Measuring Returns and CAGR

Every performance figure starts with a return, yet returns are easy to measure wrongly. A strategy that gains 50% one year and loses 50% the next has an average return of zero but has lost a quarter of its money. Deposits and withdrawals distort account growth. Monthly returns cannot simply be added to get an annual figure. This lesson covers the core return measures and when to use each, starting with the compound annual growth rate (CAGR), the standard way to express growth over several years.

Simple and log returns#

For small moves the two are almost identical: a 1% simple return is a 0.995% log return. For large moves they differ: a 50% gain is a 40.5% log return. See NumPy and Pandas for Traders.

Compound annual growth rate#

CAGR is the constant yearly rate that would turn the starting value into the ending value over the period.

CAGR = (Ending value / Starting value) ^ (1 / Years) - 1

Arithmetic versus geometric average#

The arithmetic average return is (50 plus minus 50) divided by 2, which is 0%. The geometric average, which is what CAGR measures, is the square root of (1.5 times 0.5) minus 1, about minus 13.4% a year. The account lost 25%. The gap between the two averages grows with volatility, which is why reducing volatility can raise long run compound growth even with the same average return. See Volatility and Compounding and Geometric vs Arithmetic Returns.

Volatility drag#

A useful approximation links the two averages:

Geometric mean ≈ Arithmetic mean - (Volatility ^ 2) / 2

With an arithmetic mean of 10% and volatility of 20%, the geometric mean is roughly 10% minus 2%, or 8%. With volatility of 40%, it falls to about 2%. Large swings erode compounding.

Annualising returns#

Crypto trades every day, so analysts often use 365. Annualising short track records produces dramatic but unreliable figures: a 5% gain in one month does not mean an 80% year. See Statistical Significance in Trading.

Time weighted versus money weighted returns#

If you add a large deposit just before a losing period, your money weighted return falls below the time weighted return. Fund managers report time weighted returns because clients control the timing of flows. See Investor Reporting.

What to include#

Honest returns include commissions, spreads, financing costs, fees and, for comparison with benchmarks, dividends. Pre cost returns from backtests should always be labelled as such. See Transaction Costs and Fake Performance and Track Record Verification.

Frequently asked questions#

What is CAGR?#

The compound annual growth rate: the constant yearly return that would grow a starting value into the ending value over a given number of years.

Why is average return different from CAGR?#

The arithmetic average ignores compounding; CAGR is a geometric average, which is lower when returns are volatile.

Should I use time weighted or money weighted returns?#

Use time weighted returns to judge a strategy and money weighted returns to understand your own results with deposits and withdrawals.

Next, learn the most famous risk adjusted measure in Sharpe Ratio.