From Eugene Fama and Kenneth French

“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.”

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