STOCHASTIC

76 THE REVIEW OF ECONOMICS AND STATISTICS
depend on a. Moreover, for form (i), which
shall be referred to as the "minimum a" safetyfirst
criterion, the ordering of these indifference
curves also depends on a, the probability of
disaster, while in (ii), to be called the "maximum
z" criterion, the ordering depends on 2
the disaster level. 3 Telser proposed (iii) as an
alternative to the "minimum a" criterion on
the grounds that the existence of a riskless
asset, cash, eliminated the need for minimizing
the probability of disaster. We shall refer to
this version as the "maximum /*" safety-first
criterion. 4
Both Roy and Telser obtain their criterion
by making use of the Chebychev inequality.
We will not follow them in this respect, but
instead will proceed along the following lines.
We shall assume that the distribution of the
variable z can be fully described by two parameters
st> that it can be transformed into the
standardized variable ( J which has a
distribution function F such that Pr(z == 2) =
( 2 ~ /"• \
F\ ) . Since F is monotonic, the "minimum
a" version, which involves minimizing
Pr{z =s 2), is equivalent to
min^—^-) . (2)
a
Similarly the constraint Pr(z^z) =£ a is equiv-
/ i-p. \
alent to F y / — ° an( ^ can ^ e written in
O"
the form
^ f + P-'W- (3)
where F' 1 (a) is the inverse of F and is a constant,
depending on the probability level a. 5
' By ordering, we mean the ranking of preferences as
measured by the indifference curves.
* It is worth noting here that the "maximum /t" criterion
requires investors to choose two parameters z and a. For
the two other safety-first criteria only one parameter must
be selected.
" Roy's use of the Chebychev inequality was an attempt
to avoid the two parameter restriction. Whatever the merits
of this approach in the context of our paper it is easy to
show that the results we have obtained for the "minimum
o" and "maximum ft" safety-first criteria are equally applicable
to the Roy and Telser versions of these criteria which
are derived using the Chebychev inequality. This follows
from the fact that Roy's objective function is n — z/tr
while Telser's version is max n subject to ix — aria ^ z
([8,10]).
For any distribution we can find a critical
probability level a*, such that F' 1 (a*) = 0
and F-^a) < 0 for a < a* while F-^a) > 0
for a > a*. The magnitude of a* reflects the
skewness of the distribution, with a* = 0.5 indicating
a distribution symmetrical about its
mean. Normally one would expect the investor
to choose a small value for o, so that as long as
the distribution is not too skew, 2 is less than p.
Using (2) and (3), the alternative versions
of the safety-first criteria can be restated as
follows:
(i) min
'•(^)
(4)
(ii) max p + .F- 1 (a)o- (5)
(iii) max ^ subject to (i.-f F -1 (a)o-^ 2 (6)
III Relationship Between Approaches —
No Riskless Asset
In this section a graphical demonstration of
the relationship between mean-standard deviation
analysis based on expected utility maximization
and mean-standard deviation analysis
based on the safety-first principle is given for
the no riskless asset case. Following Markowitz
and Sharpe [7, 9], we can derive the
investor's opportunity locus AB in the (/i,