I used a pattern to explain it to my 12-year-old on our car ride on Monday.
Start with:
8*8 = 64
Let’s call that a * b
It feels like if we subtract 1 from a and add 1 to b multipy it should be close to 64
7*9 = 63
Close but a tad lower.
What if we keep the 8 average between the numbers but widen the dispersion between them more:
6*10 = 60
Lower still.
When the deviation from the mean was 1, the product of a * b was just 1 lower. (63 vs 64)
When the deviation from the mean was 2, the product of a * b was 4 lower. (60 vs 64).
Hmm, I have hunch what’s gonna happen here.
5*11 = 55
When the deviation from the mean was 3, the product of a * b was 9 lower. (55 vs 64).
One more to solidify this…
4 * 12 = 48
When the deviation from the mean was 4, the product of a * b was 16 lower. (48 vs 64).
We got the pattern.
For 2 numbers, a and b:
a * b = Mean² – MAD²
where MAD = mean absolute deviation
As soon as the numbers deviate from the mean, their product is dragged down even if the mean is unchanged.
More deviation, more drag.
Note that MAD² is just variance when there are only 2 points because the mean is the midpoint, the 2 deviations must be equal.
If there are more points then MAD² < variance. We can see this from simply remembering that MAD ~ .8 * SD therefore MAD² ~ .64 * Variance
In investing, we compound or multiply returns so even if the mean of two returns is the same, the dispersion matters.
The mean of 1.1 and .9 is 1, but the geometric mean is less than 1 (ie when you multiply them together). The amount less than 1 is a function of the deviation of the 2 numbers from the mean of 1.
.8 and 1.2 have a mean of 1, but a geometric mean less than the geometric mean of .9 and 1.1.
.5 and 1.5 have a mean of 1, but a geometric mean less than the geometric mean of .8 and 1.2.
The drag is a function of squared deviation. And no deviation, no drag — the arithmetic and geometric mean are the same in that case.
Summarizing:
Then with numbers that look like returns:
Notice how the difference between the arithmetic and geometric mean is approximately half the variance.
You’ve seen this before.
r−1/2 * σ²
The median expected return (ie geometric return).
AKA the risk-neutral drift from Black-Scholes.
AKA the “volatility drain”.