STOCHASTIC

stochastic differential equations and stochastic optimal control theory, the
paper may be quite intimidating upon first reading. To make the paper
intelligible to readers unfamiliar with these mathematical concepts, we deviate
from our previous policy of keeping formalism out of the introduction:
Appendix A contains a brief, intuitive introduction to stochastic differential
equations and stochastic control theory. Although not rigorous, the arguments
are, hopefully, sufficiently plausible as to enable the reader to understand
Merton's paper with relative ease.
Merton assumes the investor's utility for lifetime consumption to be an
integral of utilities for instantaneous consumption over time. This is a natural,
continuous-time version of the additive utility assumption in discrete time.
The general formalism of the optimal control problem is outlined first for
utilities that vary arbitrarily in time and is specialized in later sections to the
pure "impatience" case, as in the Samuelson and Hakansson discrete time
models. Most of Merton's explicit solutions pertain to utility functions of the
hyperbolic absolute risk aversion (HARA) class, that is, to the class where
the reciprocal of the Arrow-Pratt absolute risk-aversion index is linear in
wealth. This class contains as special cases utilities having constant absolute
or relative risk aversion. The individual must decide at each point in time how
to allocate his existing wealth to between consumption and investment in a
number of financial assets. The risky asset returns are governed by a known
Markov process. The objective is maximization of expected utility of lifetime
consumption, including terminal bequests.
In the spirit of dynamic programming, there exists at each time t a derived
utility function for wealth w, namely, the maximum expected utility of future
lifetime consumption given that wealth is w at time t. The instantaneous
consumption-investment problem requires the maximization of utility of
consumption and terminal wealth at time t, or rather, at an "infinitesimal"
time later than t. For risky asset returns governed by a stationary log-normal
process (see Appendix A), terminal wealth at the end of the "infinitesimal"
time span is almost deterministic. The expected utility maximization thus
involves only means and variances of risky returns, as in the Samuelson paper
of Part III, Chapter 1. The portfolio selection aspect of the continuous-time
consumption-investment problem is thus solved exactly using mean-variance
analysis. As in the Lintner and Ziemba papers of Part III, this implies that the
optimal portfolio possesses the mutual fund separation pro perty: All investors
will choose a linear combination of two composite assets which are, moreover,
independent of individual preferences. If there exists a risk-free asset, it may
be chosen as one of the mutual funds, and all portfolios reduce to a mix of this
risk-free asset and a single composite risky asset which is the same for all
investors. The returns on the mutual fund are also governed by a stationary
log-normal process. Note that such a result is only true in continuous time:
450 PART V DYNAMIC MODELS