IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ The Valuaion of an Equiy-inked ife Insurance Using he Theory of Indifference ricing Jungmin Choi Absrac This sudy addresses he valuaion problem of an equiy-linked erm life insurance in wo moraliy models - a deerminisic moraliy and a sochasic moraliy. For each case, he Hamilon-Jacobi-Bellman (HJB) arial Differenial Equaion (DE) for he corresponding uiliy funcion is derived wih a coninuous ime model, and he principle of equivalen uiliy is applied o obain a DE for he indifferen price of he premium when an exponenial uiliy funcion is employed. Numerical examples are performed wih Gomperz s law of moraliy for he deerminisic model and wih a mean-revering Brownian Gomperz (MRBG) process for he sochasic model. Index Terms Equiy-inked ife Insurance, indifference pricing, sochasic moraliy. I. INTRODUCTION THIS sudy focuses on he valuaion problem of an equiy-linked erm life insurance using he heory of indifference pricing. In addiion o a moraliy risk like any oher life insurance produc, an equiy-linked erm life insurance has a marke risk from he underlying asse. In recen years, insurers have offered more flexible life insurance producs ha combine he deah benefi coverage wih an invesmen componen, o compee wih oher forms of he policy holder s savings, for example, muual funds or banks. An equiy-linked life insurance produc can offer a benefi from he performance of an underlying asse by defining he deah benefi o depend on he accoun value of he underlying asse. The pricing and hedging problem of he equiy-linked life insurance has been invesigaed exensively, and i is well summarized by Melnikov and Romanyuk [2]. As menioned in heir aricle, insurance firms do no consider he moraliy risk in valuaion of he policies nor adop adequae moraliy raes. This leads o an overpricing or underpricing of he premiums and he burden falls on he cusomers or he firm iself. Young [9] considered he same problem using he heory of indifference pricing when he moraliy rae is compued using Gomperz s law of moraliy. The purpose of his sudy is o exend heir idea o include a sochasic moraliy rae. Indifference pricing, also known as reservaion pricing or privae valuaion, is a mehod of pricing financial derivaives wih regard o a uiliy funcion. I is one of he pricing ools in incomplee financial markes, and i uses he principle of equivalen uiliy. Uiliy funcions are widely used for problems in pricing and hedging of financial derivaives, see [8] and [5], for example. Uiliy indifference pricing was firs inroduced by Hodges and Neuberger [6] when hey considered ransacion coss in replicaing coningen claims. The main idea of indifference pricing is ha by comparing he maximal expeced uiliies wih and wihou a Jungmin Choi is wih he deparmen of Mahemaics a Eas Carolina Universiy, Greenville, Norh Carolina, USA. (email: choiju@ecu.edu) coningen claim, one can find a value of he price funcion which is indifferen o he exisence of he coningen claim. This idea was used o price insurance risks in a dynamic financial marke seing by Young and Zariphopoulou [2] using an exponenial uiliy. The same idea was exended by Young [9] o sudy an equiy-indexed life insurance, and he derived DE for premiums and reserves generalized he Black-Scholes equaion by including a nonlinear erm reflecing he nonhedgeable moraliy risk. We will adop heir model o sudy he indifference pricing of an equiyindexed life insurance wih wo differen moraliy risk models: a deerminisic and a sochasic. Indifference pricing is also used in pricing problems in an incomplee marke. The valuaion of opions in a sochasic volailiy model for sock price using indifference pricing was sudied by Sircar and Surm [8] and Kumar [7]. Sochasic moraliy became