Maximizing the utility of consumption with commutable life annuities

Insurance: Mathematics and Economics 51 (2012) 352–369

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Maximizing the utility of consumption with commutable life annuities Ting Wang, Virginia R. Young ∗ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States

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Article history: Received February 2012 Received in revised form May 2012 Accepted 3 June 2012 Keywords: Commutable annuities Utility maximization Retirement Optimal investment Optimal consumption Stochastic control Impulse control Singular control Free-boundary problem

abstract The purpose of this paper is to reveal the relation between commutability of life annuities and retirees’ willingness to annuitize. To this end, we assume the existence of commutable life annuities, whose surrender charge is a proportion of their actuarial value. We model a retiree as a utility-maximizing economic agent who can invest in a financial market with a risky and a riskless asset and who can purchase or surrender commutable life annuities. We define the wealth of an individual as the total value of her risky and riskless assets, which is required to be non-negative during her lifetime. We exclude the actuarial value of her annuity income in calculating wealth; therefore, we do not allow the individual to borrow from her future annuity income because this income is contingent on her being alive. We solve this incomplete-market utility maximization problem via duality arguments and obtain semi-analytical solutions. We find that the optimal annuitization strategy depends on the size of proportional surrender charge, with lower proportional surrender charges leading to more annuitization. We also find that full annuitization is optimal when there is no surrender charge or when the retiree is very risk averse. Surprisingly, we find that in the case for which the proportional surrender charge is larger than a critical value, it is optimal for the retiree to behave as if annuities are not commutable. We provide numerical examples to illustrate our results. © 2012 Elsevier B.V. All rights reserved.

1. Introduction and motivation As a financial product designed for hedging lifetime uncertainty, a life annuity is a contract between an annuitant and an insurance company. For a single premium immediate annuity (SPIA), in exchange for a lump sum payment, the company guarantees to pay the annuitant a fix amount of money periodically until her death. Optimal investment problems in a market with life annuities have been extensively studied since the seminal paper of Yaari (1965); see, for example, the references in Milevsky and Young (2007). With the assumption that there are only bonds and annuities in the financial market, Yaari (1965), as well as Davidoff et al. (2005) among others, prove that it is optimal for an individual with no bequest motive to fully annuitize. In reality, the volume of voluntary purchases by retirees is much lower than predicted by such models, which is the so-called ‘‘annuity puzzle’’. One well-explored reason for retirees’ reluctance to annuitize is that they wish to retain their wealth in liquid form so that they can leave it to their heirs. Davidoff et al. (2005) show that if annuities are priced fairly, then people set aside what they wish to bequeath and annuitize the remainder of their wealth. Lockwood (2012) show that modest bequest motives can severely

∗

Corresponding author. E-mail addresses: [email protected] (T. Wang), [email protected] (V.R. Young). 0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2012.06.002

reduce or eliminate annuity purchasing. He finds that bequest motives that have little effect on saving or on optimal purchasing of actuarially fair annuities can have a large effect on the demand for actuarially unfair annuities. In reality, annuities are priced unfairly due to loads for risk and administrative costs, so Lockwood’s work offers an excellent explanation for the annuity puzzle. Another explanation for the annuity puzzle lies in retirees’ fear that issuers of annuities may default. Jang et al. (2009) show that this fear, indeed, affects the demand for annuities. Finally, Benartzi et al. (2011) argue that framing issues might have more of an impact on annuitization than liquidity and irreversibility; also, see the many references therein. According to a recent survey in the United Kingdom by Gardner and Wadsworth (2004), over half of the individuals in the sample chose not to annuitize given the option. The dominant reason given for not wanting to annuitize is the preference for flexibility. It is well known that annuity income is not commutable. Annuity holders can neither surrender for a refund nor short-sell (borrow against) their purchased annuities, even when they are in urgent need of money. However, if life annuities were commutable and, thus, more flexible, we expect that retirees would purchase more annuities. Our work is motivated by the potential relation between commutability of life annuities, specifically SPIAs, and retirees’ willingness to annuitize. In this paper, we investigate how commutability of life annuities affects annuitization, consumption, and investment strategies of a retiree. To this end, we assume the existence of a

T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

market of commutable life annuities, a riskless asset (bond or money market), and a risky asset (stock), and we focus on how including commutable life annuities encourages more annuitization. The commutable annuity, which is a SPIA with a surrender option, has both a purchase price and a surrender value. We assume that the purchase price of this commutable annuity is equal to the expected present value of future payments to the annuity holder, the socalled actuarial present value. The surrender value is the actuarial present value less a proportional surrender charge (denoted by p). A retiree is allowed to purchase additional annuity income or to surrender her existing annuity income, and she can invest in the riskless and risky assets in the market as well. To model the behavior of a utility-maximizing retiree in such a financial market, we formulate a continuous-time optimal consumption and asset allocation problem. We assume that the utility function of the retiree exhibits constant relative risk aversion (CRRA), and we determine the optimal strategy that maximizes the retiree’s expected discounted utility of lifetime consumption. We are especially interested in the relation between the optimal annuitization strategy and the size of the proportional surrender charge, the factor that determines the financial flexibility of life annuities. Our model is an extension of the classical asset allocation framework of Merton (1971). Merton considers the problem of optimal consumption and investment in a complete market with a riskless asset and a risky asset. Cox and Huang (1989) first extend the model to the case of an incomplete market. He and Pagès (1993) consider the case with the presence of labor income. Koo (1998) considers the case in which labor income is subject to uninsurable risk and a liquidity constraint. Davis and Norman (1990) extend the model to an imperfect market in which buying and selling of the risky asset is subject to proportional transaction costs. Øksendal and Sulem (2002) consider the case with the presence of both fixed and proportional transaction costs. See also Elie and Touzi (2008), Karatzas et al. (1997), Tahar et al. (2005), and Egami and Iwaki (2009) for other extensions. The problem treated in our paper is a direct generalization of the one in Milevsky and Young (2007), in which the life annuity is not commutable. Mathematically, our work is closely related to the literature on optimal investment under proportional transaction costs, as in Davis and Norman (1990), Shreve and Soner (1994), and the more recent survey by Cadenillas (2000). Because the surrender value is proportional to the actuarial present value of the annuity, there is a proportional transaction cost associated with selling or surrendering annuities, although we assume there is no corresponding transaction cost in buying annuities. The optimal investment strategy in Davis and Norman (1990) is one of singular and impulse control; if stock and bond holdings initially lie outside a given ‘‘wedge’’, then the investor immediately buys or sells shares of stock to reach the wedge (impulse control) and afterwards buys or sells instantaneously to remain within that wedge (singular control). We find that the resulting optimal annuitization strategy in our model is of a similar form. Indeed, if wealth and annuity income lie outside a given linearly defined region, then the retiree immediately buys annuity income to reach the region via impulse control and afterwards invests, consumes, and annuitizes to remain within that region via singular control. Our work is also related to that of liquidity constraints in the presence of (labor) income because we do not allow the individual to borrow against future annuity income, that is, (liquid) wealth must remain non-negative at all times. He and Pagès (1993) consider the problem of maximizing utility of consumption for an individual with stochastic income under borrowing constraints, which they solved via a duality method, similar to the one in this paper. Duffie et al. (1997) consider a similar problem, but generalized the stochastic income process such that its randomness was not spanned by assets in the financial market.

353

Because they assumed that preferences exhibit constant relative risk aversion, they were able to reduce the dimension of the state space from two to one, as we do in this paper. The commutability of annuities in our model complicates the optimal decisions of the retiree. It leads to a two-dimensional optimal control problem in an incomplete market. The optimal strategy depends on two state variables, wealth and existing annuity income. Taking advantage of the homogeneity of the CRRA utility, we simplify our problem to a one-dimensional equivalent problem, whose value function solves a non-linear differential equation. Via the Legendre dual, we linearize this differential equation and, we indirectly solve for the maximized utility and optimal strategies. We prove the optimality of these solutions through a verification theorem. Milevsky et al. (2006) and Milevsky and Young (2007) also apply this duality argument. We find that when the proportional surrender charge is smaller than a critical value, an individual keeps wealth to one side of a separating ray in wealth–annuity space by purchasing more annuity income. The slope of this ray increases as p decreases; that is, an individual is more willing to annuitize as the proportional surrender charge decreases. When her wealth reaches zero, the individual continues to invest in the risky asset by borrowing from the riskless account and surrenders annuity income to keep her wealth non-negative, as needed. By contrast, when the proportional surrender charge is larger than this critical value, the retiree does not invest in the risky asset when her wealth is zero. Additionally, she does not surrender her annuity income; instead, she reduces her consumption to a rate lower than her annuity income in order to accumulate wealth. For comparison’s sake, we mention that Duffie et al. (1997) show when wealth reaches zero in their problem with stochastic labor income, the individual’s investment in the risky asset is zero and consumption occurs at a rate less than income. More surprisingly, we find that in the case when the surrender charge is larger than the critical value, the optimal annuitization, investment, and consumption strategies do not depend on the size of the surrender charge. An individual behaves as if the annuity is not commutable and does not surrender existing annuity income under any circumstances. We use a variety of numerical examples to illustrate our results. The remainder of this paper is organized as follows. In Section 2, we present the financial market in which the individual invests her wealth. In addition to investing in riskless and risky assets, the individual can purchase or surrender commutable life annuities. In Section 3.1, we consider two special cases: p = 0 and p = 1. We solve the case p = 0 in the primal space by connecting it to a classical Merton problem. By analyzing the retirees’ optimal strategies in these two special cases, we gain insight for solving the more general cases. We consider the case when the proportional surrender charge is smaller than some critical value in Section 3.2, and in Section 3.3, we discuss the case when the proportional surrender charge is larger than some critical value. We present properties of the optimal strategies in Section 4 both analytically and numerically. Section 5 concludes our paper. 2. Problem formulation In this section, we first introduce the assets in the financial and annuity markets: a riskless asset (bond or money market account), a risky asset (stock), and commutable life annuities. Then, we define the maximized utility function, which is the objective function for our optimal control problem. After that, we preliminarily discuss a retiree’s optimal strategy. Finally, we prove a verification theorem, which we will use to validate our solution in the next section.

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2.1. The financial market and commutable life annuities We consider an individual with future lifetime described by the random variable τd . We assume that τd is an exponential random variable with parameter λS , also referred to as the force of mortality or hazard rate; in particular, E[τd ] = 1/λS . The superscript S indicates that the parameter equals the individual’s subjective belief as to the value of her hazard rate. We assume a frictionless financial market, which imposes no transaction costs, no taxes, and no restrictions on borrowing or selling. In this financial market, the individual can invest in or borrow from a riskless asset at interest rate r > 0. Also, she can buy or short sell a risky asset whose price follows geometric Brownian motion dXt = µ Xt dt + σ Xt dBt ,

X0 = X > 0,

in which µ > r , σ > 0, and B is a standard Brownian motion with respect to a filtration F = {Ft }t ≥0 of a probability space (Ω , F , P). We assume that B is independent of τd , the random time of death of the individual. Moreover, we assume an unrestricted life annuity market in which an individual can purchase any amount of commutable life annuity income or surrender any portion of existing annuity income at any time. The price of a life annuity (specifically, a SPIA) that pays $1 per year continuously until the individual dies is given by ∞

 a= 0

O e−rs e−λ s ds =

1 r + λO

,

in which λO > 0 is the (constant) objective hazard rate that is used to price annuities. In other words, in return for each $a the individual pays for a life annuity, she receives $1 per year of continuous annuity income until she dies. This price is also called the actuarial present value in the insurance literature, Bowers et al. (1997). It is the break-even-on-average price if the issuer of life annuities invests the proceeds from sales into the riskless account and if individuals’ lives are independently and exponentially distributed with parameter λO . Thus, if the issuer of life annuities uses a value for λO that is less than the ‘‘true’’ value, there is a risk margin in the price, as well as a margin for administrative expenses. Due to the commutability of life annuities, the retiree can surrender any amount of her existing annuity income in exchange for money from the issuer of the annuity. The surrender value of $1 of annuity income is (1 − p)a with 0 ≤ p ≤ 1, in which p is the proportional surrender charge. In other words, the individual receives $(1 − p)a from the issuer by surrendering $1 of annuity income. Notice that the surrender value is less than the actuarial present value, and the difference is the surrender charge (in dollars). In addition to the proportional surrender charge p, the company might also wish to use a lower value of λO than for non-commutable annuities as an additional risk loading for the potential loss of reserves. In Section 4.1, we explore how the insurance company might choose λO for a given value of p because with commutable annuities, the company loses reserves when the annuitant surrenders income. 2.2. Utility of lifetime consumption Following Yaari (1965), we consider a retiree without a bequest motive; therefore, utility only comes from her consumption. She chooses to consume at a rate of ct at time t. Let πt denote the amount invested in the risky asset at time t. Let Pt denote the cumulative amount of annuity income purchased on or before time t, and St the cumulative amount of annuity income surrendered on or before time t. Then, At = Pt − St equals the cumulative amount

of immediate life annuity income at time t. The wealth and annuity dynamics of the individual are given by



dWt = [rWt − + (µ − r )πt − − ct + At − ] dt + σ πt − dBt W0 = w ≥ 0; −a dPt + (1 − p)a dSt , dAt = dPt − dSt , A0 = A ≥ 0.

