MASAHIKO EGAMI AND HIDEKI IWAKI

Transcription

1 AN OPTIMAL LIFE INSURANCE PURCHASE IN THE INVESTMENT-CONSUMPTION PROBLEM IN AN INCOMPLETE MARKET MASAHIKO EGAMI AND HIDEKI IWAKI Absrac. This paper considers an opimal life insurance purchase for a household subjec o moraliy risk. The household receives wage income coninuously, which could be erminaed by unexpeced premaure loss of earning power. In order o hedge he risk of losing income sream, he household eners a life insurance conrac for he beref members. The household may also inves heir wealh ino a financial marke. If insurance paymen is made prior o he planned ime horizon, he amoun shall be used for consumpion and invesmen. Therefore, he problem is o deermine an opimal insurance/invesmen/consumpion sraegy in order o maximize he expeced oal discouned uiliy from consumpion and erminal wealh. To reflec a real-life siuaion beer, we consider an incomplee marke where he household canno rade insurance conracs coninuously. We provide explici soluions in a fairly general seup. 2 AMS Subsec Classificaion: Primary 91B28, Secondary 91B3 JEL Classificaion: C61, D91, G11, G22 Key words: Life Insurance, Invesmen/Consumpion Model, Maringale, Convex dualiy 1. Inroducion We consider a household whose income sream relies on one paricular member of he family. The household has an incenive o buy a life insurance conrac o miigae moraliy risk of he wage earner. The invesmen ime horizon of he household is [, T where T denoes he planned reiremen ime of ha person. Tha is, he household expecs o receive wage income a rae y() coninuously unil ime T, which could be erminaed before ime T by some unexpeced loss of earning power (e.g. deah). Accordingly, i is naural o assume ha he household buys an insurance ha erminaes a ime T. In oher words, he insurance coverage is effecive unil and upon ime T. The moraliy risk is modeled by a firs arrival of a cerain Poisson process N = {N(); } wih inensiy process λ = {λ(); }. We denoe he random ime of ha even by τ. The household buys n shares of an insurance policy by paying a lump-sum premium of n p a ime. We assume he premium per share, p is deermined exogenously and n is one of he decision variables. The insurance company pays insurance amoun X per share ha depends on he ime of Poisson arrival τ. Therefore, if he household purchases n shares, he paymen a τ T is n X(τ). Firs version: January 1, 28. This version: June 24, 29. M. Egami: Graduae School of Economics, Kyoo Universiy, Sakyo-Ku, Kyoo, , Japan. egami@econ.kyoou.ac.jp. H. Iwaki: Graduae School of Managemen, Kyoo Universiy, Sakyo-ku, Kyoo, , Japan. iwaki@gsm.kyoo-u.ac.jp. M. Egami is suppored in par by Gran-in-Aid for Scienific Research (C) No , Japan Sociey for he Promoion of Science. 1

2 2 MASAHIKO EGAMI AND HIDEKI IWAKI Given he iniial endowmen a ime, he household decides on he number of insurance conracs n and invess he res of he money available ino he financial marke. In he case of τ T, he household receives insurance money nx(τ) and shall use he money for consumpion and/or addiional invesmen in he financial marke. On he oher hand, if τ > T, he insurance conrac erminaes and he insured person reires a T. The household ries o maximizes is uiliy for he enire ime horizon [, T. See he nex secion for complee mahemaical formulaion. I should be emphasized here ha he decision maker in our problem is he household (i.e. he whole family), no he insured person. Hence afer ime τ, consumpion sill coninues. Alhough he financial marke (excluding insurance conracs) is assumed o be complee, he household s inabiliy o rade insurance conracs makes he whole model incomplee. We show, by using he convex dualiy mehod, ha if a cerain n solves he dual problem, ha n also solves he original uiliy maximizaion problem. We hen explicily compue he corresponding consumpion and wealh processes. Moreover, provided ha insurance benefi is a linear funcion of n, we find ha for many popular uiliy funcions, he household shall inves all he iniial endowmen eiher in he insurance conrac or in he financial marke, depending on he relaionship beween he insurance premium and he expeced discoun value of insurance benefi. This is naural because he financial marke excluding insurance conracs is complee, if hese wo quaniies are no equal, he household akes a full advanage of possible mispricing in he insurance conrac. On he oher hand, if hese quaniies are equal, he household does no have a clue as o how i should deermine he opimal number of insurance conracs. In ha case, we provide he opimal porfolio sraegy for each n. Hence his paper analyzes he household s opimal behavior when he household faces he insurance conrac and he financial marke Lieraure review. We briefly discuss his paper s posiion in he exising lieraure. While his paper considers insurance paymen (a τ) as a source of income o he beref family members for he res of he ime horizon, previous reamens of insurance in he lieraure are in essence from insurers poin of view. The main purpose is o calculae insurance premium of various conracs whose paymen is exogenously given. For example, Albizzai and Geman [1 and Persson and Aase [21 examined opion-like feaures conained in he insurance conacs. The fair premium of an equiy-linked life insurance conrac is calculaed in Brennan and Schwarz [6 and Nielsen and Sandman [2, while Marceau and Gaillardez [17 calculaed he reserves in a sochasic moraliy and ineres rae model. See Bacinello [3. See also Iwaki e al. [14 and Iwaki [13 for deermining insurance premia in a muli-period economy and in a coninuous-ime economy, respecively. In conras, his paper discusses an opimal insurance purchase from he sandpoin of households. The problem reaed in his paper can be seen as an exension of he securiy allocaion problem originally sudied by Meron [18, 19. In his model, only a riskless securiy and a risky securiy are considered and he problem is o obain an opimal porfolio rule so as o maximize he expeced uiliy from consumpion. Since hen, he model has been exended o various direcions. Richard [22 included life insurance decisions in he Moron model. He assumed a specific diffusion for he risky asse and a complee marke where he invesor can rade life insurance conracs coninuously. The invesor in [22 maximizes he uiliy unil uncerain ime of deah. Campbell [7 used a discree-ime model o derive he demand

