How this calculator works
- Step 1ArithmeticΣ r ÷ nAdd the returns and divide by the count.
- Step 2Geometric(Π (1 + r))^(1/n) − 1The steady rate that compounds to the same end.
- Step 3Gaparithmetic − geometricThe cost of ups and downs.
The arithmetic average treats each year on its own. The geometric average follows the money: it is the one steady yearly return that turns the start amount into the same end amount.
The geometric average is never higher than the arithmetic one. The more the returns swing, the bigger the gap.
Worked example: +12%, −8% and +2%
- Geometric
(1.12 × 0.92 × 1.02)^(1/3) − 1- Arithmetic
(12 − 8 + 2) ÷ 3
- Year 1× (1 + 12%)₹1,12,000.00
- Year 2× (1 − 8%)₹1,03,040.00
- Year 3× (1 + 2%)₹1,05,100.80
- Geometric average(end ÷ start)^(1/3) − 11.67%
The simple average is 2%, but the money grew like 1.67% a year. OpenStax Introductory Business Statistics works this example and reaches +1.67%.
Read the result
- +50% then −50%-13.4%Simple average 0%, yet ₹1,00,000 ends at ₹75,000.
- +100% then −50%0%Simple average 25%, yet the money is back at ₹1,00,000.
Five uneven years of +25%, −20%, +30%, −10% and +15% average 8% simply, but compound at 6.11%.
Use the geometric average to describe what a past series actually earned. It is the CAGR of the same years.
Assumptions and limits
- Full yearsEach figure is one year’s return.
- ReinvestedGains stay invested; no money is added or withdrawn.
- Before taxTaxes, fees and inflation are not included.
- Not a forecastPast returns do not predict future returns.
Other ways to measure return
An average of yearly returns ignores money added or taken out. Other measures account for it.
For one investment’s start and end values, use the CAGR calculator. When money went in or out on different dates, the XIRR calculator weighs each cash flow.
To see how fund costs eat into returns, try the expense ratio impact calculator.
Common mistakes
Mistake: Quoting the arithmetic average as what you earned.
Instead: Use the geometric average for money that compounds.
Mistake: Thinking a 50% loss is undone by a 50% gain.
Instead: After a 50% loss you need a 100% gain to break even.
Mistake: Averaging returns when money was added midway.
Instead: Use the XIRR Calculator for cash flows on dates.
Checking the method
A separate script grows the amount year by year and finds the steady rate by bisection, and reproduces the OpenStax +1.67% example.
Read how we test our calculators, or browse more money calculators.