imporan especially for he moraliy coningen claim. In [4], Milevsky and romislow sudied he pricing problem of an opion o annuiize when considering sochasic moraliy raes and sochasic ineres raes. They also sudied how o hedge an opion o annuiize using pure endowmens, defaul free bonds, and life insurance conracs. Variable annuiies under sochasic moraliy were also considered by Balloa and Haberman []. ricing, reserving and hedging of a guaraneed annuiy opion (GAO) valuaion problem was sudied when moraliy risk was incorporaed via a sochasic model of he underlying hazard raes. Assuming a sochasic moraliy ha is independen of he financial risk, a general pricing model was proposed, and he Mone Carlo mehod was used for he esimaion of he value of GAO. Their sochasic moraliy model was also used by iscopo and Haberman [6], who considered a Guaraneed ifelong Wihdrawal Benefis (GWB) conrac under he hypohesis of a predeermined wihdrawal amoun. The valuaion approach was based on he decomposiion of he produc ino living and deah benefis, and a no arbirage model was used o derive he valuaion formula, wih a fixed moraliy and a sochasic moraliy. Indifference pricing of moraliy coningen claims was invesigaed by udkovsky and Young [] wih boh sochasic hazard raes in he populaion moraliy and he sochasic ineres raes. The resuling DEs were linear for pure endowmens and emporary life annuiies in a coninuous ime model, and i was found ha he price-per-risk increases as more conracs are sold. A sudy of he indifference pricing of a radiional life insurance and pension producs porfolio wih sochasic moraliy was presened by Delong [5], when a financial marke consiss of a risk-free asse wih a consan rae of reurn and a risky asse whose price is driven by a evy process. He applied echniques from sochasic conrol heory o solve he opimizaion problems. In his paper, we consider he pricing problem of an equiy- (Advance online publicaion: 26 November 26)
IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ linked erm life insurance using he heory of indifference pricing, when he moraliy of he insured is described by wo differen moraliy models: a deerminisic moraliy and a sochasic moraliy. The remainder of his paper is organized as follows. In Secion II, we presen he pricing DE of an equiy-indexed erm life insurance using an exponenial uiliy in he coninuous-ime model wih a deerminisic moraliy funcion, as described in [9]. The numerical soluions of he DE wih Gomperz s law of moraliy and he sensiiviies of he indifference prices wih respec o various parameers are also provided. The pricing DE and numerical examples wih a sochasic moraliy model are given in Secion III. We adop he MRBG process o model he moraliy risk. Finally, Secion IV summarizes our findings and oulines fuure works o exend our model. II. INDIFFERENCE RICING OF AN EQUITY-INDEXED IFE INSURANCE WITH A DETERMINISTIC MORTAITY A. The Financial Marke: Meron s Model We presen he classical model by Meron [3] which invesigaes he opimal invesmen sraegies of an individual wih an iniial wealh, who seeks o maximize he expeced uiliy of he erminal wealh. The invesor has he opporuniy o rade beween a risky asse (sock) and a risk-free asse (U.S. reasury bond). The price of he risky asse S s for some ime s >, wih a fixed ime, follows { dss = µs s ds + σs s db s, S = S >, where B s is a sandard Brownian moion on a probabiliy space (Ω, F, ) wih a filraion F and a probabiliy measure. The rae of reurn µ and he volailiy σ are posiive consans. The price of he risk-free bond s for some ime s > follows d s = r s ds, where r is a consan rae of reurn (or force of reurn) of he risk-free bond, and we assume µ > r >. e w be he iniial wealh of he insurer a ime, and W s be he wealh of he insurer a ime s in [, T ], where T is he erminal ime. Suppose he insurer rades dynamically beween he sock and he bond. e π s be he amoun of wealh invesed in he sock a ime s. Then he amoun invesed in he bond is πs b = W s π s, and he dynamics of he wealh process becomes ( ) ( ) d dw s = πs b s dss + π s s S s = (W s π s )rds + π s (µds + σdb s ). Hence we have { dws = (rw s + (µ r)π s )ds + σπ s db s, s T, W = w. B. Expeced Uiliy Wihou he insurance risk Suppose he invesor wans o maximize he expeced uiliy of he erminal wealh, and define he value funcion V (, w) as V (, w) = sup E[u(W T ) W = w], π A where A is he se of admissible policies, and u : R R is a uiliy funcion, which is increasing, concave, and smooh. We will use an exponenial uiliy funcion o derive a DE for he indifference price. I has been shown in [2] ha V saisfies he following HJB equaion: { V + max π [(µ r)π V w + 2 σ2 π 2 V ww ] + rwv w =, V (T, w) = u(w). Since he maximum funcion is quadraic in π and he concaviy of he uiliy funcion u is inheried o he value funcion, he maximum exiss and we have he opimal invesmen process π = µ r V w (w, ) σ 2 V ww (w, ). This gives a closed form DE for V : V + rwv w (µ r)2 Vw 2 2σ 2 =, V ww V (T, w) = u(w). One of he advanages o considering an exponenial uiliy funcion is ha we can find he closed form soluion o (). Suppose u(w) = e w, for some >, hen we obain he soluion V (, w) o be V (, w) = [ ] exp we r(t ) (µ r)2 2σ 2 (T ). We also can find he corresponding opimal sraegy π (, w) = µ r σ 2 e r(t ), which is no sochasic and independen of w. I is generally observed when considering exponenial uiliy. Since he absolue risk aversion for he exponenial uiliy funcion is measured by a consan, ( u (w)/u (w) = ), one can observe ha as he invesor s risk aversion () increases, he amoun of money invesed in he risky asse (π ) decreases [2]. C. Expeced Uiliy wih he insurance risk The insurer has an opporuniy o insure a person whose age is x + a ime. The deah benefi of his life insurance is defined o be G = max(a, A τ ), where τ < T is he ime of deah of he policy holder, A is he iniial accoun value of he underlying muual fund a he ime when he conrac is made, and A s is he accoun value a ime s. This insurance policy is an equiy-indexed produc since i is ied o an accoun value hrough he funcion G. Suppose he insurer charges an insurance fee o hedge he marke risk, and we assume ha i is deduced from he accoun value as an ongoing fracion, α. The dynamics of A s follow da s = (µ α)a s ds + σa s db s, where B s is a sandard Brownian moion on (Ω, F, ), and µ (rae of reurn) and σ (volailiy) are consans. Suppose For simpliciy, we assume i is a fixed consan. () (2) (Advance online publicaion: 26 November 26)
IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ TABE I ARAMETER VAUES he insurer wans o maximize he expeced uiliy of he erminal wealh, and define he value funcion Age a incepion x Risk free ineres rae r Volailiy of he risky asse σ Insurance fee α Term of policy T U (, w, A) = sup E{u(WT ) W = w, A = A}, π A where A is he se of admissible policies, and u : R R is a uiliy funcion, which is increasing, concave, and smooh. The wealh process should follow dws = [rws + (µ r)πs ]ds + σπs dbs, W = w, Wτ + = Wτ Gτ, if τ < T. The HJB equaion for U can be obained as follows (see Appendix) U + rwuw + (µ α)aua + 2 σ 2 A2 UAA +λx ()[V (w G, ) U ] + maxπ [(µ r)π Uw + σ 2 π AUwA (3) 2 2 + σ π U ] =, ww 2 U (T, w, A) = u(w), Using his relaionship in (5), we can derive a DE for (, A) r + + (r α)aa + 2 σ 2 A2 AA r(t ) ( G) = e r(t ) λx ()[ e e ], (6) (T, A) =. D. Numerical Example and Sensiiviy Analysis where λx () is he force of moraliy of a person aged x a ime. The corresponding opimal sraegy π is π = µ r Uw UwA A. σ2 Uww Uww.8 Suppose we use he exponenial uiliy u(w) = e w, for some >. Because of he naure of he exponenial uiliy, we propose he soluion of (3) o be in he form of U (, w, A) = V (, w) ϕ(, A) [9]. Then he opimal sraegy becomes µ r Vw Vw ϕa π = 2 A. σ Vww Vww ϕ e U = V ϕ in (3). From (2), we also have.6.4.2 2. 