A policyholder's utility indifference valuation model for the guaranteed annuity option
AA policyholder’s utility indifference valuationmodel for the guaranteed annuity option
Matheus R Grasselli ∗ Sebastiano Silla † October 29, 2018
Abstract
Insurance companies often include very long-term guarantees in par-ticipating life insurance products, which can turn out to be very valuable.Under a guaranteed annuity options ( g.a.o. ), the insurer guarantees toconvert a policyholder’s accumulated funds to a life annuity at a fixedrated when the policy matures. Both financial and actuarial approacheshave been used to valuate of such options. In the present work, we presentan indifference valuation model for the guaranteed annuity option. We areinterested in the additional lump sum that the policyholder is willing topay in order to have the option to convert the accumulated funds into alifelong annuity at a guaranteed rate.
J.E.L. classification.
D91; G11; J26.
Keywords.
Indifference Valuation; Guaranteed Annuity Option; g.a.o. ; Incom-plete Markets; Insurance; Life Annuity; Annuitization; Optimal Asset Allocation;Retirement; Longevity Risky; Optimal Consumption/ Investment; Expected Utility;Stochastic Control; Hamilton-Jacobi-Bellman equation.
Insurance companies often include very long-term guarantees in participatinglife insurance products, which can turn out to be very valuable. Guaranteedannuity options ( g.a.o. ) are options available to holders of certain policies thatare common in U.S. tax-sheltered plans and U.K. retirement savings. Underthese options, the insurer guarantees to convert a policyholder’s accumulatedfunds to a life annuity at a fixed rated when the policy matures. Comprehensiveintroductions to the design of such options are offered by O’Brien [43], Boyle& Hardy [8] [7], Hardy [19] and Milevsky [30]. For concreteness, we will focuson the analysis of a particular type of policy, but the framework we use can bereadily extended to more general products in this class. ∗ McMaster University, Canada. E-mail: [email protected]. † Polytechnic University of Marche, Italy. E-mail: [email protected] a r X i v : . [ q -f i n . P M ] A ug .1 The design of the policy We analyze a standard contract designed as follows: at time t = 0 the policy-holder agrees to pay a continuous premium at a rate P for an insurance policymaturing at T . The premium is deemed to be invested in a money marketaccount with continuously compounded interest rate r , and the policyholderreceives the corresponding accumulated funds A at time T . We are interestedin the additional lump sum L that the policyholder is willing to pay at time t in order to have the option to convert the accumulated funds A into a lifelongannuity at a guaranteed rate h .Between time t and time T , the liabilities associated with such guaranteedannuity options are related to changes in economic conditions and mortalitypatterns. A rational policyholder will only exercise the option at time T if itis preferable to the annuity rates prevailing in the market at that time. Asremarked by Milevsky and Promislow [33], the company has essentially grantedthe policyholder an option on two underlying stochastic variables: future interestrates and future mortality rates. The nature of guaranteed annuity options was firstly presented in Bolton et al.[6] and O’Brien [43]. The liabilities under guaranteed annuity options representan important factor that can influence the solvency of insurance companies. Ina stochastic framework, a first pioneering approach was proposed by Milevskiand Posner [32] and Milevsky and Promislow [33]. The literature concerning thevaluation of guaranteed annuity options in life insurance contracts has grownand developed in several directions. Both financial and actuarial approacheshandle implicit (“embedded”) options: while the formers are concerned withrisk-neutral valuation and fair pricing, the others focus on shortfall risk underan objective real-world probability measure. The interaction between these twoways was analyzed by Gatzert & King [16]. The seminal approach of Milevsky& Promislow [33] considered the risk arising both from interest rates and hazardrates. In this context, the force of mortality is viewed as a forward rate randomvariable, whose expectation is the force of mortality in the classical sense. Onthe same line, the framework proposed by Dahl [14] described the mortalityintensity by an affine diffusion process. Ballotta & Haberman [2] [3] analyzedthe