By Eugene F. Fama and Kenneth R. French
The capweighted market portfolio of NYSEAmexNasdaq stocks delivered a
38.31% return for 2008. The experience was painful, but was it out of bounds? The volatility of returns also increased a lot during 2008. Was the observed volatility consistent with prior experience? These are the questions addressed here.
Returns
Figure 1 shows that the market return of 2008 is unusual  but not unprecedented. (The numbers used to create the figure are in Table 1.) The only year of the 19272008 period that produces a market return below 38.31% is 1931 (44.36%). Three additional years have returns close to or below 30%: 1930 (28.83%), 1937 (34.61%) and 1974 (27.95%). If one considers successive years of negative market returns, the cumulative return for the threeyear period 19291931 is 66.35%, the return for 19731974 is 41.47%, and the return for 20002002 is 37.54%. On the plus side, the annual market return is greater than 30% for 15 individual calendar years of 19272008. In short, extreme stock returns are common. Fortunately, extreme positives outnumber extreme negatives.
Figure 1  Annual Market Returns 19272008
Financial economists often focus on the equity premium  the extra return one gets for investing in stocks, rather than in shortterm Treasury bills. The cumulative return from rolling over one month bills in 2008 is 1.64%, so the equity premium for 2008 is 39.95 = (38.31%  1.64%). How unusual is a premium of 39.95%? For 19272008 (82 years) the average annual difference between the market return and the return from rolling over onemonth bills every month is 7.64% and the standard deviation (a statistical measure of volatility) of the annual differences is 21.04%. (See Table 2.) If we pretend that the annual premiums follow a normal distribution (in fact, they are somewhat fattailed and right skewed), and that the estimates for 19272008 are the true longterm mean and standard deviation of the distribution, the probability of a premium of 39.95 or smaller is 1.18%. (The details of this and other calculations are in the appendix.) In other words, the odds of a premium this extreme in any given year are about one in 85. Thus, the large loss for 2008 is unusual, but not out of bounds.
It is apparent from these calculations that investing in the stock market is risky. On average, stocks substantially beat bills. This is, of course, the attraction of stock market investing. But the yearbyyear equity premium is quite volatile, and there are many years when the premium is negative. (See Figure 2.) For investment purposes, the important implication of this high volatility is that the holding period for stocks must be quite long if one wants to be relatively sure of realizing a positive average equity premium.
Figure 2  Annual Equity Premium (Market Return minus TBill Return) 19272008
To illustrate, note first that the uncertainty about the average premium to be realized during a holding period is captured by the standard deviation of the average premium (statisticians call it the standard error) for the period. The standard deviation of the average equity premium for a holding period is the standard deviation of the yearbyyear premiums for the period divided by the square root of the number of years in the period. This square root rule means that the standard deviation of the average premium goes down, that is, the estimate of the average premium becomes more reliable, as one increases the holding period. This is important: it is the reason the probability of realizing a positive average equity premium during a holding period increases with the length of the period.
For example, suppose we assume future equity premiums will be drawn from a normal distribution with a mean of 7.64% per year and a standard deviation of 21.04%  the estimates for 19272008. The probability that the premium for a single future year is negative is about 36%. In other words, even though the expected annual premium (the mean of the true premium distribution) is high (7.64%), the much higher standard deviation of yearbyyear premiums (21.04%) means that singleyear premiums will be negative about 36% of the time. If one stretches the holding period to four years, the square root rule tells us that the standard deviation of the average premium drops to onehalf the standard deviation of annual premiums, from 21.04% to 10.52%. As a result, for fouryear holding periods, the probability of a negative realized average premium falls to about 23%. In other words, we expect that for fouryear holding periods the average equity premium will be negative (bills beat stocks) about 23% of the time. For 16year holding periods, the probability of observing a negative average premium drops further, to about 7%. And for 25year holding periods, the probability of a negative average premium is about 3.4%. Thus, even for quarter century holding periods, there is a 3.4% chance that bills will beat stocks.
What does all this say? The expected equity premium is compensation for bearing the high risk of equities. The risk manifests itself in highly volatile returns. This volatility means that even for long holding periods, there is some probability that less risky investments like bills beat stocks. The probability is lower for longer holding periods, but it never goes to zero.
