STOCHASTIC

MEAN-STANDARD PORTFOLIO ANALYSIS 19
The "maximum /x" formulation requires the
investor to select two parameters and as a result
the correspondence to utility maximization
is in general not unique. Referring to figure
3, we can imagine an expected utility maximizer
selecting the portfolio T. There is an infinity
of straight lines passing through T and each
one of these is consistent with the behavior of
a safety-first investor who selects T on the
"maximum /i" principle. If, however, we specify
either one of the two parameters, for example,
by requiring the investor to choose a
specific disaster level, then the correspondence
is indeed unique. The reverse correspondence
is as for the other two cases and need not be
repeated.
In summary then, as long as there is no riskless
asset, for any portfolio chosen by an expected
utility maximizing investor with concave
(/x, cr) indifference curves, we can always find
a safety-first investor who will choose the same
portfolio. For any portfolio chosen by any
variant of the safety-first principle, there is in
general an infinity of expected utility functions
that will cause an investor who maximizes expected
utility to behave in the same way.
IV Relationship Between Approaches —
Riskless Asset
As we shall now demonstrate, the corre :
spondence between the expected utility and the
safety-first approaches to mean-standard deviation
analysis breaks down seriously when the
investor has the opportunity of holding a riskless
asset. We shall assume that the riskless
asset has a rate of return r and that the investor
can either lend or borrow at this rate. Then,
following Sharpe, the investor's efficiency locus
is as drawn in figure 4.
The curved line AB is obtained as in section
III, while REF is the tangent from the point
(R,0) to AB where R = rW and W is the investor's
initial wealth. This line represents the
opportunity locus for lending (segment RE)
or borrowing (segment EF) at the riskless rate
and holding the remainder of the investment
balance (wealth less lending or plus borrowing)
in portfolio E. Since the straight line
REF dominates all parts of the curved line AB
it now becomes the efficiency locus and the
equilibrium portfolio is found where an indifference
curve is tangent to REF. 9
Assuming concave (/i, -—I,
On the other hand, a "minimum a" or a
"maximum 2" safety-first investor with straight
line indifference curves can never achieve a
tangency with REF except in the special case
that one of his indifference curves coincides
with REF, in which case an indeterminate
solution results. If the slope of his indifference
curves exceed that of REF, he will borrow an
infinite amount (assuming that there is no
limit) and move out in the direction of F. l °
Notice that although such an indifference curve
implies unlimited borrowing, the investor is
still holding a diversified portfolio of risky
assets, namely that specified by E. If the slope
is less than that of REF, the investor will move
" If we do not allow borrowing at the riskless rate, the
efficiency locus becomes REB. When the tangency of an
investor's expected utility indifference curve occurs along
the segment EB of the efficiency locus, the results are the
same as in the no riskless asset case. When the tangency
occurs along the segment RE, the results are the same as for
the case we are about to discuss in which the investor can
obtain a riskless asset.
10 This can be readily seen by superimposing the straight
line indifference curves of figure 2 on figure 4 and bearing
in mind that south easterly movements imply increasing
expected utility.
1. MEAN-VARIANCE AND SAFETY-FIRST APPROACHES 239