Investments, tenth edition

  

  All U.S.   

a

   

  Big/Value  

  Big/Growth  

  Small/Value  

  Small/Growth  

  July 1926  

  

  

  

  

  

 Number of firms 

 427  

37  

85  

90  

43 

 Average capitalization ($ mil) 

 57  

39  

108  

5  

8 

 Average B/M ratio  

b

   

 1.02  

2.36  

0.45  

3.6  

0.81 

  January 1950  

  

  

  

  

  

 Number of firms 

 899  

73  

196  

197  

75 

 Average capitalization ($ mil) 

 69  

77  

186  

7  

11 

 Average B/M ratio  

b

   

 1.18  

2.60  

0.50  

2.95  

0.67 

  January 2000  

  

  

  

  

  

 Number of firms 

 5,495  

150  

576  

1,709  

1,158 

 Average capitalization ($ mil) 

 2,545  

3,542  

18,246  

106  

299 

 Average B/M ratio  

b

   

 0.52  

1.38  

0.14  

1.70  

0.22 

  September 2012  

  

  

  

  

  

 Number of firms 

 3,383  

153  

408  

1,065  

672 

 Average capitalization ($ mil) 

 4,470  

13,325  

18,070  

297  

582 

 Average B/M ratio  

b

   

 0.68  

1.32  

0.25  

1.33  

0.26 

 Table 5.3 

 Number of firms, average capitalization, and average B/M ratios of portfolios   

Notes:  

a

  Value weighted, hence dominated by big stocks  

   

b

  B/M ratio are sampled in midyears  

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

17

 These are excess returns of the continuously compounded rates, which are the ones most appropriate to assume 

are normally distributed. We discuss this issue in the next section.  

18

 T-bills made their debut during the 1940s. For earlier dates, commercial paper is used as the closest approxima-

tion to short-term risk-free rates. In a few instances they were issued slightly above par and thus yielded slightly 

negative rates.  

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144

P A R T   I I

  Portfolio Theory and Practice

Figure 5.6 

Frequency distribution of annual rates of return, 1926–2012

Source: Prepared from data in Table 5.3.

0.00

0.02

0.04

0.06

0.08

0.10

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Normal'>Estimated 

Normal

Mean = 0.71

SD = 3.89

Negative Jumps

Average = 

215.44

SD = 17.18

Positive Jumps

Average = 15.23

SD = 16.62

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0

0.01

0.02

0.03

0.04

0.05

0.06

2

25

2

20

2

15

2

9.5

2

4.5 0.5

5.5 10.5 15.5 20.5

2

25

2

20

2

15

2

9.5

2

4.5 0.5

5.5 10.5 15.5 20.5

2

25

2

20

2

15

2

9.5

2

4.5 0.5

5.5 10.5 15.5 20.5

2

25

2

20

2

15

2

9.5

2

4.5 0.5

5.5 10.5 15.5 20.5

2

25

2

20

2

15

2

9.5

2

4.5 0.5

5.5 10.5 15.5 20.5

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0

0.01

0.02

0.03

0.04

0.05

0.06

2

0.05

0.475

0.975

Average = 0.29

SD = 0.25

A: T-Bill Rates

C:  Big/Value

E: Small/Value

F: Small/Growth

D:  Big/Growth

B:  All U.S.

1.475

Actual

Average = 0.48

SD = 5.44

Actual

Average = 0.73

SD = 7.54

Actual

Average = 0.73

SD = 6.02

Actual

Average = 1.81

SD = 8.26

Actual

Average = 0.35

SD = 8.23

Estimated 

Normal

Mean = 0.40

SD = 4.18

Negative Jumps

Average = 

217.04

SD = 19.31

Positive Jumps

Average = 14.24

SD = 14.64

Estimated 

Normal

Mean = 0.47

Stdev = 4.67

Negative Jumps

Average = 

216.93

SD = 18.08

Positive Jumps

Average = 15.62

SD = 17.52

Estimated 

Normal

Mean = 0.92

SD = 4.39

Negative Jumps

Average = 

219.41

SD = 22.98

Positive Jumps

Average = 16.24

SD = 18.32

Estimated

Normal

Mean = 0.85

Stdev = 4.15

Negative Jumps

Average = 

217.39

SD = 19.79

Positive Jumps

Average = 16.39

SD = 19.03

corresponding normal distribution using only the moderate range of excess returns, and 

estimate the distribution of extreme rates separately. Accordingly, the light-colored col-

umns in panels B through F show the expected frequency from a normal distribution with 

mean and SD matched to those of the actual returns in the range of  6 10%.  The  boxes 

to the left of the histograms show the average and SD of the actual distributions, while 

the boxes on the right show the statistics for subgroups of returns: the midrange (returns 

within  6 10% of the mean), the negative jumps (extreme returns less than  2 10%), and the 

positive jumps (returns greater than 10%). The mean and SD of the jump components are 

calculated using differences from the full-sample average. The SD of the jumps indicates 

the contribution of the positive and negative jumps to the variance of the full distribution. 

 The histograms give us a first, quite vivid look at the risk involved in owning common 

stocks. This risk is dominated by the frequency and size of negative jumps. We need more for-

mal analysis to determine whether these deviations from normality are economically decisive. 

  Table 5.4  presents a wide range of statistics for the five stock portfolios for the entire 

86-year period, as well as the three subperiods. With 1,035 monthly observations, aver-

age excess returns are all statistically significantly above zero, verifying a positive risk 

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145

  Statistic  

  All U.S.   

a

   

  Big/Value   

b

   

  Big/Growth   

c

   

  Small/Value   

d

   

  Small/Growth   

e

   

  All 1,035 Months: July 1926–September 2012  

 Average excess return 

 7.52  

12.34  

10.98  

26.28  

8.38 

 Standard deviation 

 20.46  

29.25  

20.79  

41.41  

32.80 

  Checks on normality  

  

  

  

  

  

 Lower partial SD (LPSD) 

 21.67  

26.78  

20.88  

31.57  

30.36 

 Skew  

f

   

  2  0.54 

 0.34 

  2  0.58 

 1.19  

0.17 

 Kurtosis  

f

   

 6.58  

11.40  

5.25  

13.31  

6.19 

 VaR 5% actual  

f

   

  2  8.01 

  2 10.08  

 2  7.92  

 2  8.30  

 2 11.64 

     normal 

 

f

   

  2  8.13 

  2 11.01  

 2  8.76 

  2 11.12  

 2 12.35 

 ES (CTE) 5% actual  

f

   

  2 12.40  

 2 16.35  

 2 13.05  

 2 14.85  

 2 17.39 

       normal 

 

f

   

  2 10.17  

 2 13.74  

 2 11.01  

 2 14.08  

 2 15.27 

 Negative 3-sigma (obs/1000), actual 

 7.7  

4.8  

9.7  

2.9  

3.9 

              normal 

 

0.9  

0.9  

0.8  

0.6  

1.0 

 1-month SD conditional on 10% loss, actual 

 17.18  

19.79  

19.31  

22.98  

18.08 

                                 normal  

12.82  

14.95  

13.47  

16.85  

15.16 

  Performance  

  

  

  

  

  

 Sharpe ratio (annualized) 

 0.37  

0.42  

0.53  

0.63  

0.26 

 Sortino ratio (annualized) 

 0.35  

0.46  

0.53  

0.83  

0.28 

  The 21st Century So Far: January 2000–September 2012 (153 months)  

 Average excess return 

 1.82  

8.80  

14.51  

17.89  

4.83 

 Standard deviation 

 20.08  

24.08  

20.93  

28.93  

29.49