A Portfolio Choice Problem Under Risk Capacity Constraint

Investment after retirement is significantly different from investment for (before) retirement in several respects. Retirees invest in an unknown while finite length of time because of longevity (mortality) risk. They also worry about the balance between spending and leaving wealth as an inheritance. More importantly, because of no labor income, these individuals will face living standard risk if a market downturn occurs, leading to a substantial decline in their living standards.¹¹1As Kenneth French presented at the Annual Conference for Dimensional Funds Advisors, 2016, “It is living standard risk you should know about the risk. It is what your exposure is to a major change in your standard of living during the entirely uncertain numbers of years you remain alive.”

Motivating by the asset allocation problem after retirement, this paper studies a portfolio choice problem with two distinguishing features. One is an uncertain investment time horizon; another is that the investment dollar amount in the risky asset is always bound from above by a fixed constant (capacity). We name this constraint a “risk capacity constraint”. We present the asset allocation problem after retirement as an optimal portfolio choice problem. Precisely, the retiree’s mortality risk is formulated by an uncertain investment time horizon. The length of each individual’s retirement may differ from the statistical life expectancy, and the mortality risk is virtually independent of the market risk in the financial market. Since the absence of labor income results in severe concerns about living standards, we use the risk capacity constraint to address the retiree’s concern about the living standard risk after retirement.²²2Since we focus on the risk capacity, we ignore other factors such as health care risk and real estate assets in retirement portfolios. See, for instance, Yogo (2016) about the discussion of other factors.

We first characterize the value function of the optimal portfolio choice problem. Then we characterize the region of the wealth in which the risk capacity constraint is binding. Specifically, assuming the value function is $C^{2}$ smooth, and under some technical conditions about the value function, then there exists a threshold (a positive number $W^{*}$) such that the retiree invests the capacity amount in the risky asset if and only if the portfolio wealth is greater than the threshold. Otherwise, the investment amount is strictly smaller than the capacity.

To derive the consumption-investment policy explicitly, we next investigate two particular yet critical situations in which these technical conditions of the value function can be verified. In the first situation, Problem (A), the investor (retiree) focuses on intertemporal consumption, whereas the investor concentrates on the inheritance wealth in the second situation, Problem (B). For a tractable purpose, we study the CRRA utility functions. In Problem (A) and Problem (B), if the value function is $C^{2}$ smooth, we demonstrate the existence of one number $W^{*}$ such that the risk capacity constraint is binding or not depends on whether the portfolio wealth is greater or smaller than this number. Moreover, we derive the explicit consumption-investment policy in terms of this threshold.

For Problem (A), the $C^{2}$ smooth property of the value function follows from the classical study in Zariphopoulou (1994). In general, when the utility function satisfies a global Lipschitz condition, and the invested dollar amount is bounded from below and above by two positive numbers, the value function is $C^{2}$ smooth by a recent remarkable theorem of Strulovic and Szydlowski (2015). However, the CRRA utility does not satisfy the global Lipschitz condition, and the risk capacity constraint implies that the dollar amount can be any sufficiently small positive number. A technical component in this paper is to investigate the smooth property of the value function and thus the explicit expression of the value function in Problem (B). We reduce the $C^{2}$ smooth property of the value function to solve a nonlinear equation of one variable. Briefly speaking, there exists a positive real number solution (and unique upon its existence) of the nonlinear equation if and only if the value function is $C^{2}$ smooth. Moreover, this real number solution separates the unconstrained region and constrained region of the risk capacity constraint. We also show that this method to determine the threshold $W^{*}$ and the value function explicitly can be used in Problem (A).