3 OPTIMAL LIFE INSURANCE PURCHASE 3 funcion for life insurance. Babbel and Ohsuka [2 exended o a muli-period model bu did no include risky asses in he asse porfolio. Zhu [28, in one-period model, performed a comprehensive sudy of he insurance-invesmen-consumpion problem and analyzed effecs of parameers on individuals insurance purchase, consumpion, and sock invesmen decisions by using wo differen individual groups: one wih exponenial uiliy and he oher wih power uiliy. Also, Bodie e al. [5 sudied a life ime model in which a human capial is considered, as in [22, o represen he presen value of he oal wage income o be obained in he fuure. By including he human capial in heir securiy allocaion model, hey succeeded in explaining he relaionship beween he age of an economic agen and his/her opimal invesmen sraegy. See also He and Pagès [12, Svensson and Werner [24, and Karazas and Shreve [15 as examples of such exensions. Our curren aricle conrass wih hese papers in ha we use a coninuous-ime framework wih general uiliy funcions and general underlying diffusions. Moreover, in order o make he model more realisic, we assume ha he household canno rade insurance conracs (unlike [22) and incorporae he fac ha he beref family would use he money from he insurance conrac o coninue heir consumpion unil he fixed ime T. I should be noed ha Cvianic e al. [1 showed he exisence of soluions for he invesmen-consumpion problem wih a random endowmen in a general semimaringale model. Our curren paper, hough, has disinc meris in he sense ha we obain an explici soluion (which does no follow from he said paper) for a fairly general uiliy funcion and hence make economic implicaions much clearer. Moreover, while he random endowmen in [1 is given exogenously, our random endowmen (i.e., insurance money) here can be conrolled by changing he number of shares of he insurance conrac. This paper is organized as follows. In he nex secion, we formulae our problem in a rigorous manner. We solve he problem in he following secion and discuss possible exensions afer we presen our main resuls. We defer he deailed proofs o Appendix. Throughou his paper, all he random variables considered are bounded almos surely (a.s.) o avoid unnecessary echnical difficulies. Equaliies and inequaliies for random variables hold in he sense of almos surely. 2. The Model Le us consider a complee filered probabiliy space (Ω, F, (F ) R+, P) ha hoss a Brownian moion B := {B() :, B() = } and a Poisson Process N := {N() : >, N() = }. Le F B := σ{b(s); s }, [, T. We denoe he P-augmenaion of filraion by F B := {F B ; [, T }. The Brownian moion is he source of randomness oher han he ime τ: τ := inf{ > ; N() = 1}, which denoes he ime of he insured person s loss of earning power (e.g. deah). We assume ha he inensiy process λ := {λ(); } of he Poisson process N is predicable wih respec o F B. Le F N := σ{1 {τ s} ; s } where 1 E denoes he indicaor funcion of even E F meaning ha 1 E = 1 if E is rue and 1 E = oherwise. The P-augmenaion of he filraion is denoed by F N := {F N ; [, T }. Clearly, τ is an F N -sopping ime, bu no an F B -sopping ime. Now, le F := F B F N, [, T and is P-augmenaions F := {F ; [, T }. I is assumed ha F saisfies he usual condiions regarding

4 4 MASAHIKO EGAMI AND HIDEKI IWAKI righ-coninuiy and compleeness. The condiional expecaion operaor given F is denoed by E wih E = E. Suppose ha he curren ime is, and le T > be he erminaion ime of an insurance conrac which is se o be he same as he reiremen ime. We consider a coninuous-ime economy in [, T ha consiss of he insurance conrac and a financial marke. The financial marke is assumed o be fricionless and perfecly compeiive. 1 The household may receive cash flow from various sources of income. Bu for simpliciy, we assume ha i relies on one member s income sream: y = {y(); [, T } (called income process hereafer) which is given exogenously unil ime T. To hedge he risk of loss of income flow a ime τ < T, he household buys an insurance policy described as follows: Once he household buys n shares of he policy by paying he insurance premium amouns p n a ime, he insurance company makes an insurance paymen in he amoun of (2.1) n X() = n (1 + H()) a ime = τ T. Here H : [, T R + is given exogenously, represening paymen schedule unil ime T. In case τ > T, he policy pays 1 dollar per share a ime T. In order o avoid unnecessary complicaions, we assume ha he schedule funcion saisfies he following assumpion. Assumpion 2.1. H : [, T R + is a nonincreasing coninuous funcion wih H(T ) =. Here we noe ha H(T ) = means ha he insurance amoun when τ occurs a ime T and he guaraneed insurance amoun (which is uniy) on he se {τ(ω) > T } coincide. Le c = {c(); [, T } be he consumpion process o be deermined by he household. I is assumed ha income and consumpion processes are adaped o F. In he financial marke, here is a riskless securiy whose ime price is denoed by S (). The riskless securiy evolves according o he differenial equaion; ds () = r()d, [, T, S () where r() is a posiive, predicable process wih respec o F B. The household can also inves heir wealh ino a risky securiy whose ime price is denoed by S 1 (). The risky securiy evolves according o he sochasic differenial equaion (abbreviaed SDE); (2.2) ds 1 () S 1 () = µ()d + σ()db(), [, T, where µ() and σ() are progressively measurable processes wih respec o F B. Le π() be he amoun o be invesed ino he risky securiy a ime. The process π = {π(); [, T } is referred o as a porfolio process. Now, given a porfolio process π, a consumpion process c, he number 1 A financial marke is said o be fricionless if he marke has no ransacion coss, no axes, and no resricions on shor sales (such as margin requiremens), and asse shares are divisible, while i is called perfecly compeiive if each agen believes ha he/she can buy and sell as many asses as desired wihou changing he marke price.

5 OPTIMAL LIFE INSURANCE PURCHASE 5 of shares of he insurance policy n and an income process y, he wealh process W = {W (); [, T } is defined by W np + (r(s)w (s) + y(s) c(s))ds + π(s)[(µ(s) r(s))ds + σ(s)db(s) if [, τ T ), (2.3) W () := W (τ ) + nx(τ) + τ (r(s)w (s) c(s)) ds + τ π(s)[(µ(s) r(s))ds + σ(s)db(s) if [τ T, T, where W is a given iniial wealh which is assumed o be a posiive consan. In his paper, we assume ha, given he inensiy process λ, he condiional survival probabiliy of τ is given by (2.4) P{τ > λ} = exp { Tha is, he inensiy process λ plays he role of he hazard rae, } λ(u)du, [, T. 1 λ() = lim P{ < τ + τ >, F B }. A Poisson process N driven by ha (sochasic) inensiy process is called a Cox process, which is also known as a doubly sochasic Poisson process. See, for example, Grandell [11 for deails. In his case, we have P{τ > F B } = P{τ > FT B } for T. Noe ha, in his seing, he infiniesimal incremens db() and dn() are condiionally independen given F B. Also, he process M λ = {M λ (), [, T } defined by (2.5) M λ () := is an F-maringale (i.e. [26)). 1 {N(s )=} [dn(s) λ(s)ds he inegral 1 {N(s )=}λ(s)ds is he F-compensaor (see Yashin and Arjas Definiion 2.1. A consumpion and wealh pair (c, W ) is called feasible if c(), W () > for [, T, W (T ) and i saisfies (2.3). We denoe a class of feasible pairs (c, W ) by C. Recall ha he household consumes he wage income and, if any, insurance money o maximize he expeced discouned uiliy from consumpion c and erminal wealh W (T ). Le U 1 : (, ) R be he uiliy funcion of he household from consumpion, and le U 2 : (, ) R be he uiliy funcion of he household from he erminal wealh. Assumpion 2.2. We assume ha our uiliy funcions saisfy he following: (1) U i (i = 1, 2) are sricly increasing, sricly concave and wice coninuously differeniable wih properies U i( ) := lim x U i(x) =, U i(+) := lim x U i(x) =, i = 1, 2. (2) For any c (, ) here exis real numbers a (, ) and b (, ) saisfying au i (c) U i (bc).