behavior of pension contracts with guaranteed annuity options when themortality risk is included via a stochastic component governed by an Ornstein-Uhlenbeck process. Then, Biffis & Millossovich [4] proposes a general frameworkthat examines some of the previous contributions. For an overview on stochasticmortality, longevity risk and guaranteed benefits, see also Cairns et al. [9] [10],Pitacco [51] [50] and Schrager [53]. Finally, a different approach, based on theannuity price, was offered by Wilkie [58] and Pelsser [48] [49]. In particular,Pelsser introduced a martingale approach in order to construct a replicatingportfolio of vanilla swaptions. We also mention the related contributions ofBacinello [1], Olivieri & Pitacco [45], [46], [47] and Pitacco [26], Olivieri [44].2 .3 Objective of the paper
The present paper considers, for the first time to our knowledge, an indifferencemodel to value guaranteed annuity options. The indifference model proposedhere can capture at once the incompleteness characterizing the insurance marketand the theory of the optimal asset allocation in life annuities toward the endof the life cycle.The priciple of equivalent utility is built around the investor’s attitude to-ward the risk. Approaches based on this paradigm are now common in financialliterature concerning incomplete markets. In a dynamic setting the indifferencepricing methodology was initially proposed by Hodges and Neuberger [20], whointroduced the concept of reservation price . For an overview, we address thereader to the following contributions and to the related bibliography: Carmona[11], Musiela and Zariphopoulou [42], Zariphopoulou [63]. Recently Young andZariphopoulou [62] and Young [61], applied the principle of equivalent utility todynamic insurance risk.Our argument is inspired by the theory on the optimal asset allocation inlife annuities toward the end of the life cycle. For instance, we refer to Milewsky[40], [41], Milewsky & Young [36], [37], [38], [39], Milevsky et al. [31] and Blakeet al. [5]. The model is developed in two stages. First we compare two strategiesat time T , when the policyholder is asked to decide whether or not she wantsto exercise the guaranteed annuity option. Next, we go back to t and comparethe expected utility arising from a policy with the guaranteed annuity optionagainst a policy where no implicit options are included.Assuming a utility of consumption with constant relative risk aversion andconstant interest rates, we find that the decision to exercise the option at time T and the decision to purchase a policy embedding a guaranteed annuity optionreduce to compare the guaranteed rate h and the interest rate r . It turns outthat the indifference valuation is based on two quantities: the actual value ofthe guaranteed continuous life annuity, discounted by the implicit guaranteedrate, and the actual value of a perpetuity discounted by the market interestrate. The remainder of the paper is organized as follows. Section 2 describes thefinancial and actuarial setting where the model is defined. We characterize theoptimal exercise when the policy matures and the strategies at the initial time,when the policy is purchased. The end of this section gives the definition and theexplicit formula for the indifference valuation of the guaranteed annuity option.Section 3 show numerical examples for the equivalent valuation depending ondifferent scenarios for the interest rate. It also present a discrete-time versionfor the numerical simulations. 3
The model
We assume a policyholder who invests dynamically in a market consisting of arisky asset with price given by dS t = µS t dt + σS t dW t with initial condition S >
0, where W t is a standard Brownian motion on afiltered probability space (Ω , F , F t , P ) satisfying the usual conditions of com-pleteness and right-continuity, and µ and σ > dB t = rB t dt with initial condition B = 1, where r is a constant representing the continuouslycompounded interest rate.The policyholder is assumed to consume at a instantaneous rate c t (cid:62) x andthe process X t will denote the wealth process for t (cid:62)
