On sysemaic moraliy risk and risk-minimizaion wih survivor swaps Mikkel Dahl Marin Melchior Thomas Møller Nordea Markes PFA Pension PFA Pension Srandgade 3 Sundkrogsgade 4 Sundkrogsgade 4 DK-9 Copenhagen C DK-21 Copenhagen Ø DK-21 Copenhagen Ø Denmark Denmark Denmark mikkel.dahl@nordea.com mml@pfa.dk hm@pfa.dk Absrac. A new marke for so-called moraliy derivaives is now appearing wih survivor swaps also called moraliy swaps, longeviy bonds and oher specialized soluions. The developmen of hese new financial insrumens is riggered by he increased focus on he sysemaic moraliy risk inheren in life insurance conracs, and heir main focus is hus o allow he life insurance companies o hedge heir sysemaic moraliy risk. A he same ime his new class of financial conracs is ineresing from an invesor s poin of view since hey increase he possibiliy for an invesor o diversify he invesmen porfolio. The sysemaic moraliy risk sems from he uncerainy relaed o he fuure developmen of he moraliy inensiies. Mahemaically his uncerainy is described by modeling he underlying moraliy inensiies via sochasic processes. We consider wo differen porfolios of insured lives, where he underlying moraliy inensiies are correlaed, and sudy he combined financial and moraliy risk inheren in a porfolio of general life insurance conracs. In order o hedge his risk we allow for invesmens in survivor swaps and derive risk-minimizing sraegies in markes where such conracs are available. The sraegies are evaluaed numerically. Key words: Sochasic moraliy, affine moraliy srucure, risk-minimizaion, survivor swap. JEL Classificaion: G1. Mahemaics Subec Classificaion 2: 62P5, 91B28. This version: Ocober 23, 27
1 Inroducion Life and pension insurance companies ypically use deerminisic moraliy inensiies when deermining premiums and reserves. These moraliy inensiies are ofen assessed via hisorical daa from he porfolios in he company. Given he esimaed moraliy inensiies he companies have radiionally been explicily concerned wih he unsysemaic moraliy risk, which is he risk associaed wih he randomness of deahs in an insurance porfolio wih known moraliy. According o he law of large numbers, he unsysemaic moraliy risk is diversifiable, which means ha i is eliminaed if he porfolio is sufficienly large. However, he moraliy is no deerminisic, so he companies are exposed o sysemaic moraliy risk as well. The sysemaic moraliy risk, which refers o risk associaed wih changes in he underlying moraliy inensiy or he moraliy able, is fundamenally differen han he unsysemaic moraliy risk. I is no diversifiable, and hus i is no eliminaed when he size of he porfolio is increased. In pracice, he life and pension insurance companies deal wih he sysemaic moraliy risk according o he so-called pruden man principle, i.e. hey se he moraliy inensiies o he safe side. This will, hopefully, creae a surplus which is hen redisribued o he insured using he so-called conribuion principle. During he las years, he average lifeime has increased dramaically. This fac has o some exen been negleced, since focus has been on he financial risk, which is much easier o observe and handle. I is now clear ha he moraliy ables used 2 years ago for he pricing of life annuiies were in some cases no sufficienly conservaive, since whole life annuiies are being paid ou for several years longer han expeced. The curren challenge for he life insurance companies is herefore o conrol he combined financial and insurance risk inheren in life insurance conracs. There are essenially wo ways of addressing moraliy risk: One can link or adap benefis o he curren moraliy, or one can inves in financial asses, which are correlaed wih he moraliy or number of survivors. Linking benefis o he moraliy has been called moraliy-linked conracs, see e.g. Dahl 24. In he presen paper, we focus on he possibiliy of hedging moraliy risk by invesing in moraliy-linked derivaives. The key difference beween financial risk and moraliy risk, is ha he financial marke provides a vas number of financial asses which he companies can use o hedge he financial risk, whereas his is no he case wih moraliy risk. There is, however, much focus on he subec, and some financial asses linked o he moraliy have been issued. In his paper, we consider he so-called survivor swap, see e.g. Dowd, Blake, Cairns and Dawson 26. Wih his conrac, we swap exchange a fixed rae of survivors for he acual number of survivors. Hedging wih survivor swaps has also been considered by Lin and Cox 25. Oher ypes of moraliy-linked derivaives, such as longeviy bonds, moraliy swaps and moraliy swapions, have been sudied by Blake, Cairns and Dowd 26. For a discussion on securiisaion of life insurance risk in genereal, see Cowley and Cummins 25. In he lieraure, various models for he sochasic moraliy have been proposed, see e.g. Marocco and Piacco 1998, Milevsky and Promislow 21, Dahl 24, Biffis 25, Cairns, Blake and Dowd 26, Biffis and Millossovich 26, Schrager 26, Milersen and Persson 26 and references herein. In his paper, we consider a model wih sochasic ineres raes and sochasic moraliy inensiies. The model is inspired by he one proposed in Dahl 24 and Dahl and Møller 26 and uses he so-called CIRprocesses known from he financial lieraure for he modeling of moraliy inensiies. We 1
sudy a model wih wo moraliy inensiies and wo underlying Brownian moions. Boh moraliy inensiies may be driven by boh underlying processes. The firs moraliy inensiy represens he moraliy of he insurance porfolio, and he second inensiy represens he moraliy of a populaion. We consider differen financial markes, which conain a zero coupon bond and possibly one or more survivor swaps, and sudy he possibiliies of hedging in hese markes. In all he markes we have more sources of risk financial risk and moraliy risks han financial asses, so we apply heory from incomplee markes. More precisely, we use he crierion of risk-minimizaion inroduced by Föllmer and Sondermann 1986 for coningen claims and exended o paymen processes by Møller 21 o deermine risk-minimizing sraegies. The sraegies illusrae how he combined insurance and financial risk can be hedged parly wih bonds and survivor swaps. This exends he work of Dahl and Møller 26. The paper is organized as follows: Secion 2.1 inroduces he basic financial marke, and Secions 2.2 and 2.3 review he moraliy heory of Dahl and Møller 26 wihin a wodimensional model wih wo porfolios, where he underlying moraliies are correlaed. Secion 2.4 considers he combined model. In Secion 3, we inroduce survivor swaps in he financial marke and define heir price processes. Secion 4 inroduces a porfolio of general life insurance conracs and he marke value. In Secion 5, we give a brief review of he heory of risk-minimizaion and apply hese resuls for deermining risk-minimizing sraegies in he various financial markes. Finally, he sraegies are compared numerically in Secion 6. Proofs of some echnical resuls are presened in he Appendix. 