WHAT PRACTITIONERS NEED TO KNOW...... About Higher Moents Mark P. Kritzan In financial analysis, a return distribution is coonly described by its expected return and standard deviation. For exaple, the S&P 500 Index ight have an expected return of 10 percent and a standard deviation of 15 percent. By assuing that the returns of the S&P 500 Index confor to a particular distribution, such as a noral distribution, we can infer the entire distribution of returns fro the expected return and standard deviation.^ The expected value of a distribution is referred to as the first oent of the distribution and is easured by the arithetic ean of the returns. The variance, which equals the standard deviation squared, is called the second central oent or the second oent about the ean. It easures the dispersion of the observations around the ean. The first central oent is not the ean itself, but rather zero, because central oents are easured relative to the ean. The noral distribution is syetric around the ean; hence, the edian (the iddle value of the distribution) and the ode {the ost coon value of the distribution) are both equal to the ean. Moreover, the noral distribution has a standard degree of peakedness. These properties of the noral distribution explain why just the ean and standard deviation are sufficient to estiate the entire distribution. Although investent returns usually are assued to be approxiately norally distributed, this assuption is less likely to hold for very short horizons, such as one day, and for long horizons, such as several years. Moreover, certain assets and investent strategies have properties that produce nonnoral distributions over any horizon. Thus, in soe cases, to estiate a return distribution, one ust go beyond the first oent and the second central oent to the third central oent, which is called skewness, or the fourth central oent, which is called kurtosis. Mark P. Kritzan, CFA, is a partner of Windha Capital Manageent. Skewness Skewness, which is illustrated in Figure 1, refers to the asyetry of a distribution. A distribution that is positively skewed has a long tail on the right side of the distribution and its ean is typically greater than its edian, which in turn, is greater than its ode. Because the ean exceeds the edian, ost of the returns are below the ean, but they are of saller agnitude than the few returns that are above the ean. In contrast, a distribution that is negatively skewed, which is shown in Figure 2, has a long tail on the left side of the distribution, indicating that the few outcoes that are below the ean are of greater agnitude than the larger nuber of outcoes above the ean. Hence, the ean is typically lower than the edian, which is lower than the ode. Skewness is coputed as the average of the cubed deviations fro the ean and is usuauy easured by the ratio of this value to the standard deviation cubed; that is. s = where S = easure of skewness n = nuber of returns Rj = ith return jx = arithetic ean of returns <T = standard deviation of returns (1) The distribution of long-horizon returns that arise fro copounding independent shorter horizon returns is typically skewed to the right. Suppose, for exaple, that we select 100 annual returns fro an underlying noral distribution of returns that has a ean of 10 percent and a standard deviation of 15 percent, and suppose that we repeat this selection ten ties. Hence, we generate ten saples, each consisting of 100 returns. Now, suppose we copound the returns 10 Financial Analysts Journal / Septeber-October 1994
Figure 1. Positively Skewed Distribution Median Mean fro the ten saples so that we end up with a distribution of 100 cuulative ten-year returns. Table 1 shows the results of such an experient. The first colun shows the average values of the ten saples of annual returns, and the second colun shows the values associated with the cuulative ten-year returns. Given that the annual returns were drawn Rgure 2. Negatively Skewed Distribution Financial Analysts Journal / Septeber-October 1994 11
Table 1. TTie Effect of Copounding on Skewness Mean Median Standard deviation Skewness Average of Ten Saples of Norally Distributed Annual 0.1066 0.1105 0.1496-0.1163 Cuulative Ten-Year 1.7580 1.3211 1.3073 1.3634 Figure 3. Contingent of Protective Put Strategy Protective Put Payoff Function fro a noral distribution, the average ean and the average edian are not very far apart and the average skewness of the distribution is close to zero, although slightly negative. The distribution of the cuulative ten-year returns is significantly positively skewed, however. Moreover, the ean of the cuulative ten-year returns is significantly greater than the edian, despite the fact that the average edian of the annual saples exceeds their average ean. The process of copounding introduces skewness because copounded favorable returns have a greater ipact than copounded unfavorable returns of equal agnitude. For exaple, two consecutive 10 percent returns increase an asset's value by 21 percent, whereas two consecutive -10 percent returns decrease an asset's value by only 19 percent. The skewness that arises fro copounding independent returns causes the returns to be lognorally distributed, which iplies that the logariths of the quantities 1 plus the copounded returns are norally distributed.