Figure 11.6 The choice between investments A and B.

shown in columns 4 and 5 of Table 11.7 entitled "Sum of Cumulative Probability." Applying point 3 shows clearly that A dominates B.

The reader might be curious how this analysis relates to mean-variance analysis. If returns are normally distributed, then the answer is clear. First-order stochastic dominance assumes investors prefer more to less. The first part of the efficient set theorem utilizes this same assumption and leads to the result that, at any level of standard deviation, the investor pre-k'/red a higher mean return. Thus, first-order stochastic dominance implies the first part of thii efficient set theorem if returns are normally distributed. When short sales are allowed, pi efening a higher mean for any standard deviation leads to the efficient frontier. When short vi les are disallowed, first-order stochastic dominance produces a set of portfolios that lies on the upper half of the outer boundary of the feasible set. These portfolios include the efficient set produced by mean-variance analysis, plus all portfolios that have the highest return possible for each level of risk. In Figure 11.7, the portfolios along the boundary segment BC are not in the efficient set AB but satisfy the first-order stochastic dominance condition.

Second-order stochastic dominance assumes investors are risk averse, as well as prefer more to less. These are the same assumptions that lead to the efficient set theorem. Thus, it should not surprise the reader that with normally distributed returns the only set of portfolios that is not dominated, using second-order stochastic dominance, is the mean-'.ariance efficient set.

We have just seen that, when returns are normally distributed, second-order stochastic di minance (and, with short sales, first-order stochastic dominance) leads to a definition of an optimal set of portfolios that is consistent with the efficient set produced by mean-variance analysis. The advantage of stochastic dominance is that it can be used to derive -,ers of desirable portfolios when returns follow other distributions, or when one is unwilling to assume specific utility functions.

One should be careful not to overemphasize the importance of this advantage. In genial, stochastic dominance involves pairwise comparisons of all alternatives. Since there is an infinite set of alternatives to consider in portfolio selection, the direct use of stochastic

f igure 11,7 Portfolios having first-order stochastic dominance.

a dominance becomes infeasible.12 However, if returns follow any of a variety of well-behaved distributions, then the use of stochastic dominance shows that portfolios can be selected in simpler ways. We have already seen that, when returns are normally distributed, stochastic dominance leads to the familiar mean-variance analysis. Bawa [13] has shown that when returns follow any two-parameter distribution, stochastic dominance can be used to derive simple two-parameter rules for portfolio selection. Before leaving this section, it is useful to examine third-order stochastic dominance. Figure 11.8 plots the sum of the cumulative distribution function for the example presented in Table 11.7. The sum of the cumulative frequency function can be thought of as the cumulative of the cumulative frequency function. Note that A never lies to the left of B or above B. This allows us to choose among the alternatives using second-order stochastic dominance. If the curves cross, then neither A nor B can be eliminated by second-order stochastic dominance and third-order is necessary.

Third-order stochastic dominance assumes investors exhibit decreasing absolute risk aversion. One of the properties of a function exhibiting decreasing absolute risk aversion is a positive third derivative.13 The theorem for third-order stochastic dominance utilizes this fact. Since there are other characteristics of decreasing absolute risk aversion, a more powerful theorem awaits development.14 A dominates B using third-order stochastic dominance if:

1. Investors prefer more to less,

2. Investors are risk averse,

3. The third derivative of the investors utility function is positive,

4. The mean of A is greater than the mean of B, and

5. The sum of the sum of the cumulative probability distribution for all returns is never more with A than B and sometimes less.

Columns 4 and 5 of Table 11.7 are the sums of the cumulative distributions. Columns 6 and 7 are the sums of the sum of the cumulative distribution.

As can be seen from this table, A dominates B using third-order stochastic dominance. This is to be expected since A dominates B under second-order stochastic dominance and third-order is more restrictive.

The stochastic dominance analysis may seem to the reader to be a lot of analysis with few results. Many of the results in portfolio analysis that can be obtained from using stochastic

12This drawback is not as important in other areas. For example, in looking at the selection of investment alternatives by the firm we are dealing with, a limited set of alternatives and stochastic dominance would seem to be a very useful tool.

13A utility function exhibits decreasing absolute risk aversion. IfA'(W) < 0 where U\W) and t/"(W) are the first and second derivatives of the utility function, respectively, then

The first term is positive since it is a ratio of squared terms. U'(W) > 0 by assumption. Therefore, for the second term to be negative, it is necessary for C/"'(W) > 0. Note that U' "(W) is a necessary, but not sufficient, condition for A'{W) < 0.

14For some distributions, such a theorem exists.

chapter 11 other portfolio SELECTION models

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