OPTIMAL TIMING OF THE ANNUITY PURCHASES: A

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1 OPTIMAL TIMING OF THE ANNUITY PURCHASES: A COMBINED STOCHASTIC CONTROL AND OPTIMAL STOPPING PROBLEM Gabriele Stabile 1 1 Dipartimento di Matematica per le Dec. Econ. Finanz. e Assic., Facoltà di Economia Università degli Studi La Sapienza di Roma gabriele.stabile@uniroma1.it) ABSTRACT: We consider the optimal behaviour of a retired individual, who can invest her wealth in the financial market or, in alternative, can purchase an annuity. The objective is to maximize the expected discounted utility deriving from consumptions during the investment phase and from the annuity payments flow, over the controls which are given by the time of the annuity purchase and by the consumption/portfolio policies. This is a combined stochastic control and optimal stopping problem. Using dynamic programming we give a characterization of the optimal controls for a general form of utility functions and annuity payments. Assuming power utility functions and a particular functional form for the annuity payments, we explicitely exhibit the value function and the optimal controls. For different degrees of relative risk aversion, we determine a threshold such that the individual purchases the annuity when the individual s wealth reachs it. In this case the individual s best strategy is to invest her wealth in the financial market, so to defer the annuitization until the wealth reachs the threshold. KEYWORDS: Annuities, Stochastic control, Optimal stopping, HJB equation. 1 The model This paper examines the issue of how individuals invest their wealth in the retirement age. They can optimally invest their wealth in the financial market, while withdrawing a consumption stream, until they decide to purchase a life annuity that will guarantee a life-long income. The question the paper address is: what is the optimal time to purchase the annuity for risk-averse individuals? This decision has important economic implications because it has a direct effect on how well prepared individuals are to provide consumptions in their old age. Individuals face the risk of having to reduce consumption in later years. Bequest motives will not be considered, so the problem of intergenerational transfers will not be addressed. In order to investigate this question we implement a stochastic life-cycle model seminal papers for the life-cycle models are Modigliani & Brumberg, 1955; Yaari, 1965; see also Ehrlich, 2000; Pressacco, 1984.) There are two sources of uncertainty in the model: the return of the financial market and the individual s lifetime. We analyze our problem on a complete probability space Ω,F,P) with a filtration F t) satisfying the usual condition and carrying a standard one-dimensional F t))- brownian motion Wt). We consider a complete financial market composed by two assets:

2 one riskless bond), the other risky stock). The evolution of this assets is governed respectively by the following equations: dbt) = rbt)dt B0) = B 0 1) dst) = St)µdt + σdwt)) S0) = S 0 2) The individual evaluates her life s expectation calculated from the retirement time) by means of the random variable T. We assume that lt) := P[T > t] = e θt, t > 0 where θ > 0. The annuity payment R is calculated from the insurer via the formula Rt) = R r, t p y,l y xt) ) 3) where R is C 2 R + ), with R 0) = 0 and dr dx > 0. t p y and L y are respectively the conditional probability that an individual aged y survives to age y +t, and the costs charged by the insurer. The fact that the individual and the insurer use different survival probabilities is a conseguence of the mechanism of adverse selection. At the retirement time, the individual is endowed with a certain wealth x > 0 the savings accumulated during the working age). She can imply her wealth for purchase an annuity or, in alternative, she can enter in the financial market deferring the time of the annuitization. When the individual decides to buy the annuity, we assume that she implies all the wealth she has at that time in the annuity purchase. The life annuities we consider provide constant long-life income. Between the retiment time and the time of the annuity purchase, the individual has to plan a consumption rule c ) we suppose that she has no other source of wealth) and a portfolio rule π ). From 1) and 2), the wealth of the individual evolves according to dxt) = xt) [ r + πt)µ r) ] dt + σπt)xt)dwt) ct)dt x0) = x, The individual evaluates the discounted expected utility that derives from consumption during the investments in the financial market or from the annuity payments flow. Let u and p the utility functions used by the individual to assess these two consumption streams. u, p : R + R + are supposed to be strictly increasing, strictly concave, twice continuosly differentiable and satisfying the Inada conditions. We define the expected total utility associated to c,π,τ) as ] [ZT J c,π,τ x) = E x e δt [uct))χ t τ + pr xτ)))χ τ<t ]dt 0 4)

3 ] [Rτ or equivalently J c,π,τ x) = E x 0 e βt uct))dt + e βτ gxτ)) where β = δ + θ, gx) = 1 β prx)). The individual wishes to maximize her expected discounted utility over all triples c,π,τ), and the value function is denoted by vx) = sup J c,π,τ x) x > 0. 5) c,π,τ) 2 A verification theorem and explicit solutions in the case of constant relative risk aversion From a mathematical point of view the problem is formulated as a combined stochastic control and optimal stopping problem cf. Chancelier et al., 2001; Karatzas & Wang, 2000). The Markovian setting allows us to cast the optimization problem as a free boundary problem, based on the associated Hamilton-Jacobi-Bellman HJB) equation of dynamic programming. We prove a verification theorem that provides a sufficient condition to characterize the value function and the optimal control processes, but we are not able to find the explicit solutions. Theorem 1 Let y C 1 R + ) with the second derivative continuous almost everywhere in R +, be a solution of max Lyx),gx) yx) = 0, x R +, 6) where the operator L is defined as [x ) ] 1 Lyx) := sup r + πµ r) c y x) + 2 σ2 π 2 x 2 y x) + uc) βyx), c,π) R + R such that for all x0) = x > 0 and π, the process Zt e βs σπs)xs)y xs))dws), t 0, 0 is a martingale. Then a) yx) vx), x R +. b) Moreover, assume also that y ) is strictly concave and define the set D = x R + : yx) > gx) ; 7) define the policies τ = inf t 0 : x t) D, c t) = I y x t))) ) χ t τ, π t) = µ r σ 2 y x t)) x t)y x t)) χ t τ,