0. At each time t (cid:62) π t to invest in the risky assetand, consequently, the amount X t − π t to be invested in the risk free asset.The processes c t and π t need to satisfy some admissibility conditions, which wespecify in the next sections.The assumed financial market follows the lines of Merton [27], [28], [29] andcan be generalized, at cost of less analytical tractability, following the contribu-tions provided, for example, by Trigeorgis [56], Kim and Omberg [21], Koo [22],Sørensen [54] and Wachter [57]. For example, in Grasselli & Silla [18] a shortnote with a non-stochastic labor income is considered. Consider an individual aged χ at time 0. We shall denote by s − t p Sχ + t the sub-jective conditional probability that an individual aged χ + t believes she willsurvive at least s − t years (i.e. to age χ + s ). We recall that s − t p Sχ + t canbe defined through the force of mortality. Let F χ + t ( s ) denotes the cumulativedistribution function of the time of death of an individual aged χ + t . Assumingthat F χ + t has the probability density f χ + t , the force of mortality at age χ + t + η is defied by λ Sχ + t ( η ) := f χ + t ( η )1 − F χ + t ( η ) , which leads to s − t p Sχ + t = e − R s − t λ Sχ + t ( η ) d η λ Sχ + t ( η ) = λ Sχ + t + η (0) (see Gerber [17]), it isuseful to denote λ Sχ + t ( η ) by the symbol λ Sχ + t + η . This leads to write s − t p Sχ + t = e − R st λ Sχ + η d η (1)For the numerical simulations below, we will assume a Gompertz’s specifi-cation for the force of mortality λ S : λ Sχ + η := 1 ς exp (cid:18) χ + η − mς (cid:19) Similar formulas are given for both the objective conditional probability ofsurvival s − t p Oχ + t and the objective hazard function λ O . Employing the methodproposed by Carriere [13], we estimate the parameters m and ς in Table 1 usingthe Human Mortality Database for the province of Ontario, Canada, for a femaleand a male both aged 35 in the years 1970 and 2004.Table 1: Estimated Gompertz’s parameters for a female and a male from theprovince of Ontario, Canada, conditional on survival to age 35. Source: Cana-dian Human Mortality Database available for year 1970 and 2004. Female MaleYear m ς m ς
In a continuous compounding setting with a constant interest rate r , the(present) actuarial value of a life annuity that pays at unit rate per year for anindividual who is age χ + t at time t , is given by a χ + t := (cid:90) + ∞ t e − r ( s − t ) s − t p Oχ + t ds where the survival probability is determined considering the objective mortalityassessment from the insurer’s point of view.For a given a χ + t , an individual endowed by a wealth x > x/ a χ + t unit rate annuities, corresponding to a cash–flow stream at a nominalinstantaneous rate H := x/ a χ + t . This defines a conversion rate h := 1 / a χ + t ,at which an amount x can be turned into a life long annuity with an incomestream of H = xh per annum.Notice that the conversion rate h imply a technical nominal instantaneousrate r h defined by the following expression:1 h = a ( h ) χ + t := (cid:90) + ∞ t e − r h ( s − t ) s − t p Oχ + t ds p O .Returning to our g.a.o. policy, in order to offer a given conversion rate h tobe used by the policy holder at time T , the insurer considers the interest andmortality rates based on information available at time t . However, improve-ments in mortality rates and the decline in market interest rates may representan important source of liabilities for the insurer. For instance, if at time T , theinterest rates will be below the technical rate r h and the policyholder decidesto exercise the guaranteed annuity option, the insurer has to make up the dif-ference between the two rates. Figure 1 plots the implicit rate r h with respectto different values for the conversion rate of h . The same figure also shows theimpact of the so called longevity risk : taking h = 1 / r h = 0 . r h risesto 0 . p O is made by the mortality tables availablein 2004. Hence, as remarked by Boyle & Hardy [7], if mortality rates improveso that policyholders live systematically longer, the interest rate at which theguarantee becomes effective will increase.Figure 1: Simulated implicit rate r h , assured by the an insurer in 1970 withrespect to different values for the guaranteed conversion rate h . The policyholderis supposed to be a 35 years old female, from the province of Ontario, whowill be 65 at the time T of retirement. Values for r h are compared (with anapproximation of 1E-04) using a Gompertz’s mortality function with (objective)parameters driven by survival tables available in 1970 (solid line) and in 2004(dashed line). Guaranteed conversion rate I m p li ed d i sc oun t r a t e .3 The valuation method As outlined in the introduction, our valuation method will be based on twosteps, motivated by the following two remarks:1. provided the guaranteed annuity