Merton's portfolio problem

Merton's Portfolio Problem is a well known problem in continuous-time finance and in particular intertemporal portfolio choice. An investor must choose how much to consume and must allocate his wealth between stocks and a risk-free asset so as to maximize expected utility. The problem was formulated and solved by Robert C. Merton in 1969 both for finite lifetimes and for the infinite case.[1] Research has continued to extend and generalize the model to include factors like transaction costs and bankruptcy.

Problem statement

The investor lives from time 0 to time T; his wealth at time t is denoted Wt. He starts with a known initial wealth W0 (which may include the present value of wage income). At time t he must choose what amount of his wealth to consume: ct and what fraction of wealth to invest in a stock portfolio: πt (the remaining fraction 1  πt being invested in the risk-free asset).

The objective is

 \max E \left[ \int_0^T e^{-\rho s}u(c_s) \, ds +  e^{-\rho T}u(W_T) \right]

where E is the expectation operator, u is a known utility function (which applies both to consumption and to the terminal wealth, or bequest, WT) and ρ is the subjective discount rate.

The wealth evolves according to the stochastic differential equation

d W_t = [(r + \pi_t(\mu-r))W_t - c_t ] \, dt +W_t \pi_t \sigma \, dB_t

where r is the risk-free rate, (μ, σ) are the expected return and volatility of the stock market and dBt is the increment of the Wiener process, i.e. the stochastic term of the SDE.

Additional assumptions. The utility function is of the constant relative risk aversion (CRRA) form:

 u(x) = \frac{x^{1-\gamma}}{1-\gamma}.

where \gamma is a constant which expresses the investor's risk aversion, the higher the gamma the more reluctance to own stocks.

Consumption cannot be negative: ct  0, while πt is unrestricted (that is borrowing or shorting stocks is allowed).

Investment opportunities are assumed constant, that is r, μ, σ are known and constant, in this (1969) version of the model, although Merton allowed them to change in his Intertemporal CAPM (1973).

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:

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

(Note that W and t do not appear on the right-hand side, this implies that a constant fraction of wealth is invested in stocks, no matter what the age or prosperity of the investor).

c(W,t)= \begin{cases}\nu \left(1+(\nu\epsilon-1)e^{-\nu(T-t)}\right)^{-1} W&\textrm{if}\;T<\infty\;\textrm{and}\;\nu\neq0\\(T-t+\epsilon)^{-1}W&\textrm{if}\;T<\infty\;\textrm{and}\;\nu=0\\\nu W&\textrm{if}\; T=\infty\end{cases}

where 0\le\epsilon\ll1 and

\begin{align}\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)^2/2\sigma^2+ r/\gamma)\\&=\rho/\gamma-(1-\gamma)((\mu-\gamma)\pi(W,t)/2\gamma+ r/\gamma)\end{align}

The variable \rho is the subjective utility discount rate (force of mortality.[2]:401)

Extensions

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

When there are fixed transaction costs the problem was addressed by Eastman and Hastings in 1988.[6] A numerical solution method was provided by Schroder in 1995.[7]
Finally Morton and Pliska[8] 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,[9] for example individual stocks, where it becomes difficult or intractable to give exact solutions for the problem.

References

  1. Merton, R. C. (1 August 1969). "Lifetime Portfolio Selection under Uncertainty: the Continuous-Time Case". The Review of Economics and Statistics 51 (3): 247–257. doi:10.2307/1926560. ISSN 0034-6535. JSTOR 1926560.
  2. Merton, R. C. (1971). "Optimum consumption and portfolio rules in a continuous-time model". Journal of Economic Theory 3 (4): 373–413. doi:10.1016/0022-0531(71)90038-X.
  3. Bodie, Z.; Merton, R. C.; Samuelson, W. F. (1992). "Labor supply flexibility and portfolio choice in a life cycle model". Journal of Economic Dynamics and Control 16 (3–4): 427. doi:10.1016/0165-1889(92)90044-F.
  4. Davis, M. H. A.; Norman, A. R. (1990). "Portfolio Selection with Transaction Costs". Mathematics of Operations Research 15 (4): 676. doi:10.1287/moor.15.4.676. JSTOR 3689770.
  5. Shreve, S. E.; Soner, H. M. (1994). "Optimal Investment and Consumption with Transaction Costs". The Annals of Applied Probability 4 (3): 609. doi:10.1214/aoap/1177004966. JSTOR 2245058‎.
  6. Eastham, J. F.; Hastings, K. J. (1988). "Optimal Impulse Control of Portfolios". Mathematics of Operations Research 13 (4): 588. doi:10.1287/moor.13.4.588. JSTOR 3689945‎.
  7. Schroder, M. (1995). "Optimal Portfolio Selection with Fixed Transaction Costs: Numerical Solutions". Working Paper (Michigan State University).
  8. Morton, A. J.; Pliska, S. R. (1995). "Optimal Portfolio Management with Fixed Transaction Costs". Mathematical Finance 5 (4): 337. doi:10.1111/j.1467-9965.1995.tb00071.x.
  9. http://cmcm.uni-kl.de/fileadmin/downloads/vortraege/20100329/Korn_optimal_portfolios_with_transaction_costs.pdf
  10. Karatzas, I.; Lehoczky, J. P.; Sethi, S. P.; Shreve, S. E. (1985). "Explicit solution of a general consumption/investment problem". Stochastic Differential Systems. Lecture Notes in Control and Information Sciences 78. p. 209. doi:10.1007/BFb0041165. ISBN 3-540-16228-3.
  11. Sethi, S. P. (1997). "Optimal Consumption and Investment with Bankruptcy". doi:10.1007/978-1-4615-6257-3. ISBN 978-1-4613-7871-6.