A maringale approach applied o he managemen of life insurances. Donaien Hainau Pierre Devolder 19h June 2007 Insiu des sciences acuarielles. Universié Caholique de Louvain UCL. 1348 Louvain-La-Neuve, Belgium. Absrac In his paper we deermine he opimal asse allocaion of pure endowmens insurance conracs, maximizing he expeced uiliy of erminal surplus under a budge consrain. he marke resuling from he combinaion of insurance and financial producs, is incomplee owing o he unhedgeable moraliy of he insured populaion, modelled by a Poisson process. For a given equivalen measure, he opimal wealh process is obained by he mehod of Lagrange mulipliers and he invesmen sraegy replicaing a bes his process is obained eiher by maringale decomposiion or eiher by dynamic programming. Nex, we illusrae his mehod for CARA and CRRA uiliy funcions. Keywords : Sochasic opimizaion, Opimal asse allocaion, Moraliy risk, Incomplee markes. 1 Inroducion. Due o he presence of moraliy risk, which is no ye hedgeable by radiional financial ools, he combinaion of life insurance and financial markes is incomplee and sandard uiliy opimizaion mehods have o be used wih care. he conribuion of his paper is precisely o show how he celebraed maringale approach, developed by Cox & Huang 1989 may be used o handle asse allocaion problems in life insurance business. wo classical ways are usually exploied o sudy he opimal asse allocaion of insurance conracs. he firs one is he maringale approach, already menioned above. However, he exising lieraure based on his orienaion, neglecs he moraliy risk. he ineresed reader may refer o Boulier e al. 2001, Deelsra e al. 2003, 2004 for examples of managemen and design of a pension fund. he second mehod relies on sochasic conrol and he resoluion of he Hamilon Jacobi Bellman equaion. his approach was successfully applied o he managemen and pricing of a wide variey of insurance conracs wih exponenial uiliy CARA. Some resuls of his paper will be compared wih hose obained by Young and Zariphopoulou 2002. Anoher applicaion of sochasic conrol is he managemen of one life annuiy sudied by Menoncin e al. 2004. In his paper, he maringale mehod is applied in a simplified seing: he financial marke is composed of wo asses cash, socks and he moraliy, source of incompleeness, is modelled by a Poisson process. On he liabiliy side, we have pure endowmen policies of same mauriy and guaranee. Under he assumpion ha he risk neural measure of he global marke financial and acuarial is equal o he produc of he financial risk neural measure and of he hisorical Corresponding auhor. Email: hainau@sa.ucl.ac.be 1
acuarial measure, he expeced uiliy of a erminal surplus is maximized under a budge consrain, by he approach of Lagrange mulipliers. he surplus is defined as he difference beween a arge erminal wealh and he sum of capials paid ou o alive affiliaes. he budge consrain guaranees he acuarial equilibrium beween he insurer s curren wealh and he expeced fuure benefis. he self financed invesmen sraegy replicaing a bes he opimal wealh process is nex obained eiher by maringale decomposiion of he wealh process or eiher by dynamic programming. he las par of his work is devoed o examples. In he firs applicaion, we consider an exponenial CARA uiliy. he value funcion and he opimal invesmen policy are compared wih he resuls of Young and Zariphopoulou 2002 obained by direc resoluion of he Bellman equaion. Nex, he case of a power CRRA uiliy is addressed. A our knowledge, soluions for such kind of uiliy were no ye developed in he conex of he managemen of life insurance conracs. A comparison reveals ha he choice of a power uiliy leads o an opimal asse allocaion relaively well adaped o ALM purposes. he ouline of he paper is as follows. he porfolio of life insurances and he financial marke are described in secion 2 and 3. he choice of he deflaor is discussed in secion 4. Nex, he opimizaion problem and he dynamic of he fund are presened. In secion 6, we esablish a general soluion and wo ways o infer he opimal asse allocaion. In secion 7 and 8, paricular cases of CARA and CRRA uiliies are sudied. 