Arithmetic vs geometric average return (tutorial #10)

Now that we know about multi-period returns, we can discuss average returns. Think of an investor who bought a stock for $100 and sold it after 2 years for $130. For the sake of simplicity, let's assume the stock did not pay any dividends during the investor's holding period and there were no stock splits. Then, the investor's holding period return is ($130 - $100) / $100 = 30%.

Even though the investor's holding period was 2 years, he clearly could observe returns more frequently. For instance, he could check the stock's price each month to calculate monthly returns. After 2 years, he would have 2 * 12 = 24 monthly return observations. Then, the investor could pose the following two questions:

1. What was my average return per month so that I realized a return of 30% over the past 2 years?
2. What was the stock's return in an average month over the past 2 years?

It is important to understand that these two questions are different. Answering the first and the second questions are equivalent to calculating the geometric average return (or geometric mean return) and the arithmetic average return (or arithmetic mean return) respectively.

Let's begin with the first question. We need an average monthly figure R_ga such that compounding it 24 months will be equivalent to a 24-month holding period return of HPR₂₄ = 30%. Let us use the multi-period return formula to do this:

HPR₂₄ = Π ( 1 + R_ga ) - 1 = 30%  where   t ∈ [1, 24]

Then, we have:

               Π ( 1 + R_ga ) = 1.3  where   t ∈ [1, 24]  

( 1 + R_ga )²⁴ = 1.3                

         1 + R_ga  = (1.3)^1/24            

      1 + R_ga  = 1.011                  

 R_ga  = 1.1%

This means that the holding period return of 30% over 2 years is equivalent to a return of 1.1% in each month. In general, the relationship between the holding period return HPR_T and the geometric average return R_ga over T periods is:

R_ga  = (1 + HPR_T )^1/T - 1 

Let's now turn to the second question, which is much easier to answer. To find the arithmetic average return R_aa, we simply add the returns over the past 24 months up and then divide the sum by 24:

R_aa = Σ R_t / 24  where   t ∈ [1, 24]

Σ is the summation symbol. It tells us to add the returns from the first period t=1 to the last period t=24. The general formula for the arithmetic average return when there are T periods is:

R_aa = Σ R_t / T  where   t ∈ [1, T]

Arithmetic average return and geometric average return figures will (almost) always be different than each other. It is important to understand that these figures do not conflict with each other. They simply answer the two different questions stated above.

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