A Certainty Equivalent Merton Problem

An investor must choose how to invest and consume over a lifetime of $T$ years. The investor has wealth $w_{t}>0$ at time $t$, and consumes wealth at rate $c_{t}>0$, for $t\in[0,T]$, with the remaining wealth invested in a portfolio with mean rate of return $\mu_{t}$ and volatility $\sigma_{t}$. The wealth dynamics are a geometric random walk,

where $z_{t}$ is a Brownian motion. The initial condition is $w_{0}=w_{\rm init}>0$.

The portfolio consists of $n$ assets, with an investment mix given by the fractional allocation $\theta_{t}$, with $\mathbf{1}^{T}\theta_{t}=1$ (where $\mathbf{1}$ is the vector with all entries one). Thus we invest $(w_{t}\theta_{t})_{i}$ dollars in asset $i$, with a negative value denoting a short position. The portfolio return rate and volatility are given by

where $\mu\in{\mbox{\bf R}}^{n}$ is the mean of the return process, and $\Sigma$ is the symmetric positive definite covariance. (Note that we use the time-varying scalar $\mu_{t}$ to denote the portfolio return as a function of time, and the vector $\mu$ to denote the constant expected return rates of the $n$ assets.)

The investment allocation decision $\theta_{t}$ satisfies $\mathbf{1}^{T}\theta_{t}=1$, as well as other investment constraints, which we summarize as $\theta_{t}\in\Theta$, where $\Theta$ is a convex set. These could include risk limits, sector exposure limits, or concentration limits. (See [Boy+17, §4.4] for an overview of convex investment constraints.) For notational convenience, we assume every $\theta_{t}\in\Theta$ satisfies $\mathbf{1}^{T}\theta_{t}=1$.

With the portfolio return and volatility we obtain the wealth dynamics

The investor has lifetime consumption utility $\int_{0}^{T}c_{t}^{\gamma}/\gamma\;dt$ and bequest utility $w_{T}^{\gamma}/\gamma$. The risk aversion parameter $\gamma$ satisfies $\gamma<1$ and $\gamma\neq 0$. The investor’s total expected utility is

The parameter $\beta>0$ trades off consumption and bequest utility.

At each time $t$, the investor chooses the consumption $c_{t}$ and the investment allocation $\theta_{t}$. A policy maps the time $t$ and the current wealth $w_{t}$ to the consumption $c_{t}$ and the allocation $\theta_{t}$, which we write as

where for each $t\in[0,T]$, $\pi_{t}:{\mbox{\bf R}}_{++}\to{\mbox{\bf R}}_{++}\times\Theta$. (Here ${\mbox{\bf R}}_{++}$ denotes the set of positive real numbers.) The Merton problem is to choose a policy $\pi_{t}$, $t\in[0,T]$, to maximize $U$.

The problem can be interpreted in the following way. We plan for a single outcome of the stochastic process (1). In particular, the dynamics constraint restricts the growth rate of the wealth to be no greater than the $\mu^{T}x_{t}-c_{t}$ (the mean growth rate in the stochastic process (1)), but reduced by the additional term $(1/2)(\gamma-1)x_{t}^{T}\Sigma x_{t}/w_{t}$. Because $\gamma<1$, this term is negative. With the change of variables $\theta_{t}=x_{t}/w_{t}$, we have

i.e., this adjustment term is proportional to the variance of the portfolio growth rate with investment allocation $\theta_{t}=x_{t}/w_{t}$. In other words, we are pessimistically planning for bad investment returns, with the degree of pessimism depending on the risk aversion parameter $\gamma$ and the risk of our portfolio.

In fact, in problem (9), we plan for the returns

The coefficients in front of $\Sigma x_{t}$ and $\Sigma\theta_{t}$ are negative, and the entries of $\Sigma x_{t}$ and $\Sigma\theta_{t}$ are typically positive. The vector $\Sigma\theta_{t}$ can be interpreted as the risk allocation to the individual assets in the portfolio, since

In other words, the planned asset returns are the mean returns, reduced in proportion to the marginal contribution of each asset to the portfolio variance. This is related to the concept of risk parity [BST16].

Convexity of (9) follows from the fact that the risk penalty term $x_{t}^{T}\Sigma x_{t}/w_{t}$ is a quadratic-over-linear function, with is jointly convex in $x_{t}$ and $w_{t}$ [BV04, §3.1.5]. Also, the set

is the perspective of $\Theta$, which is convex when $\Theta$ is [BV04, §2.3.3]. In fact, in most practical portfolio construction problems, $\Theta$ can described by a collection of linear and quadratic constraints [Boy+17, §4.4]. In this case, when problem (9) is discretized, it becomes an SOCP, which we describe in §7.

The Merton problem and problem (9) are equivalent in the sense that they have the same value function and optimal policy.

To see this, we first consider a modified version of (9) in which we convert the dynamics to an equality constraint using a slack variable $u_{t}\geq 0$:

The new control input $u_{t}$ can be interpreted as the rate at which we discard wealth. (We will see that at optimality $u_{t}=0$.) For this modified problem, the Hamilton-Jacobi-Bellman equation is

First note that with our value function candidate (5), we have $V^{\prime}(w)>0$, and therefore $u=0$, as expected. Now, by using the change of variables $x=\theta w$ and plugging in our value function candidate, this equation becomes (8). From this point on, the proof that this candidate value function satisfies the Hamilton-Jacobi-Bellman equation proceeds exactly as for the (stochastic) Merton problem.