option has been purchased at time t = 0,the policyholder needs to decide on whether or not to exercise it at time T ;2. assuming an optimal exercise decision at time T , the policyholder needsto decide how much she would pay at time t to embed such an option inher policy.In order to obtain a well-posed valuation for this option, we need to assumethat the purchased insurance policy does not include any other guarantees orrights. We also assume that the conversion period (i.e. the time interval inwhich the agent is asked to take a decision on whether to exercise the option)reduces to the instant T . T At time T , if the policyholder owns a guaranteed annuity option, she will beasked to take the decision to convert the accumulated funds in a life long annuityat the guaranteed rate, or to withdraw the money and invest in the market.Therefore, we need to compare the following two strategies: i ) If she decides to convert her accumulated funds A > h , she will receive a cash flow stream at a rate H = A · h per annum. In this case, we assume that, from time T , she will be allowedto trade in the financial market. Henceforth, her instantaneous incomewill be given by the rate H and by the gains she is able to realize bytrading in the financial market. ii ) On the other hand, if the policyholder decides not to convert her fundsinto the guaranteed annuity, we assume she can just withdraw funds A and go in the financial market. In this case, from time T , her incomewill be represented just by the market gains she can realize. Her totalendowment at the future time T will be then increased by the amount A .Since the accumulation phase regards the period [ t , T ), we assume thatjust before the time T the policyholder’s wealth is given by X T − = x T >
0. Ifshe decides to convert her accumulated funds exercising the guaranteed annuityoption, the problem she will seek to solve is to maximize the present value ofthe expected reward represented by value function V defined as follows: V ( x T , T ) := sup { c s , π s } E (cid:20)(cid:90) + ∞ T e − r ( s − T ) s − T p Sχ + T · u ( c s ) d s (cid:12)(cid:12)(cid:12)(cid:12) X T = x T (cid:21) u is the policyholder’s utility of consumption, which is as-sumed to be twice differentiable, strictly increasing and concave, and the wealthprocess X t satisfies, for all t (cid:62) T , the dynamics dX t = r ( X t − π t ) dt + π t ( dS t /S t ) + ( H − c t ) dt, = [ rX t + ( µ − r ) π t + H − c t ] dt + σπ t dW t , (2)with initial condition X T = x T . On the contrary, if the policyholder decidesnot to exercise the option, she withdraws the accumulated funds A at time T and to solve a standard Merton’s problem given by: U ( x T + A, T ) := sup { c s , π s } E (cid:20)(cid:90) + ∞ T e − r ( s − T ) s − T p Sχ + T u ( c s ) d s (cid:12)(cid:12)(cid:12)(cid:12) X T = x T + A (cid:21) with initial condition X T = x T + A and subject to the following dynamics: dX t = r ( X t − π t ) dt + π t ( dS t /S t ) − c t dt, = [ rX t + ( µ − r ) π t − c t ] dt + σπ t dW t , (3)We assume the control processes c t and π t are admissible, in the sense thatthey are both progressively measurable with respect to the filtration { F t } . Also,the following conditions hold a.s. for every t (cid:62) T : c t (cid:62) , (cid:90) tT c s ds < ∞ and (cid:90) tT π s ds < ∞ (4)At time T the policyholder compares the two strategies described above andthe respective expected rewards. We postulate she will decide to exercise theguaranteed annuity as long as U ( x T + A, T ) (cid:54) V ( x T , T )The previous analysis, regarding the function U , considers a policyholderthat holds a policy embedding a guaranteed annuity option. A third strategyneeds to be considered in order to describe the case in which the policyholderholds a policy with no guaranteed annuity option embedded in it: iii) If the policy does not embed a guaranteed annuity option, the policyholderdoes not have the right to convert the accumulated funds A into a lifelongannuity. In this sense, we assume that value function U will represent theexpected reward if at time t the policyholder purchased a plan withoutthe guaranteed annuity option. t After defining the optimal exercise at time T , we can formalize the policyholder’sanalysis at the initial time t , when the guaranteed annuity option may be em-bedded in her policy. We can summarize the two strategies that the policyholderfaces at time t as follows: 8 ) the policyholder purchases a policy without embedding a guaranteed an-nuity option in it. In this case, she will pay a continuous premium at anannual rate P to accumulate funds A up to time T ; ii ) the policyholder decides to embed the guaranteed annuity option in herpolicy. She will pay a lump sum L for this extra benefit immediately andthe continuous premium P for the period [ t , T ), as in previous case.In either case, the value of the accumulated funds is given by A = (cid:90) Tt e r ( T − s ) P ds
