STOCHASTIC

W. T. ZIEMBA
maximizes expected utility when the risk-free asset assumption is not made,
by utilizing standard nonlinear programming algorithms. The calculations in
this direct approach are generally much more formidable than in the two-stage
approach; hence it is generally preferable to use the latter approach if it is valid.
II. The Independent Case
We consider an investor having one dollar 4 to invest in assets i = 0,1,...,«.
Assets 1,...,» are random and they exhibit constant returns to scale so that
if Xj is invested in /, then S,i xi is returned at the end of the investment period.
The £i are assumed to have independent stable distribution functions
Fife; li,Si,P,a),i=\,...,n. A distribution function F{y) is said to be stable
if and only if for all positive numbers a1 and a2 and all real numbers bx and
b2 there exist a positive number a and a real number b such that
where the * indicates the convolution operation. Equation (1) formalizes the
statement that the stable family is precisely that class of distributions that is
closed under addition of independent and identically distributed random
variables. The F; are unimodal, absolutely continuous, and have continuous
densities j\. The parameter — 1 ^ P g 1 is related to the skewness of the
distribution. When p > 0 (< 0) the distribution is skewed to the right (left).
It is convenient for our purposes to assume that the distribution Ft is symmetric,
in which case ft = 0. 5 The parameter a is termed the characteristic
exponent and absolute moments of order < a exist, where 0 < a S 2.
When a = 2, F is the normal distribution and all moments exist. It will be
convenient to assume that 1 < a ^ 2 so that absolute first moments always
exist and the value of a is the same for each F,. The parameter £f corresponds
to the central tendency of the distribution which is the mean if a > 1. The
parameter St, assumed to be positive, refers to the dispersion of the distribution.
It will be convenient to differentiate between St and Si 1 ". We will follow
a suggestion of Beale and call this latter quantity the oc-dispersion. When
Fj is normal St is one-half the variance, in other cases it is approximately
equal to the semiinterquartile range. In general 5/ /ot is proportional to the
mean absolute deviation E\£t—1,|.
4
Without loss of generality, we may normalize the investment returns so that initial
wealth is one dollar.
5
The analysis in this section is valid though for $ # 0 as long as /? is the same for each Ft.
246 PART HI STATIC PORTFOLIO SELECTION MODELS