Our solution to the optimal portfolio choice problem provides new theoretical insights for retirement investment. First, the optimal investment strategy displays a wealth-cycle property, in contrast to the life-cycle feature which is suggested in Modigliani (1986). That is, individual’s investment and consumption decision has a life-cycle feature. See Bodie, Merton, and Samuelson (1992), Cocco, Gomes, and Maenhout (2005), Gomes and Michaelides (2005), Benzoni, Dufresne, and Goldstein (2007), and Bodie, Detemple, and Rindisbacher (2009) for life-cycle theoretical and empirical studies.³³3The life-cycle hypothesis is also used for preparing the retirement portfolio and in the retirement portfolio. For instance, a conventional rule for an agent of age $t$ is to invest (100 - t)/100 percent of the wealth in the stock market. See Malkiel (1999). Even after retirement, a time-dependent (life-cycle) investment strategy is also popular among financial advisors. Specifically, when the portfolio value is unsustainable for the entire retirement period, the retiree should invest in the market because the dollar investment in the stock will increase the expected portfolio value. Nevertheless, the percentage of wealth in the market declines in the wealth. The declining percentage of the wealth invested in risky assets is due to the retiree’ living standard concern to protect the portfolio value. This decreasing feature of the percentage becomes significant when the portfolio’s worth is sufficiently high.

Second, since the dollar invested in the stock is always a constant $L$ when the portfolio wealth is higher than a threshold $W^{*}$, this threshold $W^{*}$ measures the expected lump sum of the spending in the retirement period. Intuitively, when the portfolio is worth more than this threshold, the retiree aims to protect the portfolio by investing only a fixed amount of $L$ in the stock market without losing the living standard. This contingent constant-dollar strategy is thus a buffer-invest strategy: when the wealth is below the target, the retired invests; and if the wealth is above the target, the retire will dis-invest more on the stock market. In a classical constant-dollar strategy, the dollar invested in the risky asset is always fixed. In contrast, by a contingent constant-dollar strategy in this paper we mean a fixed dollar is invested in the risky asset only when the portfolio value is higher than a threshold. It is also different from the constant proportion portfolio insurance strategy in Black and Perold (1992) and El Karoui and Jeanblanc-Picque (1998), which is interpreting by a put option on the Merton-type portfolio and the underlying Merton-type portfolio. See Carroll (1997) for a buffer-stock saving theory under the permanent income hypothesis. Our result is consistent with the retirement portfolio’s decumulation process. In contrast, investing for (before) retirement is an accumulating asset process.

Third, the portfolio is nearly independent of the stock market when the retiree’s portfolio is worth sufficiently embracing the living standard. Moreover, the portfolio risk is also nearly zero if the retiree’s portfolio wealth is high enough. Therefore, the retiree’s living standard risk is reduced substantially under the risk capacity constraint.

Fourth, we demonstrate that the sub-optimality of the annuity, a popular insurance product for a retiree. Annuities are the perfect financial vehicle to counter only the mortality risk. However, due to unexpected costs or shock, non-annuitized wealth could be needed to cover the bill. From a portfolio choice perspective, we demonstrate that the annuity is not optimal since the optimal consumption should depend on wealth by incorporating a risk capacity constraint. The relationship between the consumption rate and wealth is highly non-linear but implementable.

Last but not least, the risk capacity constraint is significantly different from the leverage constraint. At first glance, it seems to be a particular case of leverage constraint or collateral constraint, $X_{t}\leq f(W_{t})$, where $X_{t}$ represents the dollar invested in the risky asset and $W_{t}$ the wealth at time $t$. Earlier studies on the leverage constraint include Grossman and Vila (1992), Zariphopoulou (1994), Vila and Zariphopoulou (1997). Studies on portfolio choice and asset pricing under other dynamic constraints on the control variable $c_{t}$ or the state variable, $W_{t}$, include Black and Perold (1992), Dybvig (1995), El Karoui and Jeanblanc-Picque (1998), Elie and Touzi (2008), Dybvig and Liu (2010), Chen and Tian (2016), Ahn, Choi and Lim (2019), and reference therein. However, most studies on the leverage constraint do not study the situation that $f(W_{t})$ is independent on $W_{t}$, and there are subtle differences as shown in this paper. For a comparative purpose, we solve the relevant optimal portfolio choice problem in which a dynamic leverage constraint replaces the risk capacity constraint. We demonstrate that the optimal portfolio under a leverage constraint moves precisely in the stock movement direction, which is a severe concern of the living standard risk in a stressed market period. Therefore, the risk capacity constraint is substantially different from classical leverage constraints, and it can be used to reduce the retiree’s living standard risk.

