Dec 14, 2009
Q&A
Lubos Pastor and Robert Stambaugh argue that long-horizon stock investors actually face more volatility than short-horizon investors. How should investors interpret this evidence?

KRF: The Pastor-Stambaugh result is driven by uncertainty about the true expected return. The volatility or standard deviation of returns is usually defined as the expected variation relative to the true mean of the process generating returns - as if we knew the true expected return. But, as Pastor and Stambaugh emphasize, we never actually know the true mean. When they include uncertainty about the true mean (as well as uncertainty about other true parameters) in the analysis, they find that long-run returns are indeed more volatile than short-run returns.

If we knew the true expected return, our uncertainty about long-run returns would grow with the return horizon. In essence, the unexpected return diversifies away - the average unexpected return gets smaller as N grows - so under some simplifying assumptions the variance of the cumulative unexpected return would be proportional to the holding period. But we do not know the true expected return. Because we make the same error when we forecast the one-year return one year out, two years out, and N years out, our uncertainty about long-run returns grows more quickly than the return horizon.

I have to use lots of equations to explain the logic more precisely. If you want to skip them, you can jump to Gene's answer here. OK, for the one hardy person still with me...start by splitting next year's return R1 into the true expected return T and the "true" unexpected return U1

R1 = T + U1

We usually think of the variance of next year's return as the variance of the true unexpected return,

Var(R1) = Var(U1).

But this ignores our uncertainty about T. We can include our uncertainty by splitting next year's return into our estimate of the expected return Z and the unexpected return relative to our estimate D1

R1 = Z + D1

Since our estimate of the expected return is the true expected return T plus our estimation error X, we can rewrite next year's return as,

R1 = T + X + D1

Equivalently, our unexpected return D1 is the true unexpected return U1 minus the error in our estimate of the expected return X,

D1 = U1 - X,

and the variance of our unexpected return is,

Var(D1) = Var(X) + Var(U1).

This makes sense. If we don't know the true mean, our uncertainty about next year's return is the sum of our uncertainty about the true mean and our uncertainty about how the return will differ from the true mean.

The story really gets interesting if we look at long-run returns. To simplify the analysis, let's pretend the N-year return is just the sum of the one-year returns over the next N years. (In other words, we will ignore compounding.) Also, let's assume the true expected one-year return T and the variance of the true unexpected return Var(U1) are constant through time. Finally, let's assume each year's unexpected return is unrelated to the unexpected return in any other year. (These assumptions are just to simplify our discussion. Pastor and Stambaugh do not make them.)

Then the variance of the true unexpected return over N years is just N times the variance of the true unexpected return over one year,

Var(UN)  = Var(U1,1 + U1,2 + U1,3 +...+ U1,N

= Var(U1,1) + Var(U1,2) + Var(U1,3) +...+ Var(U1,N

= N Var(U1),

where U1,1 is the one year unexpected return in year 1, etc.

Similarly, the variance of the unexpected return relative to our estimate of the expected return is,

Var(DN) = Var[(X + U1,1) + (X + U1,2) +...+ (X + U1,N)]

= Var(N X) + Var(U1,1) + Var(U1,2) + Var(U1,3) +...+ Var(U1,N

= N2 Var(X) + N Var(U1).

This equation is the key to Pastor and Stambaugh's insight. Although the uncertainty about the true unexpected return grows linearly with the number of years - the N-year variance is just N times the one year variance - the uncertainty caused by our estimation error of the mean grows with the square of the number of years; the uncertainty over two years is four times our uncertainty over one year.

Why does uncertainty caused by our estimation error grow with the square of N? Because when we are forecasting the returns for the next N years, we make the same mistake over and over again. Standing here today, our error about the expected one-year return next year is the same as our error about the expected one-year return in two years and it is the same as our error about the expected one-year return in N years. In contrast, next year's true unexpected return is unrelated to the true unexpected return in year two or in year N. As a result, the unexpected return diversifies away - the average unexpected return gets smaller as N grows - but the average estimation error does not change.