2 The model In his secion we inroduce he combined model for he financial risk and he insurance risk. The model is inspired by he one of Dahl and Møller 26. Le T be a fixed finie ime horizon and Ω, F, P a probabiliy space equipped wih a filraion IF = F T, which conains all available informaion. We define IF as he naural filraion generaed by hree independen sandard Brownian moions W µ = W µ 1, W µ 2 and W r and a 2- dimensional couning process Nx = N 1 x, N 2 x. The process Nx is used o keep rack of he number of deahs in wo porfolios, whereas W µ drives he moraliy inensiies and W r deermines he ineres rae. In addiion we consider sub-filraions IG, II and IH generaed by W r, W µ and Nx, respecively. We assume ha W r and W µ, Nx are sochasically independen. For an exension wih dependence beween he financial marke and he moraliy inensiies, see Milersen and Persson 26. 2.1 The financial marke In his secion, we inroduce he financial marke, which exiss of wo raded asses: A savings accoun and a zero coupon bond wih mauriy T wih price processes B and P, T, respecively. The shor rae is deermined by a so-called Vasiček model, i.e. he shor rae dynamics under P are dr = γ r δ r rd + σ r dw r, wih r = r. Here, γ r, δ r and σ r are consans, and W r is a sandard Brownian moion. As in Dahl and Møller 26, we assume ha [ P, T = E Qr e ] T rudu F, 2
where he measure Q r is defined via dqr dp = ΛT, wih d Λ = Λh r dw r, Λ = 1, and where c h r = σ r + cr σ r. 2.1 Here, c and c are consans. I follows by Girsanov s heorem ha he dynamics of r under Q r are given by dr = γ r,q δ r,q r d + σ r dw r,q, 2.2 wih r = r, where W r,q is a sandard Brownian moion under Q r, and where γ r,q = γ r c, δ r,q = δ r + c. The model is affine under Q r, and i is well-known ha where A r, T and B r, T are given by P, T = e Ar,T B r,t r, B r, T = 1 δ r,q 1 e δr,q T, A r, T = Br, T T + γ r,q δ r,q 1 2 σr 2 δ r,q 2 σr 2 B r, T 2 4δ r,q. The dynamics under P of he zero coupon bond price process are where dp, T = r h r, rσ p, r P, T d + σ p, rp, T dw r, σ p, r = σ r B r, T. The dynamics under Q r of he price processes are db = rbd, B = 1, dp, T = rp, T d + σ p, rp, T dw r,q. 2.3 We noe ha Q r is he unique equivalen maringale measure for he financial model. 2.2 The moraliy inensiies We consider wo porfolios and inroduce wo differen moraliies, which may be correlaed. Inspired by he model of Dahl and Møller 26, we define for each porfolio he processes dζ x, = γ x, δ x, ζ x, d + ζ x, σ x, dw µ, 2.4 ζ x, = 1, = 1, 2. The process ζ 1 is relaed o an insurance porfolio, whereas ζ 2 is relaed o a larger populaion. Each of he wo groups are assumed o exis of individuals 3
of equal age x. Here, σ is a wo-dimensional row vecor and W µ is a wo-dimensional sandard Brownian moion. The moraliy inensiy processes are given by µ x, = µ x + ζ x,, where µ are he iniial moraliy inensiies [ a ime, and he survival probabiliies are defined by S x,, T = E P e ] T µ x,τdτ [ F = E P e ] T µ x+τ ζ x,τdτ F. 2.5 For more deails, see Dahl 24 and Dahl and Møller 26. In he case where σ µ 1 x, = σ1,1, and σ µ 2 x, =, σ 2,2, he wo moraliy inensiies are independen. If insead we ake σ µ 1 x, = σ 1,1, σ 1,2, he wo moraliies are no longer independen. I follows by Iô s formula ha, for = 1, 2, he dynamics of he moraliy inensiies are dµ x, = γ µ x, δµ x, µ x, d + µ x, σ µ x, dw µ, 2.6 where γ µ x, = µ x + γ x,, 2.7 δ µ x, = δ x, d d µ x + µ, 2.8 x + σ µ x, = µ x + σ x,. 2.9 In order o ensure ha he moraliy inensiies are sricly posiive, we assume ha where a r denoes he vecor a ransposed. 2.3 The lifeimes in he porfolios 2γ µ x, σ x, σ x, r, This secion describes he lifeimes in each of he wo porfolios. For simpliciy, we assume ha he porfolios consis of n, = 1, 2, lives, all aged x years a ime. Furhermore, we assume, ha he lives in porfolio 1 are differen from he lives in porfolio 2, i.e. he wo porfolios consis of disoin lives. We adop he naural assumpion ha he lifeimes in a porfolio are muually independen and idenically disribued condiional on he moraliy inensiies. The remaining lifeimes a ime are described by a sequence of non-negaive random variables T,1,..., T,n, = 1, 2. The probabiliy of a single individual surviving o ime, given he informaion on he moraliy inensiy unil ime, is given by P T,1 > I = e µ x,sds, = 1, 2. All considered insurance conracs have expiraion ime T, so he moraliy processes are only modeled on he inerval [, T ]. We herefore inroduce censored lifeimes given by T,i = T,i T, = 1, 2, i = 1,..., n. In each porfolio he censored lifeimes are now i.i.d. given IT. We emphasize, ha even hough he moraliy inensiies for he wo porfolios may be correlaed we have condiional independence beween he lives in he wo porfolios. The number of deahs a ime [, T ] in porfolio is described by he couning process N x = N x, [,T ], where n N x, = 1 {T,i }, 2.1 i=1 4