(2.1)

The investment, consumption, and annuitization strategies {πt , ct , Pt , St }t ≥0 are said to be admissible if (i) The control processes {πt }t ≥0 , {ct }t ≥0 , {Pt }t ≥0 , and {St }t ≥0 are adapted to the filtration F. (ii) The controls ct ≥ 0, Pt ≥ 0, and St ≥ 0 almost surely for all t t≥ 0. t (iii) 0 πs2 ds < ∞ and 0 cs ds < ∞ almost surely for all t ≥ 0. (iv) The associated wealth and annuitization processes Wt ≥ 0 and At ≥ 0, respectively, almost surely for all t ≥ 0. We denote by Ap (w, A) the collection of all admissible strategies when the initial wealth and annuity is (w, A), and when the corresponding surrender charge is p. Remark 2.1. We highlight our assumption that wealth does not include the imputed value (1 − p)a A of the individual’s annuity income. In order for the individual to add any of that amount to her wealth, she surrenders the corresponding annuity income as indicated by the term (1 − p)a dSt in (2.1), so that her income is reduced in future. In particular, we prevent the retiree from borrowing against her future annuity income by requiring that her wealth remain non-negative. This assumption is reasonable because annuity income will cease when the retiree dies, so we do not allow her to die with negative wealth. On the other hand, if one were to include life insurance in the market, then an individual could borrow against her annuity income by purchasing decreasing-benefit life insurance to serve as collateral in the case that she dies before repaying her loan. We do not thus augment the model in this paper but look forward to considering this problem in future work. We assume that the individual is risk averse and that her preferences exhibit constant relative risk aversion (CRRA); that is, the utility function for the individual is given by u( c ) =

c 1−γ 1−γ

,

γ > 0 and γ ̸= 1,

in which γ is the (constant) relative risk aversion. In this paper, we assume that an individual seeks to maximize her expected utility of discounted consumption over admissible strategies {πt , ct , Pt , St }. In addition, we assume that an individual discounts her utility of consumption at the riskless rate r. Therefore, the maximized utility for such an individual is defined for (w, A) ∈ R+ × R+ by U (w, A; p) τd

 =

sup {πt ,ct ,Pt ,St }∈Ap (w,A)

−rt

e

E



u(ct ) dt | w0 = w, A0 = A . (2.2)

0

Remark 2.2. Because we assume that the hazard rates λS and λO , as well as the financial parameters r , µ, σ and p, are constant, U only depends on the state variables w and A and not upon time. We do not feel that the time independence of the parameters is a drawback of our model. Indeed, within our relatively simple market, we obtain semi-analytical expressions for the optimal annuitization, consumption, and investment strategies. Under more realistic models, we expect that our qualitative results will still hold, and perhaps the simple strategies we found will be nearly optimal. For example, Moore and Young (2006) observe that when minimizing the probability that an individual financially ruins before dying, investment strategies computed by assuming

T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

that hazard rates are constant are nearly optimal for the case of varying hazard rates. Bayraktar et al. (2011) obtain similar results in a setting of stochastic volatility. We acknowledge that using Gompertz mortality would be more realistic than constant force of mortality, but by doing so, we would lose the analytical tractability of our simple model. Because our work is the first to mathematically model commutable annuities and the corresponding annuitization decision, we wish to obtain analytical results that researchers can later extend (numerically) using more realistic models. Proposition 2.1. U (w, A; p) decreases with respect to p ∈ [0, 1]. Proof. If {πt , ct , Pt , St }t ≥0 is admissible when the surrender charge is p1 , it is also admissible when the surrender charge is p2 < p1 . In other words, Ap2 (w, A) ⊇ Ap1 (w, A) for p2 < p1 . It follows from the definition of U that U (w, A; p2 ) =

τd

 sup

E

{πt ,ct ,Pt ,St }∈Ap2 (w,A)

e−rt

 × u(ct )dt | w0 = w, A0 = A, τd > 0 ≥

sup

τd

E

{πt ,ct ,Pt ,St }∈Ap1 (w,A)

Then,

v ≥ U, on R+ × R+ . 1−γ

Remark 2.4. Note that for the utility function u(c ) = c1−γ , when 0 < γ < 1, we have u(c ) > 0 for all c > 0 and u(0) = 0. By contrast, when γ > 1, we have u(c ) < 0 for all c > 0 and limc →0 u(c ) = −∞. Thus, U ≥ 0 (< 0) when 0 < γ < 1 (γ > 1). In particular, when γ > 1, we have no reason to expect that U is bounded from below. Proof of Theorem 2.1. Suppose the function v satisfies the conditions of the theorem. We prove v ≥ U on R+ × R+ in two steps. First, we prove the theorem with two additional assumptions: (i) v is bounded from below; that is, v ≥ V > −∞ on R+ × R+ . (ii) vw (0, A) < +∞ for all A ≥ 0. Then, we remove these assumptions and show that the conclusion still holds. s Let τna , inf{s ≥ 0 : 0 πs2 ds ≥ n} and τnb , inf{s ≥ 0 : As ≥ n}. Define τn = n ∧ τna ∧ τnb , which is a stopping time with respect to the filtration F; then, using Itô’s formula for semi-martingales (see Protter, 2004), we write, for any admissible strategy {πt , ct , Pt , St },

0



355

e−rt

S e−(r +λ )τn v(Wτn , Aτn )

0



= v(w, A) +

× u(ct ) dt | w0 = w, A0 = A, τd > 0

τn



S e−(r +λ )t vw (Wt , At ) σ πt dBt

0

= U (w, A; p1 ). 

τn

 +

 −(r +λS )t

e

1−γ

L

π t , ct

v(Wt , At ) −

0

Remark 2.3. We present an equivalent form for the maximized utility that will be useful in proving our verification theorem in the next section. By taking the expectation of the right-hand side of (2.2) with respect to the time of death, we obtain U (w, A; p) =

 sup {πt ,ct ,Pt ,St }∈Ap (w,A)

∞

E

τn

 + 0



τn

+

ct



1−γ

dt

S (c ) e−(r +λ )t [vA (Wt , At ) − a vw (Wt , At )] dPt S (c ) e−(r +λ )t [(1 − p)avw (Wt , At ) − vA (Wt , At )]dSt

0 S e−(r +λ )t

+

0

S e−(r +λ )t [v(Wt , At ) − v(Wt − , At − )] .



(2.4)

0≤t ≤τn

 × u(ct ) dt | w0 = w, A0 = A .

(2.3)

For the rest of this paper, we will write U (w, A) or simply U for U (w, A; p) when the meaning is clear. 2.3. Verification theorem

(c )

(c )

Here, Pt and St are the continuous parts of Pt and St , respec (c ) = Pt − 0≤s≤t (Ps − Ps− ), and similarly for tively; that is, Pt (c )

2 St . Because v is non-decreasing and concave in w, vw (w, A) ≤ 2 vw (0, A) for w ≥ 0. Therefore,

w,A

τn



e

E

−2(r +λS )t

0



vw (Wt , At ) σ π dt < ∞, 2

2

2 t

In this section, we prove a verification theorem that we use to solve our optimization problem in Section 3. First, define, for any (π, c ) ∈ R × R+ , the functional operator Lπ,c through its action on a test function f ∈ C 2,1 (R+ × R+ ) by

which implies that

Lπ ,c f = −(r + λS )f + (r w + A)fw

Here, Ew,A denotes conditional expectation given W0 = w and A0 = A. By taking expectations of Eq. (2.4) and by using (2.5), the conditions in the statement of the theorem, and the additional assumptions at the beginning of the proof, we obtain

   1−γ  1 c + (µ − r )π fw + σ 2 π 2 fww + − cfw . 2 1−γ Theorem 2.1. Suppose the function v = v(w, A) ∈ C 2,1 (R+ × R+ ) is non-decreasing and concave with respect to w and nondecreasing with respect to A. Moreover, suppose v satisfies the following conditions on R+ × R+ : (i) Lπ,c v ≤ 0 for (π , c ) ∈ R × R+ . (ii) a vw − vA ≥ 0. (iii) (1 − p)a vw − vA ≤ 0.

E

w,A

τn



e

−(r +λS )t



vw (Wt , At ) σ πt dBt = 0.

(2.5)

0



Ew,A e−(r +λ

S )τ n

V



  S ≤ Ew,A e−(r +λ )τn v(Wτn , Aτn ) 

≤ v(w, A) − E

τn

w,A

e 0

−(r +λS )t

1−γ

ct

1−γ

 dt .

(2.6)

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T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

In deriving (2.6), we also use the fact that



e

−(r +λS )t

[v(Wt , At ) − v(Wt − , At − )] ≤ 0,

0≤t ≤τn

because conditions (ii) and (iii) in the statement of the theorem imply that v is non-increasing in the direction of jumps. Because τn ↗ ∞ as n → ∞, applying the monotonic convergence theorem to (2.6) yields

v(w, A) ≥ E

w,A





1−γ

∞

e

−(r +λS )t

0

ct

1−γ

dt .

This inequality implies that

v(w, A) ≥

w,A

sup {πt ,ct ,Pt ,St }∈Ap (w,A)



e

E



1−γ

∞

−(r +λS )t

0

ct

1−γ

dt

= U (w, A), in which we use the representation of U from (2.3). Next, we show that the conclusion holds when v is not bounded from below or when vw (0, A) is not finite. We follow an argument similar to the one in Davis and Norman (1990). For a sequence ϵn ↘ 0, define v ϵn (w, A) , v(w + ϵn , A + ϵn ). The function v ϵn is non-decreasing, twice-differentiable, and concave with respect to w and non-decreasing and differentiable with respect to A. Note that on R+ × R+ , v ϵn (w, A) is bounded from below by v(ϵn , ϵn ) ϵn and that vw (0, A) = vw (ϵn , A + ϵn ) < +∞. Since vwϵn (w, A) = vw (w + ϵn , A + ϵn ) and vAϵn (w, A) = vA (w + ϵn , A + ϵn ), we have 0 ≥ Lπ ,c v(w + ϵn , A + ϵn ) = −(r + λS )v ϵn (w, A)

 + [r (w + ϵn ) + (A +

ϵn )]vwϵn (w, A)



1

ϵn (w, A) + + σ 2 π 2 vww

2

=L



c 1−γ 1−γ

+ (µ − r )π vwϵn (w, A)

p = 1, we will obtain upper and lower bounds, respectively, for both the maximized utility function and the admissible strategy set for an arbitrary p. We first consider the case p = 0, for which the annuity is completely commutable. We solve this special case by connecting it to a classical Merton problem. When p = 0, the life annuity acts as a second money market account with a higher risk free rate of return, namely, r + λO . It is optimal for the individual to annuitize all her wealth immediately (because she can surrender annuity income without penalty) and to invest in the risky asset with money borrowed from the riskless asset at rate r. All her earnings will be used to purchase more annuity income immediately, and all her losses will be covered by surrendering existing annuity income. Although there exist two accounts growing with different rates (r versus r + λO ), there is no arbitrage opportunity to make an arbitrary amount of money, due to our restriction that wealth remain non-negative. Because the imputed value of annuity income (namely, (1 − p)aA = aA) is not included in wealth, the individual is prohibited from purchasing an arbitrarily large amount of annuity income with money borrowed from the riskless account; otherwise, wealth would become negative. Wealth Wt is always zero when the individual follows the optimal strategy described above, for t > 0, because she fully annuitizes at time t = 0. Therefore, we only need to consider the dynamics of her annuity income At . For convenience of the following discussion, let W t denote the imputed value of her annuity income At (that is, W t = a At ), and we give the dynamics of W t instead of that of At . The individual borrows π t from the riskless asset, invests this amount of money in the risky asset, and consumes at a rate of c t ; thus, the dynamics of W t are given by



=



 − c vwϵn (w, A)

π ,c ϵn

v (w, A) + (r + 1)ϵn vwϵn (w, A).

