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Compound Annual Growth Rate (CAGR)

The Compound Annual Growth Rate (CAGR) answers one question: given where an investment started and where it ended, what single annual rate, compounded, would have produced that trip? It is the number to compute yourself — from two balances and a calendar — whenever anyone quotes you a return. The formula:

CAGR = {(\frac{V_{f}}{V_{i}})}^{\frac{1}{n}} - 1

where:

is the final value of the investment.
is the initial value of the investment.
is the number of years.

Suppose you invested $10,000 in a stock, and after 5 years, the investment grew to $16,105. The CAGR would be calculated as follows:

CAGR = {(\frac{16105}{10000})}^{\frac{1}{5}} - 1 = {(1.6105)}^{0.2} - 1 \approx 0.1000 or 10 %

The $1 ∕ 5$ power is the fifth root — the rate that, compounded five times, reproduces the $1.6105 \times$ total growth. In a spreadsheet: =1.6105̂(1/5)-1.

CAGR is the geometric mean — and the other average is a lie. Insist on a compound figure whenever anyone quotes you a return. CAGR is exactly the geometric mean of the yearly return series, and it is what your account balance did. The arithmetic mean — add the annual returns, divide by the count — is a different and larger number that no investor ever receives. The two coincide only when every year’s return is identical, and they diverge in proportion to volatility, by the approximation

CAGR \approx \bar{r} - \frac{σ^{2}}{2}

where $\bar{r}$ is the arithmetic mean and $σ$ the standard deviation of annual returns. At the roughly 16% volatility of a broad equity index — plug in $σ = 0.16$ , so $0.1 6^{2} ∕ 2 = 0.0128$ — that gap is about 1.3 percentage points a year, compounded over thirty years, an enormous distance between the number in the brochure and the number in your account. section “Volatility Drag” works the mechanics; the practical instruction is to read “average annual return” as a claim requiring verification, and to compute ${(V_{f} ∕ V_{i})}^{1 ∕ n} - 1$ yourself from the beginning and ending balances.

The same asymmetry explains why losses are so expensive. Recovering from a decline of $L$ requires a gain of

r_{recover} = \frac{1}{1 - L} - 1

so a 20% drawdown needs 25% to get back to even, a 50% drawdown needs 100%, and an 80% drawdown needs 400%. Gains and losses of equal size are not equal events, which is the entire reason avoiding catastrophic drawdowns matters more than capturing marginal upside (section “Maximum Drawdown and the Recovery Asymmetry”).

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