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| '''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.<ref>{{cite doi|10.2307/1926560}}</ref> Research has continued to extend and generalize the model to include factors like [[transaction cost]]s and bankruptcy.
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| ==Problem statement==
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| The investor lives from time 0 to time ''T''; his wealth at time ''t'' is denoted ''W''<sub>''t''</sub>. He starts with a known initial wealth ''W''<sub>0</sub> (which may include the present value of wage income). At time ''t'' he must choose what amount of his wealth to consume: ''c''<sub>''t''</sub> and what fraction of wealth to invest in a stock portfolio: ''π''<sub>''t''</sub> (the remaining fraction 1 − ''π''<sub>''t''</sub> being invested in the risk-free asset).
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| The objective is
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| : <math> \max E \left[ \int_0^T e^{-\rho s}u(c_s) \, ds + e^{-\rho T}u(W_T) \right] </math>
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| where ''E'' is the expectation operator, ''u'' is a known [[utility function]] (which applies both to consumption and to the terminal wealth, or bequest, ''W''<sub>''T''</sub>) and ''ρ'' is the subjective discount rate.
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| The wealth evolves according to the [[stochastic differential equation]]
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| ::<math>d W_t = [(r + \pi_t(\mu-r))W_t - c_t ] \, dt +W_t \pi_t \sigma \, dB_t </math>
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| where ''r'' is the risk-free rate, (''μ'', ''σ'') are the expected return and volatility of the stock market and ''dB''<sub>''t''</sub> is the increment of the [[Wiener process]], i.e. the stochastic term of the SDE.
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| '''Additional assumptions'''. The utility function is of the [[constant relative risk aversion]] (CRRA) form:
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| : <math> u(x) = \frac{x^{1-\gamma}}{1-\gamma}. </math>
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| Consumption cannot be negative: ''c''<sub>''t''</sub> ≥ 0, while ''π''<sub>''t''</sub> is unrestricted (that is borrowing or shorting stocks is allowed).
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| Investment opportunities are assumed constant, that is ''r'', ''μ'', ''σ'' are known and constant, in this (1969) version of the model, although Merton later allowed{{fact|date=March 2013}} them to change.
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| ==Solution==
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| 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:
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| :<math>\pi(W,t) = \frac{\mu-r}{\sigma^2\gamma}</math>
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| (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).
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| :<math>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}</math>
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| where <math>0\le\epsilon\ll1</math> and
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| :<math>\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}</math>
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| The variable <math>\rho</math> is the subjective discount rate.
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| ==Extensions==
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| Many variations of the problem have been explored, but most do not lead to a simple closed-form solution.
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| * Flexible retirement age can be taken into account<ref>{{cite doi|10.1016/0165-1889(92)90044-F}}</ref>
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| * A utility function other than CRRA can be used.
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| * Transaction costs can be introduced. For ''proportional transaction costs'' the problem was solved by Davis and Norman in 1990.<ref>{{cite doi|10.1287/moor.15.4.676}}</ref> 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.<ref>{{cite doi|10.1214/aoap/1177004966}}</ref>
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| :When there are ''fixed transaction costs'' the problem was addressed by Eastman and Hastings in 1988.<ref>{{cite doi|10.1287/moor.13.4.588}}</ref> A numerical solution method was provided by Schroder in 1995.<ref>{{cite journal | last = Schroder | first = M. | title = Optimal Portfolio Selection with Fixed Transaction Costs: Numerical Solutions | journal = Working Paper | location = Michigan State University | year=1995 | url = https://www.msu.edu/~schrode7/numerical.pdf }}</ref>
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| :Finally Morton and Pliska<ref>{{cite doi|10.1111/j.1467-9965.1995.tb00071.x}}</ref> considered trading costs that are proportional to the wealth of the investor, as a kind of penalty to discourage frequent trading, although this cost structure seems unrepresentative of real life transaction costs. Also, they considered the case of logarithmic utility only.
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| * The assumption of constant investment opportunities can be relaxed. This requires a model for how <math>r,\mu,\sigma</math> 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.
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| * Additional assets can be added, for example individual stocks. However, the problem becomes difficult or intractable.
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| * Bankruptcy can be incorporated. This problem was solved by Karatzas, Lehoczky, Sethi and Shreve in 1986.<ref>{{cite doi|10.1007/BFb0041165}}</ref> Many models incorporating bankruptcy are collected in Sethi (1997).<ref>{{cite doi|10.1007/978-1-4615-6257-3}}</ref>
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| ==References==
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| {{reflist}}
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| *{{cite doi|10.1007/b98840}}
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| *Merton R.C.: Continuous Time Finance, Blackwell (1990).
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| [[Category:Financial economics]]
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| [[Category:Mathematical finance]]
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| [[Category:Stochastic control]]
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| [[Category:Portfolio theories]]
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