A and B he should choose. The graph shows how expeced uiliy varies wih he mix: he maximum is achieved by invesing 108% of his asses in Asse A and 8% in Asse B. Equivalenly his 14

17 Uiliy means purchasing unis of Asse A and selling 0.08 unis of Asse B, cosing a oal of 1 uni Marginal uiliy of opimal porfolio If U(x) is he uiliy funcion of wealh x, hen U'(x) measures he marginal uiliy of 1 exra uni of wealh when he wealh level is x. In he previous example, he marginal uiliy of he opimal porfolio (108% of A, -8% 0.5 of B) can be calculaed. Using U () x = gives: x Marginal uiliy of opimal porfolio Wealh of opimal porfolio Marke price now 1 Feas sae Famine sae From he marginal uiliy o a deflaor We can compare he marginal uiliies o he previously calculaed deflaors: Marginal uiliy of opimal porfolio Deflaor Feas sae Famine sae Raio of feas o famine We noice ha, up o a muliple, U'(w s * ) is a deflaor for he model, where w s * is he wealh in sae s for he opimal porfolio. This is no a coincidence, bu in fac a general propery of deflaors. 5.4 The general case We have seen in he famine & feas example ha deflaor values can be inerpreed in erms of marginal uiliy. I can be proved (see Appendix B) ha he deflaor for sae s is relaed o he marginal uiliy for he opimal porfolio in he following way: w D () s = U E w * 0 * * [ U ( w )] s s * ( w ) where w * 0 is he iniial wealh of he opimal porfolio and w * s is he wealh of he same porfolio in sae s. If a deflaor akes a higher value in one sae han anoher, hen he marginal uiliy of ha sae relaive o he oher is greaer for an invesor who is invesing opimally. This is why deflaors normally ake higher values in poor imes ('famine') han good ('feas'). Uiliy opimisaion can be used o generae deflaors for exising sochasic models which do no currenly produce deflaors. 5.5 Uiliy of he individual invesor and he marke s deflaors The uiliy funcion in he feas and famine example was for an individual invesor, ye i produced deflaors for he marke as a whole and hese deflaors can be used o s 15

18 Uiliy reproduce marke prices. Our uiliy funcion was also an arbirary funcion saisfying he condiions for a uiliy funcion. We would no assume ha every individual invesor in he marke had he same uiliy funcion and indeed i is no necessary. The mehod above produces he same deflaors whaever uiliy funcion is seleced for an individual invesor, provided he invesors agree on he probabiliies for each sae. This is possible because each invesor has a differen opimal porfolio. 5.6 Summary The individual s uiliy funcion is a funcion of wealh and enables he invesor o rank differen porfolios. Marginal uiliy is he gradien of U. The opimal porfolio is he porfolio ha maximises expeced uiliy. The sae-price deflaor for a paricular sae is a consan muliple of he marginal uiliy of he opimal porfolio in ha sae. The uiliy funcion and opimal porfolio relae o an individual invesor whereas he sae-price deflaor relaes o he marke as a whole. 16

19 Risk-neural valuaion 6 Risk-neural valuaion 6.1 Inroducion To define a sae-price deflaor in secion 4 required he use of probabiliies. In his secion, we use a simple example o illusrae he concep of changing he probabiliies from he se of real-world probabiliies. The sae prices are fixed by he marke, so changing he probabiliies changes he sae-price deflaors mainaining he same marke values. We inroduce a paricular choice of probabiliies; he risk-neural probabiliies, under which he mahemaics of valuaion becomes simpler. We consider why using deflaors under he real-world probabiliies may be preferable, alhough boh mehods will give he same values. 6.2 Feas and famine example The deflaor for he feas and famine example depends on he probabiliy measure as shown in he able below: Sae prices Probabiliy Deflaor values Feas sae 0.35 p 0.35 p Famine sae p 0.60 (1-p) When p = 0.368, he deflaor values become equal o each oher: hey ake he common value 0.95, he sum of he original sae prices. We showed in secion 4 ha E[D] is he price of he risk-free asse, for any probabiliy disribuion for he saes considered. So if we chose he probabiliies so ha he deflaor is consan, as we have done here, E[D] is precisely he consan deflaor, and so he consan deflaor mus be he price of he risk-free asse. Under hese 'risk-neural' probabiliies, A and B have he same reurn, equal o he reurn on he risk-free asse: Risk-free asse Asse A Asse B Marke price now Feas sae payou Famine sae payou Expeced fuure value Expeced reurn 5.3% 5.3% 5.3% We can show his mahemaically. Consider an asse now priced X, wih fuure cashflows a ime 1 of X 1. We know ha X =E[DX 1 ] and he expeced reurn can be wrien as E[X 1 ]/X. If Q denoes he risk-neural probabiliies and r is he risk-free rae, hen D Q =(1+r)