2 Liabiliies. We consider a porfolio of pure life endowmens, ha couns iniially n x affiliaes of age x. he insurance company will deliver a fixed capial K o each individual who aains he age x +. As in Møller 1998, remaining lifeimes are assumed independen and idenically disribued exponenial random variables, noed 1, 2,..., nx defined on a probabiliy space Ω a, F a, P a. A ime, he hazard rae of i, also called he moraliy rae of he affiliaes, is a deerminisic funcion wrien µx +. he oal number of deahs a insan is noed N and defined by: N = n x i=1 I i Where I. is an indicaor variable. he acuarial filraion F a is he one generaed by N. he sochasic inensiy of N is formally described as follows: E dn F a = n x N.µx +.d A ime, he compensaed process of N : M = N 0 n x N u.µx + u.du 2.1 is a maringale under P a. Remark ha, a mauriy, he oal paymen done by he fund worhs n x N.K and he expecaion a ime of his oal cash flow is: nx E n x N.K F a = E I i > F a.k i=1 = i> E I i > F a.k = n x N. exp µx + u.du } {{ } p x+.k 2
p x+ is he real probabiliy ha an individual of age x +, survives ill age x +. 3 Asses. We consider a financial marke composed of one risky asse, S socks and one risk-less asse, a cash accoun, which provides a consan reurn r. Socks S are driven by a geomeric Brownian moion: ds = r + ν.d + σ.dw P S Where W P f, ν, σ are respecively a Brownian moion, he consan risk premium and he consan volailiy of socks. he financial probabiliy space is noed Ω f, F f, P f where F f is he filraion generaed by W P f. he financial marke is complee and here exiss an unique equivalen measure, he risk neural measure wrien Q f, such ha he discouned asse prices are maringales under Q f. Le λ = ν σ be he cos of he risk of socks. Under Qf, he socks obey o he following SDE: ds = r.d + σ. dw P f + λ.d S } {{ } = r.d + σ.dw Qf f Where W Qf is a Brownian moion under he financial risk neural measure. 4 Deflaors. Le Ω, F, P be he probabiliy space resuling from he combinaion of he insurance and financial markes. Ω = Ω a Ω f F = F a F f N P = P a P f Where he sigma algebra N is generaed by all subses of null ses from F a F f. his global marke is incomplee owing o he presence of he moraliy, a non hedgeable risk. hen, here exiss muliple equivalen maringale measures under which he discouned prices of readable asses are maringale. For he sake of simpliciy, we assume ha he global risk neural measure, noed Q, is equal o he produc of he financial risk neural measure and of he hisorical acuarial measure P a. Noe ha his measure is known in he financial lieraure as he minimal measure, inroduced by Föllmer and Sondermann 1986. his assumpion is commonly acceped by acuaries, who relies on diversificaion o hedge he moraliy risk. he deflaor associaed o insurance producs is, in his conex, equal o he financial deflaor. A ime, for a claim occurring a, we noe i H,. H, = e R r.du. = exp dq f dp f dq f dp f r.du 1 2. λ 2.du λ.dwu P And he condiional expecaion of he deflaor is equal o he price of a zero coupon: E H, F = exp r.. Examples of deflaor defining an acuarial measure differen from he real measure may be found in our paper Hainau & Devolder 2007. 3
