the uncertainty of the reward. All the even moments (variance, M4, etc.) represent the likelihood of extreme values. Larger values for these mo- ments indicate greater uncertainty. The odd moments (M3, M5, etc.) represent measures of asymmetry. Positive numbers are associated with positive skewness and hence are desirable. We can characterize the risk aversion of any investor by the preference scheme that the investor assigns to the various moments of the distribution. In other words, we can write the utility value derived from the probability distribution as II. Portfolio Theory 6. Risk and Risk Aversion The McGraw−Hill Companies, 2001           174 PART II Portfolio Theory     U E(r) b0 2 b1M3 b2M4 b3M5 . . .   where the importance of the terms lessens as we proceed to higher moments. Notice that the "good" (odd) moments have positive coefficients, whereas the "bad" (even) moments have minus signs in front of the coefficients. How many moments are needed to describe the investors assessment of the probability distribution adequately? Samuelsons "Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances, and Higher Moments"3 proves that in many im- portant circumstances:   1. The importance of all moments beyond the variance is much smaller than that of the expected value and variance. In other words, disregarding moments higher than the variance will not affect portfolio choice. 2. The variance is as important as the mean to investor welfare.   Samuelsons proof is the major theoretical justification for mean-variance analysis. Un- der the conditions of this proof mean and variance are equally important, and we can over- look all other moments without harm. The major assumption that Samuelson makes to arrive at this conclusion concerns the "compactness" of the distribution of stock returns. The distribution of the rate of return on a portfolio is said to be compact if the risk can be controlled by the investor. Practically speaking, we test for compactness of the distribution by posing a question: Will the risk of my position in the portfolio decline if I hold it for a shorter period, and will the risk ap- proach zero if I hold the portfolio for only an instant? If the answer is yes, then the distrib- ution is compact. In general, compactness may be viewed as being equivalent to continuity of stock prices. If stock prices do not take sudden jumps, then the uncertainty of stock returns over smaller and smaller time periods decreases. Under these circumstances investors who can rebalance their portfolios frequently will act so as to make higher moments of the stock re- turn distribution so small as to be unimportant. It is not that skewness, for example, does not matter in principle. It