Investments, tenth edition

 Example  5.10 

Normal Distribution Function in Excel 

 What is the probability that the return on the index in Example 5.10 will be below  2 15%? 

 CONCEPT CHECK 

5.6 

 

12

 In 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  3  10 

 

2 15

 . The issue of negative birth weight clearly isn’t a  practical   concern.

  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 performance. 

Unfortunately, deviations from normality of asset returns are quite significant and difficult 

to ignore. 

 Deviations from normality may be discerned by calculating the higher moments of 

return distributions. The  n th central moment of a distribution of excess returns,  R,  is esti-

mated as the average value of    (R 2 R)

n

.  The first moment ( n     5  1) is necessarily zero 

    5.7 

Deviations from Normality and Risk Measures 

bod61671_ch05_117-167.indd   137

bod61671_ch05_117-167.indd   137

6/18/13   8:03 PM

6/18/13   8:03 PM

Final PDF to printer

---

138

P A R T   I I

  Portfolio Theory and Practice

(the average deviation from the sample average 

must be zero). The second moment ( n   5  2)  is 

the estimate of the variance of returns,    s

^

2

.   

13

    

 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 asymme-

try or “skewness” of a distribution.   

 Skew 5 Average 

B

(R 2 R)

3

s

^

3

R 

 (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  5.5A , the extreme posi-

tive 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 the 

skew will be negative.  

 When the distribution is positively skewed (skewed 

to the right), the standard deviation overestimates 

risk, because extreme positive surprises (which do not 

concern investors) 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, kurtosis, concerns the likelihood of extreme 

values on either side of the mean at the expense 

of a smaller likelihood of moderate deviations. 

Graphically speaking, when the tails of a distribu-

tion 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  superimposes 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.  

13

 For distributions that are symmetric about the average, as is the case for the normal distribution, all odd 

moments ( n   5  1, 3, 5, . . .) have expectations of zero. For the normal distribution, the expectations of all higher 

even moments ( n   5  4, 6, . . .) are functions  only  of the standard deviation,  s . For example, the expected fourth 

moment ( n     5  4) is 3 s  

4

 , and for  n     5  6, it is 15 s  

6

 . Thus, for normally distributed returns the standard devia-

tion,  s , provides a complete measure of risk, and portfolio performance may be measured by the Sharpe ratio, 

   R/s.  For other distributions, however, asymmetry may be measured by higher nonzero odd moments. Higher 

even moments (in excess of those consistent with the normal distribution), combined with large, negative odd 

moments, indicate higher probabilities of extreme negative outcomes.