for = 1, 2. The sochasic inensiy process λ x = λ x, [,T ] relaed o N x is given informally by λ x, d = E P [dn x, H I] = n N x, µ x, d, 2.11 = 1, 2. Hence he ransiion raes are simply he moraliy inensiy muliplied by he number of survivors us before ime. 2.4 Change of measure Inspired by Dahl and Møller 26, we consider maringale measures, wih a likelihood process Λ on he form dλ = Λ h r dw r + h µ dw µ + gdmx,, 2.12 wih Λ = 1. We assume ha E P [ΛT ] = 1 and define an equivalen maringale measure Q by dq dp = ΛT. Here, hµ and g are wo-dimensional processes. For simpliciy, we require ha g is deerminisic, coninuously differeniable and g > 1, = 1, 2. The process h r is defined in 2.1. The oher erms in 2.12 are relaed o he moraliy. We ake he Girsanov kernels on he special form h µ 1, µ 1, µ 2 = σ µ β µ 1,2 x, 2 x, µ 2 x, β µ 2 x, σ µ x, σ µ x, µ 2 x, σ µ β µ 2,2 x, 1 x, µ 1 x, σ µ x, h µ 2, µ 1, µ 2 = σ µ β µ 2,1 x, 1 x, µ 1 x, σ µ x, σ µ β µ 1,1 x, 2 x, µ 2 x, σ µ x, β µ 1 x, σ µ x, µ 1 x, β µ 1 x, σ µ x, µ 1 x, β µ 2 x, σ µ x, µ 2 x,, 2.13, 2.14 where σ µ x, = σ µ 1,2 x, σµ 2,1 x, σµ 1,1 x, σµ 2,2 x, and βµ and β µ are coninuous funcions. This ensures ha he moraliy inensiies follow CIR models under Q. Moreover, he resricions on g and h µ ensure ha he sochasic independence beween he financial marke and he insurance elemens is preserved under Q. Indeed, sraigh-forward calculaions show ha dµ x, = γ µ,q x, δ µ,q x, µ x, d + µ x, σ µ x, dw µ,q, where W µ,q is a 2-dimensional sandard Brownian moion under Q, and where γ µ,q x, = γ µ x, β µ x,, δ µ,q x, = δ µ x, βµ x,. As noed in Dahl and Møller 26, β µ mus fulfill he condiion 2γ µ x, β µ x, σ x, σ x, r, = 1, 2, in order o preven zero-valued moraliy inensiies. Finally, we define he Q-maringales M Q by dm Q x, = dn x, λ Q x, d, where λ Q x, = n N x, 1 + g µ x,, = 1, 2. We can inerpre he quaniies µ Q x, = 1 + g µ x, as he moraliy inensiies under Q. 5
2.4.1 Survival probabiliies under Q We define he survival probabiliies under [ Q by S Q x,, T = EQ e ] T µ Q x,τdτ F, and he associaed Q-maringales by [ S Q,M x,, T = E Q e ] T µq x,τdτ F = e µq x,τdτ S Q A closer inspecion of he Q-moraliy inensiies µ Q form dµ Q x, = γ µ,q,g where x, δ µ,q,g x, µ Q d x, + µ Q γ µ,q,g δ µ,q,g x, = 1 + g γ µ,q x,, x, = δ µ,q x, σ µ,q,g x, = x,, T. reveals ha he dynamics are on he d d g 1 + g, 1 + g σ µ x,. x, σµ,q,g x, dw µ,q, In his case, he diffusion erm is wo-dimensional, so in order o use he affine heory from Secion 2.1, we firs rewrie µ Q x,. I is well-known ha where W µ,q σ µ,q,g,1 x, dw µ,q 1 + σ µ,q,g,2 x, dw µ,q are sandard Brownian moions, and σ µ,q,g x, = 2 = σ µ,q,g σ µ,q,g,1 x, 2 + σ µ,q,g,2 x, 2. x, d W µ,q, The moraliy inensiies can now be wrien on a form, where he diffusion erm is onedimensional, i.e. dµ Q x, = γ µ,q,g x, δ µ,q,g x, µ Q d x, + µ Q x, σµ,q,g x, d W µ,q, such ha he drif and squared diffusion erms for µ Q x, are affine in µq x,. This is similar o he siuaion in Dahl and Møller 26, and he resuls obained here now show ha he Q-survival probabiliies S Q x,, T, = 1, 2, are given by where A µ,q Bµ,Q and B µ,q S Q x,, T = δ µ,q,g B µ,q x, T, T =, Aµ,Q x,, T = γ µ,q,g A µ,q x, T, T =. x,, T = eaµ,q x,,t B µ,q x,,t µ Q x,, are deermined from x, B µ,q x, B µ,q x,, T, x,, T + 1 2 σµ,q,g The forward moraliy inensiies under Q is for = 1, 2 given by f µ,q x,, T = T log SQ x,, T = µq x, see Dahl and Møller 26. T Bµ,Q x, 2 B µ,q x,, T 2 1, x,, T T Aµ,Q x,, T, 6
3 Survivor swaps Inspired by ineres rae swaps, so-called survivor swaps have been inroduced, where one can exchange a fixed number of survivors wih he acual number of survivors in a porfolio. The porfolio could for example be he insured lives in an insurance porfolio or he lives of a cerain age in some populaion. 3.1 The paymen process associaed wih a survivor swap When a buyer and a seller agree o ener a survivor swap, hey essenially agree on some fixed survival probabiliy, which is here deermined a ime and given by p x = e µ x,τdτ. The inensiy µ deermines he fixed paymens during he period of he conrac. A survivor swap on each of he wo porfolios can now be described by paymen processes, = 1, 2, wih dynamics A swap da swap x, = n N x, d n p x d, 3.1 and A swap x, =. The paymen rae in 3.1 is he difference beween he acual number n N x, of survivors in porfolio a ime and he expeced number n p x, which is calculaed a ime by using he survival probabiliy p x. Thus, he swap leads o a coninuous paymen if he acual number of survivors exceeds he predeermined level of survivors. If on he oher hand he predeermined level of survivors exceeds he acual number of survivors, he paymen rae 3.1 is negaive, and he buyer has o pay he difference o he seller of he conrac. 3.2 Marke values In he remaining of he paper we consider a fixed bu arbirary measure Q from he class of equivalen maringale measures inroduced in secion 2.4. Le he discouned paymens A, swap from he survivor swap be defined by da, swap x, = e rudu da swap x,, and A, swap x, =. For = 1, 2 we now inroduce he process Z,Q given by [ T Z,Q x, = E Q e ] τ rudu da swap x, τ F. 3.2 Hence Z,Q x, is he condiional expeced value a ime of discouned paymens from he survivor swap on porfolio. In his paper we adop he erminology of Föllmer and Sondermann 1986 and refer o a process on a form similar o 3.2 as an inrinsic value process. The aserisk * in Z,Q x, and A, swap indicaes ha we are working wih discouned values. We will use his noaion in he res of he paper. I follows ha Z,Q, swap x, = A x, + e [ T rudu E Q e ] τ rudu da swap x, τ F = A, swap x, + Z,Q x,, 3.3 7
where we have inroduced he noaion Z,Q x, = e [ T rudu E Q e ] τ rudu da swap x, τ F. Here, Z,Q is he discouned marke value of he fuure paymens and represens he discouned expeced value of fuure paymens given he curren informaion. Using 3.1 and he independence beween he financial marke and he insured lives, we ge ha Z,Q x, = n N x, T T P, τs Q x,, τdτ n p x P, τ τ p x+ dτ, where P, τ is he discouned price of a zero coupon bond. The firs erm is he discouned marke value of he variable paymens, and he second erm is he discouned marke value of he fixed paymens. We assume below ha asses wih discouned price processes 3.2 can be raded dynamically in he financial marke. These asses may now be used for hedging he combined insurance and financial risk inheren in he insurance porfolio. We noe ha he discouned price processes are maringales under he chosen measure Q, such ha Q is also a maringale measure in he exended markes, where he survivor swaps can be raded dynamically. 3.3 A sochasic represenaion of survivor swaps In his secion, we derive a sochasic represenaion of 3.3, which provides insigh regarding he differen ypes of risks associaed wih a survivor swap. Furhermore, i is useful for deermining risk-minimizing sraegies in he siuaion where he survivor swaps can be raded dynamically. In he remaining of he paper we work under he following assumpion. Z,Q Z,Q Assumpion 3.1 C 1,2,2, i.e. is coninuously differeniable wih respec o and wice coninuously differeniable wih respec o r and µ. Lemma 3.2 A survivor swap on porfolio wih fixed survival probabiliy p x admis he represenaion Z,Q x, = Z,Q x, + where ρ Z,Q = ρ Z,Q,1, ρz,q,2 ν Z,Q, and T ν Z,Q = P, τs Q τdm Q x, τ + η Z,Q τdw r,q τ + ρ Z,Q τdw µ,q τ, x,, τdτ, 3.4 T η Z,Q = n N x, σ r B r, τp, τs Q x,, τdτ T + n p x σ r B r, τp, τ τ p x+ dτ, 3.5 ρ Z,Q,i = σ µ,i x, µ x, n N x, 1 + g T B µ,q, τp, τs Q x,, τdτ, i = 1, 2. 3.6 8
