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Arithmetic vs geometric return

An arithmetic return averages the yearly percentages. A geometric return is the one steady rate that reproduces the ending value, which is a CAGR. Up 60 percent then down 37.5 percent averages 11.25 percent a year and compounds to 0 percent.

 Arithmetic returnGeometric return
What it averagesA list of yearly percentages.The growth factors, by multiplying them and taking a root.
Teaching sheetUp 60 percent then down 37.5 percent averages 11.25 percent a year.The factors 1.6 times 0.625 equal 1. CAGR is 0 percent. $10,000 in, $10,000 out.
What it reproducesThe average of the list, not the ending value.The ending value. That is the entire reason to quote it.
When they matchWhen every year earned the same return.Same case: a flat path has one number either way.
When they splitWhenever yearly returns bounce around. The split is volatility drag.The geometric figure is the smaller one, and the gap widens with the bounce.
When you would pick itDescribing a typical year in a sample, not a path of money.Quoting what a holding actually did over a window, which is a CAGR.
On this page

A list of percentages is not a path of money

Turn each year into a growth factor, multiply the factors, take the root. That is the geometric return, and it is the CAGR formula.

On the teaching sheet, +60 percent is a factor of 1.6. -37.5 percent is a factor of 0.625. 1.6×0.625=11.6 \times 0.625 = 1. From $10,000 you still have $10,000 two years later. The CAGR is 0 percent. The arithmetic mean of 60 and -37.5 is 11.25 percent a year, a rate the money never earned.

The same two values over a smoother path tell a different story. $10,000 to $18,000 over 6 years is a CAGR of 10.29 percent a year. Over 3 years it is 21.64 percent. Those figures already compound. Averaging six unknown yearly returns and hoping to land on 10.29 is how the 11.25 percent error gets into a report.

How CAGR works is the long form, with the CAGR calculator under the answer. Volatility drag is why the arithmetic figure sits above the geometric one whenever the path is bumpy.

Which number to put in a sentence

Use the geometric return, the CAGR, when the sentence is about what the money did. Use the arithmetic return when the sentence is about a typical year in a sample of years, which is a statistics claim rather than a compounding claim.

A quoted average return on a fund factsheet is often arithmetic. A quoted annualised return over a stated window is often geometric. The two labels are not interchangeable, and mixing them is how a bumpy path looks like a steady 11.25 percent. This is educational material, not financial advice.

Worked examples

\$10,000 to \$18,000 over 6 years

An investment was worth $10,000 at the start and $18,000 six years later, with nothing paid in and nothing taken out. What annual rate is that?

  1. Divide the ending value by the starting value: 18000/10000=1.818000 / 10000 = 1.8. The money grew to 1.8 times its starting size.
  2. Take the 6th root, which is the same as raising to the power 1/61/6: 1.81/6=1.102923571.8^{1/6} = 1.10292357.
  3. Subtract 1: 1.10292357−1=0.102923571.10292357 - 1 = 0.10292357, which is 10.29 percent a year.
  4. Check it by growing the money back up: 10000×1.102923576=1800010000 \times 1.10292357^{6} = 18000.

The CAGR is 10.29 percent a year. $10,000 compounding at that rate for 6 years lands on $18,000, which is total growth of 80 percent spread over the period.

The same money in half the time

Take the same $10,000 to $18,000, but suppose it happened over 3 years rather than 6. What is the CAGR now?

  1. The growth multiple does not change: 18000/10000=1.818000 / 10000 = 1.8.
  2. Only the exponent moves: 1.81/3=1.216441.8^{1/3} = 1.21644 where before it was 1.81/61.8^{1/6}.
  3. Subtract 1: 0.216440.21644, or 21.64 percent a year.
  4. Check it: 10000×1.21644310000 \times 1.21644^{3} comes back to 18000.

The CAGR is 21.64 percent a year, more than double the 10.29 percent that the same growth gives over 6 years. Halving the time more than doubles the rate, because each year has to carry more of the work.

The average return trap

A holding gains 60 percent in year one and loses 37.5 percent in year two. Those two returns average 11.25 percent a year. Starting from $10,000, what is the CAGR?

  1. Turn each year into a growth factor. Up 60 percent is a factor of 1.6. Down 37.5 percent is a factor of 0.625, because 100 minus 37.5 leaves 62.5 percent.
  2. Returns compound, so the factors multiply: 1.6×0.625=11.6 \times 0.625 = 1.
  3. A total factor of 1 means the ending value equals the starting value: $10,000 in, $10,000 out, 2 years apart.
  4. Put that into the formula: (10000/10000)1/2−1=1−1=0(10000/10000)^{1/2} - 1 = 1 - 1 = 0.

The CAGR is 0 percent. The holding ends at $10,000, exactly where it started, while the average of the two yearly returns says 11.25 percent a year. The average is the mean of a list of numbers. The CAGR is the rate the money grew at, and here it grew by nothing.

Common questions

Is the arithmetic return always higher?

It is always at least as large. The two are equal only when every year earned the same return. Any bounce in the yearly figures pulls the geometric return down, which is volatility drag, and the pull grows with the bounce.

Can I convert one into the other without the yearly list?

Not exactly. The gap depends on how spread out the yearly returns were, which a start value, an end value and a number of years do not record. CAGR from the two values is always available. The arithmetic mean needs the list.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.