6 6 MASAHIKO EGAMI AND HIDEKI IWAKI Also, in order o represen ime-preference of he household, we inroduce a ime-discoun facor e ρ(s)ds, [, T, where he process ρ = {ρ(), [, T } is adaped o F. A naural problem for he household is as follows: Given he iniial wealh W, he household decides how many insurance conracs o buy a ime zero o proec from he risk of he Poisson even. The res of he money W np can be invesed in he financial marke. If τ T, he household receives he insurance money nx(τ) as in (2.1) and re-solves he opimal invesmen-consumpion problem (2.7) by using he sum of he wealh a τ, W (τ ) and he insurance money nx(τ) as he iniial wealh a τ. On he oher hand, if τ > T, he problem reduces o an ordinary invesmen-consumpion problem from ime zero o T. By keeping hese possibiliies in mind, he household decides on he number of insurance conrac n a ime zero along wih he opimal consumpion-invesmen pair o maximize he overall uiliy. Mahemaically, i is saed as follows: (MP): Given he discoun process ρ and uiliy funcions U i (x), i = 1, 2, find an opimal riples consising of consumpion process, porfolio process and he number of shares of he insurance policy (ĉ, ŵ, ˆn) o solve he following maximizaion problem: (2.6) max E e ρ(s)ds U 1 (c())d + e ρ(s)ds V (W (τ T )) wih [ T (2.7) V (W (τ T )) := max E e ρ(s)ds U 1 (c())d + e T ρ(s)ds U 2 (W (T )) F where he maximum is aken over he feasible consumpion and wealh pairs, (c, W ) C under he budge consrain (2.3). In he nex secion, we shall solve he problem (MP) by applying he maringale approach in an incomplee marke (see Karazas and Shreve [15 and Kramkov and Schachermayer [16). 3. Main Resuls In order o apply he maringale approach, we need o specify a sae price densiy process firs. Le P be a class of posiive and predicable sochasic processes; { } P = ψ(); ψ()d <, [, τ T. For each ψ = {ψ(); [, τ T } P, he sae price densiy process is given by (3.1) χ() := β()χ B ()χ N (), where (3.2) β() := exp (3.3) χ N () := { } r(s)ds, ( ) ψ(τ) λ(τ) 1 τ {τ } + 1 {τ>} e (λ(s) ψ(s))ds,,

7 and (3.4) χ B () := exp wih (3.5) ξ() := OPTIMAL LIFE INSURANCE PURCHASE 7 { ξ(s)db(s) 1 2 µ() r(), [, T. σ() } ξ 2 (s)ds, Noe ha (3.1) and (3.3) say ha he sae price densiy process χ is deermined once he inensiy process ψ is specified. Here and hereafer, we denoe he condiional expecaion operaor given F under he equivalen maringale measure Q by E Q wih E Q = E Q. Remark 3.1. The following facs are well known. See for example Bellamy and Jeanblanc [4 and Yashin and Arjas [26. (1) The sae price densiy process χ = {χ(); [, T } is such a process ha χ() = 1, < χ() <, and for each [, T and for any s >, s [, T, (3.6) E [χ(s)s j (s) = χ()s j (), j =, 1, i.e. each process {χ()s j (), [, T }, j =, 1, is a maringale under P. (2) The equivalen maringale measure Q is given by dq dp = χ(t ) β(t ), (3) The process ψ represens he inensiy process under he equivalen maringale measure Q. We solve he problem (MP) in wo seps. Firs, for a given n R +, we solve he problem by applying he maringale mehods for opimal porfolio selecion problems in incomplee marke. Second, we derive he value of n which maximizes he value funcion ha is derived in he firs sep. In he following, in order o make he dependence on ψ P explici, we denoe he sae price densiy by χ ψ () = β()χ B ()χ N ψ (), [, T, where (3.7) χ N ψ () = ( ψ(τ) λ(τ) 1 {τ } + 1 {τ>} ) e τ (ψ(s) λ(s))ds, See also (3.3). Similarly, Q ψ denoes he equivalen maringale measure associaed wih he sae price densiy χ ψ, ha is given by dq ψ /dp = χ ψ (T )/β(t ). For a given consumpion and wealh pair, (c, W ), he nex resul provides a necessary condiion regarding is feasibiliy in he marke. Lemma 3.1. If a consumpion and wealh pair (c, W ) is in C (as in Definiion 2.1), hen i saisfies he following inequaliies. [ E Q ψ β() (c() y()) d + β(τ T )(W (τ T ) nx(τ T )) W np, (3.8) [ E Q ψ T β()c()d + β(t )W (T ) β(τ T )W (τ T ) for each ψ P.

8 8 MASAHIKO EGAMI AND HIDEKI IWAKI Proof. For any ψ P, suppose ha (c, W ) is in C. Then, from (2.3) and Iô s formula, we obain (3.9) W np + β()w () = β(s)π(s)σ(s)d B if [, τ T ), β(τ)(w (τ ) + nx(τ)) τ β(s)c(s)ds + τ β(s)π(s)σ(s)d B if [τ T, T, where B() := B() + ξ(s)ds is a sandard Brownian moion under Q ψ for all ψ P. Now, on he se {τ < T }, in he firs equaion, se = τ and noe ha W (τ) = W (τ ) + nx(τ) and in he second equaion, se = T. Then we obain (3.8) afer aking he expecaion for his se {τ < T }. Similarly, on he se {τ T }, in he firs equaion, we se = T and noe ha W (T ) = W (T ) + nx(t ) (recall Assumpion 2.1). In he second equaion, we se = τ = T (recall Assumpion 2.1 again). On his se we also obain (3.8) afer aking he condiional expecaion. For each uiliy funcion U i (x), i = 1, 2 and each (s, ) such ha s [, T and [s, T, we denoe by I s (i) d (x, ) he inverse funcion of dx [U i (x)e s ρ(s)ds wih respec o x. Similarly, for he funcion d V (x) defined in (2.7), we denoe by J(x) he inverse funcion of dx [V (x)e ρ(s)ds wih respec o x. Under Assumpion 2.2, for each s,, he funcions I s (i) (x, ) (i = 1, 2) and J(x) exis, are coninuous and sricly decreasing, and map (, ) ono iself. For each s,, and i, we define he Legendre ransformaion ũ s (z, ) and Ṽ by (3.1) (3.11) s (z, ) = sup [e s ρ(u)du U i (c) zc, [, T, i = 1, 2, c Ṽ (z) = sup [e ρ(s)ds V (w) zw. w ũ (i) Then we can be readily shown ha I (i) s (3.12) (3.13) (+, ) =, I (i) (, ) =, J(+, ) =, J(, ) =, and ũ (i) s (z, ) = e s ρ(u)du U i (I s (i) (z, )) zi s (i) (z, ), [, T, i = 1, 2, Ṽ (z) = e ρ(s)ds V (J(z)) zj(z). Now, in order o solve he problem (MP), we consider he following dual opimizaion problem: s (DP) max min n R + (ζ (n),ψ (n) ) R ++ P V (ζ (n), ψ (n)), where (3.14) V (ζ, ψ) = E + ζ ũ (1) (ζχ ψ(), )d + Ṽ (ζχ ψ(τ T )) ( W () np + The household s opimal consumpion/wealh process is given nex: Proposiion 3.1. (3.15) w = E [ T ) χ ψ ()y()d + χ ψ (τ T )nx(τ T ). For a given w >, Le Z(w) be a soluion of he equaion; χ(τ T, )I (1) (Z(w)χ(τ T, ))d + χ(τ T, T )I(2) (Z(w)χ(τ T, T ))

