An Intuition Behind Geometric Mean

I invested in a stock that had a 5% return in the first year. I received a better 8% return in the second year and a whopping 20% return in the third year. Last year, I received a return of 3%. Overall, I am happy with my investments. I told my friend about my returns, and she was happy that I made an average of 9% return on my stock.

I went back home and started looking at my returns. The current value of my stock was $3,504.06, and I started with $2,500. A healthy profit in four years. Just to check if everything was okay, I calculated again using the average interest of 9% and compounded it annually for four years. The result was $3,528.90!

Where did my $24 go? Did I miss something or was I cheated?

Well, the $24 went nowhere since it was never there. The major flaw in my calculation was the average rate. It had nothing to do with weightage or control factors that people often get wrong. The issue was my use of the wrong mean value. The mean I used was the Arithmetic mean, which is the sum of all values divided by the number of elements. However, what I should have used was the Geometric mean, which differs slightly. Instead of summing all the numbers, it takes the product of all the numbers and then takes the n-th root of that product, where n is the number of elements.

So what is Geometric mean and why do we use it?

Imagine a rectangle with a length of 10 meters and a breadth of 2.5 meters. What would be the length of a square with the same area?

The area of the rectangle is 10×2.5=25. Since all sides of a square are equal, and if the area of that square is supposed to be equal to the area of the rectangle, then the side length of the square would simply be the square root of 25, which is 5.

5 is the Geometric Mean of 10 and 2.5.

Multiply 10*2.5 and square root it, it’s the geometric mean between the numbers.

Let’s do some simple math to dive deeper into it.

In Arithmetic Mean, all the elements are summed and then divided by the number of elements. The Geometric Mean, on the other hand, differs slightly. Instead of summing all the numbers, it takes the product of all the numbers and then takes the n-th root of that product, where n is the number of elements.

For example, the Arithmetic mean of 3,5,7 is 3+5+7 divided by 3, which is 5. The ​Geometric mean of the same number is 3*5*7 to the power 1/3. Or,

which is 4.718.

Going back to the return on stock

So, in the case of my return on portfolio, the Arithmetic mean is a wrong estimate of average return. This is primarily because the return value is being compounded, rather than a constant return.

Let us calculate the Geometric mean first. The return each year is 1.05, 1.08,1.2 and 1.03. The Geometric mean would be the product of all four numbers and 4th root of the product.

Geometric Mean= (1.05 × 1.08 × 1.20 × 1.03​ ) ^ (1/4) ≈ 1.0887

Rate of return = 8.87 %

Arithmetic Mean = (1.05+1.08+1.20+1.03) / 4 = 1.09

Rate of return = 9%

Now, plunging the values in the Excel formula, we get

Now, you can see that the average of 9% overestimated the returns. While the calculation for the Arithmetic mean is simple, it does not factor the dependency from the compounding effect. This leads to the wrong estimate. The Geometric mean is useful when series are not independent of each other.

The use of the geometric mean is prevalent in finance and business. Comparing yield rates, returns between different stocks, and compound interest calculations all require a solid understanding of the geometric mean.

I hope this simple demonstration illustrates the difference between Arithmetic mean and its lesser-known cousin Geometric Mean. Next time someone tells you about their return on investments, make sure to do the math.