There is a similar story for the size premium (the premium in the returns of small stocks relative to big stocks) and the value premium (the premium in the returns of value stocks relative to growth stocks). Like the equity premium, the yeartoyear size and value premiums are quite volatile. This means that even for long holding periods, there is some chance that the average realized size and value premiums will be negative. And long holding periods are necessary to be relatively sure the average premiums during the holding period will be positive.
Volatility
The volatility of stock returns increased substantially during 2008. Is the resulting level of volatility unusual?
Figure 3 shows the yearbyyear standard deviations of the monthly market returns of 19272008. The standard deviation of the monthly returns for 2008, 6.67% per month, is high relative to the average for 19272008, 4.66% per month. The standard deviation for 2008 is especially high relative to the standard deviations for the preceding five years, which are all well below the historical mean and among the lowest of the 19272008 period. In other words, we are struck by the high volatility of returns during 2008 in part because they follow a fiveyear period when volatility was quite low.
Figure 3  YearbyYear Standard Deviations of Monthly Market Returns 19272008
Figure 3 suggests, however, that though the standard deviation of 2008 monthly returns is above the sample mean, it is not unusual. Even if one discards the 1930s, there are many years when the volatility of monthly market returns is close to or above the 2008 value, most recently, 1998, and the threeyear period from 2000 to 2002. What leaps out of Figure 3 is not the volatility of 2008 returns but the extreme volatility of market returns from 1929 to 1939, with only a brief respite during 1935 and 1936.
Looking at the standard deviation of monthly returns for 2008 gives a coarse picture of the increase in volatility during the year. If instead of monthly returns we examine monthly estimates of the standard deviation of daily returns, we get a finertuned and somewhat different picture of the evolution of volatility. Figure 4 show monthbymonth values of the standard deviation of daily returns for 19262008. (To fit all months on the figure, the values of the 12 monthly standard deviations of daily returns for a given year are plotted above the point for that year in Figure 4.)
The mean of the monthbymonth standard deviation of daily returns for 19262008 is 0.0085% per day. It is difficult to see in Figure 4, but from the middle of 2003 to the end of 2006 the volatility of daily returns is almost always below the longterm mean. The volatility of daily returns increases during 2007, and from July onward the monthbymonth standard deviations of daily returns are pretty consistently above 1%, somewhat but not dramatically above the longterm mean, but (as in Figure 3) a lot below the general level of volatility from 1998 to 2002. The big jump in volatility occurs in September 2008. For the last four months of 2008, the monthly standard deviations of daily returns, are 3.38%, 4.97%, 4.53%, and 3.14%. These are the four outliers plotted above 2008 in Figure 4. Figure 4 shows that volatility this high was common during the Great Depression. But after 1939, only the crash month October 1987 produces similarly high volatility (4.93%) of daily returns, and in this case the high volatility was shortlived. For perspective, a standard deviation of 4% for daily returns translates to a standard deviation of annual returns of about 56%!
Figure 4  MonthbyMonth Standard Deviations of Daily Market Returns (Plotted by Year)
In short, the high volatility of daily stock market returns during the last four months of 2008 is indeed unusual. Similarly high volatility was the rule during the Great Depression, but thereafter, there is only one month, October 1987, that rivals the volatility of the daily returns beginning in September of 2008.
Appendix
Suppose (i) the true longterm expected future equity premium is 7.64% per year (the mean for 19272008), (ii) the true longterm standard deviation of yearbyyear premiums is 21.04% per year (the standard deviation for 19272008), and (iii) the annual premiums follow a normal distribution. The equity premium for 2008, 39.95%, is then (39.95% 7.64%) / 21.04% = 2.26 standard deviations from the mean of 7.64%. Thus, the probability that the equity premium is 39.95% or less is the area to the left of 2.26 in a standard normal distribution with a mean of 0 and a standard deviation of 1. This area includes 1.18% of the distribution.
Similarly, a premium of zero is (0% 7.64%) / 21.04% = 0.36 standard deviations from the mean. The probability that the premium for a single future year is negative is the area to the left of 0.36 in a standard normal distribution, about 36%. If one increases the holding period to four years, the square root rule tells us that the standard deviation of the average premium drops to onehalf the standard deviation of annual premiums, from 21.04% to 10.52%. The probability that the average premium for a future fouryear period is negative is then the area to the left of (0% 7.64%) / 10.52% = 0.72 in a standard normal distribution, which is about 23%. Etc.
Table 1  Annual CapWeighted Market (RM), TBill (RF) and Equity Premium (RMRF) Returns 19272008: 82 YEARS
The market returns for 19271963 only cover the NYSE. Amex returns are added in 1964, and Nasdaq returns are included beginning in 1973. The annual Tbill return is the cumulative return from rolling over onemonth Tbill during the year
Year 
RM