9 OPTIMAL LIFE INSURANCE PURCHASE 9 where, by recalling (3.1), χ(τ T, ) = β()χ B () β(τ T )χ B, [τ T, T. (τ T ) Suppose ha Assumpions 2.1 and 2.2 hold. Le n be a soluion o (DP) saisfying and E [ T E Q ψ χ(τ T, )I (1) (Z(J(ζ χ ψ (τ T )))χ(τ T, ), )d +χ(τ T, T )I (2) (Z(J(ζ χ ψ (τ T )))χ(τ T, T ), T ) < β()i (1) (ζ χ ψ (), )d + β(τ T )J(ζ χ ψ (τ T )) <, where (ζ, ψ ) := argminv (ζ (n ), ψ (n ) ). Then, n agrees wih an opimal share ˆn of he insurance policy in (MP) and an opimal consumpion process ĉ and he corresponding wealh process Ŵ are given, respecively, by I (1) (ζ χ ψ (), ) [, τ T ), (3.16) ĉ() = I (1) (Z(J(ζ χ ψ ()))χ(τ T, ), ) [τ T, T, and (3.17) [ 1 β() Ŵ () = EQ ψ β(s) (ĉ(s) y(s)) ds + β(τ T )(Ŵ (τ T ) n X(τ T )) if [, τ T ), [ 1 T β(s)ĉ(s)ds + β(t )Ŵ (T ) if [τ T, T, β() EQ ψ wih Ŵ (τ T ) = J (ζ χ ψ (τ T )). Furhermore, ζ saisfies (3.18) E Q ψ = W n p + E Q ψ β()i (1) (ζ χ ψ (), ) d + β(τ T )Ŵ (τ T ) β()y()d + n β(τ T )X(τ T ). Proposiion 3.2. In addiion o Assumpions 2.1 and 2.2, suppose ha he funcions I (1) (, ), [, T, and J( ) are convex. Then, he opimal soluion (ζ, ψ ) of he problem (DP) is given by (ζ, λ). Especially, ψ is uniquely given by λ. All he proofs are given in Appendix. In summary, we have obained he following: If we assume ha I (1) and J are convex and ha n solves he dual problem (DP) along wih (ζ, ψ ), hen ζ is represened by (3.18) and ψ maches he original inensiy process λ. Moreover, n is also he soluion o he primal problem (MP), ogeher wih corresponding pair (ĉ, Ŵ ) ha is given by (3.16) and (3.17), respecively. Noe he addiional convexiy assumpion on I (1) (, ), [, T, and J( ) is saisfied by popular uiliy funcions including exponenial, logarihmic, power, ec. Now in he nex proposiion, we will show how he opimal ˆn(= n ) looks like.

10 1 MASAHIKO EGAMI AND HIDEKI IWAKI Proposiion 3.3. Suppose ha he assumpions of Proposiion 3.2 hold. The opimal share ˆn is given as follows. If E[χ λ (τ T )X(τ T ) > p hen ˆn = W /p. Oherwise, if E[χ λ (τ T )X(τ T ) < p, hen ˆn =, oherwise ˆn is indefinie. This proposiion saes ha if he money received from he insurance conrac a ime τ is linear in n, he household eiher spends all of he iniial endowmen in he insurance conrac or buys no insurance, depending on E[χ λ (τ T )X(τ T ) p. Tha is, depending on wheher he discouned expeced benefi from he insurance conrac is greaer or less han he insurance premium. This makes sense since, for a uiliy funcion ha saisfies he convexiy assumpion of I (1) (, ), [, T, and J( ), he household ries o ake a full advanage of possible mispricing in he insurance conrac. As poined above, he convexiy assumpion is saisfied by popular uiliy funcions, his resul is quie general, provided ha he household includes he insurance conrac in heir consumpion-invesmen problem and aemps o maximize heir uiliy on ha basis. On he oher hand, if he insurance premium is priced in such a way E[χ λ (τ T )X(τ T ) = p (for example, based on he law of large numbers (LLN)), he household does no have a clue as o how i should deermine an opimal ˆn. In his case, he following resul can be of heir help: Proposiion 3.4. Suppose ha he assumpions of Proposiion 3.2 hold. Provided ha all parameers r, µ, σ and λ are deerminisic funcion of ime [, T, hen he opimal porfolio process ˆπ = {ˆπ(); [, T } is explicily given by he following equaions. (3.19) ˆπ() = ζ χ()e ξ() χ(, u) 2 I (1) (ζ χ(u))du + χ(, τ T ) 2 J (ζ χ(τ T )) σ(). Noe ha dependence on n is implici hrough (3.18). The household can decide on he number of insurance conrac, for example, by he absolue amoun hey would hink necessary afer he unexpeced loss of he insured person. Given ha n, hey hold on o he opimal consumpion (3.16) and porfolio process (3.19) o maximize heir uiliy. Our framework can be discussed in oher seings. Firs, he insurance benefi was compued as a linear funcion of n bu we could assume a more general (non-linear) formula han in (2.1). In his case, he opimal number of insurance conrac can be an inerior poin beween (, W /p ). Secondly, our insurance premium p here is given exogenously and Proposiion 3.3 discusses he relaionship beween p and he expeced paymen a τ whose form is basically LLN-based pricing. In his conex, here exiss an exceedingly increasing lieraure abou pricing under disored probabiliies or Choque pricing (see, for example, Wang e al. [25 and Young and Zariphopoulou [27). In conjuncion wih hese new pricing schemes our problem can be exended o an equilibrium analysis beween he household and he insurer.