The structure of the paper is organized as follows. Section 2 introduces the model and presents a general optimal investment problem with risk capacity constraint in an infinite time horizon. In Section 3, we show that the constrained region is $(W^{*},\infty)$ under certain conditions. The explicit solution of the value function and the consumption-saving strategy are presented in Section 4 (Problem (A)) and Section 5 (Problem (B)), respectively. We present the applications to the retiree’s asset allocation problem in Section 6. The conclusion is given in Section 7, and technical proofs are given in Appendix A - Appendix B.

This section introduces a risk capacity constraint and then presents an optimal portfolio choice problem under the risk capacity constraint. Finally, we characterize the value function of this optimal portfolio choice problem in terms of the HJB equation.

In this section, we assume the $C^{2}$ smooth of the value function to characterize the constrained region under certain conditions.

Specifically, if the value function is $C^{2}$ smooth, by Proposition 1, the optimal investment strategy is

$$X^{*}=\min\left\{-\frac{\mu}{\sigma^{2}}\frac{\bar{V}^{\prime}(W)}{\bar{V}^{\prime\prime}(W)},L\right\}.$$

(7)

Following standard convention in Zariphopoulou (1994), we divide the state space $\Omega=[0,\infty)$ into two regions. On one hand, in the region

$\displaystyle{\cal U}=\left\{W>0:-\frac{\mu}{\sigma^{2}}\frac{\bar{V}^{\prime}}{\bar{V}^{\prime\prime}}\leq L\right\},$

$X^{*}<L$, we call ${\cal U}$ the unconstrained region following Vila and Zariphopoulou (1997).⁵⁵5In the region ${\cal U}$, the constrain $X^{*}\leq L$ in the HJB equation (10) becomes irrelevant. Hence, it is often called the region unconstrained. On the other hand, in the region

$\displaystyle{\cal B}=\left\{W>0:-\frac{\mu}{\sigma^{2}}\frac{\bar{V}^{\prime}}{\bar{V}^{\prime\prime}}>L\right\},$

the risk capacity constraint is binding and then $X^{*}=L$. We call ${\cal B}$ a constrained region.

For Problem (A), the smooth property of the value function follows from a difficult theorem of Zariphopoulou (1994).⁶⁶6Precisely, Zariphopoulou (1994) investigates the constraint that $X_{t}\leq f(W_{t})$ almost surely, where $f(x):[0,\infty)\rightarrow[0,\infty)$ is increasing, concave and satisfies the global Lipschitz condition. Her dedicate argument goes through if $f(\cdot)$ is a positive constant. Zariphopoulou (1994, Theorem 1.1) shows that $U(W)$ is the unique $C^{2}$ smooth function of the following HJB equation:

$$\delta U(W)=\max_{0\leq X\leq L}\left[\mu XU^{\prime}(W)+\frac{1}{2}\sigma^{2}X^{2}U^{\prime\prime}(W)\right]\\ +\max_{c\geq 0}\{u(c)-cU^{\prime}(W)\},(W>0)$$

(10)

with $U(0)=\frac{u(0)}{\delta}$.

Since the value function $U(W)$ is $C^{2}$ smooth, both the unconstrained and constrained region are well-defined. On one hand, in the unconstrained region, the HJB equation (10) reduces to the following ordinary differential equation

$$\delta U=\frac{R}{1-R}(U^{\prime})^{-(1-R)/R}-\kappa\frac{(U^{\prime})^{2}}{U^{\prime\prime}}.$$

(11)

On the other hand, in the constrained region ${\cal B}$, the HJB equation reduces to another ordinary differential equation

$$\delta U=\frac{R}{1-R}(U^{\prime})^{-(1-R)/R}+\mu LU^{\prime}+\frac{1}{2}\sigma^{2}L^{2}U^{\prime\prime}.$$

(12)

Proposition 4 is crucial to derive an explicit solution to Problem (A). It not only characterizes the constrained region but also reduces Problem (A) to determine the threshold number $W^{*}$ next.