ϵn Because vw (w, A) ≥ 0 and ϵn > 0, it follows that Lπ,c v ϵn ≤ 0 on R+ × R+ ; that is, v ϵn satisfies condition (i) of this theorem. Also, we have ϵ

ϵn a vw (w, A) − vAn (w, A) = a vw (w + ϵn , A

+ ϵn ) − vA (w + ϵn , A + ϵn ) ≥ 0,

Thus, v ϵn satisfies conditions (ii) and (iii) of this theorem. It follows that v ϵn ≥ U on R+ × R+ for all n. Since v(w, A) is continuous in both w and A, we conclude that v(w, A) = limn→∞ v ϵn (w, A) ≥ U (w, A). 



r + λO W t + (µ + λO ) − (r + λO ) π t − c t









with W 0 = w , w + a A. Therefore, the utility maximization problem becomes U (w, A; 0) ∞

 = sup E {π t ,c t }

e

−(r +λS )t



u(c t ) dt | W 0 = w + aA ,

(3.1)

0

which is equivalent to Merton’s problem of maximizing expected discounted utility of consumption over an infinite time horizon with discount rate r +λS , riskless rate of return r +λO , drift µ+λO , and volatility σ . 1−γ

It is well know that, under CRRA utility u(c ) = c1−γ , this problem is explicitly solvable under the following well-posedness condition:

3. Determining the maximized utility U

 To solve the utility maximization problem defined in the previous section, we first consider two special cases: p = 0 and p = 1, whose solutions point to the optimal strategy of a retiree in the more general case of arbitrary p ∈ [0, 1]. After analyzing these two special cases, we solve the general case for which p ∈ [0, 1].



× dt + σ π t dBt

and

(1 − p)a vwϵn (w, A) − vAϵn (w, A) = (1 − p)a vw (w + ϵn , A + ϵn ) − vA (w + ϵn , A + ϵn ) ≤ 0.

r + λO W t + (µ − r ) π t − c t dt + σ π t dBt .

dW t =

r + λS − (1 − γ ) r + λO − 2m







1−γ

γ

> 0,

(3.2)

 µ−r 2

with m = 12 σ . If this condition is not satisfied, one can obtain arbitrarily large utility, as remarked by Davis and Norman (1990). The solution is given as follows: first, define 1

 

r + λS − (1 − γ ) r + λO − m







1−γ



,

3.1. Two special cases: p = 0 and p = 1

K ,

Consider the two special cases: p = 0 and p = 1. From Proposition 2.1, we know that by solving the cases of p = 0 and

which is automatically positive if γ > 1. If 0 < γ < 1, then the well-posedness condition in (3.2) implies that K is positive. The

γ

γ

T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

(a) Optimal annuitization with a small proportional surrender charge (p < p∗ ).

357

(b) Optimal annuitization with a large proportional surrender charge (p ≥ p∗ ).

Fig. 1. The two graphs above illustrate the optimal annuitization strategies for small and large proportional surrender charges, respectively. In each case, there exists a critical ratio of wealth-to-annuity income z0 indicated by the blue ray in the graph. It is optimal for a retiree to keep herself to the left of the blue ray. Different investment strategies are applied in the two different cases when w = 0, as indicated in each graph. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

maximized utility of consumption U and optimal strategies for this version of Merton’s problem are given in feedback form by U (w) = K −γ c t = K Wt , µ−r ∗

w1−γ , 1−γ

∗

πt =

γσ2

Wt .

With this result, we directly obtain the maximized utility for (3.1) with p = 0 as follows: U (w, A; 0) = K −γ

(w + aA)1−γ . 1−γ

Remark 3.1. We assume that the well-posedness condition in (3.2) holds for the rest of this paper. From Proposition 2.1, we know that U (w, A; p) ≤ U (w, A; 0) on R+ × R+ for all p ∈ [0, 1]. Therefore, by imposing the condition in (3.2), we guarantee that the value function is finite for any p ∈ [0, 1]. When p = 1, we have a utility maximization problem with noncommutable annuities. This problem is thoroughly investigated via a duality argument by Milevsky and Young (2007). They found that an individual purchases annuity to keep herself to one side of a separating ray in wealth–annuity space and does not invest in the risky asset when her wealth is zero. Since the annuity is not commutable, she will never surrender it. The reader can refer to that paper for more details. These two extreme cases help us to better understand our optimization problem for p ∈ (0, 1). For an arbitrary value of p ∈ (0, 1), we know from Proposition 2.1 that the maximized utility is bounded by those of the two extreme cases of p = 0 and p = 1. We expect the optimal strategy for p close to 1 is similar to the one when annuities are not commutable. Similarly, for p close to 0, we expect that the optimal strategy is similar to the one when annuities are commutable with no surrender charge. Inspired by solutions for these two special cases, we solve our problem for an arbitrary proportional surrender charge p ∈ [0, 1] in the following sections. 3.2. The case for which p < p∗ Consider the utility maximization problem for an arbitrary value of p ∈ [0, 1]. For a strategy to be admissible, the wealth of the individual is required to be non-negative. In other words,

whenever one’s wealth reaches zero, the retiree is forced to keep her wealth from further decline. Through our study, we learn that an individual handles this situation in one of two ways, depending on the size of the proportional surrender charge p. If p < p∗ , a critical value to be defined later, and if wealth is zero, the retiree prefers to continue investing in the risky asset by borrowing from the riskless account. To keep her wealth from becoming negative due to a decline in the price of the risky asset, she surrenders her existing annuity income as needed. If p ≥ p∗ and if wealth is zero, then she prefers to stop investing in the risky asset because the surrender charge is large. Instead, by consuming less than the annuity income, she increases her wealth so that it is positive. We investigate the two cases of p < p∗ and p ≥ p∗ in this section and in Section 3.3, respectively. Fig. 1 illustrates the optimal annuitization strategies in both cases on the wealth–annuity (w -A) plane. Here is our plan for solving the case for which p is less than the critical value p∗ . (i) Hypothesize the form of the optimal annuitization strategy based on the one for p = 0. (ii) Define the corresponding maximum utility of expected lifetime consumption, Ψ , under admissible controls {πt , ct } and under the specific annuitization strategy from the first step. Because Ψ is the value function for a problem more restricted than the one for U, it follows that Ψ ≤ U on R+ × R+ . (iii) Determine Ψ by solving a free-boundary problem (FBP). The connection between such value functions and their corresponding FBPs (or boundary-value problems) is via verification theorems similar to Theorem 2.1, so we omit the proof of the connecting verification theorem. See, for example, Theorem 4.2 in Davis and Norman (1990). (iv) Finally, show that Ψ satisfies the conditions of Theorem 2.1 if p < p∗ . From that theorem, then deduce that Ψ ≥ U on R+ × R+ . Thus, Ψ = U, with Ψ ’s strategies optimal for the general problem, including the hypothesized (restricted) annuitization strategy. When p < p∗ , we hypothesize that there exists a critical ratio z0 of wealth-to-annuity income. If (w, A) is such that w/A > z0 , then the individual purchases annuity income to raise her annuity income to A′ and reduces her wealth to w ′ so that w ′ /A′ = z0 , an example of impulse control. Furthermore, we assume that if (w, A) is such that w/A ≤ z0 , then she purchases annuity income to keep the ratio of wealth to annuity income no greater than z0 , an example of singular control. Additionally, we hypothesize that

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T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

the retiree neither purchases nor surrenders annuity income if 0 < w/A < z0 . On the boundary w = 0, we hypothesize that she will surrender existing annuity income as needed to keep her wealth non-negative. Let Ψ denote the corresponding value function:

Ψ (w, A) =

sup

{πt ,ct },z0

w,A

τd



−rt

e

E



u(ct ) dt ,

(3.3)

0

in which the retiree follows the annuitization strategy hypothesized in this paragraph under an optimally chosen value of z0 . Partition R+ × R+ into R1 ∪ R2 , in which R1 , {(w, A) : 0 ≤ w/A ≤ z0 , A ≥ 0} and R2 , {(w, A) : w/A > z0 , A ≥ 0}. By a verification theorem similar to Theorem 2.1, if we find a function ψ ∈ C 2,1 (R1 ) that is increasing and concave with respect to w and increasing with respect to A that solves the following FBP, then ψ = Ψ on R1 :

   S  (µ − r )π ψw (r + λ )ψ = (r w + A)ψw + max  π        1−γ   1 2 2 c  + σ π ψww + max − c ψw , c ≥0 2 1−γ    0 ≤ w/A ≤ z0 ;     ψA (0, A) = (1 − p)a ψw (0, A);    ψA (z0 A, A) = a ψw (z0 A, A); ψAw (z0 A, A) = a ψww (z0 A, A).

in which 1A =

w−z0 A . z 0 +a

z ≥0

For a given y, the critical z ∗ that maximizes V (z ) − yz solves Vz (z ∗ ) − y = 0. Thus, z ∗ = I (y), in which I is the functional inverse of Vz . It follows that

(3.4)

   ≥ 0. Vzz (z ) z =Vz−1 (y) 1

Note that we can recover V from V (z ) = Vˆ (y) + yz. Define ys = Vz (0), yb = Vz (z0 ).

(3.5)

3.2.1. Dimension reduction The value function Ψ defined in (3.3) is homogeneous of degree 1 − γ with respect to wealth w and annuity A due to the homogeneity property of the CRRA utility function. More precisely, Ψ (αw, α A) = α 1−γ Ψ (w, A) for α > 0. We use this property to define V (z ) , Ψ (z , 1), from which we will recover Ψ by for A > 0.

Vˆ (y) = max [V (z ) − yz] .

Vˆ yy (y) = −

Notice that (w − a 1A, A + 1A) ∈ ∂ R1 .

Ψ (w, A) = A1−γ V (w/A),

3.2.2. Linearization via the Legendre transform The ODE in Eq. (3.7) is non-linear. Because we are looking for a concave solution to this FBP, in this section, we apply the Legendre transform to V to define its convex dual Vˆ . The function Vˆ , in turn, solves a linear ODE. Define

Vˆ y (y) = −Vz−1 (y) = −z ∗ ≤ 0,

The differential equation in (3.4) is a Hamilton–Jacobi–Bellman (HJB) equation through which we will determine the optimal consumption and investment strategies in feedback form. The first boundary condition arises from our hypothesis that the retiree surrenders annuities only when w = 0. The second boundary condition comes from our hypothesis that the retiree buys annuities only when w/A = z0 for (w, A) ∈ R1 via singular control. Finally, the last boundary condition is a smooth fit condition because the value of z0 is chosen optimally. Davis and Norman (1990) and Karatzas et al. (2000) also assume such a condition. Dixit (1991) gives an intuitive derivation of the smooth fit condition for optimal regulation problem in a discretetime setting, and Dumas (1991) further discusses the smooth fit condition. Once we have solved for Ψ on R1 , then we have for (w, A) ∈ R2 ,

Ψ (w, A) = Ψ (w − a 1A, A + 1A),

If we find a solution v to this FBP that is increasing, concave, and twice-differentiable such that (1 −γ )v(z ) ≥ z vz (z ) for 0 ≤ z ≤ z0 , then v = V , and we will have Ψ from Eq. (3.6).

(3.6)

This transform simplifies our problem by reducing it to a onedimensional problem. Davis and Norman (1990) and Koo (1998) also implement this transform in their optimal consumption and investment problems. Now, apply this dimension reduction to the FBP in (3.4) to get a FBP for V . Then, solve for V and recover Ψ from V . To that end, we obtain the following FBP for V :

   1 2 2 S   ( r + λ )v = ( rz + 1 )v + max (µ − r ) π ˆ v + σ π ˆ v z z zz   2   1−γ  πˆ    cˆ + max − cˆ vz , 0 ≤ z ≤ z0 ; (3.7) 1−γ cˆ ≥0    ( 1 − γ )v( 0 ) = ( 1 − p ) a v ( 0 ); z    (1 − γ )v(z0 ) = (z0 + a)vz (z0 );  (z0 + a)vzz (z0 ) + γ vz (z0 ) = 0.

