University:
University of Texas at Austin
Course:
ACC 311 | Fundamentals of Financial Accounting
Academic year:

2023
Views:

136

Pages:

3
Author:

Vina J.

Stochastic Dominance One of the less attractive features in assessing risk is the practical difficulty of measuring the N-M utility curve. For certain comparisons, this problem may be avoided in that preference may be established for a whole class of utility functions. In other words, it may be possible to look at the probabilities and outcomes, and declare that one option is superior to another for, say, all risk-averse decision makers. In such cases there is said to be stochastic dominance. First order stochastic dominance (FSD) applies to all positively sloped utility functions. It states that option A is preferable to B where the cumulative probabilities of A refer to a level of wealth as great as, or greater than, B. This result is demonstrated in Figure 3.17. Note that the cumulative probability of A is either equal to or lies to the right of B. The expected value in Table 3.6 also reveals the superiority of A. As U4 >U>U2> U,, then from the A-B column (U, + U4) 0.1 > (U,) 0.2, so that A must have a greater expected utility than B. In general, A is superior to B as long as the cumulative A-B column is negative or 0 at all values. In practice, there will be relatively few investments rejected on these grounds, as we are including the preferences of all decision makers. Second order stochastic dominance (SSD) applies only to risk-averse or riskneutral decision makers, and is accordingly more discerning. Under this condition, Option A is superior to Option B if the cumulative area under the cumulative probability curve for B is at least as great as, or greater than, A, for all levels of returns. Second order dominance of A over B is demonstrated in Figure 3.18. The first order dominance condition is not fulfilled because at points I and II in Figure 3.18 (b) the cumulative probability of A is higher than B. It is easier to see this point by looking at Figure 3.18 (a). Where the returns are 5, the probability of B is higher than A; the gambler may value this difference much more highly than any of the other differences and hence may prefer B to A. Second order stochastic dominance is fulfilled, as can be seen from Table where column 6 must be at all points less than or equal to column 7. Note that the fact that the returns are evenly spaced simplifies the calculation of columns 6 and 7, the full formulae are: Table : Calculation of expected values where V1 = NM utility of return I in Figure There will, of course, be many choices between investments where superiority depends upon the degree of risk aversion. In such cases stochastic dominance is likely to fail. It is instructive to look at such an example. Take the two investments in Table; casual inspection does not make it clear which is preferable. The full stochastic dominance process is given in Table. FSD implies that column 4 is not greater than column 5 for every line and is less than column 5 for at least one line. This condition is violated on lines 3 and 7. This might be because risk-seeking individuals are included. SSD