## GEOMETRIC MEAN RETURN

The geometric mean is used to determine the average compound growth rate (or average rate of return) over a given period.  The geometric mean is the most accurate method for determining average rates of return.

#### The Reason We Need The Geometric Return

To see why we need to use the geometric mean return rather than an arithmetic return, consider the following example.  Assume that the price of a stock is \$100.  Then, one year later, the price of the stock has fallen to \$50.  However, in the following year, the price rises again to \$100.  What is the average rate of return per year on the stock for the two-year period?

If we use an arithmetic return, we would say that the stock fell in price 50% during the first year and rose 100% in price the second year.  Therefore the average rate of return must be 25% per year, i.e. (-50% + 100%) divided by 2 years.

Obviously, this is not correct.  The average rate of return is zero percent per year since the price ended where it began two years earlier.  So we need a better method to calculate the average rate of return – that method is the geometric mean return.

#### Methodology

To calculate the geometric mean return, we use a five-step process:

1. Determine the rate of return for each time period,
2. Add one to each of the returns (the result is called a holding period return),
3. Multiply each of the holding period returns together (this is called “chain-linking” the returns),
4. Take the root of the product in step 3.  The root number is equal to the number of time periods.
5. Subtract one from the result.

#### An Example

For example, assume that we have already calculated the return on a common stock for each of the last four years.  We used this equation to measure the rate of return for each year:

Let us assume that those returns are as follows:

Year Annual Return
1st year 5.0%
2nd year -3.0%
3rd year 12.0%
4th year 10.0%

Steps 2 - 4 instruct us to add one to each value, take the product, and then take the nth root, where n is the number of time periods.

Annual Return Holding Period Return
5.0% 1.05
-3.0% 0.97
12.0% 1.12
10.0% 1.10
Product  =  1.254792

(If your calculator has a yx (i.e. y-to-the-x) function, the above calculation is very easy to do.

E.g.:  y = 1.254792 and x = 1/4 or 0.25)

Finally, subtract one from the result.

Geometric Mean  =  1.058383 - 1
= 0.058383 or 5.84% (rounded to two decimal places)

Now, let’s look at another example, but add one more column:  a column that shows the value of \$1.00 invested at the beginning of the period.

 Year Return HPR Value of \$1 at the end of the period 1st year 2.0% 1.02 \$1.020000 2nd year 8.0% 1.08 1.101600 3rd year -1.0% 0.99 1.090584 4th year 10.0% 1.10 1.199642

Notice the ending value (\$1.199642) at the end of the 4-year period.

Now forget that procedure for a moment and let us solve for the geometric mean return using the same data:

#### A Shortcut Using the Beginning and Ending Values

Notice that the value under the root symbol (1.199642) is the same as the value that a dollar has grown to at the end of the 4-year period.  This fact often gives us a shortcut to solving for the geometric mean.  If we only know the value at the end of the period and the value at the beginning of the period, we can divide the ending value by the beginning value.  This gives us the value that a dollar invested at the beginning of the period would have grown to by the end of the period.  Taking the root of this and subtracting one gives us the average rate of growth per period.

For example, the current price of a stock is \$15.00 per share.  You expect the stock to double in price over the next five years.  The stock pays no dividend.  What is the average rate of growth per year?

The geometric mean thus offers us a very short, convenient, and accurate method of calculating the average compound rate of return over a given period.