Since the returns on almost all assets are not fully predictable, the criterion has to take financial risk into account.
Alternatively, it may be a function of the various levels of goods and services consumption that are attained by withdrawing some funds from the portfolio after each time period.
refers to the stochastic return (the imperfectly predictable amount that the average dollar grows to) of asset i for period t, and where the shares
gives the expected utility expression to be maximized: The terms containing the choice shares
for differing t are additively separate, giving rise to the result of intertemporal independence of optimal decisions: optimizing for any particular decision period t involves taking the derivatives of one additively separate expression with respect to the various shares, and the first-order conditions for the optimal shares in a particular period do not contain the stochastic return information or the decision information for any other period.
The Kelly criterion gives rise to the same portfolio decisions as does the maximization of the expected value of the log utility function as described above.
Like the log utility function, the power utility function for any value of the power parameter exhibits constant relative risk aversion, a property that tends to cause decisions to scale up proportionately without change as initial wealth increases.
Hyperbolic absolute risk aversion (HARA) is a feature of a broad class of von Neumann-Morgenstern utility functions for choice under risk, including the log and power utility functions dealt with above.
Mossin[2] showed that under HARA utility, optimal portfolio choice involves partial time-independence of decisions if there is a risk-free asset and there is serial independence of asset returns: to find the optimal current-period portfolio, one needs to know no future distributional information about the asset returns except the future risk-free returns.
Thus the optimal portfolio for any period will depend on the probability distribution of returns for the various assets contingent on their previous-period realizations, and so cannot be determined in advance.
Dollar cost averaging is gradual entry into risky assets; it is frequently advocated by investment advisors.
[1]: ch.11 Under certain assumptions including exponential utility and a single asset with returns following an ARMA(1,1) process, a necessary but not sufficient condition for increasing conservatism (decreasing holding of the risky asset) over time (which is often advocated by investment advisors) is negative first-order serial correlation, while non-negative first-order serial correlation gives the opposite result of increased risk-taking at later points in time.
[4] Intertemporal portfolio models in which portfolio choice is conducted jointly with intertemporal labor supply decisions can lead to the age effect of conservatism increasing with age[citation needed] as advocated by many investment advisors.
This result follows from the fact that risky investments when the investor is young that turn out badly can be reacted to by supplying more labor than anticipated in subsequent time periods to at least partially offset the lost wealth; since an older person with fewer subsequent time periods is less able to offset bad investment returns in this way, it is optimal for an investor to take on less investment risk at an older age.
Robert C. Merton[5] showed that in continuous time with hyperbolic absolute risk aversion, with asset returns whose evolution is described by Brownian motion and which are independently and identically distributed through time, and with a risk-free asset, one can obtain an explicit solution for the demand for the unique optimal portfolio, and that demand is linear in initial wealth.