Assuming the policyholder’s income is given by the gains she can realizetrading in the stock market, between time t and time T , her wealth needs toobey to the following dynamics: dX t = r ( X t − π t ) dt + π t ( dS t /S t ) − ( P + c t ) dt = [ rX t + ( µ − r ) π t − P − c t ] dt + σπ t dW t (5)with initial condition X t = x >
0. Therefore, if the agent decides not toembed the guaranteed annuity option in her policy she will seek to solve thefollowing optimization problem U ( x , t ) := sup { c s , π s } E (cid:34)(cid:90) Tt e − r ( s − t ) s − t p Sχ + t · u ( c s ) ds ++ e − r ( T − t ) T − t p Sχ + t · U ( X T + A, T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t = w (cid:35) On the contrary, if she decides to embed a g.a.o. in her policy, paying thelump sum L at time t , her wealth is still given by dynamics (5), but themaximization problem will be different, namely for w − L > V ( x − L , t ) := sup { c s , π s } E (cid:34)(cid:90) Tt e − r ( s − t ) s − t p Sχ + t · u ( c s ) d s ++ e − r ( T − t ) T − t p Sχ + t max (cid:8) U ( X T + A, T ; r ) , V ( X T , T ) (cid:9) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t = x − L (cid:35) Notice that at time T the policyholder can an either exercise the option, remain-ing with the wealth X T plus the lifelong annuity obtained from converting A atthe rate h , or decide not to exercise the option and withdraw the accumulatedfunds A .The same admissibility conditions are required for the control processes c t and π t during the accumulation period, namely they are both progressivelymeasurable with respect to F t and satisfy (4)9e postulate that the agent will decide to embed a guaranteed annuityoption in her policy as long as the following inequality holds: U ( x , t ) (cid:54) V ( x − L , t ) U ( X T + A, T ) (cid:54) V ( X T , T ) Given a wealth X T at time T , Grasselli & Silla [18, App. A] find an explicit so-lution for a class of problems regarding value functions U , V , U and V , assuminga constant relative risk aversion (CRRA) utility function defined by u ( c ) = c − γ − γ , γ > , γ (cid:54) = 1 (6)and a constant interest rate satisfying the condition r > (1 − γ ) δ where δ := r + 1 / (2 γ ) · ( µ − r ) /σ . Namely, the value functions U and V are given by: U ( X T + A, T ) = 11 − γ ( X T + A ) − γ · ϕ γ ( T ) (7) V ( X T , T ) = 11 − γ (cid:18) X T + Hr (cid:19) − γ · ϕ γ ( T ) (8)where ϕ is is given by ϕ ( T ) = (cid:90) + ∞ T e − b ( s − T ) · s − T p Sχ + T ds (9)for b := − [(1 − γ ) δ − r ] /γ . Notice that for every γ > γ (cid:54) = 1, we have U ( X T + A, T ) (cid:54) V ( w T , T ) ⇔ r (cid:54) h From an economic point of view, the previous inequality tells us that, at timeof conversion T , the policyholder will find convenient to exercise the guaranteedannuity option if and only if the guaranteed rate h is greater than the prevailinginterest rate r . Moreover, recalling that 1 /h = a ( h ) χ + T , the previous inequalitycan be also written as follows: U ( X T + A, T ) (cid:54) V ( X T , T ) ⇔ a ( h ) χ + T (cid:54) /r, which says that in order to come to a decision the policyholder compares theguaranteed cost of a unit rate lifelong annuity (assured by the insurance com-pany), whose the present value is given by a guaranteed implicit rate r h , withthe market cost of a unit rate perpetuity, whose present value is determinedby the market interest rate r . Notice that the indifference point is given by a ( h ) χ + T = 1 /r , highlighting the absence of bequest motives for the policyholderafter time T . 