By Eugene F. Fama and Kenneth R. French

The cap-weighted market portfolio of NYSE-Amex-Nasdaq 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 1927-2008 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 three-year period 1929-1931 is -66.35%, the return for 1973-1974 is -41.47%, and the return for 2000-2002 is -37.54%. On the plus side, the annual market return is greater than 30% for 15 individual calendar years of 1927-2008. In short, extreme stock returns are common. Fortunately, extreme positives outnumber extreme negatives.

Figure 1 - Annual Market Returns 1927-2008

Financial economists often focus on the equity premium - the extra return one gets for investing in stocks, rather than in short-term 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 1927-2008 (82 years) the average annual difference between the market return and the return from rolling over one-month 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 fat-tailed and right skewed), and that the estimates for 1927-2008 are the true long-term 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 year-by-year 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 T-Bill Return) 1927-2008

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 year-by-year 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 1927-2008. 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 year-by-year premiums (21.04%) means that single-year 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 one-half the standard deviation of annual premiums, from 21.04% to 10.52%. As a result, for four-year holding periods, the probability of a negative realized average premium falls to about 23%. In other words, we expect that for four-year holding periods the average equity premium will be negative (bills beat stocks) about 23% of the time. For 16-year holding periods, the probability of observing a negative average premium drops further, to about 7%. And for 25-year 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 year-to-year 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 year-by-year standard deviations of the monthly market returns of 1927-2008. The standard deviation of the monthly returns for 2008, 6.67% per month, is high relative to the average for 1927-2008, 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 1927-2008 period. In other words, we are struck by the high volatility of returns during 2008 in part because they follow a five-year period when volatility was quite low.

Figure 3 - Year-by-Year Standard Deviations of Monthly Market Returns 1927-2008

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 three-year 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 finer-tuned and somewhat different picture of the evolution of volatility. Figure 4 show month-by-month values of the standard deviation of daily returns for 1926-2008. (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 month-by-month standard deviation of daily returns for 1926-2008 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 long-term mean. The volatility of daily returns increases during 2007, and from July onward the month-by-month standard deviations of daily returns are pretty consistently above 1%, somewhat but not dramatically above the long-term 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 short-lived. For perspective, a standard deviation of 4% for daily returns translates to a standard deviation of annual returns of about 56%!

Figure 4 - Month-by-Month 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 long-term expected future equity premium is 7.64% per year (the mean for 1927-2008), (ii) the true long-term standard deviation of year-by-year premiums is 21.04% per year (the standard deviation for 1927-2008), 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 one-half the standard deviation of annual premiums, from 21.04% to 10.52%. The probability that the average premium for a future four-year 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 Cap-Weighted Market (RM), T-Bill (RF) and Equity Premium (RM-RF) Returns 1927-2008: 82 YEARS

The market returns for 1927-1963 only cover the NYSE. Amex returns are added in 1964, and Nasdaq returns are included beginning in 1973. The annual T-bill return is the cumulative return from rolling over one-month T-bill during the year

Table 2 - Summary Statistics for Annual Returns

Mean is the average of the year-by-year returns during a period, Std is the standard deviation of the year-by-year 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.