Before proving he lemma, we commen briefly on he resul above. There are essenially hree ypes of risk associaed wih he value of he survivor swap. Firs we have he unsysemaic moraliy risk driven by he maringale M Q, which is he risk associaed wih a deah in he porfolio underlying he swap. Second, we have he ineres rae risk relaed o changes in he underlying process W r,q driving he ineres rae. Finally we have he sysemaic moraliy risk driven by he underlying processes W µ,q. Since W µ,q is wo-dimensional, we have boh sysemaic moraliy risk generaed by W µ,q 1 and sysemaic moraliy risk generaed by W µ,q 2. Proof of Lemma 3.2: Using Iô s formula on he maringale Z,Q defined in 3.3 and he dynamics for A swap x, from 3.1, we ge ha dz,q, swap,q x, = da x, + d Z x,,q = ψd + d Z x,. 3.7 Here, ψ is some process, whose exac form we do no need o know, since he drif of a maringale is zero. We use he processes ψ and ψ 1 as some buffers for all quaniies ha concerns he drif. Using 3.7 and Iô s formula on, we ge ha dz,q x, = ψ 1 d + σ r r + µ x, dn x, = σ r r,q Z x, dw r,q µ Z,Q T P, τs Q Z,Q x, dw r,q + Z,Q x, σ µ x, dw µ,q x,, τdτ µ x, T dm Q x, P, τs Q x,, τdτ. µ Z,Q x, σ µ x, dw µ,q In he second equaliy, we have rewrien dz,q such ha we ge a erm wih respec o he dynamics of he maringale M Q. We noice, ha he drif erm disappears in he las equaion since Z,Q is a maringale. Now, recalling ha we see ha r P, T = e Ar,T B r,t r, T,Q Z x, = n N x, B r, τp, τs Q T + n p x Furhermore, we recall ha x,, τdτ B r, τp, τ τ p x+ dτ. 3.8 S Q x,, T = eaµ,q,t B µ,q x,,t µ Q x, = e Aµ,Q,T B µ,q x,,t µ x,1+g, and ha τ p x is deerminisic a ime. Hence, we ge µ Z,Q x, = n N x, 1 + g T B µ,q, τp, τs Q x,, τdτ. 3.9 9
Collecing 3.7-3.9 and remembering ha Z,Q is a maringale, we ge dz,q x, = ν Z,Q dm Q x, + ηz,q dw r,q + ρ Z,Q dw µ,q, where ν Z,Q, η Z,Q lemma. and ρ Z,Q 4 Insurance conracs are given in 3.4, 3.5 and 3.6, respecively. This proves he An insurance conrac specifies a paymen process wih premiums paid by he policyholders and benefis paid by he insurance company. We consider a porfolio of fairly general life insurance conracs. Each conrac allows for a single premium paid a ime, coninuous premiums, a lump sum paymen upon reiremen, a single paymen upon deah and life annuiy paymens. The paymen process generaed by he porfolio of insurance conracs is formally he ne paymens o he policy-holders, which means ha premiums are negaive and benefis are posiive. 4.1 The paymen process The paymen process is described by da = n 1 π s d1 { } π c n 1 N 1 x, 1 { <T } d + a d dn 1 x, + n 1 N 1 x, T a r T d1 { T } + a p n 1 N 1 x, 1 {T T } d. 4.1 Here, n 1 is he number of people in he insurance porfolio, and N 1 x, is he number of deahs in he insurance porfolio during, ]. The erm of he conrac is T and he ime of reiremen is T T. The firs erm in 4.1 is he single premium π s paid by all n 1 policy-holders upon signing he conracs, and he second erm is coninuous premiums π c paid by he curren n 1 N 1 x, survivors unil reiremen. The hird erm is paymens a d in case of deah, and he fourh erm is he lump sum paymen a r paid o he remaining policy-holders alive a ime T. The las erm is life annuiy paymens a p o he remaining policy-holders alive in he period from reiremen unil he end of he insurance period. We assume ha π c, a d and a p are piecewise coninuous funcions. 4.2 Marke reserves The inrinsic value process associaed wih paymen process A is defined by [ T ] [ V,Q = E Q da T τ F = E Q e ] τ rudu daτ F, 4.2 for T. Using ha A and r are adaped processes, we see ha V,Q = e τ rudu daτ + E Q [ T e τ rudu daτ ] F = A + Ṽ,Q. 4.3 The process Ṽ,Q is referred o as he discouned marke reserve. I represens he discouned condiional expeced value of fuure paymens calculaed a ime. As in Dahl and Møller 26, we formulae he following proposiion. 1
Proposiion 4.1 The discouned marke reserve Ṽ,Q is given by where Ṽ,Q p Ṽ,Q p = T Ṽ,Q = n 1 N 1 x, Ṽ p,q, is he discouned marke reserve for one policy-holder and is deermined by P, τs Q 1 x,, τ a d τf µ 1,Q x,, τ π c τ1 { τ<t } + a p τ1 {T τ T } dτ + P, T S Q 1 x,, T ar T 1 {<T }. As in Dahl and Møller 26, we give a shor commen on he resul above: The marke reserve for one policy-holder alive a ime is a funcion of he curren level for he shor rae r and he insurance porfolio s moraliy inensiy µ 1 x,. The marke reserve is on he same form as usual reserves, bu i now involves he price P, τ of a zero coupon bond insead of he usual discoun facor and he sochasic survival probabiliy S Q 1 x,, τ for he porfolio insead of he usual deerminisic moraliy inensiy. Furhermore we noe ha he deerminisic moraliy inensiy is replaced by he Q-forward moraliy inensiy f µ 1,Q x,, τ in he erm involving he paymen a d upon a deah. Finally, we emphasize ha he marke reserve depends on he choice of measure Q. For a proof of Proposiion 4.1, we refer o Dahl and Møller 26. In addiion o Assumpion 3.1 we assume he following holds for he remaining of he paper. Assumpion 4.2 Ṽ p,q C 1,2,2, i.e. Ṽ p,q is coninuously differeniable wih respec o and wice coninuously differeniable wih respec o r and µ. 4.3 A sochasic represenaion of he insurance conrac Similarly o Lemma 3.2 we have he following sochasic represenaion for he inrinsic value process of he insurance paymen process. Lemma 4.3 The inrinsic value process 4.2 associaed wih he paymen process 4.1 admis he represenaion V,Q = V,Q + where ν V,Q τdm Q 1 x, τ + η V,Q τdw r,q τ + ρ V,Q τdw µ,q τ, 4.4 ν V,Q = B 1 a d Ṽ p,q, 4.5 T η V,Q = σ r n 1 N 1 x, B r, τp, τs Q 1 x,, τ a d τf µ 1,Q x,, τ π c τ1 { τ T } + a p τ1 {T τ T } dτ + B r, T P, T S Q 1 x,, T ar T 1 {<T }, 4.6 11