Define

$\displaystyle\omega^{+}={R\over\kappa}(\kappa+\delta),$

then, for any $R>0,R\neq 1$, Assumption A implies that $\omega^{+}>\max(R,1).$

Given a real number $m$, define two auxiliary function $H(C)$ and $J(C)$ as follows.

$$H(C)=\frac{1}{\lambda^{\infty}}\left(C+mC^{\omega^{+}}\right),$$

(13)

$$J(C)=\frac{1}{\lambda^{\infty}}\left\{\frac{C^{1-R}}{1-R}+m\frac{\omega^{+}}{\omega^{+}-R}C^{\omega^{+}-R}\right\},$$

(14)

Proposition 5 is understood as follows. In the unconstrained region $(0,W^{*}]$, the relationship between wealth and optimal consumption rate is characterized by $W=H(C)$, a strictly increasing function.⁷⁷7It is a well known fact the function $H(\cdot)$ is strictly increasing, since $U(\cdot)$ is concave and $U_{WW}=-RC^{-R-1}\frac{\partial C}{\partial W}=-RC^{-R-1}/H^{\prime}(C)$. Furthermore, if the optimal consumption rate is a concave function of the wealth (Carroll and Kimball (1996)), then $\frac{\partial^{2}C}{\partial W^{2}}=-\frac{H^{\prime\prime}(C)}{H^{\prime}(C)^{2}}\frac{\partial C}{\partial W}<0$ implies that $H(\cdot)$ is a convex function. The value function $U(W)$ is determined by $J(C)$. Therefore, the value function is characterized with the auxiliary parameter $C$. Moreover, the optimal investment policy is

$$X^{*}(W)=\frac{\mu}{R\sigma^{2}}CH^{\prime}(C).$$

(18)

Since $CH^{\prime}(C)$ is increasing with respect to $C$, $X^{*}(W)$ is a well-defined increasing function of the wealth in the unconstrained region. Therefore, $X^{*}(W)<L$ holds for any wealth $W<W^{*}$.

In the constrained region $(W^{*},\infty)$, the optimal investment policy is straightforward, $X^{*}=L$, the investor places $L$ dollars in the risky asset as long as the portfolio value is greater than $W^{*}$. The value function $U(W)=K(W)$ satisfies the following ordinary differential equation

$\displaystyle\frac{1}{2}\sigma^{2}L^{2}K^{\prime\prime}(x)=\delta K(x)-\frac{R}{1-R}(K^{\prime}(x))^{-(1-R)/R}-\mu LK^{\prime}(x).$

(19)

with the boundary conditions $K(W^{*})=J(C^{*})$ and $K^{\prime}(W^{*})=(C^{*})^{-R}$. These two boundary conditions of the function $K(x)$ are exactly the value-matching and smooth-fit condition at $W^{*}$. However, these two conditions alone are not sufficient yet to characterize uniquely the function $K(x)$ as $W^{*}$ is to be determined. Therefore, we need a boundary condition at $W\rightarrow\infty$, that is, the order of $K(x)$, according to standard theory of ordinary differential equation. Finally, the optimal consumption rate is $c^{*}=K^{\prime}(W)^{-\frac{1}{R}}$.

In contrast to Problem (A), the value function of Problem (B) is not known to be $C^{2}$ smooth under the risk constraint. Therefore, in solving Problem (B), we need to simultaneously investigate the smooth property of the value function. We follow three steps. First, we explicitly characterize the constrained and unconstrained region of the risk capacity constraint, assuming the value function is $C^{2}$ smooth. This characterization of the constrained region follows from Proposition 6. Second, we provide an explicit expression of the value function, still assuming the $C^{2}$ smooth property of the value function. Third, we present the sufficient and necessary condition of the $C^{2}$ smoothness of the value function, given the expression of the value function in the second step.