20 Risk-neural valuaion So we can wrie he following: Q Q 1 Q X = E[ DX1] = E D X 1 = E [ X1] 1+ r So he expeced reurn under risk-neural probabiliies is Q E [ X1] = 1+ r X i.e. he risk-free rae. Once he risk-neural probabiliies are known, asses can be priced using he formula E[DX] = DE[X]. This is similar o he radiional acuarial mehod (calculae an expeced value hen discoun), excep ha he discoun rae is no risk-adjused: he probabiliies are. The numerical example illusraes how changing he probabiliies can resul in consan deflaors and expeced reurns of any asse being equal o he risk-free rae. This change of probabiliies is known as a change of measure. Appendix D provides he mahemaical explanaion of changing he measure o generae his risk neuraliy. 6.3 Uiliy for a risk-neural invesor As menioned in secion 5, an invesor's uiliy curve is normally concave, which implies ha he is risk-averse (i.e. for a given expeced level of wealh, prefers cerainy over uncerainy): Under he change of measure above, he new probabiliies were chosen so ha he deflaors were consan in any fuure sae. Secion 5 described how deflaors were proporional o he marginal uiliy of he opimal porfolio. So we can consider he invesor o have consan marginal uiliy under hese probabiliies. An invesor having consan marginal uiliy is known as a risk-neural invesor, hence he name riskneural probabiliies. 18

21 Risk-neural valuaion 6.4 Advanages of a risk-neural valuaion A risk-neural valuaion is a valuaion carried ou using he risk-neural probabiliies as described above. I has he advanages of Sae independen discoun facor, using he risk-free rae Expeced reurns a he risk-free rae on any asse This eliminaes he need for a subjecive choice of discoun rae. I is commonly used in invesmen banks for pricing derivaives, for example, he Black Scholes call opion formula in Appendix C. 6.5 Risk-neural valuaion versus valuaion using deflaors We have shown ha here are wo ways of calculaing values of cashflows Using real-world probabiliies and deflaors Using risk-neural probabiliies and discouning a he risk-free rae. Whichever calculaion mehod is used, he same value will be placed on he cashflows. This is illusraed in Appendix C, where deflaors calculaed using real-world probabiliies are shown o reproduce a formula for he value of a call opion developed using risk-neural probabiliies. When deciding which mehod of valuaion o use, he following should be considered: The main disadvanage of he risk-neural valuaion mehod is ha he concep of risk-neural probabiliies is difficul o undersand. Anoher disadvanage is ha he risk-free rae and risk-neural law change wih currency, so muliple-currency models are harder o generae. Deflaors can cope wih muliple-currency models. 6.6 Summary We have shown ha we can change he probabiliies (or measure) o he riskneural probabiliies (or risk-neural measure), o generae: Discoun facor a he risk-free rae Expeced reurns on risky asses equal o he risk-free reurn A risk-neural valuaion uses hese changed probabiliies and places exacly he same value on a se of cashflows as a real-world valuaion using deflaors. 19