and where T ρ V,Q = σ µ 1, x, µ 1 x, n 1 N 1 x, 1 + g 1 a d τ f µ 1,Q x,, τ τ Bµ,Q 1 x,, τ B µ,q 1 x,, τ P, τb µ,q 1 x,, τs Q 1 x,, τ π c τ1 { τ T } + a p τ1 {T τ T } dτ + P, T B µ,q 1 x,, T S Q 1 x,, T ar T 1 {<T }, = 1, 2. 4.7 Before proving he lemma, we briefly explain he resul. We noe ha he represenaion of he inrinsic value process for he insurance paymen process has he same form as he represenaion of he inrinsic value process for a survivor swap in Lemma 3.2. Thus, he value process for he insurance paymen process A consiss of hree erms relaing o he unsysemaic moraliy risk, he ineres rae risk and he sysemaic moraliy risk, respecively. Proof of Lemma 4.3: The proof is similar o he one for Lemma 3.2. Recall from 4.3 ha he Q-maringale V,Q can be wrien as V,Q = A + Ṽ,Q, where Ṽ,Q is he discouned marke value of he paymen process given by T Ṽ,Q = n 1 N 1 x, P, τs Q 1 x,, τ a d τf µ 1,Q x,, τ π c τ1 { τ T } + a p τ1 {T τ T } dτ + n 1 N 1 x, P, T S Q 1 x,, T ar T 1 {<T }, see Proposiion 4.1. The dynamics of V,Q are given by dv,q = da + dṽ,q = ψ V d + B 1 a d dm Q 1 x, + dṽ,q, 4.8 where ψ V is some process. The exac form of his process is no imporan since he value process is a Q-maringale and herefore has no drif. In 4.8, we have added and subraced he quaniy B 1 a d λ Q 1 x, d in order o obain a erm involving dm Q. In order o deermine dv,q, we now need o find he dynamics of he discouned marke value dṽ,q. This is again obained via Iô s formula: dṽ,q = ψ V 1 d + σ r r Ṽ,Q dw r,q + µ 1 x, µ 1 Ṽ,Q σ µ 1 x, dw µ,q Ṽ,Q p dn 1 x, = ψ V 2 d + σ r r Ṽ,Q dw r,q + µ 1 x, µ 1 Ṽ,Q σ µ 1 x, dw µ,q Ṽ p,q dm Q 1 x,. 4.9 Again ψ1 V and ψv 2 are some processes, whose exac form we do no need o know. In 4.9 we used he same idea as in 4.8 in order o ge an expression ha involves M Q 1. We can 12
now use 4.9 in 4.8 o obain dv,q = ψ V + ψ2 V d + B 1 a d Ṽ p,q dm Q 1 x, + σ r r Ṽ,Q dw r,q + σ µ 1 x, r µ 1 x, µ 1 Ṽ,Q dw µ,q. Using he echniques from Dahl and Møller 26, i now follows ha dv,q = ν V,Q dm Q 1 x, + ηv,q dw r,q + ρ V,Q dw µ,q, 4.1 where ν V,Q and η V,Q are given by 4.5 and 4.6 and ρ V,Q = ρ V,Q 1, ρ V,Q 2 are given in 4.7. Hence we have proved he lemma. 5 Risk-minimizing sraegies 5.1 Moivaion When an insurance company signs a life insurance conrac wih a policy-holder, he company is exposed o boh financial and moraliy risk. Typically his combined risk canno be hedged perfecly. A way o handle he risk is o use he crierion of riskminimizaion suggesed by Föllmer and Sondermann 1986; see also Schweizer 21 for a survey. In he following, we firs give a brief inroducion o he crierion of riskminimizaion. We hen deermine risk-minimizing sraegies in differen financial markes. The firs marke is he one sudied in Dahl and Møller 26, which consiss of a savings accoun and a zero coupon bond. The oher markes in addiion conain survivor swaps. 5.2 Inroducion o risk-minimizaion Consider a financial marke wih a savings accoun B and a risky asse wih discouned price process X. Here, he Q-maringale X may be a vecor process. A sraegy is a process ϕ = ξ, η saisfying cerain inegrabiliy condiions, where ξ is he number of risky asses held, and η is he discouned deposi in he savings accoun. The discouned value a ime associaed wih he sraegy is V, ϕ = ξx + η. An invesmen sraegy wih V T, ϕ = is called -admissible. The cos process a ime is given by C, ϕ = V, ϕ ξudxu + A. Thus, he accumulaed coss unil ime are he discouned value of he invesmen porfolio, V, ϕ, reduced by discouned rading gains and added discouned ne paymens o he policy-holders. If a sraegy ϕ minimizes he so-called risk process R, ϕ defined by [ ] 2 R, ϕ = E Q CT, ϕ C, ϕ F, i is called risk-minimizing. The sraegy can be deermined from he Galchouk-Kunia- Waanabe decomposiion given by V,Q = E Q [A T F] = V,Q + ξ Q udxu + L Q, 13
where ξ Q is a predicable process, and L Q is a zero-mean Q-maringale which is orhogonal o X. There exiss a unique -admissible risk-minimizing sraegy ϕ = ξ, η, given by ϕ = ξ, η = ξ Q, V,Q ξ Q X A, see Møller 21, Theorem 2.1. The inrinsic risk process, which measures he minimum obainable risk, is given by R, ϕ = E Q [ L Q T L Q 2 F ]. 5.1 Typically, his quaniy has o be evaluaed numerically by simulaion. 5.3 Risk-minimizaion in a bond marke Consider he marke inroduced in Secion 2.1 wih a savings accoun and a zero coupon bond. This marke was also sudied in Dahl and Møller 26. They showed ha he risk-minimizing sraegy is given by where ϕ B = ξ B, η B = ξ Q B, Ṽ,Q ξ Q B P, T, ξ Q B = The unhedgeable risk is deermined by he process L Q wih η V,Q σ r B r, T P, T. 5.2 dl Q = ν V,Q τdm Q 1 x, τ + ρv,q 1 τdw µ,q 1 τ + ρ V,Q 2 τdw µ,q 2 τ. 5.3 By insering 5.3 in 5.1, we see ha he inrinsic risk process is given by T R, ϕ B = E Q ν V,Q τ 2 2 2 λ Q 1 x, τ + ρ V,Q τ dτ F. We evaluae he iniial inrinsic risk R, ϕ B numerically in Secion 6. 5.4 Risk-minimizaion wih an insurance porfolio survivor swap Now consider he marke B, P, Z 1, which in addiion o he savings accoun B and he zero coupon bond P, includes a survivor swap on he insurance porfolio. From Lemma 3.2 we have he following sochasic represenaion for inrinsic value process of he survivor swap dz,q 1 x, = ν Z,Q 1 dm Q 1 x, + ηz,q 1 dw r,q + ρ Z,Q 1 dw µ,q. Thus, we assume ha we can rade a survivor swap on he insurance porfolio dynamically. I would indeed be more more realisic o work wih less frequen rading of he survivor swap. However, he curren model wih coninuous ime rading of he survivor swap sill gives an idea of how he insurance company could reduce risk in he exended markes. The case wih discree ime rading of he survivor swap is posponed o fuure research. In order o deermine he risk-minimizing sraegy, i is useful o inroduce cerain processes Y,Q 1,1 and Y,Q 1,2 which are orhogonal o he raded asses. These processes will 14 =1
