Suppose I invested 100 into a risky mutual fund that has returned 50% in year 1 [that is my investment after the first year had grown to 100*(1+0.5)=100*1.5=150], 10% in year 2 and 60% in year 3. Therefore, over three years, my investment has grown to:
100*1.5*1.1*1.6 = 264
What is the average annual rate of return on my investment over three years?
That is, what is the rate of return, R, such that 100*R*R*R = 100*R3 = 264?
If we try to calculate this average rate of return as an arithmetic mean, we would get
m = (1.5+1.1+1.6)/3 = 1.4
Applying this arithmetic mean to the initial investment would result in:
100*1.43 = 274.4,
which is, of course, different from 264. How do we then calculate the average rate of return? We use the concept of a geometric mean. The geometric mean of a sequence of numbers x1, x2, … xN is defined as
GM = (x1*x2*…*xN)1/N .
In our example above, GM = (1.5*1.1*1.6)1/3 = 1.382 or 38.2%. Indeed, we can easily see that 100*1.3823 = 264.
The formula for the geometric mean is easily derived. Given the annual returns
x1, x2, … xN , after N years, the investment of 100 would grow to 100* x1*x2*…*xN. By definition, the geometric mean, GM, is a number such that
100*(GM)N=100*x1*x2*…*xN,
and, therefore, GM can be calculated as GM = (x1*x2*…*xN)1/N .
Finally, we can use the geometric mean to calculate average annual returns if we do not know each annual return, but know the initial investment and the final amount. For example, if we know that the initial investment is 100 and the amount after three years has become 264, the average annual return is:
GM=(value at end of period/value at beginning of period)1/N = (264/100)1/3 = 1.382 or 38.2%.