Rewrite (3.7) in terms of Vˆ to obtain a linear FBP for Vˆ :

 (r + λS ) vˆ (y) = λS y vˆ y (y) + m y2 vˆ yy (y) + y   γ −1 γ    y γ , 0 < yb ≤ y ≤ ys , + 1−γ (3.8) vˆ y (ys ) = 0, (1 − γ )ˆv (ys ) = (1 − p)a ys ,     vˆ y (yb ) = −z0 , (1 − γ )ˆv (yb ) + γ yb vˆ y (yb ) = a yb , vˆ y (yb ) + γ yb vˆ yy (yb ) = a,  µ−r 2 in which m = 21 σ , as before. The general solution of the ODE in (3.8) is given by

vˆ (y) = D1 yB1 + D2 yB2 +

y r

+ Cy

γ −1 γ

,

(3.9)

in which B1 = B2 = C =

1  2m 1  2m

(m − λS ) +

  (m − λS )2 + 4m(r + λS ) > 1,

(3.10)

(m − λS ) −

  (m − λS )2 + 4m(r + λS ) < 0,

(3.11)

  −1 γ λS 1−γ r+ −m 2 . 1−γ γ γ

(3.12)

Remark 3.2. If γ > 1, then (1 − γ )C is automatically positive. If 0 < γ < 1, then the well-posedness condition in (3.2) implies that r+

λS 1−γ − m 2 > 0, γ γ

(3.13)

or equivalently,  (1S−γ )C > 0.  Because [1 +γ (B1 − 1)]·[1 +γ (B2 − γ2 1−γ λ 1)] = − m r + γ − m γ 2 , the inequality in (3.13) also implies that 1 + γ (B2 − 1) < 0.

To determine the unknown values of D1 , D2 , ys , yb , and z0 , begin by expressing the boundary conditions in (3.8) in terms of the expression for vˆ in (3.9) to get B

B

D1 [1 + γ (B1 − 1)]yb1 + D2 [1 + γ (B2 − 1)]yb2

+

yb r

=

yb r + λO

,

(3.14)

T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

(1 − γ ) D1 yBs 1 + (1 − γ ) D2 yBs 2 + (1 − γ )

ys

in which x˜ > 1 is the unique solution of the following equation:

r

γ −1 γ Cys

(1 − p)ys + (1 − γ ) = , r + λO B −1 B −1 D1 B1 [1 + γ (B1 − 1)]yb1 + D2 B2 [1 + γ (B2 − 1)]yb2 +

1 r

1

=

r + λO

B −1

D1 B1 yb1

(3.15)

,

(3.16)

D1 B1 yBs 1 −1 + D2 B2 ysB2 −1 + B −1

+ D2 B2 yb2

+

1 r 1 r

+

γ − 1 − γ1 Cys = 0, γ

(3.17)

+

γ − 1 −γ Cyb = −z0 . γ

(3.18)

1

From (3.14) and (3.16), it follows that

λO r + λO

λO B1 − 1 1 1−B y 2. O r (r + λ ) B1 − B2 1 + γ (B2 − 1) b

(3.20)

1 − B2 B1 − B2

xB1 −1 +

B1 − 1 B1 − B2

xB2 −1 = 1 +

pr

λO

=

(3.21)

with x , y s . Note that (3.21) has a unique solution x ∈ [1, ∞) b because (i) when x = 1, the left-hand side of (3.21) equals 1, which is less than or equal to the right-hand side; (ii) when x → ∞, the left-hand side goes to infinity; and (iii) the left-hand side strictly increases with respect to x on (1, ∞). Given B1 and B2 , this solution x is a function of p. In our paper, when we write x or x(p), we will mean this unique solution. Substitute (3.19) and (3.20) into (3.17) to get y

1−γ

γ

− γ1

Cys

 λO B1 (1 − B2 ) xB1 −1 =− O r (r + λ ) B1 − B2 1 + γ (B1 − 1)  1 B2 (B1 − 1) xB2 −1 + . + B1 − B2 1 + γ (B2 − 1) r

(3.22)

ys x

.

(3.23)

Obtain the purchasing boundary z0 by substituting (3.19) and (3.20) into (3.18) and simplifying, z0 =

1−γ

γ



− γ1

C yb

+

λO 1 r (r + λO ) γ



r + λS − m

1−γ



γ

1

− .

(3.24)

r

For this solution to be meaningful, we require that yb > 0 (or equivalently, ys > 0), which implies that 0 < yb ≤ ys because x ≥ 1. We also require that z0 ≥ 0. In the next section, we define p∗ and show that ys > 0 for 0 ≤ p < p∗ and that z0 ≥ 0. 3.2.3. The critical proportional surrender charge p∗ Define the critical proportional surrender charge by ∗

p =

λO r



1 − B2 B1 − B2

x˜

B1 −1

+

B1 − 1 B1 − B2

x˜

B2 −1



−1 ,

+

B1 − B2

x˜

B2 −1



= 1.

(3.26)

 λO (B1 − 1)(1 − B2 )  B1 −2 x˜ − x˜ B2 −2 > 0, r B1 − B2

∀x > 1.

p =

1 λO



B1 r 1 λO B1 r



r + λO



+ (B1 − 1)˜x − B1 λO  r + λO 1 + (B1 − 1) − B1 = < 1.  λO B1 B2 −1

In the next proposition, we show that ys is a (positive) real number and that z0 is non-negative. Proposition 3.2. The well-posedness condition (3.2) implies that ys as defined in (3.22) is a (positive) real number for 0 ≤ p < p∗ . Moreover, z0 ≥ 0. Proof. The well-posedness condition implies that (1 −γ )C defined in (3.12) is positive, so ys is a (positive) real number if and only if h(x) > 0 for all 1 ≤ x < x˜ , in which

 λO B1 (1 − B2 ) xB1 −1 h(x) , 1 − r + λO B1 − B2 1 + γ (B1 − 1)  B2 (B1 − 1) xB2 −1 + . B1 − B2 1 + γ (B2 − 1) By differentiating h with respect to γ , one can show that h strictly increases with respect to γ . When γ = 0, we get

This expression gives ys in terms of x, and yb is, therefore, yb =

x˜

B2 (B1 − 1)

If x˜ = 1, we have p∗ = 0. Since x˜ > 1, we conclude that p∗ > 0. To show that p∗ < 1, we write x˜ B1 −1 in terms of x˜ B2 −1 from (3.26) and substitute it into (3.25) to get

≤ ,

B1 − B2

B1 −1

Proof. First, we show that p∗ > 0. Taking the derivative of p∗ in (3.25) with respect to x˜ , we have

∗

By substituting (3.19) and (3.20) into (3.15) and (3.17) (and eliminating the term with C in it), we obtain

B1 (1 − B2 )

Proposition 3.1. 0 < p∗ < 1.

dx˜ (3.19)



Note that p∗ is independent of γ . Moreover, x(p∗ ) = x˜ , in which x = x(p) is given by (3.21). As an aside, we discovered the critical proportional surrender charge when we determined the values of p for which the solution of (3.4) is convex; see Proposition 3.3 and the remark following its proof.

dp∗

λO 1 − B2 1 1−B D1 = − y 1, r (r + λO ) B1 − B2 1 + γ (B1 − 1) b D2 = −

359

(3.25)

h(x)|γ =0

λO =1− r + λO



B1 (1 − B2 ) B1 − B2

x

B1 −1

+

B2 (B1 − 1) B1 − B2

B 2 −1

x



.

From the definition of x˜ in (3.26) and from the fact that the lefthand side of (3.26) strictly increases with respect to x˜ for x˜ > 1, the latter expression is positive for 1 ≤ x < x˜ . Thus, h(x) > 0 for 1 ≤ x < x˜ for all γ > 0. It is straightforward to show that z0 = 0 when p = 0. From Proposition 4.1, we learn that z0 strictly increases with respect to p for 0 ≤ p < p∗ . Thus, z0 ≥ 0.  3.2.4. Constructing Ψ from vˆ In this section, we construct Ψ from vˆ ; then, we will show that Ψ satisfies the conditions of Theorem 2.1. It will follow that U = Ψ on R+ × R+ . First, we show that vˆ is decreasing and convex on (yb , ys ). Proposition 3.3. If 0 ≤ p < p∗ , then vˆ ∈ C 2 (yb , ys ), defined by Eqs. (3.9)–(3.12) and (3.19)–(3.24), is decreasing and strictly convex.

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T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

Proof. To show that vˆ is decreasing and strictly convex on (yb , ys ), it is enough to show that vˆ y (ys ) ≤ 0 and that vˆ is strictly convex on the interval. By construction, vˆ y (ys ) = 0; see the first boundary condition in (3.8). As for the convexity of vˆ , one can show that for y ∈ (yb , ys ),

Proposition 3.3 would not be convex, and our argument hinges on this important property. Indeed, if vˆ were not convex, then it would not equal Vˆ , which is the convex dual of the concave function V , which we know is concave because Ψ is concave with respect to w.

λO (B1 − 1)(1 − B2 ) vˆ yy (y) > 0 ⇐⇒ − O r (r + λ ) B1 − B2   B2 (y/yb )B2 −1 B1 (y/yb )B1 −1 − × 1 + γ (B1 − 1) 1 + γ (B2 − 1) 1−γ −1 + Cy γ > 0. γ2

Theorem 3.1. When p < p∗ , the maximized utility U defined in Eq. (2.2) in the region R+ × R+ = R1 ∪ R2 , with R1 = {(w, A) : 0 ≤ w/A ≤ z0 , A ≥ 0} and R2 = {(w, A) : w/A > z0 , A > 0}, is given by: (i) For (w, A) ∈ R1 ,



U (w, A) = A1−γ D1 yB1 + D2 yB2 +

Define the function g on (yb , ys ) by the left-hand side of the above inequality, and note that

λO (B1 − 1)(1 − B2 ) g (y) = − yb r (r + λO ) B1 − B2   B1 −2 B1 (B1 − 1) y × 1 + γ (B1 − 1) yb  B2 −2  B2 (B2 − 1) y − 1 + γ (B2 − 1) yb 1

′

1−γ

−

γ3

− γ γ+1

Cy

dp

=

λO (B1 − 1)(1 − B2 )

B 1 −2

x



> 0,

for x > 1.

Therefore, dg (ys ) dp



r

× (B1 − 1)(B2 − 1)x

B2 −2

dx dp

 < 0.

When p = p∗ ,

  p∗ r + λO λO B2 −1 − 1 )˜ x + 1 −B 1 + ( B 1 γr r + λO r + λO    λO (B1 − 1) x˜ B2 −1 − 1 + r + λO B1 1 = − γr r + λO  λO B2 −1 + (B1 − 1)˜x + 1 = 0, r + λO

g (ys ) =

r

−

1−γ

− γ1

Cy

The associated optimal annuitization, consumption, and investment strategies are as follows: w−z A

(i) Purchase additional annuity income of 1A = z +0a when 0 (Wt , At ) = (w, A) ∈ R2 . (ii) Purchase additional annuity income instantaneously to keep (Wt , At ) in the region R1 , specifically, to keep Wt /At ≤ z0 . (iii) Surrender existing annuity income instantaneously to keep wealth non-negative, as needed. (iv) Consume continuously at the following rate when (Wt , At ) = (w, A) ∈ R1 : − γ1

c ∗ (w, A) = Uw (w, A). (v) Invest the following amount of wealth in the risky asset when (Wt , At ) = (w, A) ∈ R1 :

π ∗ (w, A) = −

λO = −B1 + γr r + λO r + λO 1

1

w−z A

B 2 −2 −1

−x

in which y ∈ [yb , ys ] uniquely solves

in which 1A = z +0a . It is easy to check that (w − a 1A, A + 0 1A) ∈ ∂ R1 ⊂ R1 .

  pr + λO λO B2 −1 −B1 + ( B − 1 ) x + 1 . 1 γr r + λO r + λO



(3.27)

U (w, A) = U (w − a 1A, A + 1A),

< 0,

B1 − B2



Eqs. (3.19), (3.20), (3.10), (3.11) and (3.22)–(3.24), respectively. (ii) For (w, A) ∈ R2 ,

1

r

γ −1 γ

w =− , γ A and D1 , D2 , B1 > 1 > 0 > B2 , ys , yb , and z0 are given in

By taking the derivative of (3.21) implicitly with respect to p, we obtain dx

r

+ Cy

+ A−γ w y, D1 B1 yB1 −1 + D2 B2 yB2 −1 +

in which the inequality follows because all three terms are nonpositive, and the third is strictly negative because (1 − γ )C > 0. Therefore, to complete our proof, it is enough to show that g (ys ) ≥ 0. Note that by (3.21) and (3.22), g (ys ) =

y

1

µ − r Uw (w, A) . σ 2 Uww (w, A)

Proof. By construction, the function ψ given by the right-hand side of Eq. (3.27) is increasing and concave with respect to w , and it solves the FBP given in (3.4) on R1 . Thus, to be able to conclude that Ψ equals ψ on R1 , it is enough to show that ψ is increasing with respect to A, as stated in the discussion before (3.4). To that end, one can show that, in terms of y,

ψA (w, A) ∝ (1 − γ )ˆv (y) + γ yvˆ y (y),

(3.28)

dy dA

because the terms involving additively cancel. After substituting for vˆ from (3.9) and then substituting for D1 and D2 from (3.19) and (3.20), respectively, and canceling a factor of y/r, we learn that

in which the second line follows from expressing p∗ in terms of x˜ B2 −1 by using Eqs. (3.25) and (3.26). Therefore, g (ys ) > 0 for any p ∈ [0, p∗ ). The strict convexity of vˆ follows. 

