STOCHASTIC

utilities having constant relative risk aversion, the optimal amount of consumption
will still be proportional to wealth. The relative risk aversion of induced
utility for wealth will remain constant over time. The optimal asset proportions
at any time t will be independent of wealth, consumption, and past or future
asset returns, and will depend only on the nature of the capital market at time /.
Calculation of the optimal consumption fraction at time t will, however,
generally require the solution of all "future" consumption and portfolio
problems, and will thus be much more complex than in the iid case. In a recent
paper, Hakansson (1971c) studied a significant generalization of his model.
In Hakansson's newer paper, the individual's impatience is variable, his
lifetime is stochastic, and the capital assets (both risk free and risky) change
in time. The individual consumes, borrows (or lends), invests in risky assets,
and purchases life insurance. For additive utilities having constant relative
risk aversion, solutions are obtained for the optimal lifetime consumption,
investment, and insurance-buying program. For logarithmic utilities, Miller
(1974a) analyzed rigorously the optimal consumption strategies over an infinite
lifetime with a stochastic (possible nonstationary) income stream and a single,
safe asset. Rentz (1971-1973) has analyzed optimal consumption and portfolio
policies with life insurance, for stochastic lifetimes and changing family size.
Additional extensions are given by Hakansson (1969b), Long (1972), and
Neave (1973).
III. Models of Option Strategy
In the first two chapters of Part V, the emphasis is more on questions of
"preferences" than on "randomness": Probabilities play a role only as
weighting factors of utility through the expected value operation. The papers
in Chapters 3 and 4 seem to emphasize questions of a more "probabilistic"
nature. In particular, considerations invo ving stochastic processes and the
dynamical behavior of random occurrences over time assume an importance
lacking in previous parts of the book. Chapter 3 deals with optimal stock or
bond option strategies, and with optimal sequential strategies for divestiture
of a risky asset. The first Pye paper and the Taylor paper treat bond and stock
option strategies from a "stopping rule" viewpoint. In a stopping rule problem,
the decision maker observes (for a cost) a stochastic process and must at any
decision point choose either to continue observing for at least another period
or to stop observing and take a terminal action. Upon stopping, a terminal
reward is received which may include future as well as immediate payoffs.
The problem is to determine when to stop so as to optimize some overall
average reward function. The dynamic programming approach taken in the
first two papers of Chapter 3 is common to most stopping problem situations,
and many other similar applications can be treated using essentially the same
438 PART V DYNAMIC MODELS