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SOLUTIONS TO CONCEPT C H E C K S
APPENDIX A: A DEFENSE OF MEAN-VARIANCE ANALYSIS
The basic question is how one can best describe the uncertainty of portfolio rates of re- turn. In principle, one could list all possible outcomes for the portfolio over a given period.
If each outcome results in a payoff such as a dollar profit or rate of return, then this payoff value is the random variablein question. A list assigning a probability to all possible val- ues of a random variable is called the probability distribution of the random variable.
The reward for holding a portfolio is typically measured by the expected rate of return across all possible scenarios, which equals
where s1, . . . , nare the possible outcomes or scenarios, r(s) is the rate of return for out- come s, and Pr(s) is the probability associated with it.
Actually, the expected value or mean is not the only candidate for the central value of a probability distribution. Other candidates are the median and the mode.
The medianis defined as the outcome value that exceeds the outcome values for half the population and is exceeded by the other half. Whereas the expected rate of return is a weighted average of the outcomes, the weights being the probabilities, the median is based on the rank order of the outcomes and takes into account only the order of the outcome values.
The median differs significantly from the mean in cases where the expected value is dominated by extreme values. One example is the income (or wealth) distribution in a pop- ulation. A relatively small number of households command a disproportionate share of to- tal income (and wealth). The mean income is “pulled up” by these extreme values, which makes it nonrepresentative. The median is free of this effect, since it equals the income level that is exceeded by half the population, regardless of by how much.
Finally, a third candidate for the measure of central value is the mode, which is the most likely value of the distribution or the outcome with the highest probability. However, the expected value is by far the most widely used measure of central or average tendency.
We now turn to the characterization of the risk implied by the nature of the probability distribution of returns. In general, it is impossible to quantify risk by a single number. The idea is to describe the likelihood and magnitudes of “surprises” (deviations from the mean) with as small a set of statistics as is needed for accuracy. The easiest way to accomplish this is to answer a set of questions in order of their informational value and to stop at the point where additional questions would not affect our notion of the risk–return trade-off.
The first question is, “What is a typical deviation from the expected value?” A natural answer would be, “The expected deviation from the expected value is .” Unfortu- nately, this answer is not helpful because it is necessarily zero: Positive deviations from the mean are offset exactly by negative deviations.
There are two ways of getting around this problem. The first is to use the expected ab- solute valueof the deviation which turns all deviations into positive values. This is known as MAD (mean absolute deviation), which is given by
The second is to use the expected squared deviation from the mean, which also must be positive, and which is simply the variance of the probability distribution:
Note that the unit of measurement of the variance is “percent squared.” To return to our orig- inal units, we compute the standard deviation as the square root of the variance, which is
2
s1n Pr(s)[r(s)E(r)]2 ns1Pr(s)Absolute value[r(s)E(r)]
E(r)s1
n Pr(s)r(s)V isit us at www .mhhe.com/bkm
measured in percentage terms, as is the expected value. The variance is also called the sec- ond central momentaround the mean, with the expected return itself being the first moment.
Although the variance measures the average squared deviation from the expected value, it does not provide a full description of risk. To see why, consider the two probability dis- tributions for rates of return on a portfolio, in Figure 6A.1.
Aand Bare probability distributions with identical expected values and variances. The graphs show that the variances are identical because probability distribution Bis the mirror image of A.
What is the principal difference between Aand B? Ais characterized by more likely but small losses and less likely but extreme gains. This pattern is reversed in B. The difference is important. When we talk about risk, we really mean “badsurprises.” The bad surprises in A, although they are more likely, are small (and limited) in magnitude. The bad surprises in B are more likely to be extreme. A risk-averse investor will prefer Ato B on these grounds; hence it is worthwhile to quantify this characteristic. The asymmetry of a distrib- ution is called skewness, which we measure by the third central moment, given by
Cubing the deviations from the expected value preserves their signs, which allows us to distinguish good from bad surprises. Because this procedure gives greater weight to larger deviations, it causes the “long tail” of the distribution to dominate the measure of skewness.
Thus the skewness of the distribution will be positive for a right-skewed distribution such as Aand negative for a left-skewed distribution such as B. The asymmetry is a relevant characteristic, although it is not as important as the magnitude of the standard deviation.
To summarize, the first moment (expected value) represents the reward. The second and higher central moments characterize 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
M3s1
n Pr(s)[r(s)E(r)]3V isit us at www .mhhe.com/bkm
E(rA) Pr (r)
E(rB) Pr (r)
rB rA
A B
Figure 6A.1 Skewed probability distributions for rates of return on a portfolio.
UE(r) b02b1M3b2M4b3M5. . .
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 investor’s assessment of the probability distribution adequately? Samuelson’s “Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances, and Higher Moments”3proves 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.
Samuelson’s 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 is, instead, that the actions of investors in frequently revising their portfolios will limit higher moments to negligible levels.
Continuity or compactness is not, however, an innocuous assumption. Portfolio revi- sions entail transaction costs, meaning that rebalancing must of necessity be somewhat lim- ited and that skewness and other higher moments cannot entirely be ignored. Compactness also rules out such phenomena as the major stock price jumps that occur in response to takeover attempts. It also rules out such dramatic events as the 25% one-day decline of the stock market on October 19, 1987. Except for these relatively unusual events, however, mean-variance analysis is adequate. In most cases, if the portfolio may be revised fre- quently, we need to worry about the mean and variance only.
Portfolio theory, for the most part, is built on the assumption that the conditions for mean-variance (or mean–standard deviation) analysis are satisfied. Accordingly, we typi- cally ignore higher moments.
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3Paul A. Samuelson, “The Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances, and Higher Moments,” Review of Economic Studies37 (1970).
CONCEPT C H E C K
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QUESTION A.1
How does the simultaneous popularity of both lotteries and insurance policies confirm the notion that individuals prefer positive to negative skewness of portfolio returns?