  B1 −1 λO 1 − B2 y ψA (w, A) ∝ − O r +λ B1 − B2 yb  B2 −1  B1 − 1 y + + 1. B1 − B2 yb

Remark 3.3. At this point, we see the importance of the critical value p∗ . If p were greater than or equal to p∗ , then vˆ as given in

It is straightforward to show that the right-hand side of (3.29) is non-increasing with respect to y; thus, it is enough to show that it

(3.29)

T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

is positive for y = yb . When y = yb , the right-hand side of (3.29) r reduces to r +λ O , which is positive. Thus, ψA > 0 on R1 , and we conclude that Ψ = ψ on that region. The optimal consumption and investment strategies are given in feedback form via the first-order − γ1

necessary conditions: c ∗ (w, A) = Ψw (w, A) and π ∗ (w, A) = r Ψw (w,A) − µ− . σ 2 Ψww (w,A) We next show that Ψ is increasing and concave with respect to w and increasing with respect to A on R2 . On R2 , we have from (3.5),

Ψ (w, A) = Ψ (w ′ , A′ ), aA aA in which w ′ = z0 · w+ , and A′ = w+ . Note that (w ′ , A′ ) ∈ R1 . z 0 +a z0 +a From this relationship, we obtain the following expressions:

Ψw (w, A) = Ψw (w , A ) > 0,

(3.30)

ΨA (w, A) = ΨA (w , A ) > 0, Ψww (w, A) = Ψww (w ′ , A′ ) < 0,

(3.31)

′

′

′

′

in which we use the boundary conditions for Ψ at (z0 A, A) given in (3.4). Thus, Ψ is increasing and concave with respect to w and increasing with respect to A on R2 . Finally, observe that Ψ is continuous on the boundary between R1 and R2 , as are Ψ ’s relevant derivatives. It follows that Ψ ∈ C 2,1 (R+ × R+ ) is increasing and concave with respect to w and increasing with respect to A. To complete the proof of this theorem, it is enough to show that Ψ satisfies the three conditions of Theorem 2.1. First, consider Ψ on R1 . By construction, Lπ,c Ψ ≤ 0 on R1 for all (π , c ) ∈ R × R+ . Conditions (ii) and (iii) are equivalent to

(1 − p)a Ψw ≤ ΨA ≤ a Ψw ,

0 ≤ w/A ≤ z0 .

In terms of y, these inequalities become

(1 − p)ay ≤ (1 − γ )ˆv (y) + γ yvˆ y (y) ≤ ay

yb ≤ y ≤ ys .

Note that the middle expression above is identical to the righthand side of (3.28). After substituting for vˆ as we did earlier in this proof and simplifying, we obtain the equivalent inequalities:

λO ≤ f (y) ≤ λO + pr ,

yb ≤ y ≤ ys ,

in which f is given by

 f (y) = λ

O

1 − B2 B1 − B2



y yb

B1 −1 +

B1 − 1 B1 − B2



y

B2 −1 

yb

.

From earlier in this proof, we know f is non-decreasing with respect to y. Note that f (yb ) = λO and f (ys ) = λO + pr, in which the latter follows from (3.21). Thus, we have shown that Ψ satisfies the three conditions of Theorem 2.1 on R1 . Because of the relations in (3.30) and (3.31), it follows that Ψ satisfies conditions (ii) and (iii) of Theorem 2.1 on R2 because those conditions hold on R1 . Finally, consider condition (i) for (w, A) ∈ R2 :

Lπ ,c Ψ (w, A)

= −(r + λ )Ψ (w, A) + (r w + A)Ψw (w, A)   1 2 2 + (µ − r )π Ψw (w, A) + σ π Ψww (w, A) 2  1−γ  c − c Ψw (w, A) + 1−γ S

= −(r + λS )Ψ (w ′ , A′ ) + (r w ′ + A′ )Ψw (w ′ , A′ )   1 2 2 ′ ′ ′ ′ + (µ − r )π Ψw (w , A ) + σ π Ψww (w , A ) 2

 +

c

1−γ

361

 − c Ψw (w ′ , A′ ) + [r (w − w ′ )

1−γ + (A − A′ )]Ψw (w ′ , A′ ) = Lπ ,c Ψ (w ′ , A′ ) + [r (w − w ′ ) + (A − A′ )]Ψw (w ′ , A′ ) λO 1A Ψw (w ′ , A′ ) ≤ 0. =− r + λO Here, we use the fact that Lπ ,c Ψ (w ′ , A′ ) ≤ 0 for (w ′ , A′ ) ∈ R1 . We have shown that Ψ satisfies the conditions of Theorem 2.1 on R+ × R+ , and we conclude that Ψ ≥ U. From the definition of Ψ in (3.3), we know that U ≥ Ψ ; thus, U = Ψ on R+ × R+ , and we are done.  When p = 0, one can show that the solution in Theorem 3.1 with the optimal value of z0 = 0 is the same as the one given in Section 3.1. In Section 4, we further explore properties of the optimal strategies given in this and the next section. 3.3. The case for which p ≥ p∗ The difference between this case and the case for which p < p∗ is highlighted by the optimal strategy that an individual uses when wealth reaches zero. As we showed in the previous section, when p < p∗ , it is optimal for a retiree to keep her wealth non-negative by surrendering existing annuity income. However, when the size of the proportional surrender charge p is larger, namely, when the cost of surrendering is higher, the retiree has more incentive not to surrender annuity income. Instead, we will show in this section that, as her wealth reaches zero, the optimal strategy is for the retiree to consume less than her existing annuity income and to invest nothing in the risky asset. As before, we also hypothesize the existence of a ratio of wealth-to-annuity income z0 such that, at a point (w, A) with w/A > z0 , the individual purchases annuity income to reach (w ′ , A′ ) such that w ′ /A′ = z0 . Additionally, the retiree does not purchase any annuity income when 0 < w/A < z0 , and she does not surrender existing annuity income under any circumstance. We solve the optimization problem as we did in the previous section, and the details of the solution in this case are very similar to the case for which p < p∗ . For this reason, we omit most of the details and the proofs. Let Φ denote the value function under the restricted control strategies described in the preceding paragraph:

Φ (w, A) =

sup

{πt ,ct },z0

Ew,A

τd





e−rt u(ct ) dt .

0

If we find a function φ ∈ C 2,1 (R1 ) that is increasing and concave with respect to w and increasing with respect to A that solves the following FBP, then φ = Φ on R1 :

   S  (µ − r )π φw (r + λ )φ = (r w + A)φw + max  π       1−γ    1 2 2 c   + σ π φww + max − c φw ,  c ≥0 2 1−γ  0 ≤ w/A ≤ z0 ;    µ − r φw (0, A)   − 2 = 0;   σ φww (0, A)    φA (z0 A, A) = a φw (z0 A, A); φAw (z0 A, A) = a φww (z0 A, A).

(3.32)

The differential equation in (3.32) is an HJB equation through which we will determine the optimal consumption and investment strategies in feedback form via the first-order necessary conditions. The first boundary condition arises from our hypothesis that the retiree invests nothing in the risky asset when wealth reaches zero;

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T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

this condition is different than the boundary condition at zero wealth that we applied in (3.4). The second boundary condition comes from our hypothesis that the retiree buys annuities only when w/A = z0 for (w, A) ∈ R1 via singular control. Finally, the last boundary condition is a smooth fit condition because the value of z0 is chosen optimally. These last two boundary conditions and the HJB equation are identical to the ones in (3.4). Once we have solved for Φ on R1 , then we have for (w, A) ∈ R2 ,

Φ (w, A) = Φ (w − a 1A, A + 1A), w−z A

in which 1A = z +0a . That is, we obtain Φ on R2 from Φ on the 0 boundary of R1 as we did for Ψ because in both cases, the retiree practices impulse control if w/A > z0 . As for p < p∗ , we reduce the dimension of the problem for Φ by defining z = w/A and V (z ) = Φ (z , 1). The function V solves a non-linear ODE, which we linearize by passing to the dual space via the Legendre transform. We solve that linear problem as we did for the case in which p < p∗ , then show that by reversing the above process, we obtain a semi-analytical expression for Φ . To show that U = Φ , it is sufficient to show that Φ satisfies the conditions of Theorem 2.1. We summarize our work in the following theorem, which we state without proof because it is quite similar to the proof of Theorem 3.1. Theorem 3.2. When p ≥ p∗ , the maximized utility U defined in Eq. (2.2) in the region R+ × R+ = R1 ∪ R2 , with R1 = {(w, A) : 0 ≤ w/A ≤ z0 , A ≥ 0} and R2 = {(w, A) : w/A > z0 , A > 0}, is given by: (i) For (w, A) ∈ R1 ,  γ −1  y + A−γ w y, U (w, A) = A1−γ D1 yB1 + D2 yB2 + + Cy γ r in which y ∈ [ys , yb ] uniquely solves D1 B1 yB1 −1 + D2 B2 yB2 −1 +

1 r

−

1−γ

γ

− γ1

Cy

=−

w A

,

and D1 , D2 , and B1 > 1 > 0 > B2 are given in Eqs. (3.19), (3.20), (3.10) and (3.11), respectively, while ys , yb , and z0 are specified y as follows: x˜ = y s > 1 is the unique solution of Eq. (3.26), ys is b

as in (3.22) with x replaced by x˜ , and z0 is as in (3.24). (ii) For (w, A) ∈ R2 , U (w, A) = U (w − a 1A, A + 1A), in which 1A =

w−z0 A . z0 +a

The associated optimal annuitization, consumption, and investment strategies are as follows: w−z A

(i) Purchase additional annuity income of 1A = z +0a when 0 (Wt , At ) = (w, A) ∈ R2 . (ii) Purchase additional annuity income instantaneously to keep (Wt , At ) in the region R1 , specifically, to keep Wt /At ≤ z0 . (iii) Never surrender existing annuity income. (iv) Consume continuously at the following rate when (Wt , At ) = (w, A) ∈ R1 : − γ1

c ∗ (w, A) = Uw (w, A). (v) Invest the following amount of wealth in the risky asset when (Wt , At ) = (w, A) ∈ R1 :

π ∗ (w, A) = −

µ − r Uw (w, A) . σ 2 Uww (w, A)

Remark 3.4. Note that x˜ is independent of p in this case. Thus, the optimal annuitization, consumption, and investment strategies are independent of the value of p, other than the requirement that p ≥ p∗ . An individual behaves as if the annuity is not commutable (or equivalently, as if p = 1) and does not surrender existing annuities under any circumstances, an unexpected result.