^ This relationship is convenient because, by transforing returns into the logariths of their wealth relatives, we can ignore the skewness of the distribution of the underlying returns and use the noral distribution to estiate the probability of experiencing various logarithic results. These results, of course, ust be transfored back into their original units. ^ Now, consider another process that results in a skewed return distribution. Suppose we purchase a put option to protect an investent in an S&P 500 Index fund that is currently valued at $450.00. Furtherore, suppose that the strike price of the option is $425.00, that it expires in one year, and that it costs $10.00. Figure 3 shows the returns of such a strategy contingent upon various values for the S&P Index. If the S&P 500 Index has a 10 percent expected return (including the reinvestent of dividends) and a 15 percent standard deviation, and if its -0.10 350 400 450 S&P 500 Price 500 550 annual returns are approxiately norally distributed, we can assign probabilities that the Index will equal or fall below various values. Moreover, because we can ap these values precisely onto returns for the protective put strategy, we can generate the Index's probability distribution. Table 2 deonstrates this transforation. Colun 1 of Table 2 shows possible prices for the S&P 500 Index at the end of the one-year horizon. Colun 2 shows the corresponding probability of falling below the prices in Colun 1. Colun 3 shows the values of a protective put strategy contingent on the S&P prices in Colun 1. These values are based on an initial value of $460.00 for the protective put strategy a $450.00 investent in the S&P Index and a $10.00 investent in the put option. The protective put strategy returns shown in Colun 4 are derived by dividing the contingent values of the protective put strategy by 460 and subtracting 1. Colun 5 shows the frequency distribution of the protective put strategy returns. For exaple, there is a 16.79 percent chance that the strategy will return precisely percent because this return would obtain for any S&P value equal to or less than $.00. Also, because the probability of achieving a return between percent and 4.35 percent equals 3.97 percent, the chance of experiencing a return below 4.35 percent is 20.76 percent (16.79 + 3.97). Figure 4 plots these probabilities as a function of the protective put strategy returns. It reveals that overlaying a protective put option on an asset with a syetric distribution truncates the left tail 12 Financial Analysts Journal / Septeber-October 1994
Table 2. Probability Distribution of Put strategy (1) S&P Price (2) Probability of Falling Below S&P Price (3) Value of PP Strategy (4) PP Strategy Return (5) Probability between Successive 350 360 370 380 390 400 410 420 440 450 460 470 480 490 500 510 n 3t 530 5^ 550 560 570 580 590 600 610 620 630 640 1.59% 2.^ 3.20 4.42 5.99 7.97 10.40 13.32 16.79.76 25.25 30.20 35.54 41.2J 47.05 MM us 7i.n 79.2^ 83.21 86.^ 89.60 92.t0 94.01 95.58 96.80 97.72 98.41 4^ i 440 o 540 ^ 560 5713 ^0 590 600 610 620 640 % -6,52-6.sa -435-2.17 GM i 1^ w i$im 17.39 19.57 21.74 23.91 26.09 28.26 30.43 32.61 34.78 36.96 39.13 0.00% O.OQ O.QO 0.00 O.t» 0.00 o. 0.00 163 3.97 4.49 SM ^M $S0 $M 5.6^ 5.34 4.95 4.49 3.97 3.47 2.92 2.43 IM 1-57 IM 0.92 0.69 of the distribution and thereby iparts positive skewness. Most option strategies result in a skewed return distribution. Writing covered calls, for exaple, prevents the writer of the options fro participating in profits generated fro increases above the exercise price. Thus, the right side of the distribution is truncated at the exercise price, which results in negative skewness. Many assets include ebedded options. For exaple, convertible bonds give the owner a call option on the stock of the fir. Siilarly, callable bonds grant the issuer of the bonds a call option on its debt. Clearly, these assets have skewed returns. Diversified portfolios that include assets with ebedded options ay also exhibit significant skewness. Skewness also arises fro dynaic trading strategies such as portfolio insurance. This strategy, which reduces exposure to a risky asset as the asset's price falls and increases exposure to a risky asset as it rises, produces a positively skewed return distribution. Kurtosis The fourth central oent is called kurtosis, which refers to the peakedness of a distribution. It equals the average of the deviations fro the ean raised to the fourth power and is easured as a ratio of this quantity to the standard deviation raised to the fourth power. In equation for. where (=1 K = easure of kurtosis n = nuber of returns R, = ith return ji = arithetic ean of returns (T = standard deviation of returns (2) Financial Analysts Journal / Septeber-October 1994 13
Rgure 4. Probabiifty Distribution of Protective Put Strategy 0.20 0.15 0.10 0.05-0.0652 0.0652 0.1304 1.957 Strategy Return 0.2609 0.3261 0.3913 A noral distribution has a kurtosis value equal to 3. A distribution that has wide tails and a tall narrow peak is called leptokurtic; its kurtosis will exceed 3. Copared with a noral distribu- tion, a larger fraction of the returns are at the extrees rather than slightly above or below the ean of the distribution. This distribution is shown in Figure 5. Figure 5. ijeptoicurtic Distribution 14 Financial Analysts Journal / Septeber-October 1994
