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:
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.
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.
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