The geometric mean return formula is a way to calculate the average rate of return per period on investment that is compounded over multiple periods.

It allows understanding the effect of compounding of a portfolio of financial instruments (investments). Compounding is a process of reinvesting interest or capital gains to generate more earnings. In other words, compound interest is interest on interest.

An investment then generates profit not only from its initial principal but also from interest or capital gains from the previous periods. By using a compound interest an investor gets faster earnings than by using simple interest, where interest is not added to the original value of an investment, so there is no compounding.

The more frequent compounding is, the faster the investment increases in its value. An extreme case in which the time gap between the compounds drops to zero is called continuous compounding.

How to calculate the average rate of return using geometric mean return?

Geometric mean return is a more complicated method of calculating the average rate, but it’s more accurate than the arithmetic one. It is calculated as:

$Geometric\;return = \sqrt[n]{ (1 + r_{1} ) \times (1 + r_{2} ) \times (1 + r_{3} ) \times ... \times (1 + r_{n} ) } - 1$

Where r is the rate of return and n is the number of periods.

Geometric mean return and arithmetic mean return

Geometric mean return will always be a bit lower than the arithmetic one. Arithmetic mean return is a simpler and less accurate method because it doesn’t take compounding into account. It’s calculated by dividing the sum of returns for each period by the total number of periods.

$Arithmetic\;return = \frac{ r_{1} + r_{2} + r_{3} + ... + r_{n} }{n}$

Where r is the rate of return and n is the number of periods.

For example, the arithmetic average of rates of 5%, 6% and 7% is calculated as:

$Arithmetic\;return = \frac{5 + 6 + 7}{3} = 6\%$

Why is geometric mean return important?

Geometric mean return is a method that allows us to calculate the average rate of return on investment (or portfolio). The main advantage of this method is the fact, that we don’t have to know the original principal amount, geometric mean return method is completely focused on the rate of return.

That allows us to easily compare two or more different investments and choose the most profitable one.

Examples of the average rate of return calculation using geometric mean return

Example 1. Let’s assume $10,000 in the money market earns: • 10% in the first year, • 15% in the second year, • 5% in the third year • 7% in the fourth year. Geometric mean would be: $Geometric\;return = \sqrt[4]{ (1 + 0.10 ) \times (1 + 0.15 ) \times (1 + 0.05 ) \times (1 + 0.07 ) } - 1$$Geometric\;return = \sqrt[4]{ 1.4212 } - 1$$Geometric\;return = 1.0919 - 1 = 0.0919 = 9.19\%$ That means the average return from this investment (taking compounding into account) is 9.19%. The return every year will be equal to 9.19%. • Year 1: • Interest =$10,000 × 9.19% = $919 • Principal =$10,000 + $919 =$10,919
• Year 2:
• Interest = $10,919 × 9.19% =$1,003.46
• Principal = $10,919 +$1,003.46 = $11,922.46 • Year 3: • Interest =$11,922.46 × 9.19% = $1,095.67 • Principal =$11,922.46 + $1,095.67=$12,032.13
• Year 4:
• Interest = $12,032.13 × 9.19% =$1,105.75
• Principal = $12,032.13 +$1,105.75 = $13,137.88 The final principal is$13,137.88.

If we use the arithmetic mean return and calculate the average rate of return by simply dividing the sum of rates by 4, the result will be incorrect:

$Arithmetic\;return = \frac{10 + 15 + 5 + 7}{4} = 9.25\%$

Example 2. An investor buys a stock. The initial price of the stock is \$50. The rates of return are:

• 18% in the first year,
• 5% in the second year
• 11% in the third year.
$Geometric\;return = \sqrt[3]{ (1 + 0.18 ) \times (1 + 0.05 ) \times (1 + 0.11 ) } - 1$$Geometric\;return = \sqrt[3]{ 1.3753 } - 1$$Geometric\;return = 1.1121 - 1 = 0.1121 = 11.21\%$

That means the average return from this investment (taking compounding into account) is 11.21%.

If we used the arithmetic mean return formula, the result will be incorrect as well:

$Arithmetic\;return = \frac{18 + 5 + 11}{3} = 11.33\%$