Merton's portfolio problem

Solution

Somewhat surprisingly for an optimal control problem, a closed-form solution exists. The optimal consumption and stock allocation depend on wealth and time as follows:

${\displaystyle \pi (W,t)=\frac{\mu -r}{{\sigma }^2\gamma }.}$

This expression is commonly referred to as Merton's fraction. Because W and t do not appear on the right-hand side; a constant fraction of wealth is invested in stocks, no matter what the age or prosperity of the investor.

${\displaystyle c(W,t)=\{\begin{array}{ll}\nu {\left(1+(\nu ϵ-1)e^{-\nu (T-t)}\right)}^{-1}W& \text{if}T<\mathrm{\infty }\text{and}\nu \ne 0\\ (T-t+ϵ{)}^{-1}W& \text{if}T<\mathrm{\infty }\text{and}\nu =0\\ \nu W& \text{if}T=\mathrm{\infty }\end{array}}$

where ${\displaystyle 0\le ϵ\ll 1}$ and

${\displaystyle \begin{array}{cc}\nu & =\left(\rho -(1-\gamma )\left(\frac{(\mu -r{)}^2}{2{\sigma }^2\gamma }+r\right)\right)/\gamma \\ & =\rho /\gamma -(1-\gamma )\left(\frac{(\mu -r{)}^2}{2{\sigma }^2{\gamma }^2}+\frac{r}{\gamma }\right)\\ & =\rho /\gamma -(1-\gamma )(\pi (W,t{)}^2{\sigma }^2/2+r/\gamma )\\ & =\rho /\gamma -(1-\gamma )((\mu -r)\pi (W,t)/2\gamma +r/\gamma ).\end{array}}$

Extensions

Many variations of the problem have been explored, but most do not lead to a simple closed-form solution.

• Flexible retirement age can be taken into account.
• A utility function other than CRRA can be used.
• Transaction costs can be introduced.
  • For proportional transaction costs the problem was solved by Davis and Norman in 1990. It is one of the few cases of stochastic singular control where the solution is known. For a graphical representation, the amount invested in each of the two assets can be plotted on the x- and y-axes; three diagonal lines through the origin can be drawn: the upper boundary, the Merton line and the lower boundary. The Merton line represents portfolios having the stock/bond proportion derived by Merton in the absence of transaction costs. As long as the point which represents the current portfolio is near the Merton line, i.e. between the upper and the lower boundary, no action needs to be taken. When the portfolio crosses above the upper or below the lower boundary, one should rebalance the portfolio to bring it back to that boundary. In 1994 Shreve and Soner provided an analysis of the problem via the Hamilton–Jacobi–Bellman equation and its viscosity solutions.
  • When there are fixed transaction costs the problem was addressed by Eastman and Hastings in 1988. A numerical solution method was provided by Schroder in 1995.
  • Morton and Pliska considered trading costs that are proportional to the wealth of the investor for logarithmic utility. Although this cost structure seems unrepresentative of real life transaction costs, it can be used to find approximate solutions in cases with additional assets, for example individual stocks, where it becomes difficult or intractable to give exact solutions for the problem.
• The assumption of constant investment opportunities can be relaxed. This requires a model for how ${\displaystyle r,\mu ,\sigma }$ change over time. An interest rate model could be added and would lead to a portfolio containing bonds of different maturities. Some authors have added a stochastic volatility model of stock market returns.
• Bankruptcy can be incorporated. This problem was solved by Karatzas, Lehoczky, Sethi and Shreve in 1986. Many models incorporating bankruptcy are collected in Sethi (1997).