5 he dynamic of he fund and he opimizaion problem. he asse manager opimizes he invesmen policy so as o maximize he uiliy of he surplus a insan. his surplus is he difference beween a oal arge asse X and he oal of capials paid ou o survivors. In paricular, he value funcion V, x, n a ime, for a wealh x and for n observed deceases is: V, x, n = sup E U X n x N.K X = x, N = n 5.1 X A x } {{ } F he uiliy funcion is sricly increasing and concave. he opimal erminal wealh belongs o he se A x which is delimied by a feasibiliy condiion sipulaing ha he expecaion of he deflaed erminal wealh is a mos equal o he curren wealh. { A x = X such ha E H,. X } F x 5.2 In he sequel, his consrain is called he budge consrain. his condiion corresponds o a curren acuarial pracice which consiss in imposing ha he reserve of he company is a leas equal o he expecaion of fuure benefis. o avoid any confusion, we insis on he fac ha X is no necessary self financed he opimal arge wealh can depends on moraliy which is no hedgeable. Under he assumpion ha he fund is closed no cash in or cash ou, he opimal invesmen sraegy, replicaing a bes he opimal arge wealh, is obained by projecing he opimal arge wealh ino he space of self financed processes. If he self financed wealh process and he fracion of he fund invesed in socks are respecively noed X and π, X obeys he following SDE: dx = r + π.ν.x.d + π.σ.x.dw P As explained in he nex secion, wo mehods of projecion are available: one based on he Kunia Waanabe decomposiion of X and he oher on dynamic programming. We draw he aenion of he reader on he fac ha if he opimizaion is done on a domain of conrols resriced o he se of replicable erminal wealh, A π x raher han A x { A π x = X π F adaped : e r. π s.x } s X = x +.d e r.s.s s S s he expecaion 5.2 is no always defined for a power uiliy, widely used for ALM purposes in he seing of complee markes. Inuiively, if he available asse, x, is insufficien o hedge he survival of all affiliaes ha is nearly almos he case for life insurers, he probabiliy of having a negaive erminal surplus is no null and he he power uiliy of such negaive surplus doesn exis. Opimizing he uiliy on he enlarged se of conrols A x, allows us o avoid his drawback resuls are developed in secion 8. 6 A general soluion. Our mehod uses Lagrange mulipliers. We refer o Karazas & Shreve 1998 for a deailed presenaion of his echnique, developed by Cox Huang 1989 in complee markes. Le y R + be he Lagrange muliplier associaed o he budge consrain 5.2. he Lagrangian is defined by: L, x, n, X, y = E U X n x N.K F +y. x E H,. X F 6.1 4
he exisence of he opimal erminal wealh, noed, X, is guaraneed if we find an opimal Lagrange muliplier y > 0 such ha X is a saddle poin of he Lagrangian 6.1. he value funcion verifies: V, x, n = inf y R + sup X L, x, n, X, y Under he assumpions ha he funcion U. is C 1 sricly concave and increasing, he derivaive of U. admis a coninuous inverse funcion, noed I.. I suffices o derive he expression 6.1 wih respec o X, o obain he opimal wealh process in funcion of he Lagrange muliplier X = I y.h, + n x N.K 6.2 And he opimal Lagrange muliplier y is such ha he budge consrain is binding i.e. x = E H,. I y.h, + n x N.K F Once he opimal Lagrange muliplier deermined, he value funcion is also calculable: V, x, n = E U I y.h, F 6.3 he opimal erminal wealh 6.2, depends on he number of survivors a ime and clearly canno be replicaed by a porfolio of financial asses. However wo possibiliies are conceivable o infer he invesmen policy approaching a bes X. he firs one consiss o decompose X in a sum of an adaped process, of a Brownian maringale, and of a zero mean jump maringale. he second soluion relies on dynamic programming. 