5.5..5 5 Vw2 = V, Vww and 5.8.2. 5 years A Fig.. rice wih respec o ime and accoun value V (, w G) = V (, w) exp[ger(t ) ]. Then we obain he DE for ϕ from (3):.2 ϕ + (r α)aϕa + 2 σ 2 A2 (ϕaa Ger(T ) +λx ()(e ϕ(t, A) =. ϕ2a ϕ ) ϕ) =, By inroducing η(, A) as ϕ(, A) = eη(,a), we have 2 2 η + (r α)aηa + 2 σ A ηaa r(t ) η +λx ()(ege ) =, η(t, A) =..8 (4).6.4.2 (5) Now le (, A) be he indifference price, ha is, he minimum premium he insurer should have in exchange for insuring he person whose age is x + a ime for a erm life insurance which expires a ime T. Then (, A) should solve V (, w) = U (, w +, A) = V (, w + )ϕ(, A), and using he closed form of V in (2), we have a formula for (, A) wih respec o η as (, A) = e r(t ) η(, A)...8..5.2.25.3.35.4.45.5.55.6 Fig. 2. The relaionship beween and (, A ) In his secion, we solve (6) assuming he moraliy λx () follows Gomperz s law of moraliy λx () = B C x+, wih B =.64 5 and C =.96. These parameer esimaions were obained in [] using 959-999 moraliy daa for Sweden. To solve (6) numerically, we use he finie (Advance online publicaion: 26 November 26)
IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_.22.2.8.6.4.5.5 2 2.5 3 3.5 4 α x 3 Fig. 3. The relaionship beween insurance fee and (, A ).65.6.55.5.45.4.35.3.25.2.5 σ=. σ=.5 σ=.2 σ=.25 4 45 5 55 6 65 Age a incepion Fig. 4. The relaionship beween volailiy and (, A ) difference mehod, paricularly, backward difference scheme for he ime and he cenral difference scheme for he accoun value A. We se he minimum accoun value o be zero and he maximum accoun value o be 2, wih he iniial accoun value (A ) of. We assume he DE holds on he boundary since he domain is runcaed. For he boundary condiion when he accoun value is maximum (A = 2), we assume he lineariy of he premium in erms of he accoun value, namely, we se AA =, and solve r + +(r α)a A = e r(t ) λ[ e er(t ) ( G) ]. When he accoun value is minimum (A = ), we solve r + = e r(t ) λ[ e er(t ) ( G) ]. A ypical soluion for he premium funcion is ploed in Figure wih respec o he ime and he accoun value, using he parameers in Table I for he base case. From he plo, we can observe ha increases as he accoun value A increases, which is expeced since he higher he accoun value is, he higher he deah benefi G = max(a, A τ ) will be. The insurer should receive more premium for a higher deah benefi. The premium decreases as he ime increases,which is a common rend for an equiy linked financial derivaive (for example, he hea, he rae of change of he derivaive wih respec o he ime, is negaive for a European call opion). Figure 2 shows he relaionship beween he risk aversion and he premium. As observed in oher sudies of moraliy coningen claims ( [], [9]), when he risk aversion increases, he indifference price of he premium also increases. Figure 3 shows he impac of he insurance fee α on he premium. I is clear ha hey should have a negaive relaionship, since if one has o pay more insurance fees, he price of he produc a ime zero should decrease. A posiive relaionship beween he volailiy σ of he risky asse and he premium is refleced in Figure 4, which is consisen wih a general financial heory ha he financial produc is more expensive when he volailiy is high. The premium is ploed agains he age a incepion x for various volailiies in he same figure, and we can observe ha if a person oped o purchase he life insurance produc a laer daes (as x increases), he should pay a higher price for he benefi. This can be jusified as follows: when