A measure of asymmetry called skew uses the ratio of the average cubed deviations from the average, called the third moment, to the cubed standard deviation to measure any asymmetry or
“skewness” of a distribution. Skew is estimated as Skew5Average B(R2R)3
^3 R (5.19) Cubing deviations maintains their sign (the cube of a negative number is negative). Thus, if the distribution is “skewed to the right,” as is the dark curve in Figure 5.5A , the extreme positive values, when cubed, will dominate the third moment, resulting in a positive measure of skew. If the distribution is “skewed to the left,” the cubed extreme negative values will dominate, and the skew will be negative.
When the distribution is positively skewed (the skew is greater than zero), the standard devia-
tion overestimates risk, because extreme positive deviations from expectation (which are not a source of concern to the investor) nevertheless increase the estimate of volatility. Conversely, and more important, when the distribution is negatively skewed, the SD will underestimate risk.
Another potentially important deviation from nor- mality concerns the likelihood of extreme values on either side of the mean at the expense of a smaller fraction of moderate deviations. Graphically speaking, when the tails of a distribution are “fat,” there is more probability mass in the tails of the distribution than predicted by the normal distribution, at the expense of “slender shoulders,” that is, less probability mass near the center of the distribution. Figure 5.5B super- imposes a “fat-tailed” distribution on a normal with the same mean and SD. Although symmetry is still preserved, the SD will underestimate the likelihood of extreme events: large losses as well as large gains.
Kurtosis is a measure of the degree of fat tails. In this case, we use deviations from the average raised to the fourth power and standardize by dividing by the fourth power of the SD, that is,
Kurtosis5Average B(R2R)4
^4 R23 (5.20) We subtract 3 from the ratio in Equation 5.20 , because expected Kurtosis for a normal distribution would be 3. Thus, the kurtosis of a normal distribution is defined as zero, and any kurtosis above zero is a sign of fatter tails than would be observed in a normal dis-
tribution. The kurtosis of the distribution in Figure 5.5B , which has visible fat tails, is .35.
Higher frequency of extreme negative returns may result from negative skew and/or kur- tosis (fat tails)—an important point. Therefore, we would like a risk measure that indicates
Figure 5.5A Normal and skewed distributions (mean ⴝ 6%, SD ⴝ 17%)
.00 Rate of Return
.20 .40 .60
−.60
Probability
−.40 −.20 .030 .025 .020 .015 .010 .005 .000 Skew = −.75
Skew = .75 Negatively skewed Normal
Positively skewed
Figure 5.5B Normal and fat-tailed distributions (mean .1, SD .2)
−.4 −.2
−.6 0 .2 .4 .6
Rate of Return Kurtosis
=.35
Probability Density
.8 Normal Fat-tailed
.00 .10 .20 .30 .50 .40 .60
CONCEPT CHECK
7
Estimate the skew and kurtosis of the five rates in Spreadsheet 5.2 .
vulnerability to extreme negative returns. We discuss three such measures that are most fre- quently used in practice: value at risk, expected shortfall, and lower partial standard deviation.
Value at Risk
The value at risk (denoted VaR to distinguish it from Var, the abbreviation for variance) is the measure of loss most frequently associated with extreme negative returns. VaR is actu- ally written into regulation of banks and closely watched by risk managers. It is another name for quantile of a distribution. The quantile, q, of a distribution is the value below which lies q % of the possible values of that distribution. Thus the median is the q 50th quantile. Practitioners commonly estimate the 5% VaR, meaning that 95% of returns will exceed the VaR, and 5% will be worse. Therefore, VaR may be viewed as the best rate of return out of the 5% worst-case future scenarios.
When portfolio returns are normally distributed, the VaR may be directly derived from the mean and SD of the distribution. Recalling that 1.65 is the 5th percentile of the stan- dard normal distribution (with mean 0 and SD 1), the VaR for the normal distribution is
VaR(.05, normal distribution)5Mean1(21.65)SD
To obtain a sample estimate of VaR, we sort the observations from high to low. The VaR is the return at the 5th percentile of the sample distribution. Almost always, 5% of the num- ber of observations will not be an integer, and so we must interpolate. Suppose the sample is comprised of 84 annual returns (1926–2009), so that 5% of the number of observa- tions is 4.2. We must interpolate between the fourth and fifth observation from the bottom.
Suppose the bottom five returns are
225.03% 225.69% 233.49% 241.03% 245.64%
The VaR is therefore between 25.03% and 25.69% and would be calculated as VaR5.2(225.03)1.8(225.69)5 225.56%
Expected Shortfall
When we assess tail risk by looking at the 5% worst-case scenarios, the VaR is the most optimistic as it takes the highest return (smallest loss) of all these cases. A more realistic view of downside exposure would focus instead on the expected loss given that we find ourselves in one of the worst-case scenarios. This value, unfortunately, has two names:
either expected shortfall (ES) or conditional tail expectation (CTE); the latter terminol- ogy emphasizes that this expectation is conditioned on being in the left tail of the distribu- tion. We will use the expected shortfall terminology.
Extending the previous VaR example, we assume equal probabilities for all values.
Hence, we need to average across the bottom 5% of the observations. To interpolate as before, we assign the average of the bottom four returns weights of 4/4.2 each, and the fifth value from the bottom a weight of .2/4.2, resulting in a value of ES 35.94%, signifi- cantly worse than the 25.56% VaR. 10
10 A formula for the ES in the case of normally distributed returns is given in Jonathan Treussard, “The Non- monotonicity of Value-at-Risk and the Validity of Risk Measures over Different Horizons,” IFCAI Journal of Financial Risk Management, March 2007. The formula is
ES5 1
.05 exp( )N322F(.95)4
where is the mean of the continuously compounded returns, is the SD, N(•) is the cumulative standard normal, and F is its inverse. In the sample above, and were estimated as 5.47% and 19.54%. Assuming nor- mality, we would have ES 30.57%, suggesting that this distribution has a larger left tail than the normal. It should be noted, however, that estimates of VaR and ES from historical samples, while unbiased, are subject to
Lower Partial Standard Deviation and the Sortino Ratio
The use of standard deviation as a measure of risk when the return distribution is nonnormal presents three problems: (1) the asymmetry of the distribution suggests we should look at negative outcomes separately; (2) because an alternative to a risky portfolio is a risk-free investment vehicle, we should look at deviations of returns from the risk-free rate rather than from the sample average; and (3) fat tails should be accounted for.
A risk measure that answers the first two of these issues is the lower partial stan- dard deviation (LPSD), which is computed like the usual standard deviation, but using only “bad” returns. Specifically, it uses only negative deviations from the risk-free rate (rather than negative deviations from the sample average), squares those deviations to obtain an analog to variance, and then takes the square root to obtain a “left-tail standard deviation.” The LPSD is therefore the square root of the average squared deviation, con- ditional on a negative excess return. Notice that this measure ignores the frequency of negative excess returns, that is, portfolios with the same average squared negative excess returns will yield the same LPSD regardless of the relative frequency of negative excess returns.
Practitioners who replace standard deviation with this LPSD typically also replace the Sharpe ratio (the ratio of average excess return to standard deviation) with the ratio of average excess returns to LPSD. This variant on the Sharpe ratio is called the Sortino ratio.
In the analysis of the history of some popular investment vehicles in the next section we will show why practitioners need this plethora of statistics and performance measures to analyze risky investments. The nearby box discusses the growing popularity of these measures, and particularly the new focus on fat tails and extreme events.