6.1 Decomposiion of X. he following proposiion is he key ool o esablish he invesmen policy hedging parially X. Proposiion 6.1. Le X be an F f F a measurable non negaive random variable, such ha E X.H, F = x 6.4 hen here exiss a predicable admissible porfolio π such ha he associaed final wealh X verifies E X F = E X F 6.5 and minimizes he square of he spread beween X and X, under P : π minimizes E X X 2 F 6.6 Proof. Under he probabiliy measure Q, by he ower propery of condiional expecaions, we know ha L s = E X H, F s = E Q X.e R r.du F s wih s, is a local maringale and hence a global supermaringale. Moreover, by assumpion E Q L F = L = x and L s is in fac a maringale. hen, here exiss a Kunia Waanabe decomposiion of L s for a proof see Kunia Waanabe 1967. In paricular here exis wo progressively measurable processes ϕ 1, ϕ 2 such ha L s = L + s ϕ 1 u.dw Qf u + 5 s ϕ 2 u.dm u
Where s ϕ 2u.dM u is a maringale orhogonal o he space of sochasic Brownian inegral M u is he compensaed process of N u, see equaion 2.1. As L = x, we have ha: X.e R r.du = x + ϕ 1 u.dw Qf u + ϕ 2 u.dm u And, by comparison of his decomposiion wih he self financed wealh process, X.e R r.du = x + he sraegy π u replicaing a bes X is: π u = e R u r.dv.π u.x u.σ.dwu Qf ϕ 1 u e R u r.dv.σ. 1 u [, ] 6.7 X u By consrucion of π, asserion 6.5 is verified under P. For any oher self financed sraegy π u, such ha π u = ϕ 1u e R u r.dv.σ. 1 X u he expecaion of he quadraic spread beween X and Z under Q becomes: E X X 2 F = 2 = e 2. R r.du.e ϕ 1u ϕ 1 u.dwu Qf + ϕ 2 u.dm u F 2 = e 2. R r.du. E ϕ 1u ϕ 1 u.dwu Qf F +e 2. R r.du.e } {{ } 0 2 ϕ 2 u.dm u F 6.8 And he righ hand erm of his las expression is minimized when ϕ 1u = ϕ 1 u. As M he compensaed process of N is idenical under P and Q, π u also minimizes he expecaion of X X 2 under P. Remark 6.2. According o equaion 6.8, he expecaion E X + X 2 + F is minimized by he invesmen policy π, for any [0, ]. If + is he firs exi ime of X + X + from an inerval O round X X = 0, given ha fx = x 2 is local lipschiz, we have ha [ 0 E X + X 2 + X X ] 2 F [ E C O. X + X + X X ] F [ Where C O X+ is consan. Minimizing E X ] + F is herefore equivalen o minimizing E X + X 2 + F. his observaion will be useful o prove he equivalence beween he maringale decomposiion and he dynamic programming approach. 6
6.2 Dynamic programming. he second possibiliy o obain he opimal invesmen policy, relies on dynamic programming for an inroducion, see Fleming and Rishel 1975. Given a small sep of ime,, his principle saes ha V, x, n = E V +, X +, N + F Given ha X is he process maximizing he value funcion, any oher process X X belonging o he se A x, and in paricular a replicable process, saisfies he inequaliy: V, x, n E V +, X +, N + F 6.9 And he closes process o X is deermined by an invesmen sraegy maximizing he righ hand erm of 6.9. Indeed, he value funcion V, x, n is concave in x and hen and hen local Lipschiz in x: O R, C O R + x 1, x 2 O V, x 1, n V, x 2, n C O. x 1 x 2 And if + is he firs exi ime of X + or X + from an inerval O round x, he inequaliy 6.9 is bounded as follows: [ 0 E V +, X +, N + V +, X ] +, N + F [ E C O. X + X ] + F [ Where C O X+ is consan. Minimizing E X ] + F is herefore equivalen o minimizing he righ hand erm of 6.9. Furhermore according remark 6.2, his demonsraes he equivalence wih he maringale decomposiion. By applicaion of he Io s lemma for jump processes see Øksendal and Sulem 2005, chaper one or Duffie 2001, annex F, he expecaion of he value funcion a insan + is given by: E V +, X +, N + F = + V, x, n + E G π s, X s, N s.ds F + + E V s, X s, N s V s, X s, N s.dn s F 6.10 Where G π s, X s, N s is he generaor of he value funcion: G π s, X s, N s = V s + r + π s.