22 Exending he deflaor model 7 Exending he deflaor model 7.1 Inroducion In earlier secions we calculaed sae prices and sae-price deflaors for a single-period model; in his secion we exend he ideas over many fuure periods. We also consider he use of deflaors in he modelling of muliple currencies and he erm srucure of ineres raes. 7.2 Deflaors in a muliple-period model We assume a model wih possible fuure saes s a a paricular fuure ime { 1, 2,..,T}. As wih he single-period model, we have sae prices ψ (s) a ime 0 for 1 uni in sae s (a fuure ime ) and he probabiliy of his sae occurring a ime, p (s). As before we define a deflaor process, D, which akes he following value if sae s occurs: ψ () s D () s = p () s I is naural, paricularly in muliple-period models, o consider deflaors as a (posiive) sochasic process, aking differen values in each fuure sae. We can use deflaors o value an asse C a ime 0 wih cashflows C (s) a fuure imes : p s D s C s = E DC Value of Asse C = () () () [ ] AllTimes AllSaes AllTimes We can also value fuure cashflows a inermediae imes: he value a ime of cashflow C T payable a ime T can be calculaed using he same deflaor; a new one is no required. The expecaion operaor now becomes a condiional expecaion, E, condiional on informaion available a : E[ DTCT] Value of C T a ime = D We noe ha in earlier secions, we valued cashflows a ime 0. D 0 was shown o be uniy. Hence he formula used in hese earlier secions is a special case of his formula. A numerical example can be found in Appendix A. In a muliple-period conex, he abiliy o use he condiional form of he expecaion operaor means ha he kind of equaions which will occur are easy o sae, manipulae and undersand. Consider an asse wih uncerain marke value A a fuure imes. An invesor decides o purchase his asse a fuure ime and sell i a a laer ime T. Provided ha he asse pays no dividends, he price ha he invesor pays a ime is he marke value a ha ime, A. Bu he value mus also be considered as he value a of he fuure cashflow a ime T (A T ). So using he formulae above, we can wrie E[ DTAT] A =Value of A T a ime = D Hence we have he relaionship D A = E [D T A T ] We have shown ha for any asse wih value A, DA is a maringale because is expeced value a a fuure ime T, condiional on informaion known a ime, is is value a ime. 20

23 Exending he deflaor model In a model ha conains a coninuum of fuure saes a form of sae price densiy and a probabiliy densiy are required. The raio of he sae price densiy o he probabiliy densiy is he deflaor. 7.3 Deflaors in a muliple currency model A sae-price deflaor depends upon a currency being chosen as he uni of accoun; all cashflows are assumed o be denominaed in his currency. A differen currency choice could be chosen. Forunaely here is a simple relaionship beween he resuling deflaor and he original one, as his secion will show. Suppose ha D $ is a deflaor relaing o cashflows denominaed in. Suppose ha Y is he exchange rae beween serling and dollars, ie X pounds is he same as X $ Y dollars. I can be shown ha D Y $ is a deflaor for dollar-denominaed cashflows. This relaionship can be urned on is head. Raher han modelling one deflaor and an exchange rae wihin a sochasic model, he wo deflaors can be modelled insead; he exchange rae hen falls ou as heir raio. This approach has been advocaed for example by Rogers (1997), Smih & Speed (1998). 7.4 Deflaors in a erm srucure model In he conex of single-period models, we observed ha he sum of sae prices is jus he price of he risk-free asse. Wihin a muliple-period model, his becomes he saemen ha E [D T ]/D is he price a ime of he risk-free asse which pays ou one uni a ime T. This asse is a 'zero-coupon bond'. The zero-coupon bonds are he basic building blocks of yield curve models. Their prices, P(,T) = E [D T ]/D, ell us everyhing we need o know abou he erm srucure of ineres raes a ime. For example, he 'shor rae', r, is he heoreical rae available on very shor money-marke insrumens a ime, so for small δ: P(, + ) Hence he shor rae can be calculaed as: r δ δ r = log P(, T ) T There are many ineres-rae models widely used wihin financial markes. Several of hese are expressed as shor-rae models, which means ha he behaviour of he shor rae r is usually defined in he form of a sochasic differenial equaion. Inegraion of his equaion is normally required before he ineres rae can be used wihin a model. We sugges ha defining he behaviour of he deflaor may be a more efficien way o proceed. This deflaor echnique has been applied o many common ineres rae models; see for example Vasicek (1977), Cox e al (1985), Rogers (1997), Leippold & Wu (1999). As would be expeced from he previous secion, his approach is paricularly efficien when a modeller wishes o consider yield curves in differen currencies simulaneously. e T = 21

