Linking returns over time

To get a return over several periods you don't add the monthly returns. You compound them: multiply the growth factors together and subtract one. Up 10% then down 10% isn't zero. It's −1%, because the loss hits a bigger base.

R = (1 + r₁) × (1 + r₂) × … × (1 + r_n) − 1

Adding vs compounding

Every setting has the same average month, +0.8%. Turn up the volatility and watch compounding pull away from the simple sum.

Monthly volatility

Compounded (the real return)Monthly returns added up

Added up

Compounded

Difference

Grows with volatility

Annualizing

A 3-year and a 5-year track record can't be compared directly, so we convert each into an equivalent return per year.

R_annual = (1 + R_cumulative)^{1 ÷ years} − 1

Cumulative return (%)

Years

Annualized

Naive (divided by years)

Overstates the true rate

For practitionersThe gap between the arithmetic and geometric mean is roughly half the variance of returns: volatility drag. Linking daily or monthly sub-period returns this way is exactly how a true TWR is built. Don't annualize periods shorter than a year; GIPS prohibits it because it projects a short-run result forward.

All numbers are illustrative. The models are simplified for teaching: annual coupons, Black-Scholes options, simplified fee mechanics and no intra-period trading. Real systems add transaction-based returns, daily valuation, tax and corporate action processing, and reconciliation to the official TWR.