ν.x s.v X + 1 2.π2 s.σ 2.X 2 s.v XX 6.11 V s, V X, V XX are parial derivaives of he value funcion wih respec o ime and wealh. Deriving he generaor wih respec o π provides he invesmen sraegy maximizing he righ hand erm of 6.9: π s = ν σ 2. V X V XX. 1 X s 6.12 As he value funcion is known, see equaion 6.3, parial derivaives V X and V XX are easily calculable. 7
7 CARA uiliy. We assume ha he preferences of he asse manager are modelled by a CARA consan absolue risk aversion uiliy funcion wih a risk aversion parameer noed α. he value funcion a ime is hen rewrien: V, x, n = sup E X A x From equaion 6.2, we infer he opimal erminal wealh: he value funcion a ime is easily calculable: 1α exp α. X n x N.K F X = 1 α. ln y.h, + n x N.K 7.1 V, x, n = 1 α.y.e H, F = 1 α.y. exp r. 7.2 And he Lagrange muliplier is such ha he budge consrain is sauraed: 1 α. ln y = 1 E H, F. [x E n x N.H,.K F + 1 α. E H,. ln H, F ] 7.3 he independence beween moraliy and he financial marke enails ha: E n x N.H,.K F = K.E n x N F.E H, F Afer calculaions see annex 1 for deails, we ge ha: E H,. ln H, F = e r.. = K.e r..n x N. p x+ And insering 7.3 in 7.2, leads o he following value funcion: [ ] r λ2. 2 V, x, n = 1 α. exp α.er. x e r..k.n x n. p x+ } {{ }. exp ν2. 2.σ2 Equiy he value funcion is proporional o he difference beween he curren marke of he fund and he discouned expeced claims paid on ime. his amoun may be seen as he equiy of he fund. he opimal invesmen sraegy is buil eiher by decomposiion of X or eiher by dynamic programming. 7.1 Decomposiion of X. In his case he Kunia Waanabe decomposiion is direcly inferred from he form of X. he combinaion of expressions 7.1 and 7.3 gives ha: 7.4 X = x. e 1 + r 1 α.λ.dw Q f u + K. n x N E n x N F } {{ } R dm 8
And he opimal invesmen policy is: π u = 1 α. λ r. σ. e e. 1 u [, ] 7.5 r.u X u Conrary o he value funcion, he fracion of he fund invesed in socks a ime, is oally independen of he equiy. π = e r.. 1 α. ν σ 2. 1 X 7.6 7.2 Dynamic programming. Deriving he value he value funcion wih respec o x leads o: V, x, n = α.e r..v, x, n x 2 V, x, n x 2 = α.e r. 2.V, x, n he opimal asse allocaion a ime is obained by applicaion of he formula 6.12 and is idenical o he one calculaed by he decomposiion of X. π = ν σ 2. V X V XX. 1 X = e r.. 1 α. ν σ 2. 1 X 7.3 Relaion wih he Hamilon Jacobi Bellman equaion. V. Young and. Zariphopoulou 2002 have already solved he problem of he managemen of one pure endowmen insurance, for an exponenial uiliy funcion, in a slighly differen seing. More precisely, hey resric he se of admissible erminal wealh o he one of replicable processes. In his conex, he value funcion, ha we noe V π., is defined by: V π, x, n = sup X A π x E U X n x N.K F 7.7 Where A π x is he se of aainable erminal wealh: { A π x = X π F adaped : e r. X = x + e r.s.π s.x s.ds s } Resricing he se of admissible conrols o A π x enails ha, if an opimal soluion exiss, he value funcion is soluion of he HJB sochasic differenial equaion: 0 = Vs π + max r + π s.ν.x s.vx π + 1 π s 2.π2 s.σ 2.Xs 2.VXX π Wih he following boundary condiion: +n x N s.µx + s.e V π s, X s, N s + 1 V π s, X s, N s 7.8 V π, x, n = U x n x n.k 7.9 Whereas, as showed in paragraph 6.2, he value funcion V, x, n of he opimizaion problem on he enlarged se A x verifies only he sochasic differenial inequaliy: 0 V s + max r + π s.ν.x s.v X + 12.π2s.σ 2.X 2s.V XX π s +n x N s.µx + s.e V s, X s, N s + 1 V s, X s, N s 7.10 9