Assuming the value function $V(W)$ is $C^{2}$ smooth, the value function on the unconstrained region satisfies

$$\delta V(W)=u(W)+\kappa\frac{(V^{\prime}(W))^{2}}{-V^{\prime\prime}(W)}.$$

(20)

Similarly, on the constrained region, $V(\cdot)$ satisfies a second-order linear ODE

$$\delta V(W)=u(W)+\mu LV^{\prime}(W)+\frac{1}{2}\sigma^{2}L^{2}V^{\prime\prime}(W).$$

(21)

In the above definition of the region ${\cal U}$ and ${\cal B}$, we assume that $C^{2}$ smoothness of the value function ex ante. Without knowing the $C^{2}$ smooth property of the value function, we can still directly investigate the ordinary differential equation (20) - (21). Later, we study these two ordinary differential equations and verify the $C^{2}$ smooth property of the value function under certain conditions.

By Proposition 2, the number $W^{*}$ is a finite number. Similar to Proposition 4 for Problem (A), the characterization of ${\cal U}$ and ${\cal B}$ in Proposition 6 reduces the solution to a number $W^{*}$.

We next derive the sufficient and necessary condition of the $C^{2}$ smoothness of the value function and simultaneously characterize the threshold $W^{*}$. Define two real numbers

$$\beta_{1}=\frac{-\mu+\sqrt{\mu^{2}+2\delta\sigma^{2}}}{\sigma^{2}L},\beta_{2}=\frac{-\mu-\sqrt{\mu^{2}+2\delta\sigma^{2}}}{\sigma^{2}L},$$

(22)

where $\beta_{1}$ and $\beta_{2}$ are two roots of the following quadratic equation

$\displaystyle\frac{1}{2}\sigma^{2}L^{2}\beta^{2}+L\mu\beta-\delta=0,$

and $\beta_{1}>0>\beta_{2}$.

Let

	$\displaystyle V_{0}(W)$	$\displaystyle=$	$\displaystyle\frac{2}{(\beta_{1}-\beta_{2})(1-R)\sigma^{2}L^{2}}$
			$\displaystyle\times\left\{e^{\beta_{2}W}\int_{0}^{W}x^{1-R}e^{-\beta_{2}x}dx-e^{\beta_{1}W}\int_{0}^{W}x^{1-R}e^{-\beta_{1}x}dx\right\}.$

The function $V_{0}(W)$ is a well-defined smooth function for $W>0$. We recall the expression of lower incomplete Gamma function,

$\displaystyle\gamma(s,x)=\int_{0}^{x}t^{s-1}e^{-t}dt,Re(s)>0,$

which is well-defined for all real number $x>0$. Therefore, $V_{0}(W)$ is well-defined for any $0<R<2$.⁸⁸8We refer to Appendix B for basic properties of the incomplete Gamma function.

Given $V_{0}(\cdot)$, the following two functions $C(W)$ and $D(W)$ are well-defined.

$$C(W)={-{2\over(\beta_{1}-\beta_{2})(1-R)\sigma^{2}L^{2}}\beta_{1}^{R-1}\Gamma(2-R)e^{\beta_{1}W}[\mu+L\sigma^{2}\beta_{1}]-\mu V_{0}^{\prime}(W)-\sigma^{2}LV_{0}^{\prime\prime}(W)\over\sigma^{2}L(\beta_{2})^{2}e^{\beta_{2}W}+\mu\beta_{2}e^{\beta_{2}W}},$$

(23)

and

$$D(W)=\left\{\frac{2}{(\beta_{1}-\beta_{2})(1-R)\sigma^{2}L^{2}}(\beta_{1})^{R-1}\Gamma(2-R)e^{\beta_{1}W}+C(W)\beta_{2}e^{\beta_{2}W}+V_{0}^{\prime}(W)\right\}^{-\frac{1}{R}}.$$

(24)

Given any $W>0$, we define a function $G(g),0\leq g\leq D(W),$ by the following second-order ordinary differential equation,

$$G^{\prime\prime}(g)={R\over\kappa}g^{-1}G^{\prime}(g)\left[\delta-\rho-g^{R}(G(g))^{-R}\right]$$

(25)

with boundary condition $G(0)=0,G(D(W))=W$ and $G^{\prime}(D(W))={LR\sigma^{2}\over\mu}(D(W))^{-1}$ (See King, Billingham, and Otto (2003) for the properties of second-order ordinary differential equation).

