# Geometric Mean 2

### From Homeworkwiki

**1. 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%. What is the "average rate of return" ?**

**Solution:** The "average rate of return" is determined using the geometric mean.

The geometric mean is commonly applied in the finance sector to calculate average rates of return where dividends are reinvested (e.g. compound interest).

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).

**2. You invest $1000 and after five years it's worth $1200 (an increase of 20%). What is the compound interest return?**

**Solution:** The geometric mean must be used as the return is compounded:

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

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