MASAHIKO EGAMI AND HIDEKI IWAKI

Size: px
Start display at page:

Download "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. H. Iwaki: Graduae School of Managemen, Kyoo Universiy, Sakyo-ku, Kyoo, , Japan. 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: (H. Iwaki) Graduae School of Managemen, Kyoo Universiy, Yoshida-honmachi, Kyoo, , Japan. address:

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

Optimal Life Insurance Purchase and Consumption/Investment under Uncertain Lifetime

Optimal Life Insurance Purchase and Consumption/Investment under Uncertain Lifetime Opimal Life Insurance Purchase and Consumpion/Invesmen under Uncerain Lifeime Sanley R. Pliska a,, a Dep. of Finance, Universiy of Illinois a Chicago, Chicago, IL 667, USA Jinchun Ye b b Dep. of Mahemaics,

More information

Optimal Life Insurance Purchase, Consumption and Investment

Optimal Life Insurance Purchase, Consumption and Investment Opimal Life Insurance Purchase, Consumpion and Invesmen Jinchun Ye a, Sanley R. Pliska b, a Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA b Dep.

More information

Optimal Life Insurance, Consumption and Portfolio: A Dynamic Programming Approach

Optimal Life Insurance, Consumption and Portfolio: A Dynamic Programming Approach 28 American Conrol Conference Wesin Seale Hoel, Seale, Washingon, USA June 11-13, 28 WeA1.5 Opimal Life Insurance, Consumpion and Porfolio: A Dynamic Programming Approach Jinchun Ye (Pin: 584) Absrac A

More information

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July

More information

Time Consistency in Portfolio Management

Time Consistency in Portfolio Management 1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

More information

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619 econsor www.econsor.eu Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;

More information

Niche Market or Mass Market?

Niche Market or Mass Market? Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.

More information

ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT

ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 94-9(5)634-4 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE

More information

Dependent Interest and Transition Rates in Life Insurance

Dependent Interest and Transition Rates in Life Insurance Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies

More information

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Option Put-Call Parity Relations When the Underlying Security Pays Dividends Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

Verification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing

Verification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364-765X eissn 526-547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion

More information

OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTI-DIMENSIONAL DIFFUSIVE TERMS

OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTI-DIMENSIONAL DIFFUSIVE TERMS OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTI-DIMENSIONAL DIFFUSIVE TERMS I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA Absrac. We inroduce an exension

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

A general decomposition formula for derivative prices in stochastic volatility models

A general decomposition formula for derivative prices in stochastic volatility models A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

More information

ABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION

ABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed

More information

A martingale approach applied to the management of life insurances.

A martingale approach applied to the management of life insurances. 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.

More information

Present Value Methodology

Present Value Methodology Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

More information

Life insurance cash flows with policyholder behaviour

Life insurance cash flows with policyholder behaviour Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK-2100 Copenhagen Ø, Denmark PFA Pension,

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

More information

MULTI-PERIOD OPTIMIZATION MODEL FOR A HOUSEHOLD, AND OPTIMAL INSURANCE DESIGN

MULTI-PERIOD OPTIMIZATION MODEL FOR A HOUSEHOLD, AND OPTIMAL INSURANCE DESIGN Journal of he Operaions Research Sociey of Japan 27, Vol. 5, No. 4, 463-487 MULTI-PERIOD OPTIMIZATION MODEL FOR A HOUSEHOLD, AND OPTIMAL INSURANCE DESIGN Norio Hibiki Keio Universiy (Received Ocober 17,

More information

THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS

THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely

More information

Term Structure of Prices of Asian Options

Term Structure of Prices of Asian Options Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES.

A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES. A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES. DONATIEN HAINAUT, PIERRE DEVOLDER. Universié Caholique de Louvain. Insiue of acuarial sciences. Rue des Wallons, 6 B-1348, Louvain-La-Neuve

More information

Optimal Time to Sell in Real Estate Portfolio Management

Optimal Time to Sell in Real Estate Portfolio Management Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and Jean-Luc Prigen hema, Universiy of Cergy-Ponoise, Cergy-Ponoise, France E-mails: fabricebarhelemy@u-cergyfr; jean-lucprigen@u-cergyfr

More information

UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert

UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of Erlangen-Nuremberg Lange Gasse

More information

Longevity 11 Lyon 7-9 September 2015

Longevity 11 Lyon 7-9 September 2015 Longeviy 11 Lyon 7-9 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univ-lyon1.fr

More information

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.

