What did we notice?
a * b = Mean² − MAD² (where MAD = mean absolute deviation)
As soon as numbers deviate from the mean, their product is dragged down — even if the mean is unchanged. More deviation, more drag. And what is deviation? Volatility.
Bridging middle school math to investing math
In investing, we compound, or multiply returns. So even if the mean of two returns is identical, the dispersion between them matters. Not just matters. It matters quadratically.
No dispersion: The arithmetic mean of (8, 8) is 8. The geometric mean of (8, 8) is √(8×8) = 8.
With dispersion: The arithmetic mean of (5, 11) is still 8. But the geometric mean of (5, 11) is √(5×11) = ~7.4.
If you earn 10% on an investment and then lose 10%, your mean return is 0, but your actual compounded (geometric) return is 1 − √(1.1 × 0.9) = −0.50%.
Now increase the volatility: earn 40%, lose 40%. Mean return is still 0. Compounded return? 1 − √(1.4 × 0.6) = −8.3%.
The drag on your returns is a function of squared deviation. Put simply:
Compounded Return = Average Return − σ²/2