help us consruc a zero-mean Q-maringale, which is orhogonal wih he zero coupon bond and he survivor swap. To simplify noaion, we define χ Z,Q i, = 1 {ρ Z,Q for i, } i, {1, 2}. Le he zero-mean maringales Y,Q 1,, = 1, 2, be given by dy,q 1, = χz,q 1, dm Q 1 x, κq µ,q 1, dw, 5.4 wih Y,Q 1, =. For ρz,q 1,, we furhermore define κq 1, by κ Q 1, νz,q 1 λ Q 1 x, = ρ Z,Q 1,. 5.5 This consrucion ensures ha Y,Q 1,1 and Y,Q 1,2 are indeed orhogonal o he discouned zero coupon price process P, T and he discouned price process Z,Q 1 associaed wih he survivor swap. In he siuaion where ρ Z,Q,Q 1, =, we see ha dy1, =. In he special case wihou sysemaic moraliy risk, we have ha ρ Z,Q 1,1 = ρz,q 1,2 = for all, such ha Y1,1 and Y 1,2 are consan and equal o. In fac, his implies ha he insurance paymen process A is aainable and hus i can be hedged perfecly. The risk-minimizing sraegy in his case was essenially obained in Møller 1998, Secion 5. Proposiion 5.1 The Galchouk-Kunia-Waanabe decomposiion of V,Q in he marke B, P, Z 1 is given by V,Q = V,Q + ξ Q 1 τdp τ, T + where V,Q = n 1 π s + n 1 Ṽ Q p and wih L Q 1 = + ν V,Q τ ϑ Q 1 τνz,q 1 τ dm Q 1 x, τ 2 =1 ρ V,Q τ ϑ Q 1 τρz,q 1, τ ϑ Q 1 τdz,q 1 x, τ + L Q 1, 5.6 dw µ,q ξ Q 1 = ηv,q ϑ Q 1 ηz,q 1 σ r B r, T P, T, 5.7 ϑ Q 1 = νv,q + ρ V,Q 1 κ Q 1,1 1 χ Z,Q 1,1 + ρv,q 2 κ Q 1,2 1 χ Z,Q 1,2 ν Z,Q 1 + ρ Z,Q 1,1 κq 1,1 1 χ Z,Q 1,1 + ρz,q 1,2 κq 1,2 1 χ Z,Q 5.8 1,2. The proof is posponed o he Appendix. The opimal number of zero-coupon bonds deermined by ξ Q 1 is he opimal number ξq B from he bond marke adused by a erm originaing from he ineres rae risk inheren in he survivor swap. The opimal number of survivor swaps can be inerpreed as a raio beween risk-weighed averages of he moraliy risk associaed wih he inrinsic value process of he insurance paymen process and he price process of he survivor swap, respecively. The unhedged risk L Q 1 consiss of boh unsysemaic and sysemaic moraliy risk. The unsysemaic moraliy risk is 15 τ,
driven by he compensaed couning process M Q 1, and he sysemaic moraliy risk is driven by he 2-dimensional Brownian moion W µ,q. The unsysemaic moraliy risk is he sandard discouned sum a risk ν V,Q reduced by he discouned sum a risk relaed o he invesmen in survivor swaps. Similarly, he unhedged sysemaic moraliy risk is he original sysemaic risk, reduced by he risk from he survivor swaps. From he general heory of risk-minimizaion in Secion 5.2, we ge he unique -admissible risk-minimizing sraegy for he paymen process 4.1 ϕ 1 = ξ 1, ϑ 1, η 1 = ξ Q 1, ϑq 1, Ṽ,Q ξ Q 1 P, T ϑ Q 1 Z,Q 1, where ξ Q 1 and ϑq 1 are given in Proposiion 5.1. From 5.1 we ge he inrinsic risk process R, ϕ 1 = E Q [ T + 2 =1 ρ V,Q ν V,Q τ ϑ Q 1 τνz,q 1 τ dm Q 1 x, τ 2 ] τ ϑ Q 1 τρz,q 1, τ dw µ,q τ F [ T 2λ = E Q ν V,Q τ ϑ Q 1 τνz,q Q 1 τ 1 x, τ + 2 =1 ρ V,Q ] 2 τ ϑ Q 1 τρz,q 1, τ dτ F. Here we have used, ha M Q 1, W µ,q 1 and W µ,q 2 are muually independen and he fac ha d M Q 1 x, = λq µ,q 1 d and d W = d. 5.5 Risk-minimizaion wih a populaion survivor swap As an alernaive o he marke B, P, Z 1 considered in he previous secion, we now sudy he marke B, P, Z 2. Hence, we allow for invesmens in a survivor swap on he populaion insead of he insurance porfolio. From Lemma 3.2, we have he following represenaion for he inrinsic value process of he populaion survivor swap dz,q 2 = ν Z,Q 2 dm Q 2 x, + ηz,q 2 dw r,q + ρ Z,Q 2 dw µ,q. Here, we noe ha he unsysemaic moraliy risk is driven by he random source M Q 2, which means ha we inroduce a new random source compared o he marke B, P, Z 1 from Secion 5.4 above. Consequenly, we now need hree zero-mean maringales Y,Q 2,, = 1, 2, 3, in order o span all risk, since we have five random sources in he marke, and only wo risky asses o hedge he risk. Le he zero-mean maringales Y,Q 2,, = 1, 2, 3, be given by dy,q 2, = χz,q 2, dm Q 2 x, κq 2, dy,q 2,3 = dm Q 1 x,, µ,q dw, = 1, 2, where Y,Q 2, =, = 1, 2, 3, and for ρz,q 2,, = 1, 2, we define κq 2, by κ Q 2, νz,q 2 λ Q 2 x, = ρ Z,Q 2,. 16
Then Y,Q 2, are orhogonal o P, T and Z,Q 2. We noe ha even in he case wihou sysemaic moraliy risk, he insurance paymen process A is no aainable in he marke B, P, Z 2, since he unsysemaic moraliy risk canno be eliminaed. This is in conras o he siuaion in he previous marke B, P, Z 1. Proposiion 5.2 The Galchouk-Kunia-Waanabe decomposiion of V,Q in he marke B, P, Z 2 is given by where V,Q = V,Q + ξ Q 2 τdp τ, T + ϑ Q 2 τdz,q 2 x, τ + L Q 2, V,Q = n 1 π s + n 1 Ṽp,Q, L Q 2 = ν V,Q τdm Q 1 x, τ + 2 =1 ρ V,Q τ ϑ Q 2 τρz,q 2, ϑ Q 2 τνz,q 2 τdm Q 2 x, τ dw µ,q τ, 5.9 and ξ Q 2 = ηv,q ϑ Q 2 ηz,q 2 σ r B r, T P, T, 5.1 ϑ Q 2 = ρ V,Q 1 κ Q 2,1 1 χ Z,Q 2,1 + ρv,q 2 κ Q 2,2 1 χ Z,Q 2,2 ν Z,Q 2 + ρ Z,Q 2,1 κq 2,1 1 χ Z,Q 2,1 + ρz,q 2,2 κq 2,2 1 χ Z,Q 5.11 2,2. The inerpreaion of he Galchouk-Kunia-Waanabe decomposiion obained in Proposiion 5.2 is essenially idenical o ha of Proposiion 5.1. However in his case he unhedgeable unsysemaic moraliy risk consiss of wo erms. A erm which sems from he insurance porfolio driven by M Q 1 and a erm driven by M Q 2 originaing from invesmens in survivor swaps. The unique -admissible risk-minimizing sraegy for he paymen process 4.1 is given by ϕ 2 = ξ 2, ϑ 2, η 2 = ξ Q 2, ϑq 2, Ṽ,Q ξ Q 2 P, T ϑ Q 2 Z,Q 2, where ξ Q 2 and ϑq 2 are given in 5.1 and 5.11. The inrinsic risk process R, ϕ 2 can be deermined as in he previous secion. This leads o [ T 2λ R, ϕ 2 = E Q ν V,Q Q 2λ τ 1 x, τ + ϑ Q 2 τνz,q Q 2 τ 2 x, τ + 2 =1 ρ V,Q ] 2 τ ϑ Q 2 τρz,q 2, dτ τ F. 17
5.6 Risk-minimizaion wih survivor swaps on boh porfolios Now consider he siuaion where he marke includes boh survivor swaps. In his case he considered marke B, P, Z 1, Z 2 includes hree risky asses, whereas we have five random sources in he marke: Two random sources driving he unsysemaic moraliy risks, wo random sources driving he sysemaic moraliy risks and one random source driving he ineres rae risk. Thus, he marke is sill incomplee. Proposiion 5.3 The Galchouk-Kunia-Waanabe decomposiion of V,Q in he marke B, P, Z 1, Z 2 is given by where V,Q = V,Q + ϑ Q 3,1 = ν V,Q + ρ ϑ Q 3,2 = ρ ν Z,Q 1 + ρz,q V,Q 1 2,1 2,1 V,Q 1 1,1 1,1 1,1 1,1 1,1 + ρv,q ξ Q 3 τdp τ, T + + ρv,q + ρz,q 2 2,2 ν Z,Q 2 + ρz,q 2 1,2 1,2 1,2 2 =1 1,2 ρ Z,Q 1,2 ρ Z,Q 2,2 ϑq 3,1 ρ Z,Q 2,1 2,1 2,1 ξ Q 3 = ηv,q ϑ Q 3,1 ηz 1 ϑq 3,2 ηz 2 σ r B r, T P,, T and where ϑ Q 3, τdz,q x, τ + L Q 3, 2,1 1,1 1,1 2,1 1,1 1,1 1,1 2,1 2,1 + ρz,q 2,2 2,2 + ρz,q + ρz,q + ρz,q 2,2 1,2 2,2 1,2 1,2 2,2 1,2 ϑ Q 2, 1,2 ϑ Q 2 2,2 2,2, ϑ Q 2 = ϑ Q 2 = ρ V,Q 1 κ Q 2,1 1 χ Z,Q 2,1 + ρv,q 2 κ Q 2,2 1 χ Z,Q 2,2 ν Z,Q 2 + ρ Z,Q 2,1 κq 2,1 1 χ Z,Q 2,1 + ρz,q 2,2 κq 2,2 1 χ Z,Q 2,2, ρ Z,Q 1,1 κq 2,1 1 χ Z,Q 2,1 + ρz,q 1,2 κq 2,2 1 χ Z,Q 2,2 ν Z,Q 2 + ρ Z,Q 2,1 κq 2,1 1 χ Z,Q 2,1 + ρz,q 2,2 κq 2,2 1 χ Z,Q 2,2. Furhermore V,Q = n 1 π s + n 1 Ṽp,Q, 2 L 3 = ρ V,Q τ ϑ Q 3,1 τρz,q 1, τ ϑq 3,2 τρz,q 2, τ =1 + ν V,Q τ ϑ Q 3,1 τνz,q 1 τ dm Q 1 x, τ dw µ,q τ ϑ Q 3,2 τνz,q 2 τdm Q 2 x, τ. The proof of he proposiion, which is posponed o he Appendix, is carried ou using he same echniques as in he proof of Proposiion 5.1. We observe ha he Galchouk-Kunia-Waanabe decomposiion essenially is of he same form as in he cases wih only one survivor swap. However he coefficiens are far more 18