11 OPTIMAL LIFE INSURANCE PURCHASE 11 Appendix A. Proof of Theorem 3.1. The proof here adaps he argumens in Cvianić and Karazas [9 and Cuoco [8. Assume ha (ζ, ψ ) solves (DP) along wih n, and ha (A.1) J(ζ χ ψ (τ T )) = E [ T χ(τ T, )I (1) (Z(J(ζ χ ψ (τ T )))χ(τ T, ), )d +χ(τ T, T )I (2) (Z(J(ζ χ ψ (τ T )))χ(τ T, T ), T ) E χ ψ ()I (1) (ζ χ ψ (), )d + χ ψ (τ T )J(ζ χ ψ (τ T )) < <, holds. In order o prove ha (ĉ, Ŵ (T )) in (3.16) and (3.17) is opimal, we will proceed in wo seps; firs we will show ha (A.2) E E e ρ(s)ds U 1 (ĉ())d + e ρ()d V (Ŵ (τ T )) e ρ(s)ds U 1 (c())d + e ρ()d V (W (τ T )) and (A.3) V (Ŵ (τ T )) = E [ T E [ T e ρ(s)ds U 1 (ĉ())d + e T ρ()d U 2 (Ŵ (T )) e ρ(s)ds U 1 (c())d + e T ρ()d U 2 (W (T )) hold for all (c, W (T )) C, and hen ha (ĉ, Ŵ (T )) C. Sep 1 (opimaliy). By Assumpion 2.1-(b), here exiss a, b (, ) such ha for each i = 1, 2, au i(i s (i) (z, )) U i(bi s (i) (z, )) [s, T, s [, T. Applying I s (i) (, ) o boh sides and ieraing, show ha for all a (, ) here exiss a b (, ) such ha Hence (A.1) implies (A.4) for all ζ i (, ), i = 1, 2. I s (i) (az, ) bi s (i) (z, ), (z, ) (, ) [s, T, s [, T. E χ ψ ()I (1) (ζ 1χ ψ (), )d + χ ψ (τ T )J(ζ 2 χ ψ (τ T )) <

12 12 MASAHIKO EGAMI AND HIDEKI IWAKI By he opimaliy of ζ, we have (A.5) V (ζ + ɛ, ψ ) V (ζ, ψ ) = lim ɛ ɛ = E lim ɛ ũ 1 ((ζ + ɛ)χ ψ (), ) ũ 1 (ζ χ ψ (), ) d ɛ Ṽ ((ζ + ɛ)χ ψ (τ T ), τ T ) + lim Ṽ (ζ χ ψ (τ T ), τ T ) ɛ ɛ +W np + χ ψ ()y()d + nχ(τ T )X(τ T ) = W np E χ ψ ()(ĉ() y())d + χ ψ (τ T )(Ŵ (τ T ) nx(τ T )), where he second equaliy follows he dominaed convergence heorem, using (A.4) and he fac ha ũ (1) ((ζ + ɛ)χ ψ (), ) ũ (1) (ζ χ ψ (), ) ɛ ũ(1) ((ζ ɛ )χ ψ (), ) ũ (1) (ζ χ ψ (), ) ɛ χ ψ ()I (1) ((ζ ɛ )χ ψ (), ) ( ) χ ψ ()I (1) ζ 2 χ ψ (),, and ha Ṽ ((ζ + ɛ)χ ψ (), τ T ) Ṽ (ζ χ ψ (τ T ), τ T ) ɛ Ṽ ((ζ ɛ )χ ψ (τ T ), τ T ) ũ i (ζ χ ψ (τ T ), τ T ) ɛ χ ψ ()J((ζ ɛ )χ ψ (τ T ), τ T ) ( ) ζ χ ψ ()J 2 χ ψ (τ T ), τ T, ɛ < ζ 2, (because boh ũ (i) (, ) and Ṽ (, τ T ) are decreasing and convex, z ũ(1) (z, ) = I(i) (z, ), Ṽ (z, τ T ) = J(z, τ T ), z and boh I (1) (, ) and J(, τ T ) are decreasing). Since by concaviy e ρ(s)ds U 1 (I (1) (z, )) e ρ(s)ds U 1 (c) z[i (1) (z, ) c e ρ(s)ds V (J(z)) e ρ(s)ds V (c) z[j(z) c c >, z >,

13 OPTIMAL LIFE INSURANCE PURCHASE 13 i hen, by evaluaing he previous inequaliy a z = ζ χ ψ () and using he definiion of ĉ and Ŵ (τ T ) in (3.16) and (3.17), follows from (3.8) and (A.5) E ζ E E e ρ(s)ds U 1 (ĉ())d + e ρ()d V (Ŵ (τ T )) e ρ(s)ds U 1 (c())d + e ρ()d V (W (τ T )) χ ψ ()(ĉ() c())d + χ ψ (τ T )(Ŵ (τ T ) W (τ T )). Furhermore, we can easily confirm ha (A.3) holds since (3.12), (3.15) and ha Ŵ ( ) = J (ζ χ ψ (τ T )). Hence, (ĉ, Ŵ (τ T )) mus be opimal provided i is in C. Sep 2 (feasibiliy). We are only lef o show ha here exiss an a admissible porfolio process ˆπ financing (ĉ, Ŵ (T )). For any n R +, define a process W = { W (); [, τ T } by where W ( ) + nx() if = τ T, W () := W ( ) if (, τ T ), W ( ) := χ ψ ( ) 1 E χ ψ (s)(ĉ(s) y(s))ds + χ ψ (τ T )(Ŵ (τ T ) nx(τ T )) = β() 1 E Q ψ β(s)(ĉ(s) y(s))ds + β(τ T )(Ŵ (τ T ) nx(τ T )), Clearly, W (τ T ) = Ŵ (τ T ), and W is bounded below (because of boundedness of y and X by he ground assumpion). Also, i follows from he maringale represenaion heorem ha here exiss a process {(θ 1 (), θ 2 ()); [, τ T } wih θ1 () 2 + θ 2 () d < such ha, (A.6) β() W ( ) + = W () + β(s)(ĉ(s) y(s))ds θ 1 (s)d B(s) + θ 2 (s)(dn(s) ψ (s)ds), (, τ T. From (A.6) and he definiion of W, if (, τ T ), (A.7) d(β() W ()) = β()(ĉ() y())d + θ 1 ()d B() + θ 2 ()(dn() ψ ()d). Define a porfolio process π = {π(); [, τ T } by (A.8) π() = θ 1() β()σ().

