What is an example of a problem using geometric mean?

Answer 1

The geometric mean, by definition, is the nth root of the product of the n units in a data set. For example, the geometric mean of 5, 7, 2, 1 is (5x7x2x1)1/4 = 2.893 The geometric mean is commonly applied in the finance sector to calculate average rates of return where dividends are reinvested (e.g. compound interest).

Let's say you invest an amount of money. In the first year your investment increases 8%, in the second year it returns a further 12% but in the third year it goes down 7%. The "average rate of return" is determined using the geometric mean: Average rate of return = (1.08 x 1.12 x 0.93)1/3 = 1.04 (i.e. average 4% per annum increase over the three years).

You can use a slight variation of the formula to calculate average compound interest rate if you know the initial value and final value of your investment:

Compound interest = (final value / initial value)1/number of years invested

For example, let's say you invest $1000 and after five years it's worth $1200 (an increase of 20%). The average rate of return is not the arithmetic mean (20%/5=4.0%). The geometric mean must be used as the return is compounded:

Compound return = (1200/1000)1/5 = 1.037 (i.e. average of 3.7% per annum compounded annually).