In short, if we knew the true expected return, our uncertainty about long-run returns would grow with the return horizon. But we do not know the true expected return. Because we make the same error when we forecast the one-year return one year out, two years out, and N years out, our uncertainty about long-run returns grows more quickly than the return horizon.

EFF: It depends on whether the issue is uncertainty about the average annual return on a portfolio over many years or uncertainty about the total wealth the portfolio will produce over the same period.

To make things simple, suppose for the moment that the expected annual return on the portfolio (its true mean) is known and it doesn't change over time. In any multi-year period, the average annual return on the portfolio will deviate from the expected return. This uncertainty about the average annual return goes down like the square root of the number of years in the period. (Specifically, the standard deviation of the average return is the standard deviation of the year-by-year annual returns divided by the square root of the number of years in the period used to compute the average.) Thus, uncertainty about the average return on a portfolio goes down the longer is the investment period.

But an investor eats the cumulative wealth generated by the portfolio over the investment period, not its average return. Although the uncertainty about the portfolio's average return goes down with the number of years in the investment period, the uncertainty about cumulative wealth goes up. Here's the intuition. Suppose the investment period is 30 years. Each year the portfolio generates a return that differs from the expected return. Over 30 years the cumulative wealth generated by the portfolio bears the uncertainty about 30 drawings of the annual return. The uncertainty doesn't "wash out" because total wealth at the end of 30 years depends equally on the uncertainty about each year's return. And longer investment periods imply more uncertainty about final wealth.

We can make the argument precise if we phrase it in terms of continuously compounded (CC) returns. The cumulative wealth generated by an N-year investment in a portfolio depends on the N-year CC return on the portfolio. The N-year CC return is just the sum of the annual CC returns of the period. If the annual CC return on the portfolio has a constant variance, the variance of the N-year return is N times the variance of the annual return. The variance of the N-year return is a measure of its uncertainty. But the variance is in the wrong units (squared returns). The standard deviation - the square root of the variance - is in units of returns. Taking the square root of the variance tells us that the standard deviation of the N-year return is the square root of N times the standard deviation of the annual (year-by-year) returns. In other words, the uncertainty about the cumulative N-year return on the portfolio increases with the investment period, N.

Does this imply that long-term investment is a bad deal? Not at all. The expected value of the N-year CC return (its true mean) is N times the expected value of the annual CC return. Thus, the expected value of the N-year return increases faster than its standard deviation, which goes up like the square root of N. In other words, the uncertainty about the cumulative return increases with the investment horizon but at a much slower rate than the expected value (the true mean) of the cumulative return. The distribution of the cumulative return becomes more disperse over longer investment horizons, but it moves toward higher values at a faster rate than it becomes more disperse.

We are not done yet. To get the cumulative wealth generated by a portfolio over an N-year investment period, one must exponentiate the portfolio's N-year CC return. This exponentiation carries out the continuous compounding.  It has a big effect on the probability distribution of final wealth. Thus, suppose the annual CC return has a nice symmetric normal distribution. (In fact, the distribution is likely to be close to symmetric but somewhat fat-tailed relative to the normal.) Then the N-year CC return on the portfolio is also normal. But the exponentiation of the N-year CC return transforms its symmetric normal distribution into a distribution that is skewed to the right. And there is more right skewness for longer investment periods, N. Right skewness is a positive for the investor. It means that over longer investment periods the probability of extremely good outcomes is higher than the probability of extremely bad outcomes.

Pastor and Stambaugh point out that the analysis above is incomplete in an important way.  In particular, we don't know the true expected returns on portfolios. We typically use historical average returns to estimate expected returns, but the estimates are quite noisy, and they leave lots of uncertainty about true expected returns. The uncertainty about expected returns is an additional unavoidable risk of investing. Ken's answer above gives the details. Eugene F. Fama
The Robert R. McCormick Distinguished Service Professor of Finance at the University of Chicago Booth School of Business Kenneth R. French
The Roth Family Distinguished Professor of Finance at the Tuck School of Business at Dartmouth College
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