Friday, June 09, 2006

Long-term return is not expected return

Most investment professionals are familiar with expected return and variance of return. Through the CFA program and in their MBA courses, they have learned about the mean-variance efficient frontier and the single period Capital Asset Pricing Model. But they haven't learned that long-term return is not expected return, and is usually considerably less.

Suppose that a stock has a 50/50 chance of going up by a factor of 1.5 or down by a factor of 0.9. Then the expected return is (0.5 x 1.5 + 0.5 x 0.9) - 1 = 0.20, or 20%. Most professional investors figure that this also is the stock's long-term return. But, here is the way to see that it is not.

Just as the percentage of tails in coin flipping approaches 50% over time, the percentage of the time the stock will go up approaches 50% over time. In other words, the typical two-period long-term result must be one up and one down. This implies that $1.00 invested in the stock will, in a typical two periods, either grow by a factor of 1.5 and then decline by a factor of 0.9 or decline by a factor of 0.9 and then grow by a factor of 1.5. Either way, the typical two-period terminal value of $1.00 is (1.2 x 0.9) = $1.35, which implies a per period long-term return of 16.2%. This is a less than 20% per period.

Just to make things interesting, consider what happens if the up factor is 2.0 and the down factor is 0.4. The expected return is still 20% per period. But now the typical two period terminal wealth of a $1.00 investment is (2.0x 0.4) = $0.8. Now the per period long-term return is -10.6%. This is a lot less than 20% per period.

Evidently, long-term return is always less than expected return and the difference is large to the extent the variance is large. In fact, long-term return is approximately expected return less one-half variance of return.

One reason why leveraged investments are so dangerous is because, ultimately, leverage increases variance of return a lot faster than it increases expected return.

Moral: Leverage and volatility are dangerous.

2 comments:

Anonymous said...

Question: Couldn't we also say the expected two period return is 44%?

I would think that to get the expected two period return, one needs the probability distribution of two period returns.

TOG said...

Yes. The expected two-period return is (1.2*1.2)-1=1.44-1=0.44, or 44%.

You can get the expected two-period return using the two-period probability distribution of return. However, there is an easier way, since we are assuming each period's return is independent of the others. Denote the return in period i by Ri. Denote the compound return factor over n periods by (1+R)^n. Then:

(1+R)^n=(1+R1)(1+R2). . .(1+Rn)

Since the returns are independent from one period to another, the expected value of (1+R)^n is:

E[(1+R)^n]=[E(1+R1]...E[(1+Rn)]

E[(1+R)^n]=[E(1+Ri]^n=[1+E(Ri)]^n

This implies that the expected n-period return is obtained by simply compounding the expected one-period return for n periods.

The problem is that the return we are interested in is the expected n-period annual compound return. That is:

E(1+R)=E{[1+R1)^(1/n)]. .[1+Rn)^(1/n)]

(1+R)={E[(1+Ri)^(1/n)]}^n

This is not E(1+Ri).

I hope I have this right. It is easier to see the way I originally wrote about it.