complex here han in he previous markes. Here he unique -admissible risk-minimizing sraegy for he paymen process 4.1 is given by ϕ 3 = ξ 3, ϑ 3,1, ϑ 3,2, η 3 = ξ Q 3, ϑq 3,1, ϑq 3,2, Ṽ,Q ξ Q 3 P, T ϑ Q 3,1 Z,Q 1 ϑ Q 3,2 Z,Q 2, where ξ Q 3, ϑq 3,1 and ϑq 3,2 are given in he above proposiion. From 5.1 we ge he inrinsic risk process [ T 2λ R, ϕ 3 = E Q ν V,Q τ ϑ Q 3,1 τνz,q Q 2λ 1 τ 1 x, τ + ϑ Q 3,2 τνz,q Q 2 τ 2 τ + 2 =1 ρ V,Q τ ϑ Q 3,1 τρz,q 1, τ ϑq 3,2 τρz,q 2 2, τ ] dτ F. Remark 5.4 In he lieraure, i has been proposed o allow for umps in he underlying moraliy inensiies, see Biffis 25 who works wih affine ump-diffusions. These umps, which may be inerpreed as moraliy shocks, seem paricularly relevan for he modeling of shor erm caasrophe moraliy derivaives, such as he Swiss Re moraliy index bonds described in Blake e al. 26. I would be possible o exend he presen resuls o he case of ump-diffusion driven moraliy inensiies. However, his exension would lead o more involved formulas, which would be more difficul o inerpre and in our opinion no really lead o considerable new insighs. 6 Numerical examples In his secion, he main focus is on he risk-minimizing sraegies and heir efficiency. Firs, we explain he basic seup and illusrae relevan quaniies by considering wo scenarios generaed by he underlying sochasic model. We refer o hese scenarios as he red and blue scenario, respecively. Second, we sudy invesmens in he survivor swaps and zero-coupon bonds. Finally, we compare he efficiency of he various invesmen sraegies by deermining he inrinsic risk. This is he main obecive of he example. Since he inrinsic risk process Rϕ involves an expeced value under he maringale measure Q, we focus on he Q-dynamics for all processes involved. Hence all scenarios displayed are Q-scenarios. 6.1 Seup Unless saed oherwise we consider an insurance porfolio wih 1 policy-holders aged 3, who pay a coninuous premium of π c =.2 during [, T ], where T = 3. In case of a deah a ime he insurance company pays a lump sum of a d = 5 1 { <T }. Hence he deah benefi is paid ou upon a deah before reiremen only. A he age of reiremen, which is 6 years, a lump sum of a r T = 3 is paid o all survivors. Finally, he conrac conains a 3 year life annuiy saring a age 6 wih a rae of a p = 1, which implies ha T = 6. The populaion porfolio exiss of 1 lives aged 3. The iniial moraliy inensiies µ are aken on he Gomperz-Makeham form µ x + = a + b c x+, 19
wih parameers given in Table 1. The parameers for porfolio 1 are aken from Dahl and Møller 26. For porfolio 2, we have modified he parameers slighly, such ha here is a minor difference beween he wo iniial moraliies. The parameers for he Porfolio a b c 1.134.353 1.12 2.136.35 1.13 Table 1: Gomperz-Makeham parameers. dynamics for he moraliy inensiies are also inspired by Dahl and Møller 26 and can be found in Table 2. Here, however, he moraliy inensiies are driven by wo underlying Brownian moions. For simpliciy, we sudy he case where β = β = g =, = 1, 2. This means ha no change of measure for he moraliy is applied in he example, such ha he moraliy dynamics under he measure P and he maringale measure Q coincide. Alernaively, one could aemp o esimae hese parameers from he prices observed in moraliy securiy markes. By comparing hese wih parameers esimaed from he acual behaviour of he underlying moraliy processes, one could ge an idea of he marke price of moraliy risk. However, his is beyond he scope of he curren paper. Wih our pragmaic choices of parameers, we are able o illusrae he main echniques and o compare he differen sraegies. For reasonable choices of prices of moraliy risk, he conclusions regarding he relaive efficiency of he various sraegies remain valid. In a sudy of he absolue levels of risk, i would be of imporance o deermine he marke price of risk from empirical daa and o change measure accordingly. Porfolio µ x, γ x, δ x, σ,1 x, σ,2 x, 1 µ 1 x.18.8.6.18 2 µ 2 x.185.81..19 Table 2: Parameers for moraliy inensiies. The parameers for he financial marke are lised in Table 3. Using hese parameers in he Vasiček model, we ge a shor rae model wih Q-mean reversion level γ r,q /δ r,q =.55. The speed of mean reversion is deermined by δ r,q. Figure 1 shows he developmen of r γ r,q δ r,q σ r.3.11.2.1 Table 3: Parameers for he financial marke. he shor rae in wo sochasic scenarios red and blue lines. We observe ha he level of he shor rae is similar in he wo scenarios unil ime 2. In he ime inerval from 2 o 4, he shor rae in he blue red scenario is relaively low high, whereas he siuaion is reversed from ime 4 o 55. In he las 5 years he shor raes again lie a he same level. The corresponding inrinsic value processes for he insurance paymen process in hese 2
..4.8 1 2 3 4 5 6 Figure 1: The shor rae in wo differen sochasic scenarios red and blue lines. wo scenarios are ploed in Figure 2. We see a big difference beween he developmen of he inrinsic value process in he wo scenarios. In he blue scenario, he process aains is minimum value around ime 15. From here, i has a posiive rend unil ime 3, where i levels ou a approximaely 4,. In he red scenario, he process decreases unil ime 3 and levels ou around -1,. In paricular, we see ha he scenarios differ fundamenally from ime 2. The main reason for his difference is he magniude of he shor rae beween ime 2 and 4 in he wo scenarios. From ime 2, he main par of he remaining paymens inheren in he paymen process are benefis. Hence he marke value of fuure paymens increases decreases as he shor rae decreases increases. This behavior is mos profound unil ime 4, where he combinaion of few remaining benefis and discouning makes he impac negligible. 1 5 5 1 2 3 4 5 6 Figure 2: Inrinsic value processes for he insurance conrac he red and blue scenario. 21