14 14 MASAHIKO EGAMI AND HIDEKI IWAKI Using (A.7) and Iô s lemma shows ha W () = W () + + = W () (r(s) W (s) + y(s) ĉ(s))ds β(s) 1 θ 1 (s)d B(s) + (r(s) W (s) + y(s) ĉ(s))ds β(s) 1 θ 2 (s)(dn(s) ψ (s)ds) + nx()1 {= } π(s)[(µ(s) r(s))ds + σ(s)db(s) + nx()1 {= } β(s) 1 θ 2 (s)(dn(s) ψ (s)ds). A comparison wih (2.3) hen reveals ha only if we verify ha θ 2 () = for all [, τ T wih Ŵ () = W (), he proof will be compleed. For his purpose, fix an arbirary ψ = {ψ(); [, τ T } P and define sochasic processes κ and κ by, respecively, (A.9) κ() := and κ() := ψ (s) ψ(s) ψ (dn(s) ψ (s)ds), (s) (ψ (s) ψ(s)) ds, [, τ T, as well as he sequence of sopping imes { τ m := inf [, τ T : κ() + κ() + ψ } () ψ() ψ () m, m = 1, 2,. Then τ m τ T. Also, leing ψ ɛ,m () := ψ () + ɛ[ψ() ψ ()1 { τm} for ɛ (, 1 ) ml wih some l > 1. Clearly ψ ɛ,m ()d < a.s. Hence ψ ɛ,m = {ψ ɛ,m (); [, τ T } belongs o P. Therefore, as in (3.7), we can define χ ψɛ,m and consider where ɛζ Y ɛ m := 1 ɛζ [V (ζ, ψ ɛ,m ) V (ζ, ψ ) = E[Y ɛ m, [ũ (1) (ζ χ ψɛ,m (), ) ũ (1) (ζ χ ψ (), )d + Ṽ (ζ χ ψɛ,m (τ T )) Ṽ (ζ χ ψ (τ T )) + ζ [χ ψɛ,m () χ ψ ()y()d + ζ [χ ψɛ,m (τ T ) χ ψ (τ T )nx(τ T ). Inroduce he raio R ɛ () := χ { ψ ɛ,m () χ ψ () = exp ( ψψɛ,m (s) ln ψ ψ (s) ) dn(s) = e ɛ κ( τm) (1 ɛ(κ( τ m ) + κ( τ m ))), [, τ T. } (ψ ɛ,m (s) ψ (s))ds

15 We have hen < e 1 l OPTIMAL LIFE INSURANCE PURCHASE 15 ( 1 1 ) ( e ɛm (1 ɛm) R ɛ () e ɛm (1 + ɛm) e 1 l ), l l as well as he upper bounds for he random variable Y ɛ m: Ym ɛ Q m := χ ψ () 1 Rɛ () I (1) ɛ (ζ e sgn()ɛm (1 sgn()ɛm)χ ψ (), )d +χ ψ (τ T ) 1 Rɛ (τ T ) J(ζ e sgn( )ɛm (1 sgn(τ T )ɛm)χ ψ (τ T )) ɛ Y ɛ m Y m := sup ɛ (,1/ml) χ ψ () 1 Rɛ () ɛ 1 e ɛm (1 ɛm) ɛ ( ζ e 1 l χ ψ ()I (1) y()d χ ψ (τ T ) 1 Rɛ (τ T ) nx(τ T ), ɛ ( 1 1 ) ) ( ( χ ψ (), d + χ ψ (T )J ζ e 1 l 1 1 ) ) χ ψ (τ T ) l l + χ ψ ()y()d + χ ψ (τ T )nx(τ T ) where sgn() = 1 if 1 R ɛ () and 1 if 1 R ɛ () <. We have used he mean-value heorem applying o I (1) (y, ) = y ũ(1) (y, ), J(y) = y Ṽ (y) wih end poins χ ɛ,n() and χ ψ () and he fac boh I (1) (y, ) and J(y) are decreasing in y for all [, τ T. Since he random variable Y m is inegrable, hen by Faou s lemma, we obain lim ɛ E[Y ɛ m E[lim ɛ Y ɛ m E[lim ɛ Q ɛ m = E = E [ τ τm χ ψ ()κ( τ m )(ĉ() y())d + χ ψ (τ T )κ(τ τ m )(Ŵ (τ T ) nx(τ T )) χ ψ ()κ()(ĉ() y())d + κ(τ τ m )E τ τm χ ψ ()(ĉ() y())d + χ ψ (τ T )(Ŵ (τ T ) nx(τ T )) τ τ m [ τ τm = E χ ψ ()κ()(ĉ() y())d + κ(τ τ m )χ ψ (τ τ m )Ŵ (τ τ m). We used he definiion of Ŵ () (3.17) in he hird equaliy. On he oher hand, using (A.6)-(A.9) wih W = Ŵ and Iô s lemma shows ha, = β(τ τ m )κ(τ τ m )Ŵ (τ τ m) + τ τm τ τm β(s)κ(s)(ĉ(s) y(s))ds τ τm (s) ψ(s) β(s)ŵ (s)ψ ψ [dn(s) ψ (s)ds + β(s)κ(s)θ 2 (s)(dn(s) ψ (s)ds) (s) τ τm + β(s)κ(s)π(s)σ(s)d B(s) τ τm + β(s)θ 2 (s) ψ (s) ψ(s) ψ dn(s). (s)

16 16 MASAHIKO EGAMI AND HIDEKI IWAKI Therefore we have (A.1) = E [ τ τm E [ τ τm Subsiuing (A.1) ino (A.1) gives χ ψ (s)κ(s)(ĉ(s) y(s))ds + χ ψ (τ τ m )κ(τ τ m )Ŵ (τ τ m) χ ψ (s)θ 2 (s)(ψ (s) ψ(s))ds. V (ζ, ψ ɛ,m ) V (ζ, ψ ) lim ɛ ζ E ɛ [ τ τm Taking ψ as ψ = ψ + η wih η P, i follows ha [ τ τm (A.11) E χ ψ (s)θ 2 (s)η(s)ds. (A.11) leads o a sronger saemen: (A.12) θ 2 ()η(), [, τ T. χ ψ (s)θ 2 (s)(ψ (s) ψ(s))ds. Indeed, suppose ha, for some [, τ T, he se E = {ω Ω; θ 2 ()η() > } had posiive probabiliy for some η P such ha ψ + η P. Then by selecing η = η1 E, we have η P, ψ + η P, and [ τ τm E χ ψ (s)θ 2 (s)η (s)ds >, conradicing (A.11). From Theorem 13.1 of Rockafellar [23 and (A.12), we can conclude ha θ 2 () =, [, τ T. Appendix B. Proof of Proposiion 3.2 We firs show ha if a soluion (ζ, ψ ) exiss, hen his soluion is unique wih respec o ψ. Le ˆψ be any oher inensiy process ha minimizes V (ζ, ψ) for a given ζ, and le (B.1) M := min V (ζ, ψ) ψ P = E +ζ Since we can readily show ha (B.2) M E +ζ ũ (1) ( ũ (1) (ζχ ψ(), )d + Ṽ (ζχ ψ(τ T )) ( W np + E ) y()χ ψ ()d + nχ ψ (τ T )X(τ T ). Ṽ (y) and ũ(1) (y, ), [, T, are convex in y, (ηζχ ψ () + (1 η)ζχ ˆψ(), )d + Ṽ (ηζχ ψ (τ T ) + (1 η)ζχ ˆψ(τ T )) [ W np + E ( ) y() ηχ ψ () + (1 η)χ ˆψ() d ) +n(ηχ ψ (τ T ) + (1 η)χ ˆψ(τ T ))X(τ T ), η [, 1.