The Merton problem can be solved when $\mu$, $\Sigma$, and $\Theta$ change over time. To handle this in the certainty equivalent problem, we simply replace these parameters by $\mu_{t}$, $\Sigma_{t}$, and $\Theta_{t}$. (Here $\mu_{t}$ denotes the time-varying vector of asset expected returns, a notation clash with our previous use of $\mu_{t}$ as the scalar portfolio expected return.) Similarly, if we discount the consumption utility of the Merton problem:

where $\alpha_{t}>0$ is the discount of the consumption utility at time $t$, then the objective of the certainty equivalent problem will match $U$ (but without the expectation).

Here the terminal time $t_{f}\in[0,T]$ is random with probability density $p_{t}$ and survival function

In this case, the investor’s utility is

Here the expectation is taken over $t_{f}$ as well as the paths of the stochastic process (1).

With this modification, the objective of the certainty equivalent problem changes to

We weight the consumption utility by the probability the investor is still alive, i.e., we treat the survival function as a discount factor. We also get utility for the bequest continuously over the interval $[0,T]$, weighted by the density function $p_{t}$.

This extension is due to [Ric75]. Continuing with the previous extension, we allow the investor to purchase life insurance. The premium is $l_{t}$, which the investor can choose, and the payout of the plan is $\lambda_{t}l_{t}$, where $\lambda_{t}\geq 0$ is the payout-to-premium ratio at time $t$. When $l_{t}<0$, we interpret this as an annuity. In particular, at time $t$, the investor has $-l_{t}$ in the annuity account, which is lost on death, in return for an additional return of $-\lambda_{t}l_{t}$. The actuarially fair value of $\lambda_{t}$ is $p_{t}/s_{t}$, which is called the force of mortality. (If $\lambda_{t}>p_{t}/s_{t}$, then life insurance is favorable and annuities are unfavorable; if $\lambda_{t}<p_{t}/s_{t}$, the reverse is true.)

With this modification, the objective of the certainty-equivalant problem changes to

i.e., we add the insurance payout to the wealth in the bequest utility. The dynamics change to

Here we subtract the insurance premium from the growth rate of the wealth.

We can add a deterministic income stream, with income rate $y_{t}$ at time $t$. The stochastic dynamics are modified be the addition of $y_{t}$ to the drift term of the wealth process, i.e.,

In this case, we also assume one of the assets is risk free with return $\mu_{\rm rf}$ and volatility $0$, and that

These assumptions allow the investor to counteract the income stream by shorting the risk-free asset and investing the proceeds in a preferred portfolio of other assets. The fair value of the income stream is its net present value over $[t,T]$ at the risk-free rate:

which can be interpreted as the remaining human capital of the investor.

For this extension, the dynamics in (9) are replaced by

Note the addition of the income term $y_{t}$ and the normalization of risk by the total wealth plus the remaining human capital. In this case, the wealth $w_{t}$ need not be positive but instead satisfies $w_{t}+v_{t}>0$. Because of this, we also replace the constraint $x_{t}/w_{t}\in\Theta$ (which is not defined for $w_{t}=0$) with $\mathbf{1}^{T}x_{t}=w_{t}$.

One interesting feature of the certainty equivalent problem (9) is that the risk aversion parameter $\gamma$ appears separately in the objective and dynamics constraint. It is reasonable to ask whether, by modifying the consumption utility to be

for some $\rho\neq\gamma$ with $\rho<1$ and $\rho\neq 0$, but keeping $\gamma$ in the dynamics constraint, problem (9) is equivalent to some variant of the Merton problem. This is indeed the case, but with the expected utility $U$ replaced by Epstein–Zin preferences, where $1/\rho$ is the elasticity of intertemporal substitution and $\gamma$ is the risk aversion. For details, see [DE92].

We can change the objective of (9) to use any increasing, concave utility function for either consumption or bequest. These utility functions need not be additive over time: For example, we can maximize the minimum consumption over the interval $[0,T]$,

As a special case, we can add a minimum consumption constraint

where $c_{t}^{\rm min}$ is the minimum allowable consumption amount as a function of age. Similarly, we can enforce a minimum bequest over some time window (say, to care for underage dependents until they come of age).

We can limit consumption as a fraction of income with the constraint

for some parameter $\eta>0$. For example, when $\eta=0.7$, this constraint means that we can’t consume more than 70% of our income, i.e., we must have a savings rate of 30%.

This constraint can be adjusted to account for investment income. To see this, take $d\in{\mbox{\bf R}}^{n}$ to be the vector of dividend yields for each asset, which is constant and known in advance. The modified constraint becomes

When this constraint is tight, i.e., when we desire to consume more than $\eta$ times our income, there is added incentive to invest in assets with high dividend yield.

We can include a constraint that the amount invested in cash be above a certain level, i.e.,

where $i$ is the index of the cash asset. This is similar to an emergency fund constraint that we must keep six months worth of consumption in cash, which is expressed as