A distribution that has thin tails and a relatively flat iddle is called platykurtic. Its kurtosis will be less than 3. Relative to a noral distribution, a larger fraction of the returns are clustered around the ean, as shown in Figure 6. Return series that are characterized by jups as opposed to ore continuous changes will tend to be leptokurtic. Consider an exaple in which a country's econoic policyakers decide to anage their exchange rate relative to the currency of an iportant trading partner. Specifically, they control it such that it will not increase or decrease ore than 1 percent in any quarter. Suppose, however, that a black arket in the currency eerges aong traders; hence, the policyakers can observe the true arket prices. Although the policyakers believe they can control the shortter volatility of the currency, they recognize that periodically they ust reset the currency's value to the arket exchange rate. Therefore, they revalue the currency every two years to accord with the black arket exchange rate. Although this exaple ay see conveniently contrived, it is not qualitatively different fro any actual exchange rate systes and other regulatory echaniss such as circuit breakers on coodities and securities exchanges. Table 3 copares the black arket returns with the regulated returns. The arket exchange rates are based on rando quarterly returns drawn fro a noral distribution with a ean of zero percent and a standard deviation of 5 percent. The regulated exchange rates are based on returns that equal the iniu of the positive arket return {1 percent) or the axiu of the negative arket return (-1 percent), except for every eighth quarter. The eighth-quarter returns are derived by resetting the regulated exchange rate to the black arket exchange rate. As this experient reveals, the regulatory process induces a high level of kurtosis to the returns series: 21.52 versus 3.25 for the arket returns. (Reeber that a noral distribution has a kurtosis easure of 3.) The intuition behind this result is that the regulatory process dapens oderately deviant returns, forcing the closer to the ean than they would have been otherwise; at the end of every second year, however, the revaluation process produces highly deviant returns, thereby fattening the tails of the distribution. Daily returns also tend to be leptokurtic. Table 4 shows the kurtosis of daily and onthly currency returns for the 20-year period beginning January 1, 1974, and ending Deceber 31, 1993. Rgure 6. Piatykurtic Distribution Financial Analysts Journal / Septeber-October 1994 15
Table 3. Jup Process Introduced by Exchange Rate Manageent Period Market Market Exchange Rates Regulated Regulated Exchange Rates 1 % 3 4 5 6 7 8 9 10 11 12 13 14 u17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 0.49% 2.04 2.26 4.23-3.95 1.76-5.11-3.38-3.67-0.07 1.25-3.12 2.79-3.71 4.88-2.81-1.74 3.37 5.80 _ 0.17-1.80 6.58-1.83 8.28 1.92-4.54-9.33 1,13 0.87 12.46 "4.76-4.38-7.57-5.68-0.57-0.92 3.80-1.12 5.72-0.14 49 1.0254 1.0485 1.0928 1.0497 1.0682 1.0135 0.9793 0,9434 0.9427 0.9545 0.9246 0.9504 0.9151 0.9597 0.9327 0.9165 0.9474 23 40 0.9859 1.0507 1.0315 1.1170 1.1384 1.0868 0.9854 0.9965 52 1.1304 1.0766 1.0294 0.9515 0.8974 0.8923 0.8841 0.9177 0.9074 0.9593 0.9580 0.49% - - -4.45 - -0.07 - - -1.85-0.17 - - 19.58 - - 0.87 - -8.61 - - -0.57-0.92 - -4.56 49 1.0149 1.0251 1.0353 1.0250 1.0352 1.0249 0.9793 0.9695 0.9687 0.9784 0.9246 0.9504 0.9409 0.9503 0.9327 0.9234 0.9327 0.9420 0.9436 0.9342 0.9435 0.9341 1.1170 1.1282 1.1169 1.1057 1.1168 1.1265 1.1378 1.1264 1.0294 1.0191 89 31 0.9939 39 0.9938 38 0.9580 Kurtosis 3.25 21.52 Although onthly currency returns are only slightly leptokurtic, daily currency returns are significantly so. This epirical tendency shows up in Table 4. Currency British pound Geran ark French franc Swiss franc Japanese yen Kurtosis of Monthly and Daily Currency, January 1,1974-Oeceber 31, 1993 Monthly 4.20 3.23 3.67 3.40 3.46 Daily 7.03 23.80 16.43 31.82 106.91 other returns as well, including stock returns. This result ay arise fro price jups that occur in response to the accuulated inforation that is released during nontrading hours, especially over weekends. As the easureent interval increases, these price jups cancel out, which explains why onthly returns typically are less leptokurtic than daily returns. Although the assuption that asset returns are norally distributed is convenient, in any situations, it is inappropriate. I have shown that independent returns copounded over long horizons are lognorally distributed. Moreover, op- 16 Financial Analysts Journal / Septeber-October 1994
tion strategies and dynaic trading strategies result in skewed distributions. Finally, conditions that produce price jups typically lead to leptokurtic return distributions. The benefit of convenience ay not always outweigh the cost of iprecision. FOOTNOTES 1. For a ore detailed discussion of this notion, see M. Kritzan, "What Practitioners Need to Know About Uncertainty," Fincia! Analysts journal (March/April 1991):17-21. 2. We should also expect annual returns or returns of any periodicity to be lognorally distributed. Because short horizon returns are relatively sall, however, they do not differ significantly fro the logariths of their wealth relatives. Thus, their true lognoral distribution is well approxiated by a noral distribution. 3. For a ore in-depth review of this topic, see M. Kritzan, "What Practitioners Need to Know About Future Value," Fiticial Alysts Journal (May/June 1994):12-15. Financial Analysts Jouai / Septeber-October 1994 17