4. Properties of the optimal strategies In this section, we analyze the optimal strategies of a retiree under a variety of conditions. We also analyze the retiree’s optimal strategies as she becomes very risk averse. Finally, we illustrate the relation between the proportional surrender charge and the retiree’s optimal strategies via numerical examples. 4.1. Properties of z0 and choosing λO The optimally chosen z0 is a measure of the willingness of the retiree to annuitize. Indeed, if w/A is greater than z0 , then she immediately annuitizes enough of her wealth to bring that ratio equal to z0 . The smaller the value of z0 , the lower her wealth has to be in order for her to annuitize. Thus, we associate smaller values of z0 with greater willingness to annuitize. We expect that a retiree will be less willing to annuitize if the proportional surrender charge p is larger. It turns out that z0 is increasing with respect to p for p < p∗ , and we prove this fact in the next proposition, with the proof in the Appendix. Recall that z0 is independent of p for p ≥ p∗ . Proposition 4.1. z0 , given in (3.24), equals 0 when p = 0 and increases with respect to p for 0 ≤ p < p∗ . Because z0 = 0 when p = 0, we obtain the case of p = 0, which we considered in Section 3.1, as a special case of Theorem 3.1. As the retiree becomes more risk averse, we expect her to be more willing to annuitize her wealth in order to guarantee a particular income to fund her consumption. It turns out that z0 is decreasing with respect to γ , and we prove this fact in the next proposition; see the Appendix for the proof. In Section 4.3, we show that as γ goes to infinity, z0 goes to 0. Proposition 4.2. z0 , given in (3.24), decreases with respect to γ for all p ∈ [0, 1]. We expect the retiree to be less willing to annuitize her wealth as her own life expectancy declines, or equivalently, λS increases. Extensive numerical experiments indicate that z0 increases with respect to λS for all p ∈ [0, 1]. The insurance company can use this (numerical) result as a way of determining the health status of the individual who is annuitizing optimally. A retiree who annuitizes her wealth at a larger value of z0 has a shorter life expectancy than someone who annuitizes at a lower value of z0 , all else being equal. Thus, z0 is a proxy for health status. When the retiree surrenders annuity income, the insurance company experiences a loss of reserves, reserves that would be otherwise used to mitigate the mortality risk. Indeed, the purpose of life insurance and life annuities is to pool money so that life insurance survivors help finance death benefits for those who die and so that people who die while holding life annuities help fund the annuities of those who survive. The proportional surrender charge p serves as a risk charge by the insurance company to offset the loss of reserves when the individual surrenders annuity income. Another means of charging a risk loading is to decrease the value of λO , which is used both in pricing the life annuity and in setting the dollar amount of risk charge, namely, p/(r + λO ). We use the next result to show how an issuer of annuities might choose a value of λO for a given value of p; see the Appendix for its proof. Proposition 4.3. z0 , given in (3.24), decreases with respect to λO for all p ∈ [0, 1]. Proposition 4.3 makes sense because we expect the retiree to be less willing to annuitize her wealth as annuities become more

T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

expensive. We use this result to define a so-called indifference value for λO given the other parameters of the model, in particular, given p < p∗ . Suppose the insurance company has been issuing noncommutable annuities and at least breaking even with λO = (λO )∗ . Corresponding to these non-commutable annuities, there is a value of z0 , denote it by z0∗ . There are two individual-specific parameters that go into computing z0∗ , and these are λS and γ . Representative choices for those parameters might be λS = (λO )∗ and γ ∈ [2, 3], say, γ = 2.5. Now, suppose the insurance company wishes to issue commutable annuities with a given proportional surrender charge p < p∗ , and the company wishes to choose a lower value of λO in order to create a larger risk loading. We propose that the insurance company choose a value of λO equal to (or no lower than) λO (p) that solves z0 (λ (p), p) = z0 , O

∗

(4.1)

in which we assume that the underlying parameters are the same for both sides of this equation. Specifically, λS and γ are the same when computing z0 (λO (p), p) on the left and z0∗ on the right. If the insurance company chooses λO (p) via (4.1), the individual considering the commutable annuity will be indifferent (from the standpoint of willingness to annuitize, as measured by z0 ) between the commutable annuity and the non-commutable one. In the left-hand side of (4.1), we explicitly denote the dependence of z0 upon λO and p, in which z0 is given by (3.24), because we want to highlight the dependence of the solution of (4.1), namely λO (p), upon p. Of course, λO (p) depends upon the other parameters; in particular, it depends on the value (λO )∗ that the company uses when annuities are not commutable. In the following corollary of Proposition 4.3, we deduce that our definition of λO (p) makes sense because we expect the insurance company to increase its mortality loading by decreasing λO as the proportional surrender charge p becomes less severe. Corollary 4.1. For 0 ≤ p < p∗ , λO (p) as defined in (4.1) decreases as p decreases. It is of interest to the insurance company to compute the dollar amount of the surrender charge per dollar of annuity income surrendered, p/(r + λO ), and we graph the function f (p) = p/(r + λO (p)) in a numerical example in Section 4.4. As an alternative to defining λO (p) by (4.1), one might define it by equating the values of the maximized utilities at a specific pair (w, A), instead of equating the corresponding values of z0 . However, choosing (w, A) is difficult and makes λO (p) dependent upon (w, A), which varies in time for any given individual. On the other hand, the only individual-specific parameters underlying λO (p) in (4.1) are λS and γ , which we assume are fixed for any individual. Furthermore, one could use an equilibrium framework for pricing life annuities and their surrender values based on interactions between the seller of life annuities and potential buyers. The work in this paper provides a basis for considering such a problem because we completely determine the optimal behavior of retirees when presented with commutable life annuities. See Detemple and Serrat (2003) for an equilibrium pricing model in the presence of stochastic (labor) income under the constraint that (liquid) wealth remain non-negative; their work relies on that of He and Pagès (1993), who determine optimal consumption and investment of an individual who took asset prices as given. Numerical work indicates that p∗ decreases as λO increases. Thus, as insurance companies increase their prices (by decreasing λO ), the range of possible proportional surrender charges, namely [0, p∗ ), widens. We say that [0, p∗ ) is the range of surrender charges because if p ≥ p∗ , then the individual treats the life annuity as if it were not commutable.

363

In the next proposition, we show that if λS increases, then p∗ increases; see the Appendix for its proof. Insurance companies can use this result to compare two retirees: one who buys a commutable life annuity for a given surrender charge p and one who does not. The latter individual’s p∗ is less than the critical value of the former; thus, the company can infer that the latter individual’s λS is less than the former’s. In other words, a retiree who buys a commutable life annuity has a lower expected future lifetime than someone who does not, all else being equal. This result is intuitively pleasing because we expect people who buy non-commutable annuities to do so because they intend to receive income from them for a long time. By contrast, we expect someone who buys a commutable life annuity to do so because of the desirability of receiving a lump sum surrender amount if needed. Proposition 4.4. The critical proportional surrender charge p∗ as defined in (3.25) increases as λS increases. 4.2. Properties of the optimal consumption and investment strategies We will use the following lemma in the proof of the next proposition to show that when wealth equals zero in the case for which p ≥ p∗ , the retiree consumes at a rate less than the rate of her annuity income. Thereby, she is able (instantaneously) to return to positive wealth. (See the Appendix for a proof of the lemma.) −1/γ Lemma 4.1. When p ≥ p∗ , we have ys < 1.

Proposition 4.5. When p ≥ p∗ and when wealth is zero, the optimal rate of consumption is less than the rate of annuity income; that is, c ∗ (0, A) < A. Proof. From Theorem 3.2 and Lemma 4.1, it follows that − γ1

c ∗ (0, A) = Uw (0, A) = A−γ Vz (0)



− γ1

− γ1

= Ays

< A. 

We expect a retiree with larger wealth or with larger annuity income to consume more, and that is the case. Also, for an individual who is more risk averse than one with logarithmic utility, the optimal rate of consumption decreases with p. We omit the proof of the following proposition because it is similar to those of the preceding ones. Proposition 4.6. The optimal rate of consumption, c ∗ , increases as wealth increases and as annuity income increases. Moreover, if γ > 1, then c ∗ decreases with respect to p for 0 ≤ p < p∗ . We obtain similar results for the optimal amount invested in the risky asset, and we state them without proof in the following proposition. Note that the behavior of π ∗ differs from that of c ∗ as annuity income increases. While the retiree consumes more as annuity income increases, she invests less because she does not have to take on more risk to generate more wealth from which to consume due to her greater annuity income. Proposition 4.7. The optimal amount invested in the risky asset, π ∗ , increases as wealth increases and decreases as annuity income increases. Moreover, if γ > 1, then π ∗ decreases with respect to p for 0 ≤ p < p∗ . 4.3. Optimal strategies as γ → ∞ In this section, we examine the optimal consumption, investment, and annuitization strategies as the individual becomes highly risk averse. Proposition 4.8. As γ → ∞, c ∗ (0, A) → A for all p ∈ [0, 1].

364

T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

Proof. From the proof of Proposition 4.5, we know that c ∗ (0, A) = − γ1

− γ1

Ays , so it sufficient to show that limγ →∞ ys follows from Eqs. (3.12) and (3.22). 

= 1. The latter

Proposition 4.9. As γ → ∞, π ∗ (0, A) → 0 for all A > 0 and p ∈ [0, 1]. Proof. When p ≥ p∗ , from Theorem 3.2, we know that π ∗ (0, A) = 0 regardless of the value of γ . When 0 ≤ p < p∗ , from Theorem 3.1 and the work in Section 3.2, we have

π ∗ (0, A) =

µ−r A · ys Vˆ yy (ys ). σ2 Fig. 2. This graph illustrates how λO (p), as defined in Eq. (4.1), varies with p. In this example, we set r = 0.04, µ = 0.08, λS = (λO )∗ = 0.04, σ = 0.2, and γ = 2.5.

Using (3.9), (3.19) and (3.20), we obtain

 lim ys Vˆ yy (ys ) = lim

γ →∞

γ →∞

−

 × +

λO (B1 − 1)(1 − B2 ) r ( r + λO ) B1 − B2

B1 xB1 −1 1 + γ (B1 − 1)

1−γ

γ2

−

B2 xB2 −1



1 + γ (B2 − 1)

− γ1

Cys

= 0. 

Proposition 4.10. As γ → ∞, z0 → 0 for all p ∈ [0, 1]. Proof. We prove this property for the case p < p∗ . The proof for the case p ≥ p∗ is similar. First, consider the definition of z0 in (3.24). Recall that yb = ys /x, in which x is independent of γ . Thus, limγ →∞ x1/γ = 1, and as shown in Proposition 4.8, −1/γ

= 1. By taking the limit in (3.24), we obtain  − γ1 1−γ λO 1 lim z0 = lim C yb + γ →∞ γ →∞ γ r ( r + λO ) γ   1 1 − γ − × r + λS − m γ r

limγ →∞ ys

=

1 r

−

1 r

= 0. 

From Propositions 4.8–4.10, we deduce the following corollary. Corollary 4.2. The optimal strategy for a highly risk averse retiree is to annuitize fully and to consume her annuity income thereafter. Her wealth is always zero because z0 = 0 in the limit as γ → ∞. 4.4. Numerical examples We provide numerous numerical examples to illustrate the analytical results of Section 3. We focus our attention on the effects of the proportional surrender charge p and risk aversion γ on the optimal annuitization, consumption, and investment strategies of an individual. We use the following parameter values for our computations:

• λS = λO = 0.04; the hazard rate is such that the expected future lifetime is 25 years.

• r = 0.04; the riskless rate of return is 4% over inflation. • µ = 0.08; the drift of the risky asset is 8% over inflation. • σ = 0.20; the volatility of the risky asset is 20%.

