STOCHASTIC

SAFETY-FIRST AND EXPECTED UTILITY MAXIMIZATION
IN MEAN-STANDARD DEVIATION PORTFOLIO ANALYSIS
David H. Pyle and Stephen J. Turnovsky *
I Introduction
DATING back to the original work by
Markowitz and Tobin [7, 11], portfolio
theory has usually followed the mean-standard
deviation approach in which the investor is
assumed to choose among alternative portfolios
on the basis of a utility function defined in
terms of the mean and standard deviation of
the portfolio return. Because of the arbitrary
nature of utility functions, more or less parallel
with these developments there have been attempts
to depart from the utility framework
altogether and to invoke criteria based on more
objective concepts. As the first of such objective
criteria, Roy [8] suggested that investors
have in mind some disaster level of returns
and that they behave so as to minimize the
probability of disaster. This criterion, along
with some variants of it, since developed by
others (Telser [10],Kataoka [5]), has become
known as the safety-first criterion.
As we shall see, the various safety-first criteria
also lead to optimization of expressions
involving the mean and standard deviation.
The objective of this paper is to compare this
justification of mean-standard deviation analysis
with the more conventional approach based
on expected utility maximization. This relationship,
although hinted at, has not been systematically
studied in the literature (Lintner
[6],Farrar [3]).
In particular we shall show that:
1) In the absence of a riskless asset, a correspondence
can be established between the
safety-first criterion and expected utility maximization
when that maximization results in
concave indifference curves in the mean-standard
deviation space.
2) If a riskless asset is available then except
in one special case the safety-first criterion does
not lead to the traditional liquidity preference
behavior.
* We thank Gordon Pye and Paul Cootner for helpful
comments on an earlier draft of this paper. Needless to say
they are not responsible for any errors which may remain.
II Expected Utility Maximization and the
Safety-First Criteria
According to the expected utility approach to
mean-standard deviation portfolio analysis, the
investor's problem is to select a portfolio of
the available assets so as to maximize some
expected utility function of the form
£(«) = V(>,o-) (1)
where n is the expected value and cr is the standard
deviation of z, the total one-period return
on the portfolio. Furthermore, this utility function
has the properties
^ > 0 , ^ < 0 .
1. MEAN-VARIANCE AND SAFETY-FIRST APPROACHES
Given positive but diminishing marginal utility
of wealth and a multivariate normal distribution
of returns to the available assets, V is a
concave function in the (/J., O-) plane. 1
Turning to the safety-first principle, we shall
state the following forms where i is the subsistence
or disaster level of returns and a is the
probability of disaster. 2
(i) min Pr(z -= z)
(ii) max s subject to Pr(z ^ z) == a
(m) max *i subject to Pr(z ^s) £«
where Pr denotes probability.
Form (i) is the objective as originally stated
by Roy, while (ii) is a later version proposed
in a somewhat different context by Kataoka.
These alternative forms both lead to indifference
curves in the (/*, o-) plane whose slopes
1 The original proof that V is a concave function under
these conditions was given by Tobin [11]. As Fama [2]
and others have pointed out, Tobin's proof based on any
two parameter distribution is only valid for the normal.
Feldstein [4] has presented a counter-example of a twoparameter
distribution for which V is not concave. It should
also be pointed out that only in special cases will £(u)
depend only on /* and a. In general it will depend on higher
moments as well.
'Recently Baumol [1] has suggested a related criterion
in which the investor compares portfolios on the basis of
M, z rather than /i, 9. Since this approach involves specifying
a probability level of disaster for the individual, it effectively
incorporates to some extent his preferences and therefore
succeeds in narrowing down the Markowitz efficiency
frontier.