4 where x t) is the solution of 4) with controls c,π ), and I is the inverse of u x). If [ ] lim yx t))χ τ >t = 0, t e βt E then c,π,τ ) is the optimal controls triple and The proof of this theorem is omitted. vx) = yx) x R +. Notice that D, defined in 7), is the so-called continuation set since it is not optimal to stop the diffusion process until xt) exits from D. From 6) we have Lvx) = 0, x D and vx) = gx), x R + \ D Identifying the value function in C 1 R + ) implies the so called smooth fit principle: v x) = g x), x D. 8) We note that, in force of the lower semicontinuity of vx) and the strictly concavity of the obstacle function gx), the continuation region D is an open and connected set in R +. In order to investigate the form of D, we make the following assumption: HARA Hyperbolic Absolut Risk Aversion) utility functions and the function Rx) defined in 3) is linear with respect to the wealth. ux) = x α, px) = x ν α,ν 0,1), Rx) = Fx, F > 0. Define U = x R + : Lgx) > 0. Observe that U D, then it is never optimal to stop the process before it exits from U. [ ] Let M = Fν β β rν γ ν 1 ν, N = νf ν αβ ) α α 1 1 α) and h = MN ) α 1 ν α, where γ := 1 2 µ r σ )2. The degrees of relative risk aversion determine the form of the set U: ν = α = U = R + se N > M, altrimenti U = /0; ν < α = ν α α 1 > 0, U = h,+ ); 9) 2.1 Equal degree of risk aversion α = ν) In this case the individual has the same preferences to assess the consumption streams from the financial market and from the annuity market. From 9) we see that, depending on the values of the quantities M and N, there are two different cases which we study separately.

5 2.1.1 The continuation region coincides with R + the case N > M) From 9) we know that D = R + if N > M. This implies that τ = +, so the diffusion is never arrested and the problem 5) becomes the classical Merton s infinite horizon consumption/investment problem. The value function and the optimal control policies are respectively 1 α ) 1 α vx) = x β rα γ 1 α α α, x 0, and c x) := Cx) = 1 π := µ r 1 α)σ 2 1 α ) β rα γ 1 α α x, x > The continuation region coincides with the empty set the case N M) In this case we have that D = /0 then the optimal stopping time is τ = 0. Thus the individual optimizes her utility function by immediately purchasing the annuity. The parameters of the economy interest rate, subjective discount rate, etc...) are such that the returns from the financial market are not attractive for the individual. 2.2 Higher degree of relative risk aversion in the annuity assesment phase the case α > ν) We have that the continuation region D = ˆx, ) is an open interval unbounded on the right. We solve the control problem 5) by the same method as in Karatzas et al 1986). We guess that the value function 5) is given by yx) se x > ˆx vx) = gx) otherwise, where y ) C 2 D) is a strictly concave function satisfying the smooth fit principle 8). We define the processes τ = inf t 0 : x t) ˆx, ), c t) = Cx t)) ) χ t τ, π t) = µ r c t) 1 α)σ 2 x t)c t) χ t τ, where C ) is a twice continuously differentiable function, increasing and convex respect to the individual s wealth. By applying Theorem 1, we verify that these are respectively the value function and the optimal control processes. Notice that if β r + γ = Pτ < ) = 1, otherwise Pτ < ) 0,1).

6 It is interesting to qualitative describe the individual s optimal behavior. If the initial wealth is greater than ˆx, it is optimal for the individual to invest in the financial market according to the portfolio rule π, and to consume at rate c. If the returns from the financial market are such that the wealth increases, then the rate of consume increases and the individual rebalances her portfolio continuously in favour of the riskless asset. In the limit, when the individual s wealth goes to infinity, the portfolio rule converges to the constant described by Merton, Viceversa, if the retuns from the financial market are such that the wealth decreases, the rate of consume decreases as well, and the individual increases the portfolio s exposure to the risky asset, accepting so doing an increasing financial risk. When the wealth reachs the threshold ˆx, the individual has a low rate of consume but, because of her risk aversion, she does not want to increase the exposure to the risky asset. Then, the phase of investment in the financial market ends, and the individual implies her wealth in the annuity purchase. If the initial wealth is lower that ˆx, the individual immediately buys the annuity. References CHANCELIER, J-P., OKSENDAL, B., & SULEM, A Combined stochastic control and optimal stopping, and application to numerical approximation of combined stochastic and impulse control. Page 26 of: Proceedings of the Steklov Mathematical Institute on Mathematical Finance. EHRLICH, I Uncertain lifetime, life protection, and the value of life saving. Journal of Health Economics., 19, KARATZAS, I, & WANG, H A barrier option of american type. Applied Mathematics & Optimization., 642, KARATZAS, I, LEHOCZKY, J P, SETHI, S P, & SHREVE, S E Explicit solution of a general consumption/investment problem. Mathematics of Operations Research., 11, MERTON, R Optimum consumption and portfolio rules in a continuos-time model. Journal of Economic Theory., 3, MODIGLIANI, F, & BRUMBERG, R Utility analysis and the consumption function: an interpretation of cross-section data. K.K Kurihara ed., Post-Keynesian economics Allen and Unwin, London). PRESSACCO, F A life-cycle model of life insurance purchases. Metroeconomica., 36, YAARI, M E Uncertain lifetime, life insurance and the theory ot the consumer. Review of Economic Studies., 32,