Finally, we define one equation of the variable $W$ as follows.

$$\frac{u(0)}{\delta}+\int_{0}^{D(W)}g^{-R}G^{\prime}(g)dg={2\over(\beta_{1}-\beta_{2})(1-R)\sigma^{2}L^{2}}(\beta_{1})^{R-2}\Gamma(2-R)e^{\beta_{1}W}+C(W)e^{\beta_{2}W}+V_{0}(W).$$

(26)

In what follows, we are interested in one particular function $G(\cdot)$ that is defined by one specific number $W^{*}$.

Proposition 7 presents a sufficient and necessary condition of the $C^{2}$ smoothness of the value function and solves Problem (B) explicitly. It also presents a closed-form expression of the value function and the optimal strategy in terms of $W^{*}$ and the auxiliary function $G(\cdot)$. By its construction, if there is one positive solution of Equation (26), then the solution $W^{*}$ is unique, and the unconstrained region and the constrained region are separated by $W^{*}$. Moreover, the value function $V(W,L)$ is a $C^{2}$ smooth function of the HJB equation and given by the expression (27) in Proposition 7. Conversely, if $V(W,L)$ is $C^{2}$ smooth, then there exists a unique solution of Equation (26). The $C^{2}$ smoothness of the value function is essentially the existence of a positive solution of the nonlinear equation (26) of one variable.

If $L=\infty$, it reduces ${\cal U}=(0,\infty)$ and an empty region ${\cal B}$. By Proposition 2, the function $G(\cdot)$ is a linear function. For any $L<\infty$, the function $G(\cdot)$ is highly non-linear, and its non-linearity is equivalent to the non-myopic property of the optimal strategy, as will be explained in the next section.

In this section, we present applications to the optimal portfolio for retirees. Our objective is to explain the implications of the risk capacity constraint and demonstrate its substantial difference from the leverage constraint.

There are several distinct features in the retiree’s portfolio choice problem compared with a standard investor before retirement. (1) The retiree has a fixed cash flow from his social security account post-retirement.⁹⁹9 See www.ssa.gov for the social security system in the U.S.A. There are similar social security systems in Europe and Canada. (2) He has no labor income. (3) He has a mortality risk, and (4) he becomes more risk-averse than before retirement because he has concerns about the market downturn and has no sufficient time to wait for the market return, leading to a substantial decline in his living standards. We formally incorporate these features into the optimal portfolio choice problems as follows.

The retirement date is zero. The retiree’s initial wealth is $W_{0}$ at the retirement date. Since the retiree faces his mortality risk, the investment time-horizon is uncertain, neither a fixed finite time nor infinity. We assume that the investor’s death time $\tau$ has an exponential distribution with mean $\lambda$, that is, the probability of the retiree surviving in the next $t$ years is $e^{-\lambda t}$. The investor’s average lifetime is $\frac{1}{\lambda}$, and the variance of his lifetime is $\frac{1}{\lambda^{2}}$. For example, if $\lambda=0.05$, an ordinary retiree who retires at 65 is likely to die at 85 years old. $\tau$ is independent of the information set ${\cal F}_{\infty}$.

We first consider an optimal portfolio choice problem with random maturity (see Blanchet-Scaillet, et al. 2008; Chen, Fu, and Zhou, 2020) as follows,

$$\max_{X,c}\mathbb{E}\left[\int_{0}^{\tau}e^{-\delta t}u(c_{t})dt\right].$$

(28)

