Merton's portfolio problem
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Merton's portfolio problem is a problem in continuous-time
finance Finance refers to monetary resources and to the study and Academic discipline, discipline of money, currency, assets and Liability (financial accounting), liabilities. As a subject of study, is a field of Business administration, Business Admin ...
and in particular
intertemporal portfolio choice Intertemporal portfolio choice is the process of allocating one's investable wealth to various assets, especially financial assets, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines ...
. An investor must choose how much to consume and must allocate their wealth between stocks and a risk-free asset so as to maximize expected
utility In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a normative context, utility refers to a goal or objective that we wish ...
. The problem was formulated and solved by
Robert C. Merton Robert Cox Merton (born July 31, 1944) is an American economist, Nobel Memorial Prize in Economic Sciences laureate, and professor at the MIT Sloan School of Management, known for his pioneering contributions to continuous-time finance, especia ...
in 1969 both for finite lifetimes and for the infinite case.Sethi, S.P. and Taksar, M.I., “A Note on Merton's 'Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” ''Journal of Economic Theory'', 46, 1988, 395-401. Research has continued to extend and generalize the model to include factors like
transaction cost In economics, a transaction cost is a cost incurred when making an economic trade when participating in a market. The idea that transactions form the basis of economic thinking was introduced by the institutional economist John R. Commons in 1 ...
s and bankruptcy.


Problem statement

The investor lives from time 0 to time ''T''; their wealth at time ''T'' is denoted ''W''''T''. He starts with a known initial wealth ''W''0 (which may include the present value of wage income). At time ''t'' he must choose what amount of his wealth to consume, ''c''''t'', and what fraction of wealth to invest in a stock portfolio, ''π''''t'' (the remaining fraction 1 − ''π''''t'' being invested in the risk-free asset). The objective is : \max E \left \int_0^T e^u(c_t) \, dt + \epsilon^\gamma e^u(W_T) \right where ''E'' is the
expectation operator In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected val ...
, ''u'' is a known
utility function In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a Normative economics, normative context, utility refers to a goal or ob ...
(which applies both to consumption and to the terminal wealth, or bequest, ''W''''T''), ''ε'' parameterizes the desired level of bequest, ''ρ'' is the subjective discount rate, and \gamma is a constant which expresses the investor's risk aversion: the higher the gamma, the more reluctance to own stocks. The wealth evolves according to the
stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics an ...
:dW_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 ''dB''''t'' is the increment of the
Wiener process In mathematics, the Wiener process (or Brownian motion, due to its historical connection with Brownian motion, the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one o ...
, i.e. the stochastic term of the SDE. The utility function is of the
constant relative risk aversion In economics, the isoelastic function for utility, also known as the isoelastic utility function, or power utility function, is used to express utility in terms of consumption or some other economic variable that a decision-maker is concerned wit ...
(CRRA) form: : u(x) = \frac. Consumption cannot be negative: ''c''''t'' ≥ 0, while ''π''''t'' 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 In mathematical finance, the intertemporal capital asset pricing model, or ICAPM, created by Robert C. Merton, is an alternative to the Capital Asset Pricing Model (CAPM). It is a linear factor model with wealth as state variable that forecasts cha ...
(1973).


Solution

Somewhat surprisingly for an
optimal control Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...
problem, a closed-form solution exists. The optimal consumption and stock allocation depend on wealth and time as follows: :\pi(W,t) = \frac. 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. :c(W,t)= \begin\nu \left(1+(\nu\epsilon-1)e^\right)^ W&\textrm\;T<\infty\;\textrm\;\nu\neq0\\(T-t+\epsilon)^W&\textrm\;T<\infty\;\textrm\;\nu=0\\\nu W&\textrm\; T=\infty\end where 0\le\epsilon\ll1 and :\begin\nu&=\left(\rho-(1-\gamma)\left(\frac+r\right)\right)/\gamma \\&=\rho/\gamma-(1-\gamma)\left(\frac+\frac r\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


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 The Hamilton-Jacobi-Bellman (HJB) equation is a nonlinear partial differential equation that provides necessary and sufficient conditions for optimality of a control with respect to a loss function. Its solution is the value function of the opti ...
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 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).


References


Further reading

*{{Cite book , first1 = Ioannis, last1 = Karatzas, first2 = Steven E., last2 = Shreve, author-link2 = Steven E. Shreve, doi = 10.1007/b98840 , title = Methods of Mathematical Finance , series = Stochastic Modelling and Applied Probability , volume = 39 , year = 1998 , isbn = 978-0-387-94839-3 *Merton R.C.: ''Continuous Time Finance'', Blackwell (1990). Financial economics Stochastic control Portfolio theories Intertemporal economics