Investments, tenth edition

  Checks on normality  

  

  

  

  

  

 Lower partial SD (LPSD) 

 23.02  

26.02  

19.46  

27.40  

28.05 

 Skew 

  2  0.74 

  2  0.59 

  2  0.34 

  2  0.32 

 0.10 

 Kutosis  

1.21  

2.60  

1.43  

1.62  

1.72 

 VaR 5%, actual 

  2  8.86 

  2 10.68  

 2  8.11 

  2 9.44  

 2 13.77 

     normal 

  2  7.85 

  2  8.97 

  2  8.45 

  2  8.77 

  2 13.40 

 ES 5%, actual 

  2 11.09  

 2 14.02  

 2 11.74  

 2 12.36  

 2 17.46 

     

normal 

  2  9.73 

  2 11.22  

 2 10.69  

 2 11.14  

 2 16.47 

  Performance  

  

  

  

  

  

 Sharpe ratio (annualized) 

 0.09  

0.37  

0.69  

0.62  

0.16 

 Sortino ratio (annualized) 

 0.08  

0.34  

0.75  

0.65  

0.17 

 Table 5.4 

 Annualized statistics from the history of monthly excess returns on common stocks, July 1926–September 2012 

continued

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146

Statistic

All U.S.

a

Big/Value

b

Big/Growth

c

Small/Value

d

Small/Growth

e

The 20th Century, Second Half: January 1950–December 1999 (600 months)

Average excess return

8.44

11.50

9.83

17.05

7.20

Standard deviation

14.99

17.21

16.51

21.41

25.60

Checks on normality

Lower partial SD (LPSD)

15.87

16.39

16.69

20.14

26.40

Skew

2 

0.81

2

 0.15

2

 0.70

2

 0.22

2 

0.77

Kutosis

3.50

2.28

3.76

5.09

4.23

VaR 5%, actual

2 

6.02

2

 6.67

2 

6.94

2

 6.86

2 

9.51

    normal

2

 6.08

2

 6.48

2

 7.13

2

 7.33

2 

9.85

ES 5%, actual

2

 9.06

2

 8.98

2

10.07

2

10.36

2

14.30

   normal

2

 7.70

2 

8.24

2

 9.01

2

 9.37

2

12.25

Performance

Sharpe ratio (annualized)

0.56

0.67

0.60

0.80

0.28

Sortino ratio (annualized)

0.53

0.70

0.59

0.85

0.27

The 20th Century, Second Quarter: July 1926–December 1949 (282 months)

Average excess return

8.64

16.02

11.49

50.48

12.81

Standard deviation

28.72

46.59

27.61

63.74

45.08

Checks on normality

Lower partial SD (LPSD)

29.92

40.28

28.43

44.04

37.54

Skew

2 

0.30

0.40

2 

0.50

0.96

0.61

Kutosis

4.60

4.88

4.41

6.25

5.36

VaR 5%, actual

2

12.55

2

17.54

2

11.68

2

16.73

2

15.70

    normal

2

11.39

2

17.46

2

11.60

2

16.34

2

15.88

ES 5%, actual

2

17.36

2

24.16

2

18.22

2

22.61

2

21.22

   normal

2

14.14

2

21.41

2

14.43

2

20.59

2

19.52

Performance

Sharpe ratio (annualized)

0.30

0.34

0.42

0.79

0.28

Sortino ratio (annualized)

0.29

0.40

0.40

1.15

0.34

Table 5.4

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concluded

Annualized statistics from the history of monthly excess returns on common stocks, July 1926–September 2012

Notes: 

a

 Stocks trading on NYSE, AMEX, and NASDAQ, value weighted

b

 Firms in the top 1/2 by market capitalization of equity and top 1/3 by ratio of book equity/market equity (B/M), equally weighted

c

 Firms in the top 1/2 by capitalization and bottom 1/3 by B/M ratio, equally weighted

d

 Firms in the bottom 1/2 by capitalization and top 1/3 by B/M ratio, equally weighted

e

 Firms in the bottom 1/2 by capitalization and bottom 1/3 by B/M ratio, equally weighted

f

 Calculated from monthly, continuously compounded rates

Source: Author’s calculations, using data from Professor Kenneth French’s Web site, http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

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  C H A P T E R  

5

  Risk, Return, and the Historical Record 

147

  

  All U.S.  

  Big/

Value  

  Big/

Growth  

  Small/

Value  

  Small/

Growth  

  Average of Four 

Comparison Portfolios  

 All years 

 20.46  

29.25  

20.79  

41.41  

32.80  

31.06 

 21st century 

 20.08  

24.08  

20.93  

28.93  

29.49  

25.86 

 20th cent. 2 

nd

  half 

 14.99  

17.21  

16.51  

21.41  

25.60  

20.18 

 20th cent. 2 

nd

  quarter 

 28.72  

46.59  

27.61  

63.74  

45.08  

45.76 

 Table 5.4B 

 Standard deviations over time 

  

  All U.S.  

