Compound annual growth rate (CAGR) is the constant annual rate that, through compounding, connects an investment's starting wealth to its ending wealth over a stated horizon. It compresses cumulative growth into one comparable yearly figure. For a quant evaluating a backtest, that compression is useful—and incomplete: identical CAGR can accompany very different losses along the way.
The rate that reconstructs the ending value
For starting wealth , ending wealth and a horizon of years, with no external deposits or withdrawals,
The defining check is . The rate is an equivalent growth schedule, not a statement that each year's realised return equalled .
Take a synthetic wealth index that grows from 100 to 121 over two exact years. All investment income stays inside the account; assume no external flows, fees or taxes. Then
The holding-period return is 21%. Dividing that gain by two gives 10.5%, which spreads the gain relative to initial capital; it does not solve the compound-growth equation. CAGR uses a geometric root of the growth factor.
Equal CAGR does not mean equal experience
Consider two possible sets of annual observations for that same investment horizon:
| Wealth index | Year 0 | Year 1 | Year 2 | CAGR |
|---|---|---|---|---|
| Path A | 100 | 110 | 121 | 10% |
| Path B | 100 | 80 | 121 | 10% |
Path A earns 10% in each year. Path B loses 20%, then gains 51.25% on the smaller capital base. Both compound to the same ending value.
Original synthetic annual snapshots with all income retained, no external flows, fees or taxes. Solid and dashed lines identify paths A and B; filled points are supplied values. Connecting lines do not reveal within-year losses.
Measured at these snapshots, maximum drawdown is 0% for A and 20% for B. Finer observations could reveal deeper losses in either path. CAGR retains neither the intermediate observations nor their ordering, so it cannot describe volatility, recovery time or the capital needed to survive an interim loss. It is also a historical summary, not a forecast of the next year's return.
Backtest Evaluation develops the next step: assess compound performance alongside drawdown, costs, concentration and uncertainty. Its public preview is open; the full lesson requires membership.
What has to be settled before taking the root
Choose the performance series. An account contribution increases wealth without being an investment gain. With external flows, first construct a time-weighted return or a corresponding flow-adjusted wealth index, then annualise its growth factor using elapsed years. If the question is the investor's experience including the timing and size of contributions, money-weighted return answers a different question.
Choose the wealth basis. Price growth excludes distributions; a reinvested total-return series includes their effect. State the currency and whether fees, financing costs and taxes have been deducted. CAGR inherits those choices. As a specific reporting example, SEC Form N-1A, Item 26(b)(1) uses a compound equation for average annual total return and prescribes reinvestment and charge assumptions for the funds it covers.
Choose the clock. is elapsed time in years, not automatically the number of price observations. For fractional years, disclose the date and day-count convention; different year fractions change the exponent. The example stipulates two exact years.
CAGR is the compound endpoint case of annualised return, a broader label that can also refer to other scaling conventions. Keeping that distinction explicit prevents an annual-scale arithmetic mean from being mistaken for realised wealth growth.
The usual formula requires a positive starting base and positive ending wealth. A zero ending value is the complete-loss boundary, giving for any positive horizon. A zero starting base or negative wealth needs a separately defined performance framework; an unqualified CAGR is unsuitable.
Sources
- Edwin J. Elton, Martin J. Gruber, Stephen J. Brown and William N. Goetzmann, Modern Portfolio Theory and Investment Analysis, 9th edition (Wiley, 2014), Chapter 11, pp. 232–233: geometric averaging and return-factor products. The wealth paths here are original synthetic examples.
- US Securities and Exchange Commission, Form N-1A, Item 26(b)(1), printed p. 53: average annual total return as a compound rate connecting initial payment and ending redeemable value, with specified reinvestment and charge assumptions.