When he porfolio couns one insurance policy, n x = 1, he value funcion soluion of he HJB equaion found by V. Young and. Zariphopoulou 2002 is slighly differen from he one obained by he maringale approach, equaion 7.4 wih n x = 1. If he affiliae is sill alive a ime, heir soluion is: V π, x, 0 = 1 α. exp α.e r.. 1 α. exp α.e r..x x e r..k ν2 ν2.. 2.σ2 q x+. 2.σ2. p x+ If he individual deceases before, he value funcion is he soluion found by Meron 1969 and 1971, ha maximizes he expeced uiliy of erminal wealh. V π, x, 1 = 1 α. exp α.e r..x ν2. 2.σ2 However, he opimal invesmen sraegy is idenical o he one obained by maximizaion on he enlarged se A x equaion 7.6. o close his paragraph, we calculae he residue a ime s, noed ɛ s, of he Bellman equaion for he soluion found by maringale approach and check ha i is negaive. he combinaion of he opimal asse allocaion 6.12 wih he righ hand erm of 7.10, leads o he following definiion of ɛ s : ɛ s = V s + r.x s.v X 1 2. ν2 σ 2. V 2 X V XX +n x N s.µx + s.e V s, X s, N s + 1 V s, X s, N s 7.11 Afer simplificaion, we ge ha: ɛ s = α.k. s p x+s.n x N s.µx + s.v s, X s, N s + e α.k. sp x+s 1 n x N s.µx + s.v s, X s, N s And by a developmen of aylor of he exponenial e α.k. sp x+s residue of he ype: round 0, we finally obain a ɛ s = e φ. α.k. s p x+s 2.n x N s.µx + s.v s, X s, N s Where φ belongs o he inerval [ α.k. s p x+s, 0] and clearly he residue is negaive. 8 CRRA uiliy. In his secion, preferences of he asse manager are refleced by a CRRA consan relaive risk aversion of parameer 0 < γ < 1. he opimizaion problem is only defined for posiive erminal surplus. 1 γ V, x, n = sup E X n x N.K F X A x γ Again, we maximize he uiliy of he erminal surplus on he enlarged se of conrols A x and projec he opimal wealh process in he space of aainable wealh. We ge ha: X = y.h, 1 γ 1 + n x N.K 8.1 10
he value funcion a ime is rewrien as a funcion of he opimal Lagrange muliplier: V, x, n = 1 γ γ.y γ 1.E H, γ γ 1 F And he Lagrange muliplier is such ha he budge consrain is sauraed: y 1 γ 1 1 =. x K.e E H, r..n γ x N. p x+ γ 1 F 8.2 8.3 Afer calculaions see annex 2 for deails, we obain ha: E H, γ γ 1 F = e γ r. γ 1 1 2.λ2 γ. γ 1 2. And insering 8.3 in 8.2, leads o he following expression for he value funcion: V, x, n = 1 γ. x K.e r..n x n. p x+ } {{ } equiy.e H, γ 1 γ γ 1 F 8.4 As in he CARA case, he value funcion depends on he equiy of he fund. However, if he oal asse is insufficien o cover he expeced value of fuure claims, he value funcion is no defined and he opimizaion problem doesn admi as soluion. In he nex wo subsecions, we build he opimal invesmen sraegy by decomposiion and dynamic programming. 8.1 Decomposiion of X. he Kunia Waanabe decomposiion of X requires more calculaions han for a CARA uiliy. Firsly, we develop he discouned opimal erminal wealh a ime : e r.. X = x K.e r..n x N. p x+ +K.e r..n x N And afer simplificaions, we ge ha: e r..h, 1 γ 1 E H, γ γ 1 F = exp 1 2.λ2. = d Q dq 1 γ 1 2. λ. 1 / d Q dq γ. e r..h, 1 γ 1 E H, γ γ 1 F γ 1. dw Qf u Where d Q dq defines a change of measure from he risk neural measure Q o Q, a measure under which dw Q u = dwu Qf + λ γ 1.du is a Brownian moion. he nex sep consiss o derive he condiional expecaion of he discouned erminal wealh X. his condiional expecaion is noed L s, for s : L s = E Q e r.. X F s = x K.e r..n x N. p x+. exp 1 1 2.λ2. γ 1 2.s λ. 1 s γ 1. dwu Qf +K.e r..e Q n x N F s 11