With the parameter values given above, the critical proportional surrender charge is p∗ = 0.308. Recall that this is the critical value of p above which the optimal annuitization, investment, consumption strategies, as well as the maximized utility, of a retiree do not depend on the size of the surrender charge. This feature along with others are demonstrated in the tables and figures in this section. In Table 1, we give the value of z0 = w/A for various values of surrender charge p and relative risk aversion γ . The value of z0 is the critical ratio of wealth to annuity income above which an individual will purchase more annuity income; it indicates the willingness of an individual to annuitize. For example, assuming γ = 2.5 and p = 0.3, a retiree with $100,000 of assets and with $25,000 of existing annuity income would immediately trade $47,116 of assets for $3773 of additional annuity. By so doing, the critical ratio of wealth-to-annuity income becomes z0 = 1.8362. By comparison, a retiree with γ = 0.8 and all other parameters the same, would not purchase additional annuities since her critical ratio of wealth-to-annuity income of 4.0 is already below the level of 7.1591. A higher value of z0 indicates less interest in annuitization. In Table 1, we observe that z0 increases with respect to p for a fixed value of γ and decreases with respect to γ for a fixed value of p, as expected from Propositions 4.1 and 4.2, respectively. In other words, a lower proportional surrender charge encourages retirees to annuitize, and those who are more risk averse are more willing to purchase annuities. This result is consistent with our proposition that lack of flexibility discourages retirees from purchasing (non-commutable) immediate life annuities. In the case for which p ≥ p∗ , z0 does not change with p. An individual treats commutable annuities with a surrender charge greater than p∗ as a non-commutable one (p = 1) regardless of the size of p ∈ [p∗ , 1]. Figs. 2 and 3 examine the indifference value λO (p), as defined in Eq. (4.1). In Fig. 2, note that λO (p) decreases as p decreases, which confirms the result of Corollary 4.1. Then, in Fig. 3, we plot the dollar amount of the surrender charge (per dollar of annuity income surrendered), p/(r + λO (p)), as a function of p ∈ (0, p∗ ). It is interesting that p/(r + λO (p))is nearly linear with respect to p. Tables 2 and 3 illustrate, respectively, the impact of the surrender charge on the individual’s optimal investment and consumption strategies when wealth is zero. In these cases, we assume that the existing annuity income is two units, that is, A = 2. This assumption of existing annuity income is reasonable due to the fact that most American retirees have annuity income at the time of retirement, say, from Social Security or a private pension. From the tables, we observe that the optimal strategy is divided into two categories: In the cases for which p < p∗, an individual continues to invest in the risky asset; in the cases for which p ≥ p∗ ,

T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

365

Table 1 How do the proportional surrender charge p and risk aversion γ affect annuitization? p

z0 for various levels of p and γ

(p∗ = 0.308)

γ = 0.8

γ = 1.5

γ = 2.0

γ = 2.5

γ = 3.0

γ = 5.0

0.01 0.02 0.04 0.08 0.10 0.20 0.30 0.40 0.60 1.00

1.5794 2.2400 3.1723 4.4593 4.9542 6.5875 7.1591 7.1622 7.1622 7.1622

0.8173 1.1443 1.5904 2.1759 2.3914 3.0593 3.2723 3.2734 3.2734 3.2734

0.6078 0.8478 1.1722 1.5919 1.7444 2.2088 2.3530 2.3537 2.3537 2.3537

0.4838 0.6733 0.9280 1.2548 1.3726 1.7276 1.8362 1.8367 1.8367 1.8367

0.4018 0.5584 0.7680 1.0355 1.1314 1.4184 1.5052 1.5057 1.5057 1.5057

0.2394 0.3318 0.4545 0.6093 0.6642 0.8263 0.8743 0.8746 0.8746 0.8746

Table 2 How do the proportional surrender charge p and risk aversion γ affect investment when w = 0? p

Investment in the risky asset for various levels of p and γ when w = 0 (A = 2)

(p∗ = 0.308)

γ = 0.8

γ = 1.5

γ = 2.0

γ = 2.5

γ = 3.0

γ = 5.0

0.01 0.02 0.04 0.08 0.10 0.20 0.30 0.40 0.60 1.00

25.2800 22.9429 19.7310 15.2923 13.5110 6.3519 0.4460 0 0 0

13.4827 12.2362 10.5232 8.1559 7.2059 3.3877 0.2378 0 0 0

10.1120 9.1772 7.8924 6.1169 5.4044 2.5408 0.1784 0 0 0

8.0896 7.3417 6.3139 4.8935 4.3235 2.0326 0.1427 0 0 0

6.7413 6.1181 5.2616 4.0780 3.6029 1.6938 0.1189 0 0 0

4.0448 3.6709 3.1570 2.4468 2.1618 1.0163 0.0714 0 0 0

Table 3 How do the proportional surrender charge p and risk aversion γ affect consumption when w = 0? p

Rate of consumption for various levels of p and γ when w = 0 (A = 2)

(p∗ = 0.308)

γ = 0.8

γ = 1.5

γ = 2.0

γ = 2.5

γ = 3.0

γ = 5.0

0.01 0.02 0.04 0.08 0.10 0.20 0.30 0.40 0.60 1.00

1.8486 1.8353 1.8013 1.7235 1.6824 1.4682 1.2478 1.2300 1.2300 1.2300

2.0766 2.0549 2.0168 1.9477 1.9147 1.7559 1.6016 1.5893 1.5893 1.5893

2.0911 2.0718 2.0389 1.9812 1.9540 1.8254 1.7018 1.6920 1.6920 1.6920

2.0891 2.0721 2.0438 1.9947 1.9719 1.8644 1.7617 1.7536 1.7536 1.7536

2.0832 2.0682 2.0435 2.0010 1.9814 1.8893 1.8016 1.7947 1.7947 1.7947

2.0607 2.0507 2.0345 2.0071 1.9946 1.9363 1.8811 1.8768 1.8768 1.8768

Fig. 3. This graph illustrates how the amount of the surrender charge, p/(r +λO (p)), varies with p, in which λO (p) is defined in Eq. (4.1). In this example, we set r = 0.04, µ = 0.08, λS = (λO )∗ = 0.04, σ = 0.2, and γ = 2.5.

it is optimal not to invest in the risky asset and to consume less than the annuity income. When p < p∗ , the optimal strategy is to continue investing in the risky asset by borrowing from the riskless account and to surrender just enough annuity income to keep wealth non-negative when needed. In contrast, when p ≥ p∗ , it is optimal not to surrender the annuity income under

any circumstances, as we showed in Section 3.3. Therefore, when w = 0, the corresponding investment and consumption strategies guarantee that wealth will not decrease farther. These numerical results are consistent with our analytical results in Theorems 3.1 and 3.2. Note that the amount invested in the risky asset at w = 0 decreases with respect to p (and γ ), other parameters fixed. In fact, we can show that investment decreases with respect to p at any level of wealth, but for the sake of brevity, we omitted that proposition and the corresponding proof from this paper. Similarly, we can show that investment decreases with respect to γ when w = 0. Also, the rate of consumption at w = 0 decreases with p given γ . However, the relationship between the rate of consumption and risk aversion is not monotonic. Both Tables 2 and 3 confirm our analytical conclusion that when p ≥ p∗ , an individual behaves as if the annuity is not commutable at all; compare with Theorem 3.2. Figs. 4–6 illustrate our results from a different point of view. In these figures, we plot the optimal investment, optimal consumption, and maximized utility, respectively, for wealth w ranging from 0 to 1. We choose the parameters as described above and assume that γ = 2.5 and A = 1. In each figure, we plot two curves representing the cases p = 0.05 and p = 0.6. The graphs show that a larger value of p leads to lower

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T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

Fig. 4. This graph illustrates the optimal consumption strategies for wealth equal to zero. We set the existing annuity income A to be 1 in this example. The blue dotted line represents the case p ≥ p∗ ; recall that in this case, the optimal rate of consumption at w = 0 is less that the rate of annuity income. By contrast, the optimal rate of consumption is allowed to be above A when p < p∗ . In this example, we set r = 0.04, µ = 0.08, λS = λO = 0.04, σ = 0.2, γ = 2.5, and A = 1.

Fig. 5. This graph illustrates the optimal investment strategies for wealth equal to zero. When p ≥ p∗ , a retiree invests nothing in the risky asset when w = 0. This case is represented by the blue dotted line. By contrast, when p < p∗ , it is optimal to invest in a risky asset when w = 0. In this example, we set r = 0.04, µ = 0.08, λS = λO = 0.04, σ = 0.2, γ = 2.5, and A = 1.

Fig. 6. This graph compares the maximized utilities when the proportional surrender charges differ. In this example, we set r = 0.04, µ = 0.08, λS = λO = 0.04, σ = 0.2, γ = 2.5, and A = 1.

investment, consumption, and utility for all w . Namely, both the investment and consumption strategies are more conservative when the surrender charge is higher. Lastly, Fig. 7 displays the maximized utility as a function of p ∈ [0, 1] for fixed (w, A) = (100, 0) and γ = 2.5. This representative graph shows that the maximized utility is a monotonically decreasing function of p and is of the same value for all p ≥ p∗ . 5. Conclusion In this paper, we considered a utility maximization problem with commutable life annuities. In an incomplete financial market

Fig. 7. This graph illustrates how the proportional surrender charge p affects the maximized utility. The shape of the curve above is representative for situations with different given initial wealth and annuity income. In this example, we set r = 0.04, µ = 0.08, λS = λO = 0.04, σ = 0.2, γ = 2.5, w = 100, and A = 0.

with a riskless asset, a risky asset, and commutable annuities, we investigated a retiree’s optimal annuitization, consumption, and investment strategies. In our model, the commutability of an annuity is quantified by a proportional surrender charge, which ranges from 0% to 100%. A retiree’s willingness to annuitize is indicated by the critical ratio of wealth-to-annuity income above which she would immediately purchase more annuity income if the surrender charge were low enough. We proved that a smaller surrender charge leads to a smaller critical ratio of wealth-toannuity income. This result indicates that the commutability of annuities encourages retirees to annuitize. We found that the individual’s optimal annuitization, consumption, and investment strategies depends on the size of the proportional surrender charge. When the surrender charge is larger than the so-called critical value, a retiree does not surrender her existing annuity income under any circumstance. She stops investing in the risky asset and consumes less than her annuity income as her wealth approaches zero. When the surrender charge is smaller than the critical value, a retiree surrenders enough annuity income to keep her wealth non-negative as needed. She continues to invest in the risky asset as her wealth approaches zero. One might think that a smaller surrender charge is always better than a larger one. But unexpectedly, we found that in the case for which the surrender charge is larger than the critical value, the optimal strategies and maximized utility of a retiree do not depend on the size of surrender charge. A commutable annuity with a proportional surrender charge above the critical value is equivalent to a non-commutable one for an optimally behaving retiree. We also found that, for a retiree without a bequest motive, full annuitization is optimal when the surrender charge is zero or as the risk aversion of the retiree approaches infinity. In our paper, we assumed a constant hazard rate and a constant interest rate in our analysis as simplifications of reality. However, we believe that the main qualitative insights will hold in general, as argued in Moore and Young (2006) and further discussed above in Remark 2.2. To investigate commutable annuities, an alternative to the utility metric is the probability of lifetime ruin. As a risk metric, it is some times used to investigate optimization problems faced by retirees in a financial market. The interested reader can find a study of commutable annuities within that framework in Wang and Young (2012). Wang and Young assumed that a retiree consumes at an exogenously given level, and they determined the optimal investment strategy, as well as the optimal time to annuitize or to surrender, in order to minimize the probability that wealth reaches zero before her death. By contrast, in this paper, consumption is determined by the retiree herself. If one were to include life insurance in the market, then an individual could borrow against her annuity income by purchasing

T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

decreasing-benefit life insurance to serve as collateral in the case that she dies before repaying her loan. We look forward to considering this problem in future work. Framing the problem in this paper in an equilibrium setting is the next step. In that setting, one would not take annuity prices nor surrender values as given, but rather, have them set based on interactions between the annuity seller and buyers. We also acknowledge the importance of allowing for more realistic assumptions within equilibrium models, assumptions such as Gompertz mortality or the age and health status of buyers. Traditionally, one considers equilibrium models only after understanding individuals’ behavior when prices are given because that optimal behavior gives one information about how to find an equilibrium, and we hope this paper gives a basis for considering the problem of pricing life annuities.