x increases, he moraliy of he person increases, and, hence, he price of he life insurance produc should also increase. III. INDIFFERENCE RICING OF AN EQUITY-INDEXED IFE INSURANCE WITH A STOCHASTIC MORTAITY A. ricing DE wih sochasic moraliy Now we consider a sochasic model for he force of moraliy for an individual or a se of individuals of he same age. We adop he model proposed by udkovsky and Young [] and assume he force of moraliy λ follows a diffusion process as dλ s = µ(s, λ s )ds + σ(s)λ s db λ s, (7) where Bs λ is a Brownian moion on a probabiliy space (Ω, F, ) which is independen of B s in he previous secion. The volailiy σ is a nonzero coninuous funcion of ime s bounded below by a posiive consan κ on [, T ]. The drif µ(s, λ) is a coninuous funcion of s and λ which is posiive for all s in [, T ]. We will use he mean-revering Brownian Gomperz model in [4] for he numerical examples. Suppose he accoun value A s and he wealh process W s are defined as in Secion II. The insurer agrees o pay G τ = max(a, A τ ) upon deah a τ < T given a person aged x a = purchased he life insurance produc. Suppose he insurer wans o maximize he expeced uiliy of he erminal wealh, and define he value funcion U(, w, A, λ) = sup {u(w T ) W = w, A = A, λ = λ}, π A where A is he se of admissible policies, and u : R R is a uiliy funcion, which is increasing, concave, and smooh. The HJB equaion for U can be obained as follows (see Appendix) U + rwu w + (µ α)au A + 2 σ2 A 2 U AA +λ[v (, w G) U] + µλu λ + 2 σ2 λ 2 Uλλ + max π [(µ r)πu w + σ 2 πau wa + 2 σ2 π 2 U ww ] =, U(T, w, A) = u(w). Since he opimized erms are he same as in (3), i will assume he same opimal sraegy π = µ r U w σ 2 A U wa. U ww U ww (8) (Advance online publicaion: 26 November 26)
IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ Following he idea in Secion II and using he same exponenial uiliy u(w) = e w, we assume he soluion of (8) o be in he form of U(, w, A, λ) = V (, w) ϕ(, A, λ). Then he opimal sraegy becomes π = µ r V w σ 2 A V w ϕa V ww V ww ϕ. e U = V ϕ in (8). Afer collecing V erms and ϕ erms, (8) becomes V [ϕ s + (r α)aϕ A + 2 σ2 A 2 ϕ AA +λ[exp(ge r(t ) ) ϕ] +µλϕ λ + 2 σ2 λ 2 ϕ λλ 2 σ2 A 2 ϕ 2 A ϕ ] +ϕ[v s + rwv w (µ r)2 2σ 2 V ] =. The muliple o ϕ in he las erm is zero because of (), hence we obain he DE for ϕ ϕ + (r α)aϕ A + 2 σ2 A 2 (ϕ AA ϕ2 A ϕ ) +µλϕ λ + 2 σ2 λ 2 ϕ λλ + λ(e Ger(T ) ϕ) =, ϕ(t, A, λ) =. To eliminae he nonlinear erm ϕ2 A in (4), le us define ϕ η(, A, λ) by ϕ(, A, λ) = e η(,a,λ). Then we have a DE for η(, A, λ): (9) η + (r α)aη A + 2 σ2 A 2 η AA + µλη λ + 2 σ2 λ 2 (η 2 λ + η λλ) + λ(e Ger(T ) η ) =, η(t, A, λ) =. () Now le (, A, λ) be he indifference price, ha is, he minimum premium he insurer should have in exchange for a erm life insurance which expires a ime T. Then (, A, λ) should solve V (, w) = U(, w +, A, λ) = V (, w + )ϕ(, A, λ), and using he closed form of V in (2), we have a formula for (, A, λ) wih respec o η as (, A, λ) = e r(t ) η(, A, λ). Using his relaionship in (), we can derive a DE for (, A, λ) r + + (r α)a A + 2 σ2 A 2 AA +µλ λ + 2 σ2 λ 2 (e r(t r) 2 λ + λλ ) = e r(t ) λ[ e ) ( er(t G) () ], (T, A, λ) =. B. Numerical Example For numerical examples ha solves (), we use he following MRBG process proposed in [4]: ( dλ s = g + ) 2 σ2 + κ(gs + ln λ ln λ s ) λ s ds+σλ s dbs λ, wih κ =.5. The process ln λ s follows an Ornsein- Uhlenbeck model wih a linear drif g. The parameer values.4.2..8.6.4.2 TABE II ARAMETER VAUES Risk free ineres rae r.8 Volailiy of he risky asse σ.2 Insurance fee α. Term of policy T years Volailiy of he force of moraliy σ.2 Force of moraliy a incepion λ.3 Gomperz parameer g..5..5.2 Fig. 5. 