What’s a Geometric Mean
Return?
What is a geometric mean return?
A geometric mean return is an average return that considers compounding and is the
standard metric for conveying return performance for investments.
When investment
professional refer to the average annual return, they are referring to the geometric average
annual return.
Calculating the geometric mean return
The calculation of the average annual return is straightforward: you calculate the geometric
average of the product of one plus the period returns.
Step 1: Calculate the return for each period
Year
Price at the end of
the period
1
$100
2
$110
3
$105
4
$112
Year
Price at the end of
the period
Return
1
$100
2
$110
$110-100
=0.10 or 10%
$100
3
$105
$105-110
=-0.0455 or -4.55%
$110
4
$112
$112-105
=0.0667 or 6.67%
$105
Step 2: Add 1.0 to each return
Year
Return
2
1.1000
3
0.9545
4
1.0667
This is an adjustment to consider that the investment’s value is compounded such that the initial
value is invested and the return is an adjustment to that value. For example, if you invest $1 and
earn 6% on that $1 during the period, the value at the end of the period is $1 x (1 + 0.06) =
$1.06.
Note: There are only 3 periods’ returns in this analysis because we have prices for four periods.
Because we need two prices to calculate a return, there are only three returns for this series.
Step 3: Multiply the 1+ return terms for the n periods
Product = (1.100) (0.9545) (1.0667) = 1.1200
This is the adjustment needed to compound values from one period to the next.
Step 4: Take the nth root
The 3rd root of 1.1200 = 1.12001/3 = 3 1.12 = 1.038499
This step involves the geometric mean of the compounded values over the n periods.
Step 5: Subtract 1.0
Average annual return = 1.038499 – 1.0 = 0.038499 or 3.8499%
The term 1.038499 is the value of $1 invested in the investment at the beginning of the three
periods of growth. Hence, the return is 1.038499 less 1.0.
An alternative
Another way of calculating this, since we have a starting and an ending value is to consider the
following:
PV = $100
FV = $112
n=3
and solve for i. The i in this calculation is the compounded growth rate over the three-year
period (a.k.a. the geometric average return). 1
i = 3.8499%
How does this compare with the arithmetic average?
The arithmetic average, which ignores the compounding of values, is 4.04%. It is recognized
that the arithmetic average return is biased and that the geometric average return is the most
appropriate to use in evaluating returns on investments. 2
1
Some rounding was used to simplify this example. The precise answers for both methods will be the same
in all cases, but you may see slight differences if you round in the calculation of the period returns.
2
Blume, M.E., “Unbiased Estimators of Long Run Expected Rates of Return,” Journal of the American
Statistical Association, Col. 69, No. 347 (September 1974), and Indro, D. C., and W. Y. Lee, “Biases in
Arithmetic and Geometric Averages as Estimates of Long Run Expected Returns and Risk Premia,” Financial
Management, Vol. 26, No. 4 (Winter 1997).