10 .7 A closed form for value functions U and V Combining the results of the previous two sections, we have that the valuefunction U at time T needs to be equal to g ( X T ) = ( X T + A ) − γ − γ ϕ γ ( T ) . Using the change of variables technique proposed in Grasselli & Silla [18, App.B], we find that the value function U is given by U ( x , t ) = 11 − γ (cid:16) x − (cid:98) ξ U ( t ) (cid:17) − γ ϕ γ ( t ) (10)where (cid:98) ξ U is defined by (cid:98) ξ U ( t ) = Pr (cid:16) − e r ( t − T ) (cid:17) − A · e r ( t − T ) . (11)Similarly, we have that the value function V , at time T , needs to be equal to G ( x T ) := max (cid:8) U ( x T + A, T ) , V ( x T , T ) (cid:9) = ( X T + A ) − γ − γ ϕ γ ( T ) , if r (cid:62) h ( X T + H/r ) − γ − γ ϕ γ ( T ) , if r < h Using the same change of variables technique, we arrive at the following expres-sion for the value function V : V ( w , t ) = U ( w , t ) if r (cid:62) h − γ (cid:16) w − (cid:98) ξ V ( t ) (cid:17) − γ ϕ γ ( t ) if r < h where (cid:98) ξ V is given by (cid:98) ξ V ( t ) = Pr (cid:16) − e r ( t − T ) (cid:17) − Hr · e r ( t − T ) (12) Consider the policyholder that, at time t , compares the two expected rewardsarising from the value functions U and V , and define the indifference value forthe guaranteed annuity option by L ∗ := sup (cid:8) L : U ( w , t ) (cid:54) V ( w − L , t ) , w − L > (cid:9) If the indifference value exists, it is straightforward to deduce that it is given by L ∗ = (cid:18) Hr − A (cid:19) e − r ( T − t ) .9 Stochastic interest and mortality rates As we mentioned in the introduction, the liabilities associated with guaranteedannuity options depends on the variations of interests rates and mortality ratesover the time. In this sense a richer model has to take into account and toformalize these rates as stochastic processes. In present section we will justoffer a sketch for stochastic models for mortality intensity.The debate over the stochastic mortality is very prolific and the literatureconcerning this problem is huge. For what concerns in particular mortalitytrends and estimation procedure, we recall for example: Carriere [12] and [13],Frees, Carriere and Valdez [15], Stallard [55], Willets [59] and [60], Macdonaldet al. [24] and Ruttermann [52]. In what concern stochastic diffusion processesto model the force of mortality, excellent contributions are offered by: Lee [23],Pitacco [51], [50], Olivieri and Pitacco [46], [47], [45], Olivieri [44], Dahl [14],Schrager [53], Cairns et al. [9], Marceau Gaillardetz [25], Milevsky, Promislowand Young [34], [35]. We also recall that the approach followed by Milevskyand Promislow [33] was the pioneering contribution that consider at one timeboth the stochastic mortality and a financial market model, in order to pricethe embedded option to annuitise (what we call guarantee annuity option).The contribution by Dahl [14], propose to model the mortality intensityby a fairly general diffusion process, which include the mean reverting modelproposed by Milevsky and Promislow [33]. Precisely the author consider a Pdynamics for the mortality intensity given byd λ χ + s = α λ ( s, λ χ + s ) d s + σ λ ( s, λ χ + s ) d (cid:102) W s (13)where α λ and σ λ are non-negative and { (cid:102) W s } is a standar Wiener process withrespect to the same filtration { F s } , defined above, for s (cid:62) t . { (cid:102) W s } is assumeduncorrelated with { W s } .In order to avoid analytical difficulties, we investigate the effect of varyingmortality rates by comparing different scenarios for different survival proba-bilities. In particular, the next section highlights the effect of different pa-rameterized functions describing different specifications concerning the force ofmortality. Consider t = 0 and, at this time, a female aged χ = 35 who is willing topurchase a policy. Also, suppose that this plan will accumulate, until time T : = 30 (i.e. when the policyholder will be aged χ + T = 65) an amount A : = $350 , T may coincidewith her retirement time and that the purchase takes place in 1975. In thiscontext, the g.a.o. (if the agent decides to embed such an option in her policy)12igure 2: Value function U (solid) and value function V (dashed), for an indi-vidual characterized by γ = 1 .
4, that observes a financial market described by r = 0 . µ = 0 . σ = 0 .
12. The value of r and µ are taken large enough tosimulate the 1970’s financial market. In this setting we find L ∗ = 25 , g.a.o. exercisable in 2005, for a female in year 1970, fromthe province of Ontario, assuming a (subjective) mortality specification givenby the survival table available in 1970, see Table 1. Initial wealth E x pe c t ed u t ili y could be exercised in 2005. We would like to stress that these calendar datesare not necessary to implement a numerical experiment. However they give astronger economic meaning for a contract designed as follows: we assume thatthe agent is asked to decide whether to include a guaranteed annuity optionwith a conversion rate h := 1 / H ≈ $38 , .
