# Geometric Mean

Applications in finance and portfolio management

## What is Geometric Mean?

The geometric mean is the average growth of an investment computed by multiplying n variables and then taking the n square root.  In other words, it is the average return of an investment over time, a metric used to evaluate the performance of an investment portfolio.

### Why use Geometric Mean?

The arithmetic mean is the calculated average of the middle value of a data series; it is accurate to take an average of independent data, but weakness exists in a continuous data series calculation.

Example: An investor has annual return of 5%, 10%, 20%, -50%, and 20%.

Using the arithmetic mean, the investor’s total return is (5%+10%+20%-50%+20%)/5 = 1%

By comparing the result with the actual data shown on the table, the investor will find a 1% return is misleading.

 Year Starting Equity Return % Return \$ Closing equity 1 \$1,000 5% \$50 \$1,050 2 \$1,050 10% \$105 \$1,155 3 \$1,155 20% \$231 \$1,386 4 \$1,386 -50% -\$693 \$693 5 \$693 20% \$138.6 \$831.6

The actual 5 year return on the account is (\$831.6 – \$1,000)/\$1,000 = -16.84%

The geometric mean is used to tackle continuous data series which the arithmetic mean is unable to accurately reflect.

### Geometric Mean Formula for Investments

#### Geometric Mean = [Product of (1 + Rn)] ^ (1/n) -1

Where:

• Rn = growth rate for year N

Using the same example as we did for the arithmetic mean, the geometric mean calculation equals:

##### 5th Square Root of ((1 + 0.05)(1 + 0.1)(1 + 0.2)(1 – 0.5)(1 + 0.2)) – 1 = -0.03621

Multiply the result by 100 to calculate the percentage. This results in a -3.62% annual return.

### Example of the Geometric Mean in Finance

Return, or growth, is one of the important parameters used to determine the profitability of an investment, either in the present or the future. When the return or growth amount is compounded, the investor needs to use the geometric mean to calculate the final value of the investment.

Case example: an investor is offered two different investment options. The first option is a \$20,000 initial deposit with a 3% interest rate for each year over 25 years. The second option is a \$20,000 initial deposit, and after 25 years the investor will get \$40,000. Which investment should the investor choose?

The investor will use the future value or the present value formula, which is derived from the geometric mean. Here are the formulas used to calculate each:

#### Future value = E*(1+r)^n                           Present value = FV*(1/(1+r)^n)

Where:

• E = Initial equity
• r = interest rate
• FV = Future value
• n = number of years

The investor will compare both investment options by analyzing the interest rate or the final equity value with the same initial equity.

Option 1 – Future value

Future value = E*(1+r)^n

= \$20,000*(1+0.03)^25

= \$20,000*2.0937

= \$41,875.56

Option 2 – Present value

Present value = FV*(1/(1+r)^n)

\$20,000 = \$40,000*(1/(1+r)^25)

0.5 = (1/(1+r)^25)

0.973 = 1/(1+r)

r = 0.028 or 2.8%

From the calculation, the investor should choose option one because it is a better investment option based on the following:

It offers a better future value of \$41,875.56 vs. \$40,000 or higher interest rate of 3% vs. 2.8%.

### More resources

We hope this has been a helpful guide to understanding geometric mean as it applies to finance and portfolio management.  To keep learning, we recommend exploring these relevant resources below:

• What does a portfolio manager do?