How Prescribed Consumption Affects Merton Portfolio Allocation

Summary

The document asks whether the Merton investor’s optimal risky asset allocation changes over time when the investor must consume wealth at a prescribed rate. It gives a Hamilton-Jacobi-Bellman equation for value as a function of time and wealth, with a consumption term, and identifies the allocation-relevant expression involving the first and second derivatives of value.

For the no-consumption case, it recalls the familiar power utility allocation determined by excess return, volatility, and risk aversion. The question is whether that allocation ratio remains time independent under a nonzero, potentially time-varying consumption schedule, including when terminal utility is logarithmic. No solution or numerical evidence is supplied, and the setup omits further model details such as the exact utility specification beyond the optional terminal condition. The document is useful as a formulation of a continuous-time portfolio optimization problem, but it does not establish the answer or provide a general allocation rule.

Key ideas

• The setup uses an HJB equation for wealth with investment and prescribed consumption.
• The investor maximizes expected terminal utility in a Merton market model.
• The time dependence of the value function’s curvature ratio determines whether allocation varies through time.
• With no consumption, the document states the standard power utility allocation formula.
• The question is posed but no solution is provided for nonzero consumption.

Tags

# Continous-time portfolio allocation optimization for a given consumption rate


# Continous-time portfolio allocation optimization for a given consumption rate












I have the following PDE $0 = V_t - c(t)V_x - \lambda^2 V_x^2/V_{xx} + rxV_x + 1/2\lambda^2x^2V_{xx}$ where $t\mapsto c(t)$ is some given function and $r,\lambda$ are given constants. If necessary, one can impose $V(T,x) = \log(x)$ for some finite $T$.

What I would like to know is whether the quantity $V_x/(xV_{xx})$ depends on $t$.

The short background to this question is that there is an agent who wants to maximize his terminal wealth utility at $T$ and this agent consumes at a rate $c(t)$. Dependence of the quantity above on $t$ would imply dependence of optimal allocation strategy (stocks versus bonds) on $t$ and that is what I would like to find out.

The PDE above comes from the HJB equation. Namely, $$0 = V_t +\sup_{\delta}\left((\delta(\mu-r)+r)xV_x - cV_x + \frac{1}{2}\delta^2x^2\sigma^2V_{xx}\right)$$ with $V(T,x) = U_2(x)$ where $\delta$ is the allocation to stock and $c$ is the consumption rate.

The underlying model is the Merton model and the agent wants to maximize $E[U_2(X_T)]$ where $U_2$ is some utility function and $(X_t)_{t\in[0,T]}$ is the wealth process. I could write more details but this is a very standard problem. The only thing that makes it non-standard is that there is a prescribed consumption. If $c(t) = 0$ for all $t \in [0,T]$, then the optimal $\delta = \frac{\mu -r}{\gamma\sigma^2}$ where $\gamma$ is the risk-aversion coefficient in power utility.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.