Investors presumably are interested in the expected excess return they can earn by replac- ing T-bills with a risky portfolio, as well as the risk they would thereby incur. Even if the T-bill rate is not constant, we still know with certainty what nominal rate we will earn in any period if we purchase a bill and hold it to maturity. Other investments typically entail accepting some risk in return for the prospect of earning more than the safe T-bill rate.
Investors price risky assets so that the risk premium will be commensurate with the risk of that expected excess return, and hence it’s best to measure risk by the standard deviation of excess, not total, returns.
The importance of the trade-off between reward (the risk premium) and risk (as mea- sured by standard deviation or SD) suggests that we measure the attraction of a portfolio by the ratio of its risk premium to the SD of its excess returns. This reward-to-volatility measure was first used extensively by William Sharpe and hence is commonly known as the Sharpe ratio. It is widely used to evaluate the performance of investment managers.
Sharpe ratio = Risk premium ________________SD of excess return (5.18)
7When returns are uncorrelated, we do not have to worry about covariances among them. Therefore, the variance of the sum of 12 monthly returns (i.e., the variance of the annual return) is the sum of the 12 monthly variances.
If returns are correlated across months, annualizing is more involved and requires adjusting for the structure of serial correlation.
Take another look at Spreadsheet 5.1. The scenario analysis for the proposed invest- ment in the stock-index fund resulted in a risk premium of 5.76% and standard devia- tion of excess returns of 19.49%. This implies a Sharpe ratio of .30, a value roughly in line with the historical performance of stock-index funds. We will see that while the Sharpe ratio is an adequate measure of the risk–return trade-off for diversified portfo- lios (the subject of this chapter), it is inadequate when applied to individual assets such as shares of stock.
Example 5.9
Sharpe RatioUsing the annual returns for years 3–5 in Spreadsheet 5.2, a. Compute the arithmetic average return.
b. Compute the geometric average return.
c. Compute the standard deviation of returns.
d. Compute the Sharpe ratio, assuming the risk-free rate was 6% per year.
Concept Check 5.5
5.6 The Normal Distribution
The bell-shaped normal distribution appears naturally in many applications. For example, the heights and weights of newborns, the lifespans of many consumer items such as light bulbs, and many standardized test scores are well described by the normal distribution.
Variables that are the end result of multiple random influences tend to exhibit a normal distribution, for example, the error of a machine that aims to fill containers with exactly 1 gallon of liquid. By the same logic, because rates of return are affected by a multiplicity of unanticipated factors, they also might be expected to be at least approximately normally distributed.
To see why the normal curve is “normal,” consider a newspaper stand that turns a profit of $100 on a good day and breaks even on a bad day, with equal probabilities of .5. Thus, the mean daily profit is $50 dollars. We can build a tree that compiles all the possible outcomes at the end of any period. Here is an event tree showing outcomes after 2 days:
Two good days, profit = $200
Two bad days, profit = 0
One good and one bad day, profit = $100
Notice that 2 days can produce three different outcomes and, in general, n days can produce n + 1 possible outcomes. The most likely 2-day outcome is “one good and one bad day,” which can happen in two ways (first a good day, or first a bad day). The probability of this outcome is .5. Less likely are the two extreme outcomes (both good days or both bad days) with probability .25 each.
What is the distribution of profits at the end of 200 days? There are 201 possible outcomes and, again, the midrange outcomes are the most likely because there are more sequences that lead to them. While only one sequence can result in 200 consecutive bad days, an enormous number of sequences of good and bad days result in 100 good
days and 100 bad days. Therefore, midrange outcomes are far more likely than are either extremely good or extremely bad outcomes, just as described by the familiar bell-shaped curve.
Figure 5.3 shows a graph of the normal curve with mean of 10% and standard devia- tion of 20%. A smaller SD means that possible outcomes cluster more tightly around the mean, while a higher SD implies more diffuse distributions. The likelihood of real- izing any particular outcome when sampling from a normal distribution is fully deter- mined by the number of standard deviations that separate that outcome from the mean.
