STOCHASTIC

The Asymptotic Validity of Quadratic Utility
as the Trading Interval Approaches Zero
James A. Ohlson*
STANFORD UNIVERSITY
STANFORD, CALIFORNIA
I. Introduction and Summary
Samuelson [7] has developed the notion that the mean and variance of
wealth are approximately sufficient parameters for the standard portfolio
selection model when the probability distribution of wealth is "compact."
The concept of a "compact" distribution refers to a decision context involving
a family of distributions where the limiting moments of order 3 and higher are
of a smaller magnitude relative to the first two moments of the portfolio
return; in a limiting and approximating sense this would indicate that only
the first two moments are relevant when selecting an optimal portfolio. One
place where this mode of analysis arises is in models in which the portfolio
revision horizon is small, and the limiting situation is one where the period
spacing approaches zero. Models of these types are of increasing importance
in portfolio theory, as is evident in the significant work by Merton [3-5].
The Samuelson [7] exposition is heuristic, and no rigorous weak conditions
are presented as to when quadratic utility is valid in the asymptotic sense
indicated above. 1 The formal analysis set forth by Samuelson requires unnecessarily
sharp constraints on the utility of wealth function and the probability
distribution. In fact, the important case of log-normally distributed
returns will not involve convergent infinite Taylor expansions, and the same
applies when utility is power or logarithmic.
In this paper a set of weak conditions is developed with respect to utility
and probability distributions such that expected utility converges toward
quadratic utility in the limit. It is shown that few restrictions are required on
the probability distribution; essentially, it is sufficient if the third (absolute)
moment vanishes at a faster rate than the first two moments, and moments of
order 4 and higher need not be finite. Few assumptions are made about utility,
aside from differentiability conditions. A not surprising result, therefore, is
* Present address: University of California, Berkely, California.
1 Another heuristic discussion of the problem at hand is provided in a paper by Merton
and Samuelson [6].
1. MEAN-VARIANCE AND SAFETY-FIRST APPROACHES 221