---
tag: DZKBF576W5
kind: problem
name: Merton's portfolio problem
msc: [91G10, "93E20", 49L20]
uses: [B24T6DQCDR, D48DR46B97, SJP9HW82VX]
about: [B24T6DQCDR]
sources:
  - url: https://en.wikipedia.org/wiki/Merton%27s_portfolio_problem
same_as:
  wikidata: Q3467003
  wikipedia: Merton's portfolio problem
history:
  posed_by: "Robert C. Merton"
  first_stated: "1969"
---

## Problem

Consider a market with a riskless asset with interest rate $r$ and a stock.
The stock price follows the [[geometric Brownian motion|D48DR46B97]] $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$, where $\mu > r$, $\sigma > 0$ and $W$ is a Wiener process.
An investor with initial wealth $x > 0$ chooses, at each time $t$, the fraction $\pi_t$ of wealth held in the stock and a consumption rate $c_t \ge 0$.
Both are adapted to the information generated by $W$.
The wealth then evolves by
$$dX_t = \left( r X_t + \pi_t (\mu - r) X_t - c_t \right) dt + \pi_t \sigma X_t \, dW_t, \qquad X_0 = x.$$
Let $\rho > 0$ be a discount rate and let $U(c) = \dfrac{c^{1 - \gamma}}{1 - \gamma}$ with $\gamma > 0$ and $\gamma \ne 1$.
The problem is to find the strategies $(\pi, c)$ with $X_t \ge 0$ for all $t$ that maximise the expected discounted utility of consumption
$$E \int_0^\infty e^{-\rho t} U(c_t) \, dt.$$

## Remarks

The problem is solved, and it is a standard example of [[stochastic optimal control|B24T6DQCDR]].
Write $\theta = (\mu - r) / \sigma$ and assume $\nu = \dfrac{\rho - (1 - \gamma)\left( r + \theta^2 / (2 \gamma) \right)}{\gamma} > 0$.
The value function is $V(x) = \nu^{-\gamma} U(x)$.
The optimal strategy holds the constant fraction $\pi^* = \dfrac{\mu - r}{\gamma \sigma^2}$ in the stock and consumes at the rate $c_t = \nu X_t$.
These follow from the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] $\rho V = \max_{\pi, c} \left\{ U(c) + V'(x) \left( r x + \pi (\mu - r) x - c \right) + \frac{1}{2} \sigma^2 \pi^2 x^2 V''(x) \right\}$ with the guess $V = A U$ for a constant $A > 0$.
Merton also treated a finite horizon with utility of final wealth, and the logarithmic utility $U = \ln$, which is the limit case $\gamma = 1$ and gives $\pi^* = (\mu - r) / \sigma^2$ and $c_t = \rho X_t$.

## Explanations

### L1 human

How much of your savings should you keep in risky stocks, and how fast should you spend?
Merton's answer, for this model, is to keep a fixed share of your wealth in stocks and to spend a fixed share of it per unit time.
The share in stocks grows with the excess return $\mu - r$ and falls with the risk $\sigma^2$ and with your aversion to risk $\gamma$.