STOCHASTIC

JAMES A. OHLSON
that quadratic utility is in general valid asymptotically whenever utility is
negative exponential, power, logarithmic, and some obvious extensions of
such functions.
In the second part of the paper exact compactness properties of the lognormal
distribution are derived, assuming that expected growth rates and
variance of growth rates of the individual assets are linear in time. This model
will not imply a distribution which is compact in the strong Samuelson sense;
however, it is sufficiently well behaved to satisfy the sufficient conditions
evolved in this article.
II. The General Problem
Let X = (Xl,X2,...,Xm)' denote the vector of random asset returns with a
distribution function G(x1,...,xm; 0(t)) = Pr{Xt^xu ...,Xm%xm\ 9(t)}.
6(t) expresses the parameters of the distribution G as a function of t, where t
denotes the time span of the decision horizon. G, therefore, is determined by
the value of t, and t > 0. Let A = (A1;..., Am)' denote the vector of fractions
of initial wealth invested in each asset, where £jA; = 1 by convention. Further,
let U(W) = UCEXiXj) denote the investor's utility of wealth function, wealth
being denoted by W, and suppose the investor wishes to maximize his expected
utility. The optimal portfolio for any t, A*( 0. For notational simplicity,
expected utility can be expressed as
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EU(W) = U(W) dF{W), (2)
keeping firmly in mind that the distribution of wealth, F(W), and expected
utility depend on both A and t. Without loss of generality, suppose
Pr{|W— 11 < e} -* 1 as t -> 0, i.e., the portfolio return is approximately equal
to unity for the short-period investment horizon.
The concern is to obtain a set of weak assumptions on lim,^0 F an d U, so
that the general problem (2) is equivalent to the quadratic utility problem
and where
EU(W) = C/(l) + f/ (1) (l) E{W-1) + iC/ (2) (l) E{W-1) 2 + EQ3, (3)
lim EQ3IE(W- \) n = 0 for n = 1,2, and all A e D.
r-»o
222 PART III STATIC PORTFOLIO SELECTION MODELS