3D plo of he soluion (, A, λ) λ.25 in Table II are used o obain he soluions unless noed oherwise. The DE for he premium (, A, λ) is solved in he domain [, T ] [, 2] [,.25] wih an iniial accoun value (A ), using he backward in ime finie difference mehod. As in he previous secion, we assume he DE holds on he boundary, and assuming lineariy when A = A max and λ = λ max. The boundary condiions imposed are ) A = : r + + +µλ λ + 2 σ2 λ 2 (e r(t r) 2 λ + λλ ) = e r(t ) λ[ e er(t ) ( G) ] 2) A = A max : 3) λ = : 4) λ = λ max : r + + (r α)a A +µλ λ + 2 σ2 λ 2 (e r(t r) 2 λ + λλ ) = e r(t ) λ[ e er(t ) ( G) ] r + + (r α)a A + 2 σ2 A 2 AA = r + + (r α)a A + 2 σ2 A 2 AA +µλ λ + 2 σ2 λ 2 (e r(t r) 2 λ) = e r(t ) λ[ e er(t ) ( G) ] A ypical soluion for he premium funcion (, A, λ) is given in Figure 5 a he ime of incepion ( = ). I.5. A.5 2. (Advance online publicaion: 26 November 26)
IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_.2..8.6 λ=.25 λ=.5 λ=. λ=.5 λ=.2.3.25.2.5 λ=.25 λ=.5 λ=. λ=.5.4..2.5.2.4.6.8.2.4.6.8 2 A Fig. 6. The relaionship beween A and (, A, λ) for various values of λ 2 3 4 5 6 7 8 9 Time o Expiraion Fig. 8. The relaionship beween ime o expiraion T and (T,., λ) for various values of λ.7.234.6.5.232.23.4.3.2.228.226.224.222..22.5.5 2 2.5 Fig. 7. The relaionship beween λ and (, A, λ) for various values of A λ x 3.28.26..2.3.4.5 Fig. 9. The plo of (,,.) wih respec o σ σ shows ha is an increasing funcion in A and λ, which is expeced since he price of a life insurance produc should increase when he accoun value increases or he moraliy rae increases. We can observe his more clearly in Figures 6 and 7. The premium funcion is ploed agains A for various λ in Figure 6. The premium is an increasing convex funcion in he accoun value A for various values of λ; he rae of change is more significan wih a higher value of A. I is clearer wih a higher force of moraliy λ. The premium funcion is ploed agains λ for various values of A in Figure 7. The premium is an increasing concave funcion in λ; he rae of change is more significan wih a smaller value of λ. I is clearer wih a higher value of A. As in he deerminisic moraliy model, he premium funcion has a posiive relaionship wih he ime o expiraion (T ) in Figure 8. Similar rends can be observed in oher lieraure, for example, in he descripion of he deah benefi in he sudy by iscopo and Haberman [6]. To see he effec of he volailiy of he force of moraliy in (7), he value of (,,.) is ploed for differen values of σ. We observe ha as he volailiy increases, he premium also increases in Figure 9, which reflecs he common phenomenon in he financial markes ha he price of a moraliy coningen produc increases when here is more risk in moraliy. The effecs of he risk aversion rae and he insurance fee α on he premium is similar as in he case wih a deerminisic moraliy model in Secion II. The premium (,,.) wih respec o he risk aversion raes is ploed in Figure, and we observe ha he premium increases as he risk aversion rae increases. The premium (,,.) wih respec o he insurance fee α is ploed in Figure, which shows ha he premium of a life insurance produc should decrease when he policy holder pays a higher insurance fee α. IV. CONCUSION We have considered he valuaion problem of an equiyindexed erm life insurance wih wo differen moraliy models, a deerminisic moraliy and a sochasic moraliy. For he financial marke, we employ Meron s model and use an exponenial uiliy o obain HJB equaions for he uiliy funcions wih and wihou he life insurance risks. By using he heory of equivalen uiliy, we derive he DEs for he indifference price of he premium for boh moraliy models. The DE wih a deerminisic moraliy is solved numerically using Gomperz law of moraliy, while he MRBG process is adoped for he sochasic moraliy case. The derived DEs are no simple and closed form soluions canno be