Investments, tenth edition

 Figure 5.5A 

Normal and skewed distributions 

(mean  5  6%, SD  5  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  5  .1, SD  5  .2)    

−.4

−.2

−.6

.2

.4

.6

0

Rate of Return

Kurtosis

=.35

Probability Density

.8

Normal

Fat-Tailed

.00

.10

.20

.30

.50

.40

.60

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  C H A P T E R  

5

  Risk, Return, and the Historical Record 

139

    Kurtosis    measures the degree of fat tails. We use deviations from the average raised to 

the  fourth  power, scaled by the fourth power of the SD,   

 Kurtosis 5 Average 

B

(R 2 R)

4

s

^

4

R 2 3 

 (5.20)   

 We subtract 3 in Equation 5.20, because the ratio for a 

normal distribution is 3. Thus, the kurtosis of a normal 

distribution is defined as zero, and any kurtosis above 

zero is a sign of fatter tails. The kurtosis of the distribu-

tion in  Figure 5.5B , which has visible fat tails, is .35. 

 

In addition to a shift of observations between the 

shoulders and the tails, kurtosis can be affected by a shift 

from the shoulders to the center of the distribution (which decreases kurtosis), or vice 

versa. This element of kurtosis is called  peakedness,  as it affects the height of the peak of 

the distribution at its center. It is not shown in  Figure 5.5B , but we often encounter peaked-

ness in histograms of actual distributions. 

 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.