Put differently, the normal distribution is completely characterized by two parameters, the mean and SD.
Investment management is far more tractable when rates of return can be well approxi- mated by the normal distribution. First, the normal distribution is symmetric, that is, the probability of any positive deviation above the mean is equal to that of a negative deviation of the same magnitude. Absent symmetry, the standard deviation is an incomplete mea- sure of risk. Second, the normal distribution belongs to a special family of distributions characterized as “stable” because of the following property: When assets with normally distributed returns are mixed to construct a portfolio, the portfolio return also is normally distributed. Third, scenario analysis is greatly simplified when only two parameters (mean and SD) need to be estimated to obtain the probabilities of future scenarios. Fourth, when constructing portfolios of securities, we must account for the statistical dependence of returns across securities. Generally, such dependence is a complex, multilayered relation- ship. But when securities are normally distributed, the statistical relation between returns can be summarized with a single correlation coefficient.
How closely must actual return distributions fit the normal curve to justify its use in investment management? Clearly, the normal curve cannot be a perfect description of real- ity. For example, actual returns cannot be less than −100%, which the normal distribution would not rule out. But this does not mean that the normal curve cannot still be useful.
A similar issue arises in many other contexts. For example, birth weight is typically evalu- ated in comparison to a normal curve of newborn weights, although no baby is born with a negative weight. The normal distribution still is useful here because the SD of the weight
-3σ -2σ -1σ 0 +1σ +2σ +3σ
-50% -30% -10% 30% 50% 70%
68.26%
95.44%
99.74%
10%
Figure 5.3 The normal distribution with mean 10% and standard deviation 20%.
is small relative to its mean, and the predicted likelihood of a negative weight would be too trivial to matter.8 In a similar spirit, we must identify criteria to determine the adequacy of the normality assumption for rates of return.
8In fact, the standard deviation is 511 grams while the mean is 3,958 grams. A negative weight would therefore be 7.74 standard deviations below the mean, and according to the normal distribution would have probability of only 4.97 × 10−15. The issue of negative birth weight clearly isn’t a practical concern.
9In older versions of Excel, this function is NORMDIST(cutoff, mean, standard deviation).
10For distributions that are symmetric about the average, as is the case for the normal distribution, all odd moments (n = 1, 3, 5, . . .) have expectations of zero. For the normal distribution, the expectations of all higher even moments (n = 4, 6, . . .) are functions only of the standard deviation, σ. For example, the expected fourth moment (n = 4) is 3σ4, and for n = 6, it is 15σ6. Thus, for normally distributed returns the standard deviation, σ, provides a complete measure of risk, and portfolio performance may be measured by the Sharpe ratio. For other distributions, however, asymmetry may be measured by the higher odd moments. Larger even moments (in excess of those consistent with the normal distribution), combined with large, negative odd moments, indicate higher probabilities of extreme negative outcomes.
Suppose the monthly rate of return on the S&P 500 is approximately normally distributed with a mean of 1% and standard deviation of 6%. What is the probability that the return on the index in any month will be negative? We can use Excel’s built-in functions to quickly answer this question. The probability of observing an outcome less than some cutoff according to the normal distribution function is given as NORM.DIST(cutoff, mean, standard deviation, TRUE).9 In this case, we want to know the probability of an outcome below zero, when the mean is 1% and the standard deviation is 6%, so we compute NORM.DIST(0, 1, 6, TRUE) = .4338. We could also use Excel’s built-in standard normal function, NORM.S.DIST, which uses a mean of 0 and a standard deviation of 1, and ask for the probability of an outcome 1/6 of a standard deviation below the mean: NORM.S.DIST(−1/6, TRUE) = .4338.
Example 5.10
Normal Distribution Function in ExcelWhat is the probability that the return on the index in Example 5.10 will be below −15%?
