Average Return Calculator
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Results
Over 5 years the arithmetic mean return is 9% and the geometric mean (CAGR) is 8.497%
- Year 1
- +12%
- Year 2
- -8%
- Year 3
- +15%
- Year 4
- +4%
- Year 5
- +22%
| Year | Return | Value at year end |
|---|---|---|
| 1 | +12% | $11,200.00 |
| 2 | -8% | $10,304.00 |
| 3 | +15% | $11,849.60 |
| 4 | +4% | $12,323.58 |
| 5 | +22% | $15,034.77 |
How to use the average return calculator
- To average a series, enter each year's return in percent, in order, for example 12, -8, 15. Add a starting value to see how the money would have grown.
- To find a CAGR, choose the start and end value option and enter both values and the number of years between them. Fractions of a year work.
- Compare the geometric mean (what you actually earned per year, compounded) with the arithmetic mean (the plain average).
Average return formulas
Geometric mean = [(1 + r1) × (1 + r2) × … × (1 + rn)]1 ÷ n − 1
CAGR = (End value ÷ Start value)1 ÷ years − 1
Each r is one year's return as a decimal (12% = 0.12) and n the number of years. The geometric mean multiplies the yearly growth factors together, which is how returns really combine, and then finds the single steady rate that gives the same result. Applied to start and end values, that same idea is the CAGR. A rough rule of thumb links the two averages: geometric ≈ arithmetic − σ² ÷ 2, where σ is the standard deviation of the returns.
Worked example
An investment returns 12%, −8%, 15%, 4% and 22% over five years.
- Arithmetic mean: (12 − 8 + 15 + 4 + 22) ÷ 5 = 9%.
- Geometric mean: (1.12 × 0.92 × 1.15 × 1.04 × 1.22)1/5 − 1 = 8.5% a year. $10,000 grows to $15,034.77, and $10,000 × (1 + 0.08497)5 gives the same figure.
- The standard deviation is 11.49 points, so the rule of thumb estimates 8.34%, close to the exact answer.
- CAGR from values: growing from $10,000 to $18,000 in 6 years is (1.8)1/6 − 1 = 10.29% a year.
Same average, different results
Four return patterns, each with an arithmetic mean of 10% a year over four years, starting from $10,000:
| Yearly returns | Arithmetic mean | Geometric mean | $10,000 becomes |
|---|---|---|---|
| 10%, 10%, 10%, 10% | 10% | 10% | $14,641 |
| 25%, -5%, 25%, -5% | 10% | 8.97% | $14,102 |
| 40%, -20%, 40%, -20% | 10% | 5.83% | $12,544 |
| 60%, -40%, 60%, -40% | 10% | -2.02% | $9,216 |
The bigger the swings, the further the geometric mean falls below the arithmetic mean. In the extreme case, a 50% gain followed by a 50% loss averages 0%, yet $100 ends at $75.00, a geometric mean of -13.4% a year.
Which average should you use?
- To describe what happened, use the geometric mean or CAGR. It is the rate that actually turned the starting amount into the ending amount, and it is what fund reports usually call the annualized return.
- To describe a typical single year, the arithmetic mean is the better summary, but it overstates long-run compound growth when returns vary.
- For projections, a geometric figure is the safer input to the investment calculator, which assumes the same return every year. Past returns do not predict future returns.
- Include every year in order, and include losses. Leaving out a bad year, or using returns before and after fees inconsistently, changes the answer a lot. For a guaranteed rate rather than a varying one, the compound interest calculator is simpler.
Frequently asked questions
What is the difference between arithmetic and geometric average return?
The arithmetic mean is the plain average of the yearly returns; the geometric mean is the steady yearly rate that gives the same final value. For returns of 12%, −8%, 15%, 4% and 22%, the arithmetic mean is 9% but the geometric mean is 8.50%.
How do I calculate CAGR?
CAGR = (end value ÷ start value)^(1 ÷ years) − 1. An investment that grows from $10,000 to $18,000 in 6 years has a CAGR of 1.8^(1/6) − 1 = 10.29% a year.
Why is the geometric mean lower than the arithmetic mean?
Because losses hurt more than equal gains help: after a 50% gain and a 50% loss, $100 is $75, even though the returns average 0%. The geometric mean, −13.40% a year here, reflects that. It equals the arithmetic mean only when every year’s return is the same.
Which average return should I use for planning?
For long-term projections, the geometric mean or CAGR is the safer figure, since it describes compound growth. Treat any past return as an assumption for the future, not a prediction.
How should I enter negative returns?
Type them with a minus sign, such as -8 or −8%. Separate returns with commas, spaces or new lines, one per year in order. A return of −100% means the investment was wiped out, so the geometric mean is −100%.