One formula. Geometric is what compounds. Variance is what eats it.

The brochure prints arithmetic mean because it is the bigger number. Your bank account prints geometric mean because that is what compounded. The wedge between the two is set by variance, and it grows faster than the volatility you feel.

Variance drag . geometric vs arithmetic return

Geometric ~ Arithmetic - sigma squared / 2

Sigma is annualised standard deviation. The penalty scales with vol squared, not vol. Double the swing, four times the drag. The arithmetic mean ignores the drag. The chart of your folio does not.

Same arithmetic 12 pct . 10Y on Rs 10 lakh

Vol

Geom

End balance

10 pct

11.5pct

29.7L

14 pct

11.0pct

28.4L

20 pct

10.0pct

25.9L

28 pct

8.1pct

21.8L

35 pct

5.9pct

17.8L

Recovery asymmetry . the hole that bites

Drawdown

Gain to break even

-10 pct

+11 pct

-20 pct

+25 pct

-30 pct

+43 pct

-40 pct

+67 pct

-50 pct

+100 pct

Vol 14 pct costs ~ 1.0 pp of compounding. Vol 28 pct costs ~ 3.9 pp. Double the swing, four times the drag. And every drawdown carries a non-linear recovery bill on the way back. You do not lose what you swing through. You lose what you stay below.

Numbers illustrative. Geometric approximation g = mu - sigma squared / 2 from log-normal assumption. End balance = 10L . (1 + g) ^ 10. Recovery gain = 1 / (1 - DD) - 1. Vol assumed annualised, returns IID. Real world frictions (taxes, costs, fat tails) widen the wedge further. Past performance . backtest only . not a guarantee.