  Big/

Value  

  Big/

Growth  

  Small/

Value  

  Small/

Growth  

  Average of Four 

Comparison Portfolios  

 All years 

 0.37  

0.42  

0.53  

0.63  

0.26  

0.46 

 21st century 

 0.09  

0.37  

0.69  

0.62  

0.16  

0.46 

 20th cent. 2 

nd

  half 

 0.56  

0.67  

0.60  

0.80  

0.28  

0.59 

 20th cent. 2 

nd

  quarter 

 0.30  

0.34  

0.42  

0.79  

0.28  

0.46 

 Table 5.4C 

 Sharpe ratios over time 

premium. Compared to subperiod averages, the 21st century so far has been particularly 

hard on very large firms, as we see from the value-weighted All U.S. portfolio. Not sur-

prisingly, the second half of the 20th century, politically and economically the most stable 

subperiod, offered the highest average returns, particularly for the equally-weighted port-

folios.   Table 5.4A ,  which  reports  a  subset  of   Table 5.4 ,  shows  these  average  returns.        

 As we would expect, the second quarter of the 20th century, dominated by the Great 

Depression and legendary for upheaval in stock values, exhibits the highest standard devia-

tions  ( Table 5.4B ).   

 All portfolios attained their highest Sharpe ratios over the second half of the 20th 

century ( Table 5.4C ). The 21st century has witnessed the lowest performance from the large 

cap-weighted All U.S. portfolio and a middling performance from the equally weighted 

portfolios. More surprising is the fact that average returns were not particularly low over 

the second quarter of the 20th century, despite the deep setbacks of the Depression period. 

  

  All U.S.  

  Big/

Value  

  Big/

Growth  

  Small/

Value  

  Small/

Growth  

  Average of Four 

Comparison Portfolios  

 All years 

 7.52  

12.34  

10.98  

26.28  

8.38  

14.49 

 21st century 

 1.82  

8.80  

14.51  

17.89  

4.83  

11.51 

 20th cent. 2 

nd

  half 

 8.64  

16.02  

11.49  

50.48  

12.81  

22.70 

 20th cent. 2 

nd

  quarter 

 8.44  

11.50  

9.83  

17.05  

7.20  

11.40 

 Table 5.4A 

 Average excess returns over time 

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148 

P A R T   I I

  Portfolio Theory and Practice

However, given the considerable imprecision of these estimates (standard errors of around 

0.20 or 20%), we cannot be sure Sharpe ratios are all that different either across subperiods 

or  across  portfolios.    

  Portfolio Returns 

 The major objective is to compare the five equity portfolios. We started with the prem-

ise that the value-weighted All U.S. portfolio is a natural choice for passive investors. 

We chose the other four portfolios because empirical evidence suggests that size (big vs. 

small) and B/M ratios (value vs. growth) are important drivers of performance. 

 The average returns in  Table 5.4A  show that the Small/Value portfolio did in fact offer 

a higher average return in all periods, and the differences from the averages of the other 

portfolios are all statistically significant.  

19

   In addition, the average of the returns on the 

equally weighted comparison portfolios (the right-most column in  Table 5.4A ) was signif-

icantly higher than that of the All U.S. portfolio. But before deeming these performances 

superior or inferior, it must be shown that the differences in their average returns cannot 

be explained by differences in risk. Here, we must question the use of standard deviation 

as a measure of risk for any particular asset or portfolio. Standard deviation measures 

overall volatility and hence is a legitimate risk measure only for portfolios that are consid-

ered appropriate for an investor’s entire wealth-at-risk, that is, for broad capital allocation. 

Assets or portfolios that are considered to be  added  to the rest of an investor’s entire-

wealth portfolio must be judged on the basis of  incremental  risk. This distinction requires 

risk measures other than standard deviation, and we will return to this issue in great detail 

in later chapters.

 

  Table  5.4B  shows the large standard deviation involved in these broad-based stock 

investments. Annual SD ranges from 15% to as much as 63%. Even using the smallest SD 

suggests that losing 15% of portfolio value in one year would not be so unusual. Apparently, 

size is correlated with volatility, as suggested by the higher SD of the two small portfolios, 

and the lowest volatility of the large-cap All U.S. portfolio. While it appears that value 

portfolios generally are more volatile than growth portfolios, the difference is not suffi-

cient to make us confident of this assertion. 

 Regardless of how we resolve the question of performance of these portfolios, we 

must determine whether SD is an adequate measure of risk in the first place, in view of 

deviations from normality.  Table 5.4  shows that negative skew is present in some of the 

portfolios some of the time, and positive kurtosis is present in  all   portfolios   all   the  time. 

This implies that we must carefully evaluate the effect of these deviations on value-at-risk 

(VaR), expected shortfall (ES), and negative 3-sigma frequencies. Finally, since  Figure 5.6  

separates the distributions of monthly excess returns to those within a range of  6 10%  and 

those outside that range, we can quantify the implication of extreme returns. 

 We start with the difference between VaR from the actual distribution of returns and the 

equivalent normal distribution (with the same mean and variance). Recall that the 5% VaR 

is the loss corresponding to the 5th percentile of the rate of return distribution. It is one 

measure of the risk of extreme outcomes, often called  tail risk  because it focuses on out-

comes in the far left tail of the distribution. We compare historical tail risk to that predicted 

by the normal distribution by comparing actual VaR to the VaR of the equivalent normal 

distribution. The excess VaR is the VaR of the historical distribution versus the VaR of the 

corresponding normal, where negative numbers indicate greater losses. 

  Table 5.4D  shows that for the overall period, VaR indicates no greater tail risk than is 

characteristic of the equivalent normal. The worst excess VaR compared to the normal 

19

The t-statistic of the difference in average return is: Average difference/SD(Difference).

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