89 per year.Notice that, in this situation, if we refer to survival tables available in 1970(see table 1), the implicit discount rate is r h ≈ . U and V are plotted infigure 2, where we assume a Gompertz’s mortality specification. We estimatethe parameters ς and m , minimizing a loss function using the method proposedby Carriere [13]. We refer to the Human Mortality Database for the province ofOntario, Canada, for a female and a male both aged 35 using tables availablein 1970 or in 2004. The results of our estimations are summarized in Table 1.For some values of the market interest rate r , Table 2 shows the premium P and the equivalent valuation L ∗ for this policy. Figure 3 depicts the dependencyof L ∗ on both the guaranteed conversion rate h and the interest rate r . Asexpected, the greater the interest rate, the lower the policyholder’s indifferenceprice for the option. Also, the analysis remains consistent with respect to h :the lower the guaranteed rate, the lower the agent’s indifference price.13able 2: Premium and indifference valuation associated to the policy, dependingon the current interest rate. r P L ∗ p l Total0.035 $6,594 $266,342 $550 $419 $9690.050 $5,026 $ 95,450 $420 $115 $5350.085 $2,519 $ 8,395 $211 $ 5 $216
Depending on r , Table 2, shows the nominal instantaneous rate for thepremium P (that the policyholder needs to pay to in order to accumulate A = $350 , L ∗ for the g.a.o. . Notice thatit is not immediately possible to compare L ∗ and P since the former denotesa lump sum, while the latter refers to a nominal instantaneous rate to be paidover time.In order to better understand the meaning of P and L ∗ , it can be usefulto think of an auxiliary problem. This problem is independent of the previousindifference model, but will offer a way to validate the previous results. To dothis, consider a premium to be payed monthly for a pension or an insuranceplan. We can ask two questions: what is the value p of a monthly paymentwhose the future value, after 30 years, is exactly A ; and what is the value ofa monthly payment l necessary to amortize, after 30 years, the lump-sum L ∗ payed at t = 0.In order to compute l , consider a horizon of T ×
12 months. Thus l isgiven by the following relation: L ∗ = l · a T × i where i := e r/ − e r , and where in general we define a n i := 1 − (1 + i ) − n i as the present value of an annuity that pays one dollar for n periods, discountedby the effective interest rate i compounded each period. Similarly, define p such that A = p · s T × i where s n i := (1 + i ) n − i = (1 + i ) n · a n i represents the future value after n periods, of an annuity that pays one dollarper period, under an effective interest rate i compounded each period.Coming back to Table 2 it is interesting to see that for r = 0 . L ∗ . Set-ting r = 0 . L ∗ depending on the guaranteed conversion rate h and the market interest rate r . The valuation is given for a g.a.o. exercisablein 2005, for a female in year 1970, from the province of Ontario, assuming a(subjective) mortality specification given by the survival table available in 1970,see Table 1. Conversion rate.Interest rate I nd i ff e r e v a l ua t i n f o r t he g . a . o . and a monthly stream of only $5. These intuitive results are consistent with theliterature concerning the guaranteed annuity option. As mentioned by Boyle &Hardy [8], these guarantees were popular in U.K. retirement savings contractsissued in the 1970’s and 1980’s, when long-term interest rates were high. Thesame authors also write that at that time, the options were very far out–of–the–money and insurance companies apparently assumed that interest rates wouldremain high and thus the guarantees would never become active. As a result,from the indifference model discussed in the present paper, when the interestrate is very high - as was the case in the 1970’s and 1980’s - the guaranteedannuity option’s value, from the point of view of the policyholder, is very small.Interestingly, in the same period, empirically it was observed that a very smallvaluation was also given by insurers.These facts are confirmed by the extremely low value of L ∗ = $8 ,
395 (over T − t = 30 years), against the yearly nominal premium P = $2 , L ∗ can be amortized by a monthly cash flow of $5, against a monthly equiv-alent premium of $211. Moreover, p and l by construction are homogeneousquantities. Their sum gives an idea of the equivalent monthly value associatedto the policy the agent is willing to buy at time t . This sum is showed in thelast column of Table 2. It is interesting to note the large difference between thetotal value corresponding to r = 0 .