By use of he Io s lemma for jumps processes, he dynamic of L s is inferred: d Q dl s = x K.e r. λ.n x N. p x+. γ 1. dq s s p x+s.k.e r..dm s d Q dq.dw Qf s And, by formula 6.7, he opimal invesmen sraegy a ime s is: π s = λ σ. 1 x K.e r. γ 1..n x N. p x+. X s d Q dq d Q dq s 8.5 Conrary o he opimal asse allocaion obained wih a CARA uiliy, he sraegy is here funcion of he fund equiy insead of he oal asses. 8.2 Dynamic programming. For he CRRA uiliy, o obain he opimal asse allocaion by dynamic programming is relaively easier han by decomposiion. Indeed, i suffices o calculae he firs and second order derivaives of he value funcion wih respec o x. V, x, n x = 2 V, x, n x 2 = γ 1. x K.e r..n x n. p x+ γ 1.E H, γ 1 γ γ 1 F γ 2 x K.e r..n x n. p x+.e H, γ γ 1 F 1 γ he opimal asse allocaion a ime is nex obained by applicaion of he formula 6.12 and is idenical o he one calculaed by he decomposiion of X. π = ν σ 2. V X. 1 V XX X = ν σ 2. 1 x K.e r. γ 1..n x N. p x+ 8.3 Residue of he Bellman equaion. A our knowledge, he Bellman equaion corresponding o he CRRA uiliy has no explici soluion. Bu i is sill possible o check ha he residue of Bellman equaion is well negaive. Afer calculaions, he combinaion of he residue 7.11 and of he value funcion 8.4 gives ha: ɛ s = n x N s.µx + s. [ X Hs, γ γ 1 Fs 1 γ. E γ 1 x e r s.k.n x N s. s p x+s.k.e r s. s p x+s + 1 γ. x e r s.k.n x N s 1. s p x+s γ 1 γ. x e r s.k.n x N s. s p x+s γ ] 12
And by a aylor s developmen of x e r s γ.k.n x N s 1. s p x+s round he value x e r s.k.n x N s. s p x+s, we obain afer simplificaion a residue of he form: ɛ s = n x N s.µx + s. Where φ is a consan ha belongs o he inerval: φ. Hs, γ γ 1 Fs 1 γ E 2 K.e r s. s p x+s.γ 1.φ γ 1 [ x e r s.k.n x N s. s p x+s ; x e r s.k.n x N s 1. s p x+s ] I may be easily checked ha he residue is sricly negaive. 9 Conclusion. he main conribuion of his paper is o show ha he maringale mehod, widely used in he seing of complee financial markes, can be applied o he managemen of insurance producs and in paricular of endowmens. We have addressed he maximizaion of he expeced uiliy of erminal surplus, under a budge consrain, by an adaped invesmen sraegy. Our approach is based on wo assumpions. Firsly, he deflaor of he insurer is well deermined. his implies o choose a risk neural measure amongs he se of equivalen measures, which is non empy owing o he incompleeness of he insurance marke. In his work, he chosen risk neural measure is equal o he produc of he he financial risk neural measure and of he acuarial hisorical measure. Such assumpion is commonly acceped by acuaries o price insurance risks and he hedging of liabiliies relies on diversificaion. he second condiion required o apply he maringale mehod is o maximize he uiliy on a se of admissible erminal wealhs larger han he one of aainable wealhs. More precisely, his se is delimied by a budge consrain which ensures ha he curren richness is a leas equal o he deflaed erminal wealh. he main consequence of enlarging he se of conrols is ha he corresponding value funcion is no anymore soluion of he Bellman equaion. his approach is mainly moivaed by he hope of finding he opimal ALM policy coupled o he power uiliy of he surplus. Once ha he opimal erminal wealh is deermined by he mehod of Lagrange mulipliers, one projecs i in he space of processes, replicable by an adaped invesmen policy. his operaion is done eiher by Kunia Waanabe decomposiion or eiher by dynamic programming. his second possibiliy is relaively simpler han he decomposiion. I suffices indeed o calculae he firs and second order derivaives of he value funcion, wih respec o he wealh, o infer he opimal asse allocaion. Finally, resuls are