24 Exending he deflaor model 7.5 Summary The deflaor, D, is a posiive process wih he propery ha a sochasic cashflow X T payable a ime T has value a ime <T given by he formula E[ DTXT] D Deflaors have he maringale propery; where A is he value of an asse and D is he deflaor a ime, hen for T> DA [ ] = E D A T T The deflaor echnique is useful when modelling muliple currencies and he erm srucure of ineres raes. 22

25 Valuaion using deflaors 8 Valuaion using deflaors 8.1 Inroducion In his secion we look a radiional acuarial valuaion mehod and illusrae he difficulies of using his mehod o produce a marke consisen valuaion. We hen explore he advanages of he deflaor echnique and we consider when he use of deflaors is appropriae. 8.2 Tradiional acuarial valuaion mehod Proposed accouning sandards are forcing he profession o consider how radiional acuarial valuaion mehodology can be reconciled o marke values. The radiional valuaion mehodology would place a value, C, a ime of an uncerain fuure cashflow, C T, a ime T> using a discoun rae d. 1 C = E[ CT] 1+ d One of he acuary s asks is o find a suiable value for d. I has been shown ha d is he risk-free rae if C T is he same in all fuure saes, i.e. he cashflows are guaraneed. The discoun rae, d, should be changed o value riskier cashflows. However here is no objecive mehod for deermining he marke-consisen risk discoun rae for an arbirary se of cash flows. 8.3 Illusraion of he difficulies The difficuly in deermining a marke-consisen discoun rae o value a series of cashflows is illusraed by he use of deflaors in a simple, hypoheical, life insurance model. The simple model projecs an annual premium wih-profis policy and calculaes he paymens made o shareholders as a resul of bonus declaraions and he paymens required, if any, a he erminaion of he policy o mee benefis in excess of he asse share. These paymens were valued using deflaors. An implied discoun rae was hen deermined by equaing he value of he paymens using deflaors o he discouned expeced value of he paymens a each fuure ime. The changes in he implied discoun rae when he proporion of asses invesed in equiies increases are illusraed in he following graph. More deails are in secion 12. T 23

26 Valuaion using deflaors The radiional acuarial valuaion mehodology requires he acuary o choose a discoun rae using his judgemen. The difficuly of his choice, especially where complex conracs are concerned, lead o inconsisen advice being given. The use of a deflaor-based model removes his source of error. 8.4 The advanages of he deflaor valuaion mehod Deflaors can be used o place a marke value on any se of cashflows generaed by he model. The deflaors are model dependen, bu are independen of he cashflows being valued. No consideraion of he riskiness of he cashflows needs o be made, he same deflaors are always used. In he possible fuure saes, differen values may be aached o an increase in wealh of a uni of currency. For example, an exra pound would have greaer value in a recession han in a boom. These values are he marginal uiliy, and in secion 5, we saw ha deflaors are proporional o he marginal uiliy of he opimal porfolio in each sae. This means ha he value o he invesor of he cashflows in each fuure sae is quanified in he valuaion using deflaors. 8.5 Appropriae use of deflaors Sysemaic risk in financial heory is risk correlaed wih financial markes. The deflaor echnique can only value risk associaed wih sysemaic risk. Deflaors are an appropriae echnique when he value of he cash flows is largely influenced by oher economic eniies. For example, he value of a pension increasing in line wih Limied Price Inflaion (LPI) is compleely dependen on fuure value of Reail Price Inflaion (RPI). Unsysemaic risks such as he rae of improving moraliy are no shown in he deflaor. They are aken a heir expeced value: here is a nil risk premium for hese risks. Therefore hey may have o be allowed for separaely in he liabiliy projecion. Some hybrid risks, such as lapse raes, should have heir unsysemaic pars allowed for separaely. In heory hese do no affec value, bu hey may affec fricional coss. See he general insurance example, secion Summary Tradiional acuarial mehodology does no provide a mehod for deermining a marke-consisen risk discoun rae o use in a valuaion. The deflaor echnique places a marke value on any cashflow generaed wihin he deflaor model because i applies sae-dependen discouning. 24

