Choose annual rates or holding-period returns
Returns are quoted in two different ways, and mixing them up is the commonest source of wrong answers. An annualized rate is a yearly figure: 10% means 10% a year, so holding for 14 months applies it for 14/12 of a year. A holding-period return is the whole gain over the interval: 10% over 14 months means the balance grew 10% in total across those 14 months.
This calculator makes you choose. The default treats every entry as an annualized rate for its interval, which matches how long-run fund and index figures are usually quoted. The alternative treats each entry as a total return, which matches statements that show performance for a specific span. Whichever you choose, the table shows the growth factor and the annual equivalent for every row, so the conversion is never hidden.
To tell which basis you have, look for the words annualized, per year or average annual next to the figure. A statement that reads 3-year return: 30% is normally a total unless it says otherwise. The same number quoted for different spans, such as 10% for six months and 10% for three years, can only be read one way or the other, never both, so choose the basis that matches the source.
Cumulative return and annualized return
The cumulative return is the total change over the whole sequence: multiply the growth factors of the rows and subtract 1. The annualized return is the constant yearly rate that would give the same cumulative result over the same total time. It is found by taking the product of the growth factors to the power of 1 divided by the total number of years.
The page's example starts with a hypothetical 100 and applies 10% a year for 14 months, -2% a year for 5 months and 15% a year for 27 months. The balance grows to 151.78, a cumulative return of 51.7769% over 46 months. The annualized return is 11.4990%.
Here is the arithmetic behind it. Each interval multiplies the balance by its growth factor, which is (1 + rate) raised to the number of years held. The three default rows give factors of 1.1176, 0.9916 and 1.3695. Their product is 1.5178, so 100 becomes 151.78. The annualized figure is that product taken to the power of one over the total years.
Arithmetic versus geometric average
The arithmetic average adds the rates and divides by how many there are, weighting by length of time when the intervals differ. For the example it is 11.6304%, higher than the annualized 11.4990%. The geometric figure is lower because losses hurt more than equal gains help.
The clearest case is +50% in one year followed by -50% in the next, starting from 100:
| Year | Return | Opening balance | Ending balance |
|---|---|---|---|
| 1 | 50% | 100.00 | 150.00 |
| 2 | -50% | 150.00 | 75.00 |
The arithmetic average is 0%, which suggests breaking even. The balance is actually 25% lower: a cumulative return of -25%, or -13.39746% a year compounded. Only the compound figure describes what happened to the money.
Volatility is what separates the two averages. A steady 5% a year for two years has an arithmetic and a geometric average that are both 5.00%. Alternate +10% and -10% and the geometric figure falls to -0.5013% a year while the arithmetic stays at 0%. Make the swings +20% and -20% and it falls to -2.0204%. The larger the swings, the wider the gap between what the rates average and what your money earned.
Different holding lengths
Intervals rarely have the same length, and a fair average has to respect that. In the example the three rows last 14, 5 and 27 months, 46 months in all. The arithmetic average weights each annual rate by its months, so the 27-month row counts more than five times as much as the 5-month row. The compound calculation weights them the same way automatically, through the growth factors.
With total returns the arithmetic comparison is shown only when every interval has the same length, because a plain mean of returns over different spans mixes unlike things. Two returns of 10% over 14 months each compound to 21.0000% over 28 months, an annualized 8.5124%; with unequal lengths the page withholds the arithmetic figure and says why.
Order does not matter when no money moves in or out. Multiplying growth factors gives the same product in any order, so reordering the same intervals leaves the cumulative and annualized returns unchanged. Order does matter once deposits or withdrawals enter, because a loss after a deposit hurts more than a loss before it, which is exactly what this page does not model.
What this sequence cannot measure
The page compounds returns that you supply. It has no deposits, withdrawals, fees or taxes, so it cannot tell you what your account earned if money moved in and out; for that, calculate a dated money-weighted return with XIRR. It is not a time-weighted return built from account flows, and it is not an accounting rate of return. For the effect of volatility on growth, explore why average and compound returns differ; for one investment's gain, calculate a single investment's ROI; and to account for inflation, adjust a return for inflation. Figures are hypothetical.
Nor does the page adjust for inflation or taxes, and it does not turn a return into a forecast.
Average return questions
Why doesn't +50% cancel -50%?
Each percentage applies to a different balance. 100 grows 50% to 150, and a 50% loss on 150 takes off 75, leaving 75. The two moves leave you 25% down, even though the plain average of +50% and -50% is zero.
Is the arithmetic mean what I earned?
No. The arithmetic mean summarises the rates themselves, not the compounded growth of your money. What you actually earned is the cumulative return, and the geometric figure is the constant yearly rate that would produce it.
Are entries annual or period returns?
You choose. Select annualized rates if each figure is a yearly rate, or total returns if each figure is the whole gain over its interval. The page shows how each entry was converted, so you can check the basis.
Can holding periods differ?
Yes. Each row has its own length in years and months, and annualization uses the total elapsed time with the correct growth factor for every interval, so a short strong period is not given the same weight as a long one.
Is this the accounting average rate of return?
No. Accounting ARR divides average annual profit by an investment cost and ignores timing. This page compounds investment returns over time, so it answers a different question about growth.
Can deposits be included?
Not here. Supplied returns alone cannot identify when money was added. Use the XIRR calculator for dated deposits and withdrawals, which finds a money-weighted rate from the actual cash flows.