035 compared to r = 0 . .2 Valuation under different mortality scenarios Through our analysis we consider a deterministic process for the force of mor-tality. Under this assumption, the indifference valuation – in line with theprevious literature – depends on the difference between the interest rate r andthe guaranteed rate h . However it is interesting to simulate the effect arisingfrom different mortality rates, because even if the indifference value (at time t ) is still given by L ∗ , the value functions U and V change. Figure 4 show theeffect of assuming different mortality specifications.Figure 4: Value function U (solid) and value function V (dashed), for a guar-anteed annuity option maturing in 2005, for a 35 years old female in year 1975,from the province of Ontario, comparing a (subjective) mortality specificationgiven by the survival table available in 1970 (light lines) and in 2004 (demi-boldlines), see Table 1. The policyholder is characterized by γ = 1 .
4, observing afinancial market described by r = 0 . µ = 0 . σ = 0 .
12. The value of r and µ are taken large enough to simulate the 1970’s financial market.
0 125 250-28-19-10
Initial wealth E x pe c t ed u t ili t y V U (a) Value functions U and V .
35 45 55 65 75 85 95 1050 0.005 0.010 0.015 0.020
Age P r obab ili t y den s i t y o f da t h (b) Density of death. For instance, we compare the optimal expecter reward considering the samepolicyholder under a different subjective assessment of the survival probability– notice that in both cases we assume deterministic process for p S , consideringdifferent scenarios. To make things easy, we keep referring to Table 1, comparingthe estimations available in 1970 and in 2004. The same value functions plottedin figure 2 are compared with the ones calculated using data available in 2004.In other words, if the policyholder could use a more optimistic assessmentfor the survival probability, the convenience to by the option remains the same(i.e. V > U ) but the expected utility is affected in the change in the value of ϕ and the negative value of 1 − γ . The coefficient γ expresses the policyholder’srisk aversion over a larger trading horizon. For instance, the gap between the“new” V and the “old” V (as well as the “new” U and the “old” U ) reduces for16maller values of γ . Eventually, for γ →
0, the policyholder does not suffer anyimpact from different mortality scenarios. Notice, however, that this claim istrue because different scenarios do not change during the trading period, thatis, there is no stochasticity other than the perturbation considered just at theinitial time t . We value the guaranteed annuity option using an equivalent utility approach.The valuation is made from the policyholder’s point of view. In a setting whereinterest rates are constant, we find an explicit solution for the indifference prob-lem, where power–law utility of consumption is assumed. In this setting wecompare two strategies when the policy matures, and two strategies at the ini-tial time. For the former we assume that, if the annuitant does not exercise theoption, she first withdraws her accumulated funds and then she seeks to solve astandard Merton problem under an infinite time horizon case. At the time whenthe policy matures, we compare the policyholder’s expected reward associatedto a policy embedding a guaranteed annuity option, and the one which arisefrom a policy that does not embed such an option. We find that the option’s in-difference price depends on the difference between the market interest rate r andthe guaranteed conversion rate h . Numerical experiments reveals that in periodscharacterized by high market interest rates, the value of the g.a.o. turns out tobe very small. Finally, we also consider an auxiliary (and independent) problemin which we compare the pure premium asked by the insurance company (foraccumulating the funds up to the time of conversion) and the indifference pricefor the embedded option.For future research, the present model can be generalized in several ways.First, the policyholder can be allowed to annuitize her wealth more than onceduring her retirement period. This fact leads us to consider an unrestrictedmarket where the policyholder can annuitize anything at anytime, as defined byMilevsky & Young [39]. Second, the financial market can be modeled consideringa richer setting: stochastic interest rates and stochastic labor income. To thisend, we recall the work of Koo [22]. Third, and most important, in the presentframework, the longevity risk is considered by comparing different scenarios,given by the survival tables available in 1970 and in 2004. For this, a moregeneral stochastic approach, as proposed by Dahl [14], can be considered instead. References [1] Anna Rita Bacinello. Fair valuation of life insurance contracts with embed-ded options.
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