Quantitative Methods & RiskExpectancy, Edge & Sample Size

Compounding & the Volatility Tax

12 min read

The average return you earn is not the average return you see

Two strategies can have the same arithmetic average return and end up with wildly different wealth, because returns compound multiplicatively, not additively. The number that actually governs how your capital grows is the geometric mean, and it is always less than or equal to the arithmetic mean. The gap between them is driven by volatility — a drag sometimes called the volatility tax. Understanding it ties together expectancy, risk, and the Kelly sizing of Module 4.

Formula

Arithmetic vs. geometric mean return

Geometric: g = (Π(1 + rᵢ))^(1/n) − 1        Arithmetic: a = (1/n)·Σ rᵢ

Heads up

+50% then −50% is not break-even — it is a 25% loss

Start with $100, gain 50% to $150, then lose 50% to $75. The arithmetic mean of +50% and −50% is zero, but you are down 25%. Losses and gains do not cancel under compounding; a percentage loss bites a larger base back than the same percentage gain restored. This is the single most under-appreciated fact in retail trading.

Formula

The volatility drag approximation

g ≈ a − σ²/2

Annual volatility20%

0%60%

Holding the arithmetic mean return at 10%, a volatility of 20% imposes a drag of about 2.0 percentage points, leaving a compounded growth rate near 8.0%. Notice the drag grows with the SQUARE of volatility — doubling vol quadruples the tax. Two strategies with identical average returns are not equally rich.

geometric_mean.py

python

1import numpy as np
2 
3def geometric_return(returns):
4    r = np.asarray(returns, dtype=float)
5    return np.prod(1 + r) ** (1 / len(r)) - 1
6 
7rets = np.array([0.50, -0.50])
8print(f"arithmetic = {rets.mean():.2%}")          # 0.00%
9print(f"geometric  = {geometric_return(rets):.2%}")  # -13.40%
10# The geometric mean is what your ACCOUNT experiences. Volatility
11# pulls it below the arithmetic mean by roughly sigma^2 / 2.

Always judge a strategy by its geometric (compounded) growth, not its arithmetic average. The two diverge exactly as volatility rises — which is why a smoother equity curve at the same average return is genuinely worth more.

Tip

Why this connects to sizing

Because compounded growth is arithmetic mean minus a volatility penalty, there is an optimal amount of risk to take — too little leaves growth on the table, too much lets the σ²/2 drag (and ruin) dominate. That optimum is the Kelly fraction of Module 4. Compounding math and position sizing are the same problem viewed from two angles.

Knowledge check

An account gains 30% one year and loses 30% the next. What is the two-year compounded result?