Investments, tenth edition

  

  All U.S.  

  Big/

Value  

  Big/

Growth  

  Small/

Value  

  Small/

Growth  

 Number of negative jumps 

 7.7  

4.8  

9.7  

2.9  

3.9 

 Expected jumps for equivalent normal 

 0.9  

0.9  

0.8  

0.6  

1.0 

 Difference  5  excess jumps 

 6.8  

4.0  

8.9  

2.3  

2.8 

 Average years between excess jumps 

 12.24  

21.06  

9.42  

36.23  

29.37 

 Expected excess returns between extra 

jumps (in units of SD) 

 16.90  

32.91  

17.16  

104.23  

28.98 

 Table 5.4F 

 Incidence of negative 3-sigma returns 

  

  All U.S.  

  Big/

Value  

  Big/

Growth  

  Small/

Value  

  Small/

Growth  

  Average of Four 

Comparison Portfolios  

 All years 

 0.12  

0.93  

0.85  

2.81  

0.72  

1.33 

 21st century 

  2 1.02  

 2 1.71 

 0.34 

  2 0.67  

 2 0.37  

 2 0.60 

 20th cent. 2 

nd

  quarter 

 0.06 

  2 0.19 

 0.19  

0.47  

0.34  

0.20 

 20th cent. 2 

nd

  half 

  2 1.16  

 2 0.09  

 2 0.08  

 2 0.39 

 0.18 

  2 0.09 

 Table 5.4D 

 VaR (excess over normal distribution) expressed as a fraction of monthly standard deviation 

  

  All U.S.  

  Big/

Value  

  Big/

Growth  

  Small/

Value  

  Small/

Growth  

  Average of Four 

Comparison Portfolios  

 All years 

  2 0.41  

 2 0.33  

 2 0.34  

 2 0.08  

 2 0.25  

 2 0.25 

 21st century 

  2 0.28  

 2 0.47  

 2 0.17  

 2 0.19  

 2 0.11  

 2 0.24 

 20th cent. 2 

nd

  quarter 

  2 0.18  

 2 0.06  

 2 0.13  

 2 0.07  

 2 0.18  

 2 0.11 

 20th cent. 2 

nd

  half 

  2 0.77  

 2 0.59  

 2 0.77  

 2 0.37  

 2 0.26  

 2 0.50 

 Table 5.4E 

 Expected shortfall (excess over normal distribution) expressed as a fraction of monthly standard deviation 

( 2 1.71% for the Big/Value portfolio in the 21st century) is less than a third of the monthly 

SD of this portfolio, 6.01%. Hence, VaR figures indicate that the normal is a decent 

approximation to the actual return distribution.   

 However, other measures indicate that tail risk may be somewhat greater than in the 

normal distribution. The expected shortfall (ES) figures in  Table  5.4  are more negative 

for the actual than for the equivalent normal excess returns (consistent with the fat tails 

indicated by the positive kurtosis). To assess the economic significance of the differ-

ences from normal, we present them in  Table  5.4E  as fractions of the monthly SDs of 

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150

P A R T   I I

  Portfolio Theory and Practice

the various portfolios. The negative signs tell us that while the most negative 5% of the 

actual observations are always worse than the equivalent normal, the differences are not 

substantial: The magnitudes are never larger than 0.77 of the portfolio SD. Measured over 

the entire period, the excess shortfall does not exceed 0.41 of the monthly standard devia-

tion. Here, again, we don’t see evidence that seriously undermines the adequacy of the 

normality assumption.   

  Table 5.4F  shows the actual number of negative monthly returns or “jumps” of magni-

tude greater than 3 standard deviations, compared with the expected number correspond-

ing to the equivalent normal distributions. The actual numbers range from 2.9 to 9.7 per 

1,000 months, compared with only 0.6 to 1.0 for equivalent normal distributions. What are 

we to make of this? Negative 3-sigma returns are very bad surprises indeed. To help inter-

pret these differences, we compute the expected length of time (number of years) between 

“extra jumps,” i.e., jumps beyond the expected number based on the normal distribution. 

We also calculate the expected total return over this period, also in units of standard devia-

tion of the actual distribution. 

  Table  5.4F  shows the results of these calculations. We see that one excess jump is 

observed every 9 to 36 years, and that over such periods, the portfolios are expected to 

yield excess returns of 16 to 104 standard deviations compared with the loss of 3 SD or 

more due to these jumps. Thus, jump risk does not appear large enough to affect the risk 

and return of long-term stock returns.   

 Finally, we interpret the  size  of the jumps outside the range of  6 10% that appear so 

ominous in  Figure 5.6 . To quantify this risk, we ask: “When we look at all excess returns 

below  2 10% in our history of 1,035 months, what is the SD of all these (extremely bad) 

returns?” And a follow-up question: “What would be the tail SD of a normal return with 

the same mean and overall SD as our sample, conditional on return falling below  2 10%?” 

 Table  5.4G  answers these two questions. It is evident that the actual history suggests a 

larger SD than a normal distribution would imply, consistent with  Figure 5.6 . The differ-

ence can be as large as 43% of the SD of the equivalent normal in the extreme negative 

range. Of all the statistics we have examined so far, this is the most damning for a straight-

forward approximation of actual distributions by the normal.   

 We can conclude from all this that a simple normal distribution is generally not a bad 

approximation of portfolio returns, despite the fact that in some circumstances it may 

understate investment risk. However, we can make up for this pitfall by more careful esti-

mation of the SD of extreme returns. Nevertheless, we should be cautious about application 

of theories and inferences that require normality of returns. In general, one should verify 

that standard deviations assumed for assets or portfolios adequately represent tail risk. 

 In the next chapters we will return to these portfolios and ask whether the All U.S. 

portfolio is the most efficient in terms of its risk-return trade-off. We will also con-

sider adjustments in view of the performance of the size-B/M portfolios as well as other