17 OPTIMAL LIFE INSURANCE PURCHASE 17 Since M is he minimum, we have equaliy in (B.2). By he previous inequaliy, we conclude ha ψ = ˆψ a.s. To idenify he opimal ψ, we consider an equivalen problem. Le us define (B.3) G(η) := E +ζ ũ (1) ( (ηζχ ψ () + (1 η)ζχ ψ(), )d + Ṽ (ηζχ ψ (τ T ) + (1 η)ζχ ψ(τ T )) [ W np + E y (ηχ ψ () + (1 η)χ ψ ()) d +n(ηχ ψ (τ T ) + (1 η)χ ψ (τ T ))X(τ T ) ), η [, 1. Then, by using he definiion of I (1) and J, we have [ G (η) = ζe I (1) (ηζχ ψ () + (1 η)ζχ ψ(), )(χ ψ () χ ψ ())d and G (η) = ζ 2 E +J(ηζχ ψ (τ T ) + (1 η)ζχ ψ (τ T ))(χ ψ (τ T ) χ ψ (τ T )) [ y (χ ψ () χ ψ ())d nx(τ T )(χ ψ (τ T ) χ ψ (τ T )) I (1) (ηζχ ψ () + (1 η)ζχ ψ (), )(χ ψ () χ ψ ()) 2 d +J (ηζχ ψ (τ T ) + (1 η)ζχ ψ (τ T ))(χ ψ (τ T ) χ ψ (τ T )) 2. Thus G(η) is a convex funcion of η. If ψ is a soluion of he original problem, hen G(η) achieves is minimum a η = 1. This is possible if and only if G (1). Explicily, ψ is a soluion if and only if, for every oher ψ ha saisfies [ E we have (B.4) [ E E [ E I (1) (ζχ ψ())χ ψ ()d + J(ζχ ψ (τ T ))χ ψ (τ T ) [ y()χ ψ ()d + nχ ψ (τ T )X(τ T ) I (1) (ζχ ψ (), )χ ψ ()d + J(ζχ ψ (τ T )χ ψ (τ T ) + W np, y()χ ψ ()d nχ ψ (τ T )X(τ T ) I (1) (ζχ ψ (), )χ ψ()d + J(ζχ ψ (τ T ))χ ψ (τ T ) y()χ ψ ()d nχ ψ (τ T )X(τ T )).

18 18 MASAHIKO EGAMI AND HIDEKI IWAKI Since E[ϕ() = 1 for each [, T, if P({ϕ ψ () > 1}) >, hen P({ϕ ψ () < 1}) >. Accordingly, for any inensiy process ψ, since ζ >, χ ψ > a.s., and ha I (1) and J are convex, we have (B.5) [ E E [ T I (1) (ζχ ψ(), )χ ψ ()d + J(ζχ ψ (τ T ))χ ψ (τ T ) y()χ ψ ()d nχ ψ (τ T )X(τ T ) I (1) (ζχ λ(), )χ ψ ()d + J(ζχ λ (τ T ))χ ψ (τ T ) y()χ ψ ()d nχ ψ (τ T )X(τ T ) So ha, if we choose a ζ so ha i saisfies (B.6) [ E = W np, hen we have (B.7) [ E E [ I (1) (ζχ λ(), )χ λ ()d + J(ζχ λ (τ T ))χ λ (τ T ) y()χ λ ()d nχ λ (τ T )X(τ T ) I (1) (ζχ λ(), )χ λ ()d + J(ζχ λ (τ T ))χ λ (τ T ) y()χ λ ()d nχ λ (τ T )X(τ T ) I (1) (ζχ λ(), )χ ψ ()d + J(ζχ λ (τ T ))χ ψ (τ T ) y()χ ψ ()d nχ ψ (τ T )X(τ T ) holds by (B.4), replacing is LHS by λ. If we compare (B.7) wih (B.4), λ mus be ψ and his argumen complees he proof. Finally, we noe ha here exiss a unique ζ saisfying (B.6) since we have. and lim U x i(x) =, lim U i(x) =, U i (x) <, i = 1, 2, x >, x + V (x) = Z(x), V (x) = Z (x) <, x >, from Assumpion 2.2 and he definiion of U i, i = 1, 2, and V. For each n R +, le ˆV (n) be defined by Then, we can readily show ha ˆV (n) = E Appendix C. Proof of Proposiion 3.3 ˆV (n) = min V (ζ (n), λ). ζ (n) R ++ ũ (1) (ˆζ (n) χ λ (), )d + Ṽ (ˆζ (n) χ λ (τ T )) + ˆζ (n) (W () np + ) χ λ ()y()d + χ λ (τ T )nx(τ T ).

19 OPTIMAL LIFE INSURANCE PURCHASE 19 Here, noing ha, from (3.18), E χ λ ()I (1) (ˆζ(n) χ λ (), ) d + χ λ (τ T )Ŵ (τ T ) (C.1) = W np + E χ λ ()y()d + nχ λ (τ T )X(τ T ) holds, we obain (C.2) ˆV (n) := d ˆV (n) = dn ˆζ (n) (E[χ λ (τ T )X(τ T ) p ). Since ˆζ (n) >, (C.2) immediaely leads o he resul. Firs, we define some noaions as follows. (D.1) (D.2) (D.3) (D.4) Appendix D. Proof of Proposiion 3.4 P (τ ds) = λ(s)e s λ(v)dv ds, E (u, z) := e α(u)+γ(u)z, α (u) := γ (u) := u ( u r(s)ds 1 2 ) 1 ξ(s) 2 2 ds. u P (τ > u) = e u λ(v)dv, ξ(s) 2 ds, We noe ha if Z N(, 1), ha is, if Z is a random variable ha follows he sandard normal disribuion, a process {E (u, Z); u [, T } has an idenical disribuion wih ha of he sae price densiy process {χ(u); u [, T } condiioned by F. Now, we consider he value of he household s wealh Ŵ a ime. If τ, hen Ŵ ( ) + nx() if τ =, (D.5) Ŵ () = Ŵ ( ) if τ >. On he oher, considering oal ne value a ime of opimal fuure consumpion of he household, if τ =, (D.6) Ŵ () = J(ζ χ()), oherwise, if τ >, (D.7) Ŵ () = (T C) (T Y ) (T X), where (D.8)(T C) := E E (u, Z)I (1) (ζ χ()e (u, Z), u)du + E (τ T, Z)J(ζ χ()e (τ T, Z)), [ T T (T Y ) := E E (u, Z)y(u)1 {τ>u} du = y(u)e u (D.9) (r(s)+λ(s))ds du,