27 Inroducion o he examples 9 Inroducion o he examples In secion 4 we showed ha an arbirary cash flow could be valued using he following equaion: [ ] E CTDT C = D This mehod could jus as easily be applied o any funcion, f, of economic oupus, U T using he following equaion: ( ) E f UT DT f ( U ) = D This permis he valuaion of a very large range of producs. In his paper so far we have been using simple models. The inenion was o give he reader a feeling for he underlying conceps and build confidence in he deflaor echnique. To be useful in real-world valuaions deflaors need o be a par of a more complex model wih oucomes generaed by a coninuous disribuion. A coninuous disribuion is needed o invesigae a wide range of fuure saes and for convergence of valuaion equaions. The mahemaics required o build and calibrae coninuous disribuion deflaor models can be involved. For hose ineresed in he deails we recommend Duffie (1996) or Shiryaev (1999). Though difficul, proprieary models wih coninuous disribuion oucomes and deflaors have been buil and calibraed. The nex hree chapers use coninuous disribuion models o illusrae issues in pensions, general insurance and life insurance. 25

28 General insurance example 10 General insurance example 10.1 Inroducion Deflaors and sysemaic risk As we have seen deflaors are adep a valuing fuure cash flows. They work paricularly well for valuing financial insrumens, where he cash flows are conracually linked o indices for which we have observable hisories. Such cash flows consis principally of risks correlaed o he marke as a whole, ha is, sysemaic (or non-diversifiable) risk. If we use deflaors o value a firm, we likewise obain a good measure of how invesors are likely o value he firm allowing for is sysemaic risk. Modern finance heory ells us ha he marke only rewards invesors for exposure o sysemaic risk Deflaors and unsysemaic risk Deflaor valuaions are no affeced by variabiliy uncorrelaed o capial markes. These risks are someimes called unsysemaic. The marke does no reward unsysemaic risk because of he invesor s abiliy o diversify away he risk by holding a porfolio of asses. There is a curren debae as o wheher hese unsysemaic risks should affec valuaions, for example in reserving for general insurance liabiliies (see, for example, Hindley e al (2000)). In heir ground-breaking paper of 1958, Modigliani & Miller demonsraed ha he way a firm was financed, eiher using deb or equiy, made no fundamenal difference o is marke value. Their argumen showed ha swapping equiy capial for bond capial jus increased he gearing of he firm and hence he reurn required by equiy holders. They concluded ha he capial srucure of he firm was irrelevan o he firm s valuaion. Deflaors respec his conclusion. The managemen usually have he objecive of maximising he sock value. The Modigliani & Miller paper conclusions suggesed ha paying managers o choose a capial srucure reduced he value of he firm by he combined value of he salaries involved. Modigliani and Miller had considered a simple model of a company, which assumed ha coss were linear wih profi. Because firms do sill devoe resources o finding an opimal capial srucure, we assume his is no he whole sory. Second order effecs, also known as fricional coss need o be considered. They are he non-linear expenses incurred as a resul of doing business. Examples include: Fuure business dealings being sensiive o credi risk. Projec disrupion and wasage of unbudgeed flows. Opimisic plans survive longer in uncerain world. Convex ax formulas no able o use ax losses. Exra back office or processing expense. Capial raising, disribuion, resrucuring coss. Double axaion of income on invesed risk capial. Operaional risk of cash misuse. Managemen ime opporuniy cos. 26