More information

Research Article Optimal Geometric Mean Returns of Stocks and Their Options

Research Article Optimal Geometric Mean Returns of Stocks and Their Options Inernaional Journal of Sochasic Analysis Volume 2012, Aricle ID 498050, 8 pages doi:10.1155/2012/498050 Research Aricle Opimal Geomeric Mean Reurns of Socks and Their Opions Guoyi Zhang Deparmen of Mahemaics

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

Jump-Diffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach

Jump-Diffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach ump-diffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,

More information

Option Pricing Under Stochastic Interest Rates

Option Pricing Under Stochastic Interest Rates I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp://www.mecs-press.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecs-press.ne/ijem Opion ricing Under Sochasic Ineres

More information

On the Role of the Growth Optimal Portfolio in Finance

On the Role of the Growth Optimal Portfolio in Finance QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 1441-8010 www.qfrc.us.edu.au

More information

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

More information

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se

More information

Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Solutions Question 5 Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

More information

Differential Equations in Finance and Life Insurance

Differential Equations in Finance and Life Insurance Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange

More information

Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach

Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach Working Paper 5-81 Business Economics Series 21 January 25 Deparameno de Economía de la Empresa Universidad Carlos III de Madrid Calle Madrid, 126 2893 Geafe (Spain) Fax (34) 91 624 968 Opion-Pricing in

More information

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical

More information

Optimal Consumption and Insurance: A Continuous-Time Markov Chain Approach

Optimal Consumption and Insurance: A Continuous-Time Markov Chain Approach Opimal Consumpion and Insurance: A Coninuous-Time Markov Chain Approach Holger Kraf and Mogens Seffensen Absrac Personal financial decision making plays an imporan role in modern finance. Decision problems

More information

Dynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract

Dynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium

More information

2.5 Life tables, force of mortality and standard life insurance products

2.5 Life tables, force of mortality and standard life insurance products Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n

More information

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work

More information

Working Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits

Working Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion

More information

Optimal Investment, Consumption and Life Insurance under Mean-Reverting Returns: The Complete Market Solution

Optimal Investment, Consumption and Life Insurance under Mean-Reverting Returns: The Complete Market Solution Opimal Invesmen, Consumpion and Life Insurance under Mean-Revering Reurns: The Complee Marke Soluion Traian A. Pirvu Dep of Mahemaics & Saisics McMaser Universiy 180 Main Sree Wes Hamilon, ON, L8S 4K1

More information

Distributing Human Resources among Software Development Projects 1

Distributing Human Resources among Software Development Projects 1 Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources

More information

Annuity Decisions with Systematic Longevity Risk

Annuity Decisions with Systematic Longevity Risk Annuiy Decisions wih Sysemaic Longeviy Risk Ralph Sevens This draf: November, 2009 ABSTRACT In his paper we invesigae he effec of sysemaic longeviy risk, i.e., he risk arising from uncerain fuure survival

More information

A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul

A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul universiy of copenhagen Universiy of Copenhagen A Two-Accoun Life Insurance Model for Scenario-Based Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

An Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price

An Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor

More information

The Uncertain Mortality Intensity Framework: Pricing and Hedging Unit-Linked Life Insurance Contracts

The Uncertain Mortality Intensity Framework: Pricing and Hedging Unit-Linked Life Insurance Contracts The Uncerain Moraliy Inensiy Framework: Pricing and Hedging Uni-Linked Life Insurance Conracs Jing Li Alexander Szimayer Bonn Graduae School of Economics School of Economics Universiy of Bonn Universiy

More information

Efficient Risk Sharing with Limited Commitment and Hidden Storage

Efficient Risk Sharing with Limited Commitment and Hidden Storage Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing

More information

The Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies

The Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies 1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz- und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany

More information

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

More information

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments BALANCE OF PAYMENTS DATE: 2008-05-30 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se

More information

Chapter 1.6 Financial Management

Chapter 1.6 Financial Management Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1

More information

Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? Life-Cycle Asse Allocaion wih Annuiy Markes: Is Longeviy Insurance a Good Deal? by WOLFRAM HORNEFF, RAIMOND MAURER, and MICHAEL STAMOS November 25 Compeiive Paper Wolfram Horneff Goehe Universiy of Frankfur

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility

Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility Scand. Acuarial J. 00; 4: 46 79 ORIGINAL ARTICLE Pricing Dynamic Insurance Risks Using he Principle of Equivalen Uiliy VIRGINIA R. YOUNG and THALEIA ZARIPHOPOULOU Young VR, Zariphopoulou T. Pricing dynamic

More information

The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.

The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees. The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, 89081

More information

The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees

The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees 1 The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, 89081

More information

PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM

PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu

More information

On Valuing Equity-Linked Insurance and Reinsurance Contracts

On Valuing Equity-Linked Insurance and Reinsurance Contracts On Valuing Equiy-Linked Insurance and Reinsurance Conracs Sebasian Jaimungal a and Suhas Nayak b a Deparmen of Saisics, Universiy of Torono, 100 S. George Sree, Torono, Canada M5S 3G3 b Deparmen of Mahemaics,

More information

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1 Absrac number: 05-0407 Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy

More information

The Grantor Retained Annuity Trust (GRAT)

The Grantor Retained Annuity Trust (GRAT) WEALTH ADVISORY Esae Planning Sraegies for closely-held, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business

More information

OptimalCompensationwithHiddenAction and Lump-Sum Payment in a Continuous-Time Model

OptimalCompensationwithHiddenAction and Lump-Sum Payment in a Continuous-Time Model Appl Mah Opim (9) 59: 99 46 DOI.7/s45-8-95- OpimalCompensaionwihHiddenAcion and Lump-Sum Paymen in a Coninuous-Time Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

RISK-SHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION. 1. Introduction

RISK-SHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION. 1. Introduction RISK-SHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION AN CHEN AND PETER HIEBER Absrac. In a ypical paricipaing life insurance conrac, he insurance company is eniled o a

More information

Introduction to Arbitrage Pricing

Introduction to Arbitrage Pricing Inroducion o Arbirage Pricing Marek Musiela 1 School of Mahemaics, Universiy of New Souh Wales, 252 Sydney, Ausralia Marek Rukowski 2 Insiue of Mahemaics, Poliechnika Warszawska, -661 Warszawa, Poland

More information

Applied Intertemporal Optimization

Applied Intertemporal Optimization . Applied Ineremporal Opimizaion Klaus Wälde Universiy of Mainz CESifo, Universiy of Brisol, UCL Louvain la Neuve www.waelde.com These lecure noes can freely be downloaded from www.waelde.com/aio. A prin

More information

Individual Health Insurance April 30, 2008 Pages 167-170

Individual Health Insurance April 30, 2008 Pages 167-170 Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

More information

Technical Appendix to Risk, Return, and Dividends

Technical Appendix to Risk, Return, and Dividends Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,

More information

I. Basic Concepts (Ch. 1-4)

I. Basic Concepts (Ch. 1-4) (Ch. 1-4) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

On the degrees of irreducible factors of higher order Bernoulli polynomials

On the degrees of irreducible factors of higher order Bernoulli polynomials ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on

More information

A Production-Inventory System with Markovian Capacity and Outsourcing Option

A Production-Inventory System with Markovian Capacity and Outsourcing Option OPERATIONS RESEARCH Vol. 53, No. 2, March April 2005, pp. 328 349 issn 0030-364X eissn 1526-5463 05 5302 0328 informs doi 10.1287/opre.1040.0165 2005 INFORMS A Producion-Invenory Sysem wih Markovian Capaciy

More information

Cash-Lock Comparison of Portfolio Insurance Strategies

Cash-Lock Comparison of Portfolio Insurance Strategies Cash-Lock Comparison of Porfolio Insurance Sraegies Sven Balder Anje B. Mahayni This version: May 3, 28 Deparmen of Banking and Finance, Universiy of Bonn, Adenauerallee 24 42, 533 Bonn. E-mail: sven.balder@uni-bonn.de

More information

IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **

IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß ** IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include so-called implici or embedded opions.