367



λO B1 (1 − B2 ) B1 −1 x r + λO B1 − B2  B2 (B1 − 1) B2 −1 x . + B1 − B2

∝ 1−

(A.4)

This last expression is positive for 1 ≤ x < x˜ because the left-hand side of Eq. (3.26) is increasing with respect to x˜ for x˜ > 1.  Proof of Proposition 4.2. First, consider the case for which p < p∗ . From Eqs. (3.24), (3.23) and (3.22), we can express z0 as



λ

z0 =

Acknowledgments VRY thanks the Nesbitt Professorship of Actuarial Mathematics for financial support. We both thank a reviewer for helpful comments.



r 1 + 1 − γ − 1 γ r + λS − m 1−γ r γ γ2   1 λO B1 (1 − B2 ) x B 1 −1 + xγ − O r (r + λ ) B1 − B2 1 + γ (B1 − 1)   B2 (B1 − 1) xB2 −1 1 + + . B1 − B2 1 + γ (B2 − 1) r O

r (r + λO )

(A.5)

It follows that Appendix

 ∂ z0 λ 1 − γ =   ∂γ r + λO γ O

Proof of Proposition 4.1. From Eqs. (3.24), (3.12) and (3.23), it follows that

∂ yb ∂x ∂ ys ∂ z0 ∝− ∝ ys −x . ∂p ∂p ∂p ∂p

r+

(A.1)

×

From (3.21), we get

−

 ∂x (B1 − 1)(1 − B2 )  B1 −2 r x − xB2 −2 = O; B1 − B2 ∂p λ

(A.2)

which implies that ∂∂ px > 0 for x > 1. Next, from (3.22), we obtain

+

1 − 1 −1 ∂ ys − ys γ γ ∂p

+

  λO λS 1 − γ (B1 − 1)(1 − B2 ) =− r + − m r (r + λO ) γ γ2 (B1 − B2 )   B1 −2 B2 −2 B1 x B2 x ∂x × − . 1 + γ (B1 − 1) 1 + γ (B2 − 1) ∂ p

× (A.3)

× ×

(B1 − 1)(1 − B2 ) B1 − B2  B1 −1 B1 x

1 + γ (B1 − 1)

−

B2 xB2 −1 1 + γ (B2 − 1)



− m 1γ−γ 2 

2

  2 1 λS − 2 −m − 3 + 2 γ γ γ  1

1

γ2 r + 1

γ

2

λS γ

1

x γ ln x

−m 

γ2

 

λO r ( r + λO )

B2 (B1 − 1) B1 − B2

1−γ

x



B1 (1 − B2 )

xB1 −1

B1 − B2

1 + γ (B1 − 1)

B2 −1

1 + γ (B2 − 1)



 −

1 r



1

+ xγ

λO r ( r + λO )

(B1 − 1)(1 − B2 ) B1 x B1 − B2 (1 + γ (B1 − 1))2  B 1 −1

B2 xB2 −1

−

(1 + γ (B2 − 1))2   2  −1 1 1 − γ ∝ − r + λS + m − ys γ x γ ln x γ 1   λS 1 − γ r + λO m xγ × r+ −m + 2 γ γ2 λO γ B1 − B2

Thus,

  1 ∂ z0 λO λS 1−γ γ +1 ∝ ys − γ x ys r+ −m 2 ∂p r (r + λO ) γ γ (B1 − 1)(1 − B2 ) × B1 − B2   B1 xB1 −2 B2 xB2 −2 × − 1 + γ (B1 − 1) 1 + γ (B2 − 1)   O λ B1 (1 − B2 ) 1 xB1 −1 ∝ − O r r (r + λ ) B1 − B2 1 + γ (B1 − 1)  B2 (B1 − 1) xB2 −1 λO + −γ B1 − B2 1 + γ (B2 − 1) r ( r + λO )

−1 λS γ

× [B1 (1 + γ (B2 − 1))2 xB1 −1 − B2 × (1 + γ (B1 − 1))2 xB2 −1 ] =: g (x).

(A.6)

It is straightforward to show that g (1) = −r < 0. Thus, if we ∂z prove g ′ (x) ≤ 0 for 1 ≤ x < x˜ , then it will follow that ∂γ0 < 0 ∗ for p < p . Additionally, because z0 is independent of p for p ≥ p∗ , ∂z ∂z from continuity of ∂γ0 , we will be able to conclude that ∂γ0 < 0 for all p. To that end, −1

∂ ys γ γ1 r + λO g (x) = x ln x ∂x λO ′



λS 1−γ r+ −m γ γ2



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T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369 1

−γ



− ys

−m

1

γ 1−γ

1−γ

x

γ

γ

r + λO



λO

r+

λS γ

   S  λS 1−γ λ 1−γ × r+ −m 2 + −m 2 γ γ γ γ × [1 + γ (B1 − 1)] · [1 + γ (B2 − 1)] > 0.

1−γ



γ2

1−γ

ln x + x



+

m

x

γ

γ 3 B1 − B2

(1 + γ (B1 − 1))2

× (1 + γ (B2 − 1))   B1 xB1 −1 B2 xB2 −1 × − 1 + γ (B1 − 1) 1 + γ (B2 − 1)   B1 (1 − B2 ) B1 −1 ln x x +1 ∝ γ B1 − B2  B2 (B1 − 1) B2 −1 r + λO + x − < 0, B1 − B2 λO

Use (3.26) to simplify the above expression to obtain

2

− γ1

ys

 1−γ λS −m 2 ⇐⇒ −(1 − γ ) r + γ γ   O λ B1 − 1 B2 −1 1 − B2 B1 −1 − x˜ + x˜ γ B1 B2 r + λO B1 − B2 B1 − B2     λS 1−γ λS 1−γ × r+ −m 2 + −m 2 γ γ γ γ × [1 + γ (B1 − 1)] · [1 + γ (B2 − 1)] > 0   λO 1 − B2 B1 −1 B1 − 1 B2 −1 ⇐⇒ − γ B1 B2 x˜ + x˜ r + λO B1 − B2 B1 − B2   λS 1−γ × r+ −m 2 γ γ   S S λ λ 1−γ r+ >0 −γ −m 2 m γ γ   λO 1 − B2 B1 −1 B1 − 1 B2 −1 ˜ ˜ ⇐⇒ − B B x + x 1 2 r + λO B1 − B2 B1 − B2 S λ > 0. −

(A.7)

Proof of Lemma 4.1. − γ1

<

1

 B1 (1 − B2 ) λO 1 x˜ B1 −1 O r (r + λ ) B1 − B2 1 + γ (B1 − 1)   1 B2 (B1 − 1) 1 B2 −1 x˜ + + B1 − B2 1 + γ (B2 − 1) r γ × <1 (1 − γ ) C   λO B1 (1 − B2 ) 1 ⇐⇒ − x˜ B1 −1 r ( r + λO ) B1 − B2 1 + γ (B1 − 1)   1 B2 (B1 − 1) 1 x˜ B2 −1 + + B1 − B2 1 + γ (B2 − 1) r   S λ 1−γ × r+ −m 2 <1 γ γ  B1 (1 − B2 ) 1 λO x˜ B1 −1 ⇐⇒ − r + λO B1 − B2 1 + γ (B1 − 1)   B2 (B1 − 1) 1 λS + x˜ B2 −1 · r + B1 − B2 1 + γ (B2 − 1) γ   S  1−γ λ 1−γ −m 2 + −m 2 < 0. γ γ γ   S γ2 1−γ Recall that [1+γ (B1 −1)]·[1+γ (B2 −1)] = − m r + λγ − m γ 2 < 0; multiply the above inequality by this factor to get 

⇐⇒

− γ1

ys

<

1



for 1 ≤ x < x˜ , in which negativity follows from the definition of x˜ in (3.26) and from the fact that the left-hand side of (3.26) strictly increases with respect to x˜ for x˜ > 1. 

ys

<

−

m

We demonstrate this last inequality by using (3.26) and the equality B1 + B2 − 1 = − λm : S

−

  1 − B2 B1 −1 B1 − 1 B2 −1 λS λO ˜ ˜ B B x + x − 1 2 r + λO B1 − B2 B1 − B2 m S λO λ = −B2 − B2 (B1 − 1)˜xB2 −1 − r + λO m   λO B2 −1 > 0.  = (B1 − 1) 1 − B2 x˜ r + λO

Proof of Proposition 4.3. It is enough to consider the case for which 0 < p < p∗ , or equivalently, 1 < x < x˜ . From (3.24), we can write z0 =

1−γ

γ

− γ1

Cys

1

xγ + A

λO 1 − , r (r + λO ) r

in which 1

 λO B1 (1 − B2 ) [1 + γ (B2 − 1)] x˜ B1 −1 ⇐⇒ − r + λO B1 − B2  B2 (B1 − 1) [1 + γ (B1 − 1)] x˜ B2 −1 + B1 − B2   S   λS 1−γ λ 1−γ × r+ −m 2 + −m 2 γ γ γ γ × [1 + γ (B1 − 1)] · [1 + γ (B2 − 1)] > 0  λO B1 (1 − B2 ) B1 −1 ⇐⇒ − ( 1 − γ ) x˜ O r +λ B1 − B2    B2 (B1 − 1) B2 −1 λS 1−γ + x˜ · r+ −m 2 B1 − B2 γ γ   λO 1 − B2 B1 −1 B1 − 1 B2 −1 − γ B1 B2 x˜ + x˜ r + λO B1 − B2 B1 − B2

r + λS − m γ

1−γ

A=

r γ + λS − m γ

1−γ

> 0.

Differentiate z0 with respect to λO to obtain 1 ∂ z0 1 − γ − γ1 1 1 1 ∂ x ∂ = Cys xγ + xγ O O ∂λ γ γ x ∂λ ∂λO   1 1 − γ −γ A × Cys + . γ (r + λO )2

Use (3.21) and (3.22) to rewrite the above expression: 1

∂ z0 xγ = B −1 O 1 ∂λ x − xB2 −1



1 − B2 B1 − B2

xB1 −1 +

B1 − 1 B1 − B2

λO × · r γ (B1 − 1)(1 − B2 ) r + λO 1

B1 − B2







xB2 −1 − 1

B1 (1 − B2 ) B1 − B2

xB1 −1

T. Wang, V.R. Young / Insurance: Mathematics and Economics 51 (2012) 352–369

B2 (B1 − 1)

+

B1 − B2



B1 (1 − B2 )

×

+

xB2 −1





A

−1 + x

1

(r + λO )2

−

xγ

(r + λO )2

B1 −1

1 + γ (B1 − 1)

B1 − B2 B2 (B1 − 1)

xB2 −1

B1 − B2

1 + γ (B2 − 1)

 .

The factors in the first line are all positive, while the factor in the second line is negative for x ∈ (1, x˜ ); thus, the first term in the above expression is negative. It follows that if we can show that the sum of second and third terms is negative for x ∈ (1, x˜ ), then we are done. That sum is negative if h(x) < 0 for x ∈ (1, x˜ ), in which h is defined by

 h(x) = A −

+

1

B1 (1 − B2 )

xγ

1 + γ (B1 − 1)

B1 − B2

1

B2 (B1 − 1) B1 − B2

+B1 −1

xγ

+B2 −1

1 + γ (B2 − 1)

 .

One can show that h(1) = 0 and h′ (x) ∝ − B1 (1 − B2 ) xB1 −B2 + B2 (B1 − 1) ,





which is negative for x > 1, so our proof is complete.



Proof of Proposition 4.4. One can show that

∂ B1 ∂λS ∂ ∂λS ∂ ∂λS ∂ ∂λS ∂ ∂λS

B1 − 1 ∂ B2 1 − B2 =− , =− , m(B1 − B2 ) ∂λS m(B1 − B2 )   1 − B2 2(B1 − 1)(1 − B2 ) = , B1 − B2 m(B1 − B2 )3   B1 (1 − B2 ) (B1 − 1)(1 − B2 )(B1 + B2 ) = , B1 − B2 m(B1 − B2 )3   B1 − 1 2(B1 − 1)(1 − B2 ) =− , and B1 − B2 m(B1 − B2 )3   B2 (B1 − 1) (B1 − 1)(1 − B2 )(B1 + B2 ) =− . B1 − B2 m(B1 − B2 )3

By differentiating the defining equation for x˜ , (3.26), and by using the above expressions, we obtain 1 ∂ x˜ x˜ ∂λ

S

=

ln(˜x)

B1 x˜ B1 −1 + B2 x˜ B2 −1

m(B1 − B2 ) B1 x˜ B1 −1 − B2 x˜ B2 −1

−

x˜ B1 −1 − x˜ B2 −1

B1 + B2 m(B1 − B2 )

2

B1 x˜ B1 −1 − B2 x˜ B2 −1

.

Then, by differentiating the defining equation for p∗ , (3.25), and by substituting the above expressions and simplifying, it follows that

∂ p∗ x˜ B1 −B2 − x˜ −(B1 −B2 ) ∝ − ln(˜x). S ∂λ 2(B1 − B2 ) q

−q

It is straightforward to show that ln x ≤ x −2qx for all q > 0 and all x ≥ 1. Indeed, the derivatives of the two sides of this inequality are ordered. The proposition follows. 

369

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