29 General insurance example Wha his means for managing a firm In so far as hey can minimize he fricional coss he managemen can hen influence he marke value of he firm. Mos acuarial models do no allow explicily for fricional coss. Insead fricional coss are implicily modelled using crude echniques, such as adjusmen o a liabiliy or embedded value discoun rae. Can we use deflaors o value oher cash flows such as he profi sreams o shareholders, o come up wih an esimae of he value of a company? We can, bu he cash flow model needs o be good enough. Financial producs such as sock opions have a conracual formula linking heir value o an observable marke value. There is no conracual formula linking profi sreams o capial marke inpus. The links are via acuarial formulas conaining all sors of esimaed parameers and approximaions. An efficien sock marke will value he firm on is rue cash flow. A rue cash flow model is realisic enough o reconcile o he marke value of he firm. A managemen wishing o maximize heir shareholder value mus allow for fricional coss when seing heir sraegy. In oher words heir cash flow model mus bridge he gap beween he oupus of an acuarial model and hose of a rue cash flow model by modelling fricional coss Theory Inroducion In his secion we will use a simple model of fricional cos and examine he resuls when applied o a simplified general insurer. The example illusraes wha is possible and i is undersood ha a pracical analysis requires a more sophisicaed fricional cos model. Such models exis bu go beyond he scope of his paper. Furher informaion can be found in Chrisofides (1998) and Wicraf (2001) A fricional cos funcion One way of modelling fricional coss is o consider hem as a funcion of he profi generaed. We would usually choose a posiive, convex funcion. We choose a quadraic funcion for simpliciy. We propose ha he cos funcion for he firm be defined as follows: Fricional Cos = K α 1 ( P β) 2 + α We assume ha he sraegy chosen by he managemen can be encapsulaed in he parameers of he funcion: α This parameer conrols he variabiliy of fricional coss as a funcion of profi. If α is large hen he fricional cos does no depend srongly on profi. I also affecs he minimal fricional cos. β This parameer ses he level of profi a which he minimum cos is aained. The value of his parameer would ypically be chosen close o he expeced profi level 27

30 General insurance example The following parameer represens he inheren coss associaed wih he marke, beyond managemen conrol: K A higher value indicaes higher marke fricional coss. This figure would need o be derived empirically. We define he pure profi of he company as he profi according o he sochasic model. I does no ake accoun of fricional coss. I is represened by he sochasic variable: P This is disinc from rue profi which is he pure profi less fricional coss. Le us consider wo firm sraegies called Confiden & Cauious. The parameers used for he wo sraegies are given in he following able. Alpha Bea K Confiden Cauious The Confiden sraegy is no forgiving if profis are far from he expeced value. Is advanage is ha i has lower coss if profiabiliy is close o he expeced value. The Cauious sraegy makes more provision for volaile profis a he expense of a higher fricional coss close o he expeced profi. In he figure below we show how he fricional cos curves look for wo possible managemen sraegies. I is clear ha he Confiden sraegy is cheaper if he realized profi does no deviae oo far from he expeced profi of 35. By selecing appropriae parameers he funcion we have shown ha our simple funcion can be used o approximae profi-dependen fricional coss. The funcions are convex because here are fricional coss associaed wih doing beer han expeced as well as doing worse Finding he opimum sraegy The managemen of a general insurer will wan o know wha sraegy hey should pursue o maximize he marke value of he firm for he shareholders. This would mean finding he righ balance beween confidence abou he expeced profi and coningency in case he arge is no me. 28

31 General insurance example In our simple model his is he same as finding he parameer values, which minimize he marke value of he fricional coss, given by he equaion below. We seek parameer values, α and β, ha will solve his equaion. min E DK α + 1 α ( P β) 2 Afer some inermediae working we ge he following expressions for opimum parameers α and β, which will minimize he expeced fricional cos: E[ DP] β = E D [ ] [ ] α 1 = E D [ ] [ ] 2 2 E DP E D E DP If he profi is purely deerminisic we find ha he β parameer reduces o he deerminisic pure profi. The α parameer reduces o zero as shown in he expression below. P α = E[ D] E[ D] E[ D] 2 = 0 E[ D] We can show ha as he pure profi becomes deerminisic he fricional coss in his model come down o zero In Pracice Inroducion We have used oupu from a Dynamic Financial Analysis (DFA) model of a general insurer. DFA models can, of course, produce huge volumes of oupu, bu in our example we focus paricularly on he surplus and is value o shareholders. The profi is jus he increase in surplus aribuable o shareholders. The assumpions used by our model are given in he able below. The combined raio is defined as he raio of claims and expenses o he premium. Assumpion Value Premium income 100 Iniial invesmen porfolio value 30 Combined raio mean 102% Combined raio sandard deviaion 8% Tax rae 30% on Profi. Equiy gains deferrable unil realized. We used he model o illusrae how adjusmen of he asse mix of equiy and bonds can maximize he value of a firm hrough he second order effecs of fricional coss and axaion. We considered five differen invesmen sraegies, each invesing a differen proporion of he porfolio in equiy. The balance was invesed in 10-year bonds. The model was run for one year and used simulaions. 29

