Convert a measured return to a yearly scale, distinguish compound growth from arithmetic scaling, and keep the horizon, income, costs and cash-flow basis explicit.
Annualised return
Annualised return expresses a return on a one-year scale under a stated calculation convention. For realised investment performance, the usual compound version is the constant yearly rate that reproduces the investment's growth over the full measurement period. A quant comparing backtests needs that common scale, but also the original horizon: a yearly unit does not turn a short record into a year of evidence.
The word annualised is incomplete without the return type. A compounded growth rate and an annual-scale arithmetic mean answer different questions, even when both carry a “per year” label.
From an endpoint result to a yearly rate
Let RT be the holding-period return over T>0 years. Its effective compound annual equivalent is
rann=(1+RT)1/T−1.
Equivalently, for positive initial and ending wealth and no external deposits or withdrawals,
rann=(W0WT)1/T−1,WT=W0(1+rann)T.
Here WT must include the holdings and retained investment income on the chosen basis. If distributions are reinvested, their subsequent gains and losses belong in ending wealth; do not add the original distributions again. A total return and a price-only return remain different after annualisation.
Take a synthetic wealth index that starts at 100 and ends at 121 over two equal years. All investment income is retained within the account; assume no external flows, costs or taxes. The full-period return is
RT=100121−1=21%,
while the compound annual equivalent is
rann=(100121)1/2−1=10%.
The check is economic as well as algebraic: 100 compounded at 10% becomes 110, then 121. The actual intermediate wealth is not supplied. The calculation gives an equivalent schedule rather than a claim that the investment earned exactly 10% in each year.
Synthetic wealth grows from 100 to 121 over two equal years. Total return is 21%; the compound annual equivalent is 10%, with an implied year-one value of 110. Simple scaling gives 10.5%. The actual intermediate value is unknown.
Original synthetic wealth-index example: two equal years, income retained, no external flows, costs or taxes. Filled points are supplied endpoints; the hollow midpoint belongs only to the equivalent compound schedule.
This endpoint growth calculation is also called the compound annual growth rate, or CAGR. Neither label reveals the intervening losses, recoveries or drawdown.
Why dividing by the years gives another answer
Simple scaling gives 21%/2=10.5% per year. It spreads the total percentage gain evenly across years relative to the original capital. It does not preserve the terminal wealth under annual compounding: compounding 100 at 10.5% for two years produces 122.1025, rather than 121.
There is a separate arithmetic convention for a series of equal-length periodic simple returns. If rˉ is their arithmetic mean and there are m such periods per year, the annual-scale arithmetic mean is mrˉ. This scales the mean in yearly units; it is not the realised compound annual growth rate. In particular, dividing a multi-year total return by the number of years does not recover the mean of the individual annual returns when those returns are unknown.
Choose the statistic first, then annualise it. Compounding a wealth record, scaling an arithmetic return mean, and scaling volatility are separate operations.
What must travel with the percentage
A useful performance label states the measurement dates and original horizon, the definition of a year or fractional year, the currency, the return type, the income treatment, and which costs and taxes have been deducted. Calendar dates and trading observations are not interchangeable clocks. In the example above, “two equal years” is stipulated; no trading-day or calendar-day convention is being inferred.
Annualisation also comes after the treatment of external cash flows. A contribution is new capital, not a gain. If deposits or withdrawals occur during the interval, do not apply the raw endpoint formula to account balances. A time-weighted return chains flow-adjusted subperiod returns before converting the resulting growth factor to a yearly rate. A money-weighted return uses the timing and size of investor cash flows; its period and compounding convention must also be stated.
The Risk-Adjusted Performance Measures lesson develops this return-measurement step before Sharpe, Treynor and Jensen measures, including why the elapsed years need not equal the number of cash-flow subperiods. A public preview is available; the full lesson requires membership.
The endpoint formula assumes positive starting wealth. Zero ending wealth gives a complete loss and an annual equivalent of −100% for any positive horizon. Negative ending wealth or a zero starting capital base requires a different, explicitly defined framework rather than an unqualified fractional power of a wealth ratio.
A scale conversion is not a forecast
Annualising a historical result does not estimate how likely the result is to repeat. Short records can produce striking annual equivalents while containing little information about investment risk or sustainable returns. Keep the original-period result visible beside any annual equivalent.
Reporting rules may also restrict the conversion. For firms claiming GIPS compliance, the 2020 GIPS Standards for Firms, provision 2.A.12 prohibits annualising returns for periods shorter than one year. That is a specific performance-reporting requirement, separate from the mathematical ability to calculate a growth-equivalent rate.
For another concrete reporting convention, SEC Form N-1A, Item 26(b)(1) defines a fund's average annual total return through a compounded initial-to-ending-value equation, with specified treatment of distributions and charges. “Annualised” alone does not supply those underlying assumptions.
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 means multiply gross returns; the local text also discusses their limitations as portfolio-selection criteria.