Here the wealth process $W_{t}$ satisfies $dW_{t}=X_{t}(\mu dt+\sigma dZ_{t})-c_{t}dt,W_{0}=W$, $W_{t}\geq 0\ a.s.,\forall t\geq 0$, and the risk capacity constraint $0\leq X_{t}\leq L,\forall t\leq\tau$.¹⁰¹⁰10When the retiree receives a constant social security stream, the wealth equation becomes $dW_{t}=X_{t}(\mu dt+\sigma dZ_{t})-(c_{t}-y_{0})dt$, $y_{0}$ represents the social security with continuously compounding, the discussion in Problem (A) can be applied if we consider $u(c_{t}-y_{0})$ in Problem (A). The reason is as follows. If the social security is sufficient, there is no need to withdraw from the pension portfolio. Therefore, the retiree focuses on the difference $c_{t}-y_{0}$ in the withdraw decision. To sharpen our intuition of the risk capacity constraint, we omit the social security or other fixed cash-flow income. We also ignore medical costs or other costs in the portfolio choice decision. In this problem, the investor finds the best withdraw (consumption) rate whereas the inheritance wealth is not a concern. Because the stopping time $\tau$ is independent from the equity market information,

	$\displaystyle\mathbb{E}\left[\int_{0}^{\tau}e^{-\delta t}u(c_{t})dt\right]$	$\displaystyle=$	$\displaystyle\mathbb{E}\left[\int_{0}^{\infty}e^{-\delta t}1_{\tau\geq t}u(c_{t})dt\right]$
		$\displaystyle=$	$\displaystyle\mathbb{E}\left[\int_{0}^{\infty}\mathbb{E}[e^{-\delta t}1_{\tau\geq t}u(c_{t})|{\cal F}_{\infty}]dt\right]$
		$\displaystyle=$	$\displaystyle\mathbb{E}\left[\int_{0}^{\infty}e^{-\delta t}u(c_{t})e^{-\lambda t}dt\right]$

where we make use of the fact that $P(\tau\geq t)=e^{-\lambda t}$. Therefore, this optimal retirement portfolio reduces to Problem (A) in which the the subjective discount factor is replaced by $\delta+\lambda$. It is thus natural to assume that $\delta+\lambda<1$ and then $\delta+\lambda>\rho$.

Alternatively, the second optimal retirement portfolio problem for the retiree at time zero is as follows (see Liu and Loewenstein, 2002),

$$J(W,L)=\max_{(X)}\mathbb{E}\left[e^{-\delta\tau}u((1-\alpha)W_{\tau})\right]$$

(29)

where $\delta$ is the retiree’s subjective discount factor, $\alpha$ is the inheritance tax rate of the wealth. $X_{t}$ satisfies the risk capacity constraint. Given the distribution of $\tau$, and the independent assumption between $\tau$ and ${\cal F}_{\infty}$, using the same derivation as in the first problem, the retiree’s optimal retirement portfolio problem (29) is reduced to

$$J(W,L)=\max_{(X)}\mathbb{E}\left[\int_{0}^{\infty}e^{-(\lambda+\delta)t}u((1-\alpha)W_{t})dt\right].$$

(30)

Assuming $u(W)=\frac{W^{1-R}}{1-R},R>0,R\neq 1$, and using its scaling property,

$$u((1-\alpha)W_{t})=(1-\alpha)^{1-R}u(W_{t}),$$

we have

$$J(W,L)=(1-\alpha)^{1-R}\max_{(X)}\mathbb{E}\left[\int_{0}^{\infty}e^{-(\lambda+\delta)t}u(W_{t})dt\right].$$

(31)

A general expression of the retirement portfolio problem elaborates these two cases. If both the consumption/withdraw and the inheritance wealth are considered together, ¹¹¹¹11In practice, a constant consumption rate is often fixed. For instance, a standard withdrawal rate is between 4% to 5%. See Bengen (1994). Then the problem reduces to Problem (B).

$\displaystyle\mathbb{E}\left[\int_{0}^{\tau}e^{-\delta t}u(c_{t})dt+e^{-\delta\tau}u((1-\alpha)W_{\tau})\right]=\mathbb{E}\left[\int_{0}^{\infty}e^{-(\lambda+\delta)t}\left\{u(c_{t})+(1-\alpha)^{1-R}u(W_{t})\right\}dt\right].$

It reduces to the optimization problem (4) studied in Section 2 and Section 3.