STOCHASTIC

JAMES A. OHLSON
analysis here is "one short period," while the Merton analysis is "many short
periods" (continuous portfolio revision is the limiting case). The many-shortperiods
context will, under appropriate assumptions, generate the product of
a "large" number of independently distributed wealth relatives. The limiting
distribution of the latter process will then be a log-normal distribution.
Finally, it is interesting to note that the third and fourth moments are of
the same order. Referring to Theorem II in the cited Samuelson paper, this
implies that a higher order approximation is not attained by adding a third
moment and maximizing a cubic utility function.
ACKNOWLEDGMENT
Without implicating him, the author wishes to thank Markku Kallio for valuable
discussions.
REFERENCES
1. FELLER, W., An Introduction to Probability Theory and Its Applications, Second edition,
Volume 1. New York, 1957.
2. HALMOS, P. R., Measure Theory. Van Nostrand-Reinhold, Princeton, New Jersey, 1950.
3. MERTON, R. C, "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time
Case." Review of Economics and Statistics 51 (1969), 247-257.
4. MERTON, R. C, "Optimum Consumption and Portfolio Rules in a Continuous-Time
Model." Journal of Economic Theory 3 (1971).
5. MERTON, R. C, "An Intertemporal Capital Asset Pricing Model." Econometrica 40 (1972).
6. MERTON, R. C, and SAMUELSON, P. A., "Fallacy of the LogNormal Approximation to
Optimal Portfolio Decision-Making over many Periods." Journal of Financial Economics
1 (1974).
7. SAMUELSON, P. A., "The Fundamental Approximation Theorem of Portfolio Analysis
in Terms of Means, Variances, and Higher Moments." Review of Economic Studies
37 (1970).
234 PART HI STATIC PORTFOLIO SELECTION MODELS