Manual Stochastic Dominance: Investment Decision Making under Uncertainty (Studies in Risk and Uncertainty)

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To see how this is possible, let us first assume, for simplicity only, that U' x is strictly declining.


Therefore, the weight of the negative area in utility terms is smaller than the weight of the positive area in utility terms. To introduce this idea, consider the previous example see Figure 3. Of course, the larger negative area that is allowed is a function of the positive area that precedes it as well as the relative location of these two areas on the horizontal axis. We elaborate below on the relationship between the convexity of U', the location of the various areas and TSD. The above example is overly simplistic, serving merely to introduce the relationship between the convexity of U' x and TSD.

In the example below, we discuss the importance of the location of the various areas for the existence of TSD. We illustrate this by means of Figure 3.

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We show below, and intuitively explain, the difference between F and G and the relationship to TSD criterion. However, because U' is convex, the gain from the difference between the two areas "a" and "b" in utility terms is larger than the loss from the blocks c and d in utility terms. This explanation allows for U' x to be zero and not strictly declining as long as the function U' x is even Figure 3. In such a case, U' will be convex albeit only weakly so in some segments.

A convex U' x and the location of the various blocks is critical for the existence of such dominance. In such a case, even though U' is convex, we do not have TSD: U' a - U' b may be smaller than U' c - U' d because the distance corresponding to the various blocks is not the same as in Figure 3. The convexity of U' x and the location of the various blocks are crucial for the existence of TSD.

However, with such a shift, dominance is not guaranteed and l3 x should be calculated to confirm the existence of TSD. However, here too, we can establish various sufficient rules and necessary rules for UGU3 dominance. Of course, many more sufficient rules are possible e. This condition on the expected values is a necessary condition for dominance in U3. Note that for FSD and SSD we had to prove that this condition was necessary for dominance but for TSD, there is nothing to prove because it is explicitly required by the dominance condition.

Hence, it is a necessary condition for dominance in U3. To see this, suppose that the necessary condition does not hold. It 37l w is claimed that investor's behavior reveals that — 0. A natural way to analyze DSD is to write the utility function in terms of 7c x. Indeed, it is possible to express U x in terms of 7i x. Therefore, if n x is known for all wealth levels, then U x will be fully known up to multiplicative and additive constants.

Indeed, Hammond used this formulation to reach conclusions regarding preference of one investment over an other under various restrictions on 7i x. However, in this case, integration by parts such that n' x will appear in the expression of U x does not lead to a clear rule as it did with the three stochastic dominance rules; hence, the DSD is very difficult to analyze.

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Rather we need to employ a relatively complicated algorithm procedure to prove or disprove DSD in specific given cases. These algorithms can be found in Vickson who provides several necessary conditions and several sufficient conditions for DSD. Proof: Suppose that FD3G. Now we have to show is that the opposite also holds.

Savage, "The utility of analysis of choices involving risk," The Journal of Political Economics, 56, , pp. Markowitz, H.

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This will be followed by an intuitive explanation and graphical explanations. Sufficiency By eq. Therefore, it is necessary for SSD that the condition of Theorem 3. However, this time the area accumulation is not done from the lower bound, a, to x but rather from the upper bound, b, to x, i.

Stochastic Dominance

Thus, the integral condition of Theorem 3. This is not true and one counter example is sufficient to show this claim: in Figure 3. Yet such opposite dominance is possible in some specific cases. For example, Figure 3. The intuitive explanation of SSD is similar to the intuitive explanation of SSD but this time U' x is increasing with x rather than decreasing with x.

The answer is positive and one example is sufficient to show this claim. Consider, once again, F and G, as drawn in Figure 3. The proof is a trivial extension of the previous discussion of U4. However, there are some important utility functions with positive odd derivatives and negative even derivatives. In such a case, simply add these two values with zero probability to obtain the same formulation as for F. Using this formula for discrete random variables, integration by parts can be carried out to prove the SD rules in exactly the same manner as with the continuous random variables.

In the following example, we conduct a direct calculation of expected utility and show that the same results are obtained by employing eq. Note that eq. Thus, using eq. In the proof of eq. What is the effect on the proof of eq. To illustrate, suppose that alternatively we select ai Cumulative probability 0. Does the extension of the bound from [a,b] to [ai,bi] affect the value given in eq.

Generally, for the calculation of EU x , we take a to be the lower value and b the higher value.

However, if we were to take a lower value than a as a lower bound or a higher value than b as an upper bound, the results of the expected utility calculation would remain unchanged. Thus, the lower and upper bounds [a,b] can be selected arbitrarily without affecting the results, and without affecting eq.

It is natural to ask whether there is a condition on the variances which is also a necessary condition for dominance. The answer is generally negative. Therefore, we can conclude that the superior investment does not necessarily have a lower variance. Suppose that x corresponds to distribution F and y to distribution G. It can be easily verified that Uk x e U3. Not really. Take the utility function " Hanoch, G.

Hence the preferred investment by this specific risk averse investor may have a lower mean and a higher variance. As we can see, the more information we assume on U, the smaller the class of preferences and the smaller the size of the resulting efficient set.

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Yet, the U2 set is only a subset of Ui and hence SSD efficient set is a subset of FSD and as there is no relationship between U2 and other sets of preferences there is no relationship between the SSD efficient set and the other efficient set. The inefficient set corresponding to each SD rule is the feasible set less the corresponding efficient set. Both methods yield the same partition of the feasible set into efficient and inefficient sets.