More information

Network Effects, Pricing Strategies, and Optimal Upgrade Time in Software Provision.

Network Effects, Pricing Strategies, and Optimal Upgrade Time in Software Provision. Nework Effecs, Pricing Sraegies, and Opimal Upgrade Time in Sofware Provision. Yi-Nung Yang* Deparmen of Economics Uah Sae Universiy Logan, UT 84322-353 April 3, 995 (curren version Feb, 996) JEL codes:

More information

Optimal Reinsurance/Investment Problems for General Insurance Models

Optimal Reinsurance/Investment Problems for General Insurance Models Opimal Reinsurance/Invesmen Problems for General Insurance Models Yuping Liu and Jin Ma Absrac. In his paper he uiliy opimizaion problem for a general insurance model is sudied. he reserve process of he

More information

This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore.

This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore. This documen is downloaded from DR-NTU, Nanyang Technological Universiy Library, Singapore. Tile A Bayesian mulivariae risk-neural mehod for pricing reverse morgages Auhor(s) Kogure, Asuyuki; Li, Jackie;

More information

Markit Excess Return Credit Indices Guide for price based indices

Markit Excess Return Credit Indices Guide for price based indices Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semi-annual

More information

Time-Inconsistent Portfolio Investment Problems

Time-Inconsistent Portfolio Investment Problems Time-Inconsisen Porfolio Invesmen Problems Yidong Dong Ronnie Sircar April 2014; revised Ocober 6, 2014 Absrac The explici resuls for he classical Meron opimal invesmen/consumpion problem rely on he use

More information

THE IMPACT OF THE SECONDARY MARKET ON LIFE INSURERS SURRENDER PROFITS

THE IMPACT OF THE SECONDARY MARKET ON LIFE INSURERS SURRENDER PROFITS THE IPACT OF THE ECONDARY ARKET ON LIFE INURER URRENDER PROFIT Nadine Gazer, Gudrun Hoermann, Hao chmeiser Insiue of Insurance Economics, Universiy of. Gallen (wizerland), Email: nadine.gazer@unisg.ch,

More information

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives Sochasic Calculus, Week 10 Term-Srucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Term-srucure models 3. Ineres rae derivaives Definiions and Noaion Zero-coupon

More information

Investment under Uncertainty, Debt and Taxes

Investment under Uncertainty, Debt and Taxes Invesmen under Uncerainy, Deb and Taxes curren version, Ocober 16, 2006 Andrea Gamba Deparmen of Economics, Universiy of Verona Verona, Ialy email:andrea.gamba@univr.i Gordon A. Sick Haskayne School of

More information

Chapter 6: Business Valuation (Income Approach)

Chapter 6: Business Valuation (Income Approach) Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he

More information

Credit risk. T. Bielecki, M. Jeanblanc and M. Rutkowski. Lecture of M. Jeanblanc. Preliminary Version LISBONN JUNE 2006

Credit risk. T. Bielecki, M. Jeanblanc and M. Rutkowski. Lecture of M. Jeanblanc. Preliminary Version LISBONN JUNE 2006 i Credi risk T. Bielecki, M. Jeanblanc and M. Rukowski Lecure of M. Jeanblanc Preliminary Version LISBONN JUNE 26 ii Conens Noaion vii 1 Srucural Approach 3 1.1 Basic Assumpions.....................................

More information

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his

More information

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear

More information

Market Completion and Robust Utility Maximization

Market Completion and Robust Utility Maximization Marke Compleion and Robus Uiliy Maximizaion DISSERTATION zur Erlangung des akademischen Grades docor rerum nauralium (Dr. rer. na.) im Fach Mahemaik eingereich an der Mahemaisch-Naurwissenschaflichen Fakulä

More information