presened when uiliies are exponenial CARA or power CRRA. For an exponenial uiliy, he value funcion found by maringale approach is compared wih he soluion of he Bellman equaion. hose wo approaches lead o he same opimal invesmen sraegy, which is independen of liabiliies. On he conrary, he opimal asse allocaion found by he maringale approach for a power uiliy funcion, depends on he equiy, defined as he difference beween asses and expeced discouned liabiliies. he choice of a power uiliy funcion seems herefore more adaped o ALM purposes han he exponenial. 13
Appendix 1. he calculaion of E H,. ln H, F is relaively direc. Indeed, we have ha And, by change of measure: E H,. ln H, F = E Q e r. ln H, F = e r.e Q ln H, F E Q ln H, F = E r Q + 12 λ2. λdw u F = E r Q 12 λ2. = r 12 λ2. λdw Qf u F Appendix 2. he calculaion of E Where d P dp dw P u = dw u + λ γ 1 References H, γ γ 1 F is easily performed by an adaped change of measure: E H, γ γ 1 F γ = E exp γ 1. γ = exp γ 1. r.du 1 2 r.du + 1 2. 1 γ 1. λ 2.du λ 2.du λdw u F. E d P dp d P dp F } {{ } =1 defines a change of measure from he real measure P o P, a measure under which.du is a Brownian moion. d P dp d P dp = exp 1 2 2 γ γ 1 λ.du γ γ 1 λdw u [1] J.F. Boulier, S. Huang, G. aillard 2001. Opimal managemen under sochasic ineres raes: he case of a proeced defined conribuion pension fund. Insurance: Mahemaics and Economics 28, 173-189. [2] J. Cox, C.F. Huang 1989. Opimal consumpion and porfolio policies when asse prices follow a diffusion process. Journal of economic heory, vol 49, 33-83. [3] G. Deelsra, M. Grasselli, P.F. Koehl, 2003. Opimal invesmen sraegies in he presence of a minimum guaranee. Insurance: Mahemaics and Economics, vol 33, 189-207. [4] G. Deelsra, M. Grasselli, P.F. Koehl, 2004. Opimal design of he guaranee for defined conribuion funds. Journal of economic dynamics and conrol, vol 28, 2239-2260.J. 14
[5] D. Duffie, 2001. Dynamic asse pricing heory. hird ediion. Princeon Universiy Press. [6] W. Fleming, R. Rishel, 1975. Deerminisic and Sochasic opimal conrol. Springer. [7] H. Föllmer, D. Sondermann 1986. Hedging of non redundan coningen claims. In Hildebrand W. Mas-Colell A. eds, Conribuions o mahemaical economics in honor of G. Debreux. Norh-Holland, Amserdam. [8] D. Hainau, P. Devolder 2007. Managemen of a pension fund under sochasic moraliy and ineres raes. Insurance: Mahemaics and economics, Vol 41, 134-155. [9] I. Karazas, S. Shreve, 1998. Mehods of mahemaical finance. Springer. [10] H. Kunia S. Waanabe, 1967. On square inegrable maringales. Nagoya Mah. J. Vol 30, 209-245. [11] R. Meron 1969. Lifeime porfolio selecion under uncerainy: he coninuous-ime case. Review of Economics and Saisics, Vol 51 247-257. [12] R. Meron 1971. Opimum consumpion and porfolio rules in a coninuous-ime model. Journal of Economic heory, Vol 3 373-413. [13] F. Menoncin, O. Scaille, P. Baochio, 2004. Opimal asse allocaion for pension funds under moraliy risk during he accumulaion and decumulaion phases. Forhcoming in Annals of operaions research. [14]. Møller,1998. Risk minimizing hedging sraegies for uni-linked life insurance conracs. ASIN Bullein, Vol 28-1, 17-47. [15] B. Øksendal, A. Sulem, 2005. Applied sochasic conrol of jump diffusions. Springer. [16] V.R. Young. Zariphopoulou, 2002. Pricing dynamic insurance risks using he principle of equivalen uiliy. Scandinavian Acuarial Journal, Vol 4 246-279. I graefully acknowledge he financial suppor of he Communaué française de Belgique under he Proje d Acion de Recherches Concerées. I also hanks my colleague Jérôme Barbarin for his consrucive commens. 15