Concept Check 5.6
As we noted earlier (but you can’t repeat it too often!), normality of excess returns hugely simplifies portfolio selection. Normality assures us that standard deviation is a complete measure of risk and, hence, the Sharpe ratio is a complete measure of portfolio perfor- mance. Unfortunately, deviations from normality of asset returns are potentially significant and dangerous to ignore.
Statisticians often characterize probability distributions by what they call their
“moments,” and deviations from normality may be discerned by calculating the higher moments of return distributions. The nth central moment of a distribution of excess returns, R, is estimated as the average value of (R − ¯ R ) n . The first moment (n = 1) is necessarily zero (the average deviation from the sample average must be zero). The second moment (n = 2) is the estimate of the variance of returns, σ ^ 2 .10
5.7 Deviations from Normality and Alternative Risk Measures
A measure of asymmetry called skew is the ratio of the average cubed deviations from the sample average, called the third moment, to the cubed standard deviation:
Skew = Average [ (________ R − ¯ R ) 3
σ ^ 3 ] (5.19) Cubing deviations maintains their sign (the cube of a negative number is negative).
When a distribution is “skewed to the right,” as is the dark curve in Figure 5.4A, the extreme positive values, when
cubed, dominate the third moment, resulting in a positive skew. When a distribution is “skewed to the left,” the cubed extreme negative values dominate, and skew will be negative.
When the distribution is posi- tively skewed (skewed to the right), the standard deviation overesti- mates risk, because extreme posi- tive surprises (which do not concern investors) nevertheless increase the estimate of volatility. Conversely, and more important, when the distribu- tion is negatively skewed, the SD will underestimate risk.
Another potentially important deviation from normality, kurtosis, concerns the likelihood of extreme values on either side of the mean at the expense of a smaller likelihood of moderate deviations. Graphi- cally 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. That extra probability is taken at the expense of “slender shoulders,” that is, there is less probability mass near the center of the distribution.
Figure 5.4B superimposes a “fat- tailed” distribution on a normal distribution with the same mean and SD. Although symmetry is still preserved, the SD will under- estimate the likelihood of extreme events: large losses as well as large gains.
Kurtosis measures the degree of fat tails. We use deviations from the
Figure 5.4A Normal and skewed distributions (mean = 6%, SD = 17%)
0 Rate of Return
0.20 0.40 0.60
-0.60
Probability
-0.40 -0.20 0.030
0.025 0.020 0.015 0.010 0.005 0
Skew =-0.75
Skew = 0.75 Negatively Skewed Normal
Positively Skewed
Figure 5.4B Normal and fat-tailed distributions (mean = .1, SD = .2)
-0.4 -0.2
-0.6 0 0.2 0.4 0.6
Rate of Return Kurtosis
=0.35
Probability Density
0.8 Normal Fat-Tailed
0 0.10 0.20 0.30 0.50 0.40 0.60
average raised to the fourth power (the fourth moment of the distribution), scaled by the fourth power of the SD,
Kurtosis = Average [ (________R − ¯ R ) 4
σ ^ 4 ] − 3 (5.20) We subtract 3 in Equation 5.20 because the expected value of the ratio for a normal distri- bution is 3. Thus, this formula for kurtosis uses the normal distribution as a benchmark:
The kurtosis for the normal distribution is, in effect, defined as zero, so kurtosis above zero is a sign of fatter tails. The kurtosis of the distribution in Figure 5.4B, which has visible fat tails, is .35.
Estimate the skew and kurtosis of the five holding period returns in Spreadsheet 5.2.
Concept Check 5.7
Notice that both skew and kurtosis are pure numbers. They do not change when annual- ized from higher frequency observations.
Higher frequency of extreme negative returns may result from negative skew and/or kurtosis (fat tails). Therefore, we would like a risk measure that indicates vulnerability to extreme negative returns. We discuss four such measures that are most frequently used in practice: value at risk, expected shortfall, lower partial standard deviation, and the frequency of extreme (3-sigma) returns.