found, bu sraighforward applicaions of he finie difference mehod wih proper boundary condiions produce soluions ha are reasonable for a life insurance conrac. The sensiiviy analysis shows ha he models are appropriae o explain he premiums of he equiy-indexed erm life insurance. Fuure research should consider he effec of sochasic ineres raes, since i is unreasonable o assume he risk-free rae is consan for a long period of ime. We can also apply he indifference pricing heory o variable annuiy producs exposed o similar risks, for example, wih a Guaraneed Minimum Deah Benefi opion or a Guaraneed ifelong Wihdrawal Benefi opion. (Advance online publicaion: 26 November 26)
IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_.4.35.3.25.2.5...3.5.7.9 Fig.. The plo of (,,.) wih respec o.28.28.28 [4] Milevsky, M. A., and S. D. romislow, Moraliy Derivaives and he Opion o Annuiise. Insurance: Mahemaics and Economics, 29:299-38, 2. [5] Mukupa G. M., and Offen E. R., The Impac of Uiliy Funcions on The Equilibrium Equiy remium In A roducion Economy Wih Jump Diffusion, IAENG Inernaional Journal of Applied Mahemaics, vol. 45, no. 2, pp2-27, 25 [6] iscopo, G., and S. Haberman, The Valuaion of Guaraneed ifelong Wihdrawal Benefi Opions in Variable Annuiy Conracs and he Impac of Moraliy Riak. Norh American Acuarial Journal, 5(): 59-76, 2. [7] Sihole, T. Z., S. Haberman, and R. J. Verrall, An Invesigaion ino arameric Models for Moraliy rojecions, wih Applicaions o Immediae Annuians and ife Office ensioners daa. Insurance: Mahemaics and Economics, 27: 285-32, 2. [8] Sircar, S. and S. Surm, From Smile Asympoics o Marke Risk Measures. Mahemaical Finance, :259-276, 22. doi:./mafi.25. [9] Young, V. R., Equiy-Indexed ife Insurance: ricing and Reserving Using he rinciple of Equivalen Uiliy. Norh American Acuarial Journal, 7(): 68-86, 23. [2] Young, V. R., and T. Zariphopoulou, ricing Dynamic Insurance Risks Using he rinciple of Equivalen Uiliy. Scandinavian Acuarial Journal, 4:246-279, 22..28.28.27.27.27.27.5.5 2 2.5 α x 3 Fig.. The plo of (,,.) wih respec o α AENDIX A HJB EQUATION FOR U WITH THE INSURANCE RISK Here we derive he HJB equaion for U. Assume ha he insurer follows an arbirary invesmen policy {π s } beween and + h, and afer + h, he insurer follows he opimal invesmen policy {π s}. If he insured aged x + survives unil ime + h, he conrac goes on. If he insured aged x+ dies before + h, he insurer pays G +h, and coninues under V, he value funcion wihou he claim. Thus REFERENCES [] Balloa,., and S. Haberman, The Fair Valuaion roblem of Guaraneed Annuiy Opions: The Sochasic Moraliy Environmen Case. Insurance: Mahemaics and Economics, 38:95-24, 26. [2] Bjork, T. Arbirage Theory in Coninuous Time. Oxford Universiy ress, 998. [3] Carmona, R. Indifference ricing. rinceon Universiy ress, 29. [4] Choi, J., and M. Gunzburger, Opion ricing in he resence of Random Arbirage Reurn. Inernaional Journal of Compuer Mahemaics, 86(6): 68-8. [5] Delong,., Indifference ricing of a ife Insurance orfolio wih Sysemaic Moraliy Risk in a Marke wih an Asse Driven by a evy rocess. Scadinavian Acuarial Journal, :-26, 29. [6] Hodges, S. D., and A. Neuberger, Opimal Replicaion of Coningen Claims under Transacion Coss. Review of Fuures Markes, 8:222-239, 989. [7] Kumar, R., Effec of Volailiy Clusering on Indifference ricing of Opions by Convex Risk Measures. Applied Mahemaical Finance 22():63-82, 25, doi:.8/35486x.24.94985. [8] Qiang i, and ap Keung Chu, A Muli-sage Financial Hedging Sraegy for a Risk-averse Firm wih Coningen aymen, IAENG Inernaional Journal of Applied Mahemaics, vol. 45, no., pp7-76, 25 [9] oeve, M. robabiliiy Theory. New York: Springer, 977. [] udkovski, M. and V.R. Young, Indifference ricing of ure Endowmens and ife Annuiies