RMRF

RF

Year

RM

RMRF

RF


1927 
33.40

30.27

3.13

1968

14.16

8.94

5.22

1928 
39.07

35.53

3.54

1969

10.85

17.42

6.57

1929 
15.02

19.76

4.74

1970

0.06

6.45

6.52

1930 
28.83

31.25

2.43

1971

16.19

11.80

4.39

1931 
44.36

45.44

1.09

1972

17.33

13.50

3.84

1932 
8.47

9.42

0.95

1973

18.77

25.70

6.93

1933 
57.52

57.22

0.30

1974

27.95

35.96

8.01

1934 
4.29

4.11

0.18

1975

37.35

31.55

5.80

1935 
44.85

44.71

0.14

1976

26.77

21.68

5.08

1936 
32.15

31.97

0.18

1977

2.97

8.10

5.13

1937 
34.61

34.90

0.29

1978

8.53

1.34

7.19

1938 
28.17

28.21

0.04

1979

24.39

14.01

10.38

1939 
2.12

2.11

0.01

1980

33.24

21.98

11.26

1940 
7.44

7.42

0.02

1981

3.98

18.70

14.72

1941 
9.63

9.67

0.04

1982

20.43

9.90

10.53

1942 
16.31

16.03

0.28

1983

22.66

13.87

8.80

1943 
28.06

27.70

0.36

1984

3.17

6.67

9.84

1944 
21.36

21.03

0.33

1985

31.41

23.69

7.72

1945 
38.45

38.13

0.32

1986

15.55

9.40

6.16

1946 
5.91

6.27

0.36

1987

1.81

3.66

5.47

1947 
3.37

2.87

0.50

1988

17.56

11.20

6.36

1948 
2.36

1.55

0.81

1989

28.42

20.04

8.38

1949 
20.08

18.96

1.12

1990

6.09

13.93

7.84

1950 
30.03

28.81

1.22

1991

33.63

28.03

5.60

1951 
20.83

19.34

1.49

1992

9.06

5.56

3.50

1952 
13.29

11.64

1.65

1993

11.58

8.68

2.90

1953 
0.36

1.47

1.83

1994

0.75

4.66

3.91

1954 
50.22

49.36

0.86

1995

35.67

30.07

5.60

1955 
25.33

23.76

1.57

1996

21.15

15.95

5.20

1956 
8.48

6.01

2.47

1997

30.33

25.08

5.25

1957 
10.36

13.52

3.15

1998

22.28

17.42

4.85

1958 
44.84

43.31

1.53

1999

25.27

20.58

4.69

1959 
12.61

9.63

2.98

2000

11.09

16.97

5.88

1960 
1.17

1.50

2.67

2001

11.27

15.13

3.86

1961 
26.95

24.83

2.12

2002

20.83

22.46

1.63

1962 
10.33

13.06

2.73

2003

33.13

32.10

1.02

1963 
20.89

17.78

3.11

2004

13.01

11.82

1.19

1964 
16.30

12.78

3.53

2005

7.32

4.34

2.98

1965 
14.39

10.47

3.92

2006

16.24

11.42

4.81

1966 
8.69

13.44

4.75

2007

7.27

2.61

4.67

1967 
28.56

24.35

4.21

2008

38.31

39.96

1.64

Table 2  Summary Statistics for Annual Returns
Mean is the average of the yearbyyear returns during a period, Std is the standard deviation of the yearbyyear returns, and t(mn)is the ratio of the mean to its standard deviation, which is Std divided by the square root of the number of years in the period.
RM

RMRF

RF
 

19272008: 82 Years  
Mean 
11.39

7.64

3.76

Std 
20.75

21.01

3.10

t(mn) 
4.97

3.29

10.99

192762: 36 Years  
Mean 
11.96

10.65

1.31

Std 
24.14

24.35

1.23

t(mn) 
2.97

2.62

6.41

19632008: 46 Years  
Mean 
10.95

5.28

5.67

Std 
17.92

17.91

2.57

t(mn) 
4.14

2.00

13.97

Behavioral Finance (1)
Diversification (1)
Economic Policy (4)
Financial Markets (2)
Hedge Funds (2)
Investments (3)
Market Efficiency (5)
Eugene Fama and Ken French are members of the Board of Directors of the general partner of, and provide consulting services to Dimensional Fund Advisors LP.