6.2 Risk sensiiviy of he insurance conrac Figure 3 shows he oucomes of he so-called risk sensiiviies ν V,Q, η V,Q and ρ V,Q in he red and blue scenario. We observe ha due he relaively large number of insured lives ν V, Q η V, Q 2 2 4 1 2 3 4 5 6 15 5 1 2 3 4 5 6 V, Q ρ 1 V, Q ρ 2 35 2 5 1 2 3 4 5 6 1 4 1 2 3 4 5 6 Figure 3: Risk sensiiviies for he insurance conrac in he red and blue scenario. in he insurance porfolio, he magniude of he sensiiviy ν V,Q o unsysemaic moraliy risk is insignifican compared o he oher sensiiviies. Iniially ν V,Q is posiive reflecing ha he discouned deah benefi is larger han he discouned individual marke reserve. From ime o he ime of reiremen, ν V,Q decreases due o he premiums paid. Afer approximaely 15 years ν V,Q becomes negaive. A he ime of reiremen we observe a negaive ump in ν V,Q since he deah benefi is larger han he sum a reiremen. A he age of reiremen, only paymens from he life annuiy remains, and here ν V,Q aains is minimal value. Hereafer i increases seadily owards he value a he expiry of he conrac. We see ha ν V,Q aain lower values in he blue scenario han in he red scenario. This difference is mos prominen beween imes 2 and 4, where we also observe considerable differences for he shor raes. The sensiiviy o ineres rae risk η V,Q is he mos significan risk due o he large ineres rae volailiy. As noed already his fac is especially eviden if we focus on he value process in Figure 2 in he period from 2 o 35 years. The differences in he sensiiviies o ineres rae risk in he wo scenarios can be explained by he developmen in he value processes. Finally, we recall from Table 2 ha he dependence of µ 1 on W µ,q 2 is hree imes ha of he dependence on W µ,q 1. This relaionship essenially carries hrough o he risk sensiiviies ρ V,Q 1 and ρ V,Q 2. Once again he differences in he sensiiviies o sysemaic moraliy risk in he wo scenarios can be explained by he developmen of he value processes. 6.3 Invesmen sraegies The risk-minimizing sraegy minimizes he oal risk, which consiss of unsysemaic and sysemaic moraliy risk and ineres rae risk. Here we firs focus on invesmens in survivor swaps and hen we urn o invesmens in bonds. In his secion we diverge 22
.2.4.6.8 1. 1 2 3 4 5 6 Figure 4: Number of survivor swaps on he insurance porfolio held a ime in he B, P, Z 1 marke in he red scenario. from he sandard seup and consider an insurance porfolio of 1, individuals and a populaion of 1, individuals. In he B, P, Z 1 marke, he survivor swap can be used o hedge boh unsysemaic moraliy risk and sysemaic moraliy risk generaed by W µ,q 1 and W µ,q 2. The large number of insured implies ha he unsysemaic moraliy risk is insignifican compared o he sysemaic moraliy risks. Unil he age of reiremen he invesmens in he survivor swap is herefore essenially he raio beween a weighed average of he sensiiviies o sysemaic moraliy risk from he insurance conrac and he sensiiviies o sysemaic moraliy risk from he survivor swap. A he age of reiremen, he only remaining paymens from he insurance conrac is a life annuiy, which has he exac same sensiiviy o moraliy risk as a survivor swap on he insurance porfolio. Hence, from he ime of reiremen he moraliy risk can be hedged perfecly by holding exacly one survivor swap. The invesmen sraegy is seen in Figure 4. In he B, P, Z 2 marke, he survivor swap can be used o hedge sysemaic moraliy risk generaed by W µ,q 2 only. Furhermore, he survivor swap inroduces an addiional source of unsysemaic moraliy risk, M Q 2. Since he unsysemaic moraliy risk is insignifican compared o he W µ,q 2 -sysemaic moraliy risk, he invesmens in he survivor swap is essenially he raio beween a weighed average of he sensiiviy o W µ,q 2 -sysemaic moraliy risk from he insurance porfolio and he sensiiviy o W µ,q 2 -sysemaic moraliy risk from he survivor swap. The invesmen sraegy is seen in Figure 5. Comparing Figures 4 and 5 we observe a large similariy beween he shapes of ϑ 1 and ϑ 2. However, we noice, ha he invesmen in he B, P, Z 2 marke ends o zero a he end of he insurance period. This can be explained by he fac ha he sensiiviy o sysemaic moraliy risk from he insurance conrac converges o, whereas he swap s sensiiviy o unsysemaic moraliy risk does no converge o. Since he number of individuals in he populaion is 1 imes he number in he insurance porfolio he magniude of ϑ 1 is essenially 1 imes he magniude of ϑ 2. In he B, P, Z 1, Z 2 marke, he survivor swaps are used o hedge boh unsysemaic 23
..2.4.6.8.1 1 2 3 4 5 6 Figure 5: Number of survivor swaps on populaion held a ime in he B, P, Z 2 marke. moraliy risk relaed o he insurance porfolio and he sysemaic moraliy risk. Furhermore, he survivor swap on he populaion inroduces he addiional unsysemaic moraliy risk. Firs we recall ha, from he ime of reiremen, a survivor swap on he porfolio eliminaes all moraliy risk, which in urn means ha no invesmens in he survivor swap on he populaion is needed afer he ime of reiremen. Prior o he ime of reiremen we use boh survivor swaps o hedge he moraliy risks. From he sraegies in Figure 6, we observe ha, upon correcing for he larger number of individuals in he populaion, he oal number of swaps is slighly higher han he number of swaps held in he B, P, Z 1 marke. In he four markes considered in Secions 5.3-5.6 all ineres rae risk can be hedged. When we inves in survivor swaps, we inroduce addiional ineres rae risk, and he invesmens in he bond eliminae he ineres rae risk from he insurance paymen process correced for he ineres sensiiviy from he survivor swaps. Thus, he invesmen sraegies for he zero-coupon bond will differ slighly, depending on he invesmens in survivor swaps. The invesmen sraegies are ploed in Figure 7. 6.4 Efficiency of he sraegies The efficiency of he sraegies is assessed by calculaing he iniial inrinsic risks R,. We use Mone Carlo simulaion wih 1, simulaions o calculae R, in differen markes wih porfolios of differen sizes. In order o quanify he oal risk inheren in he insurance conrac we consider he iniial inrinsic risk in he marke including only he risk free asse B. In his case he inrinsic risk process for he insurance conrac is given by R, ϕ V = E Q T 2λ 2 2 ν V,Q Q τ 1 η x, τ + V,Q τ + =1 ρ V,Q 2 τ dτ F. The resuls of he Mone Carlo simulaions are colleced Table 4. Firs of all, we observe ha inroducing a zero coupon bond eliminaes approximaely 8% of he risk, so he ineres rae is wihou doub he mos significan source of risk. Invesigaing he markes wihou survivor swaps we observe how he unsysemaic moraliy risk is diversified 24