32 General insurance example Sraegy % Equiy 0% 10% 20% 30% 40% % Bonds 100% 90% 80% 70% 60% We analysed hree oupus from he model: Expeced Value of Gross Surplus Presen Value of Gross Surplus Cos of Capial For each of above we considered he breakdown beween: Ne Surplus aribuable o shareholders. Tax Fricional Coss We assumed ha he firm s managemen sraegy could be described by he quadraic cos funcion given earlier in his chaper. We also assumed ha he firm s managemen opimised heir managemen sraegy for each of he asse managemen sraegies considered. For each invesmen sraegy we firs ran a se of simulaions o calculae he expecaions used in he opimal managemen sraegy formulae given earlier. The differen opimal managemen sraegies required for each invesmen sraegy are ploed below. We assumed he marke-wide fricional coss, K, were 4%. Using he erminology of he Confiden and Cauious sraegies earlier we see ha he 0% equiy invesmen sraegy produces he mos confiden opimal managemen sraegy. Meanwhile he 40% equiy invesmen sraegy produces he mos cauious opimal managemen sraegy. These resuls make sense o our inuiion, as more equiy invesmen is likely o decrease he confidence in achieving a paricular surplus figure Expeced values We invesigaed he breakdown of he expeced value of he gross surplus. The breakdown is ploed below for each of he invesmen sraegies. 30

33 General insurance example We see ha he expeced value of he ne surplus increases wih he equiy invesmen. If we were only o consider expeced reurn as he crieria for our invesmen, mos of our funds would be invesed in equiy. Clearly his would be unwise, as his would ake no accoun of risk Presen values Nex we invesigaed he breakdown of he presen value of he gross surplus. The presen values were calculaed using he deflaor echnique discussed in Chaper 9. The breakdown is ploed below for each of he invesmen sraegies. As we would expec, he presen value of he gross surplus is unaffeced by he differen invesmen sraegies. This demonsraes clearly he resul of Modigliani & Miller. The presen value of he gross surplus is unaffeced by he invesmen sraegy. The sakeholders are he ax auhoriy, he shareholders and a hird group who benefi from fricional coss. The hird group would consis of employees, recruimen agencies, suppliers ec. The invesmen sraegy does change he spli beween he sakeholders, and in paricular he ne surplus aribuable o he shareholders, which will affec he profiabiliy and consequenly he marke value of he firm. These are he second order effecs discussed earlier in he paper. 31

34 General insurance example If we zoom in on he ne surplus figures, we can see he effec of he invesmen sraegy on he presen value of he surplus aribuable o he shareholders. I is clear ha here is an opimum level of equiy invesmen ha balances he exra fricional coss of equiy invesmen agains he ax advanage of holding unrealised equiy gains Cos of capial Finally we looked a he cos of capial. Cos of capial is a measure used by many firms o communicae he rae of reurn required on new projecs if hey are o be allocaed capial. Fricional coss are usually ignored when he presen value is calculaed by applying cos of capial o cash flow projecions. If he presen values are o be comparable wih marke value he fricional coss need o be refleced in he cos of capial rae. We hough i would be of ineres o see how he cos of capial breaks down ino hree componens: Risk free componen. Sysemaic risk componen. Unsysemaic risk componen. 32