20 2 MASAHIKO EGAMI AND HIDEKI IWAKI and (D.1) (T X) := ne [E (τ T, Z)X(τ T ) [ T = nxe E (s, Z)(1 + H(s))P (τ ds) + E (T, Z)P (τ > T ) ( T = nx (1 + H(s))λ(s)e s (r(u)+λ(u))du ds + e ) T (r(s)+λ(s))ds. Here, we noe ha (T Y ) denoes ime value of he household s income o be gained in he fuure and ha (T X) denoes ime value of he money from insurance paid if τ occurs before ime T. Therefore, from (D.5) and (D.7), if τ, (D.11) dŵ () = dŵ ( ) + nx()dn() = d(t C) d(t Y ) d(t X) + nx()dn(). Nex, we derive differenial, d(t C), d(t Y ) and d(t X), explicily. Since (D.12) (T C) = T + ( s ( T E (u, z)i (1) (ζ χ()e (u, z), u)du + E (s, z)j(ζ χ()e (s, z)) E (u, z)i (1) (ζ χ()e (u, z), u)du + E (T, z)j(ζ χ()e (T, z)) ) ) dφ(z)p (τ ds) dφ(z)p (τ > T ), where Φ(z) is he c.d.f. of he sandard normal disribuion. Noe ha he las equaion holds by Fubini s heorem. A sraighforward bu long algebra leads o (D.13) d(t C) = I (1) (ζ χ(), )d + (T C) r()d ζ χ()e [ (J(ζ χ()) (T C) ) λ()d. E (u, Z) 2 I (1) (ζ χ()e (u, Z), u)du +E (τ T, Z) 2 J (ζ χ()e (τ T, Z)) Similarly, from (D.1) and (D.9), we can readily confirm ha (ξ() 2 d + ξ()db()) (D.14) d(t X) = nx(1 + H())λ()d + (T X) (r() + λ())d = (T X) r()d (nx() (T X) )λ()d, and (D.15) d(t Y ) = (T Y ) (r() + λ())d y()d,

21 OPTIMAL LIFE INSURANCE PURCHASE 21 hold. Therefore, from (D.11), (D.13), (D.14) and (D.15), we obain (D.16) dŵ () = d(t C) d(t X) d(t Y ) + nx()dn() = y()d ĉ()d + Ŵ ( )r()d +ˆπ()((µ() r())d + σ()db()) [J(ζ χ()) nx Ŵ ( )λ() + nx()dn(). From (D.5), since J(ζ χ()) = Ŵ ( ) + nx(), we can conclude ha he proposiion holds. References [1 M. O. Albizzai and H. Geman. Ineres rae risk managemen and valuaion of he surrender opion in life insurance policies. Journal of Risk and Insurance, 61: , [2 D. F. Babbel and E. Ohsuka. Aspecs of opimal muli-period life insurance. Journal of Risk and Insurance, 56:46 481, [3 A.R. Bacinello. Equiy linked life insurance, Encyclopedia of Quaniaive Risk Analysis and Assessmen, E. Melnick and B. Everi (eds.). John Wiley & Sons, 28. [4 N. Bellamy and M. Jeanblanc. Incompleeness of markes driven by a mixed diffusion. Finance and Sochasics, 4(2):29 222, 2. [5 Z. Bodie, R. C. Meron, and W. Samuelson. Labor supply flexibiliy and porfolio choice in a life cycle model. Journal of Economic Dynamics and Conrol, 18: , [6 M. J. Brennan and E. S. Schwarz. The pricing of equiy-linked life insurance policies wih an asse value guaranee. Journal of Financial Economics, 3: , [7 R. A. Campbell. The demand for life insurance: An applicaion of he economics of uncerainy. Journal of Finance, 35: , 198. [8 D. Cuoco. Opimal consumpion and equilibrium prices wih porfolio consrains and sochasic income. Journal of Economic Theory, 72:33 73, [9 J. Cvianić and I. Karazas. Convex dualiy in consrained porfolio opimizaion. Ann. Appl. Probab., 2: , [1 J. Cvianić, W. Schachermayer, and H. Wang. Uiliy maximizaion in incomplee markes wih random endowmen. Finance and Sochasics, 5 (2): , 21. [11 J. Grandell. Double sochasic poisson processes, Lecure noes in mahemaics 529. Springer-Verlag, New York, [12 H. He and H. F. Pagès. Labor income, borrowing consrains and equilibrium asse prices; A dualiy approach. Economic Theory, 3: , [13 H. Iwaki. An economic premium principle in a coninuous-ime economy. Journal of he Operaions Research Sociey of Japan, 45: , 22. [14 H. Iwaki, M. Kijima, and M. Morimoo. An economic premium principle in a muliperiod economy. Insurance: Mahemaics and Insurance, 28: , 21. [15 I. Karazas and S. E. Shreve. Mehods of Mahemaical Finance. Springer-Verlag, New York, [16 D. Kramkov and W. Schachermayer. The asympoic elasiciy of uiliy funcions and opimal invesmen in incomplee markes. Ann. Appl. Probab., 9:94 95, [17 E. Marceau and P. Gaillardez. On life insurance reserves in a sochasic moraliy and ineres raes environmen. Insurance: Mahemaics and Economics, :261 28, 25.

22 22 MASAHIKO EGAMI AND HIDEKI IWAKI [18 R. C. Meron. Life ime porfolio selecion under uncerainy. Review of Economics and Saisics, 25: , [19 R. C. Meron. Opimum consumpion and porfolio rules in a coninuous-ime model. Journal of Economic Theory, 3: , [2 J. A. Nielsen and K. Sandman. Equiy-linked life insurance: A model wih sochasic ineres raes. Insurance: Mahemaics and Economics, 16: , [21 S. A. Persson and K. K. Aase. Valuaion of he minimum guaraneed reurn embedded in a life insurance producs. Journal of Risk and Insurance, 64: , [22 S. F. Richard. Opimal consumpion, porfolio and life insurance rules for an uncerain lived individual in a coninuous ime model. Journal of Financial Economics, 2:187 23, [23 R. T. Rockafellar. Convex Analysis. Princeon Universiy Press, Princeon, N.J., 197. [24 L. E. O. Svensson and I. M. Werner. Nonradable asses in incomplee markes: Pricing and porfolio choice. European Economic Review, 37: , [25 S. Wang, V. R. Young, and H. Panjier. Axiomaic characerizaion of insurance prices. Insurence: Mahemaics and Economics, 21: , [26 A. Yashin and E. Arjas. A noe on random inensiies and condiional survival funcions. Journal of Applied Probabiliy, 25:63 635, [27 V. R. Young and T. Zariphopoulou. Compuaion of disored probabiliies for diffusion processes via sochasic conrol mehods. Insurance: Mahemaics and Economics, 27:1 18, 2. [28 Y. Zhu. One-period model of individual consumpion, life insurance, and invesmen decisions. Journal of Risk and Insurance, 74(3): , 27. (M. Egami) Graduae School of Economics, Kyoo Universiy, Yoshida-honmachi, Kyoo, , Japan. address: egami@econ.kyoo-u.ac.jp (H. Iwaki) Graduae School of Managemen, Kyoo Universiy, Yoshida-honmachi, Kyoo, , Japan. address: iwaki@gsm.kyoo-u.ac.jp