under Sochasic Hazard and Ineres Raes. Insurance: Mahemaics and Economics, 42:4-3, 28. [] Melnikov, A. and Y. Romaniuk, Evaluaing he erformance of Gomperz, Makeham and ee-carer moraliy models for risk managemen wih uni-linked conracs. Insurance: Mahemaics and Economics, 39:3-329, 26. [2] Melnikov, A. and Y. Romaniuk, Efficien Hedging and ricing of Eruiy-inked ife Insurance Conracs on Several Risky Asses. Inernaional Journal of Theoreical and Applied Finance, (3): 295-323, 28. [3] Meron, R. C., ifeime orfolio Selecion under uncerainy: he Coninuous-ime Case. The Review of Economics and Saisics, 5(3): 247-257, 969. U(, w, A) E[ + h, U(W +h, A +h ) W = w, A = A] h p x+ +E[V ( + h, W +h G +h ) W = w] h q x+, (2) where h p x+ is he probabiliy of a person aged x+ survives unil x + + h, and h q x+ = h p x+. This will have an equaliy if and only if he invesmen policy is opimal beween and + h. Assuming U and V are smooh enough o have all he derivaives, we have U( + h, W +h, A +h ) = U(, w, A) + +h du, (3) where du is he differenial of U. Using Iô s formula, du = [U s + U w (rw + (µ r)π) + (µ α)au A + 2 σ2 π 2 U ww + 2 σ2 A 2 U AA + σ 2 πau wa ]ds +σπu w db + σau A db. The righ hand side of (3) becomes U(, w, A) + +h Uds + +h σπu w db + σau A db, where +h U = U s + U w (rw + (µ r)π) + (µ α)au A + 2 σ2 π 2 U ww + 2 σ2 A 2 U AA + σ 2 πau wa, and aking expecaions yields he las wo inegrals zero. Similarly, for he value funcion wihou he claim V, we have V (+h, w +h ) = V (, w)+ +h 2 V + +h σπv w dbds, where 2 V = V s + (rw + (µ r)π)v w + 2 σ2 π 2 V ww. (Advance online publicaion: 26 November 26)
IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ or Now (2) becomes U(, w, A) E,w,A [U(, w, A) + +h Uds] h p x+ +E w, [V (, w G) + +h 2 V ds] h q x+, U(, w, A) h q x+ E,w,A [ +h Uds] h p x+ +E w, [V (, w G) + +h 2 V ds] h q x+h. If we divide boh sides by h and ake limi h, hen λ x () U U + V (w G, ) λ x (), since h q x+ λ x (), he force of moraliy of a person h aged x a ime, and h q x+ as h. If he invesmen policy is opimal, we have an equaliy, which gives he following HJB equaion of U; U + rwu w + (µ α)au A + 2 σ2 A 2 U AA +λ x ()[V (w G, ) U] + max π [(µ r)πu w + σ 2 πau wa + 2 σ2 π 2 U ww ] =, U(T, w, A) = u(w). (4) or U(, w, A, λ) h q x+ E,w,A,λ [ +h 3 Uds] h p x+ +E,w [V (, w G) + +h 2 V ds] h q x+h. If we divide boh sides by h and ake limi h, hen λ U 3 U + V (, w G) λ. If he invesmen policy is opimal, we have an equaliy, which gives he following HJB equaion of U; U + rwu w + (µ α)au A + 2 σ2 A 2 U AA +λ[v (w G, ) U] + µλu λ + 2 σ2 λ 2 Uλλ + max π [(µ r)πu w + σ 2 πau wa (7) + 2 σ2 π 2 U ww ] =, U(w, A, T ) = u(w). AENDIX B HJB EQUATION FOR U WITH THE INSURANCE RISK The derivaion of HJB equaion for U is similar o he process for U. Wih he same assumpion in he previous secion, we have U(, w, A, λ) E[U( + h, W +h, A +h, λ +h ) W = w, A = A, λ = λ] h p x+ +E[V ( + h, W +h G +h ) W = w] h q x+. (5) This will have an equaliy if and only if he invesmen policy is opimal beween and + h. Assuming U and V are smooh enough o have all he derivaives, we have U( + h, W +h, A +h, λ +h ) = U(, w, A, λ) + +h du, (6) where du is he differenial of U. Using Iô s formula, du = [U s + U w (rw + (µ r)π) + (µ α)au A +µλu λ + 2 σ2 π 2 U ww + 2 σ2 A 2 U AA + σ 2 πau wa + 2 σ2 λ 2 U λλ ]ds +(σπu w + σau A )db + σλu λ db λ. The righ hand side of (6) becomes where 3 U = U(, w, A, λ) + +h + +h 3 Uds + +h σπu w db σau A db + +h σλu λ db λ, U s + U w (rw + (µ r)π) + (µ α)au A +µλu λ + 2 σ2 π 2 U ww + 2 σ2 A 2 U AA + σ 2 πau wa + 2 σ2 λ 2 U λλ, and aking expecaions yields he las hree inegrals zero. Now (5) becomes U(, w, A, λ) E,w,A,λ [U(, w, A, λ) + +h 3 Uds] h p x+ +E,w [V (, w G) + +h 2 V ds] h q x+, (Advance online publicaion: 26 November 26)