Theoretically the standard deviation of average annual returns is
inversely proportional to the holding period if asset returns follow a ran-
dom walk.^3 Arandom walk is a process whereby future returns are
considered completely independent of past returns. The dashed bars
in Figure 2-3 show the decline in risk predicted under the random walk
assumption.
But the historical data show that the random walk hypothesis can-
not be maintained for equities. This is because the actual risk of stocks de-
clines far faster than the predicted rate, indicated by the dashed bars. This
occurs because of the mean reversion of equity returns that I described in
Chapter 1.CHAPTER 2 Risk, Return, and Portfolio Allocation 29
FIGURE 2–3
Risk for Average Real Return over Various Holding Periods, 1802 through December 2006 (Historical
Risk versus Risk Based on Random Walk Hypothesis)(^3) In particular, the standard deviation of average returns falls as the square root of the length of the
holding period.