Geometric Average Return (GAR) Calculator

Understanding Geometric Average Return

In statistical and business terms, a geometric average return (a.k.a. geometric mean return) represents the rate of return on investment per year, averaged over a specified time period. When assets increase in value year on year, a geometric average return will let you know what the increase in value would look like if represented by an annual interest rate.

Geometric Average vs. Arithmetic Average

Imagine you put $500 into zero-coupon bonds for one year with 6% interest. You take this profit and reinvest at 14% for the next year. How can you calculate what your average return is for the two years together?

The simple way of doing this is to average out the interest rates for the two years, i.e. 6% + 14% = 20%, divided by two years = 10%. However, this does not take account of the compound interest factor which comes into play when the profits and in the first year are reinvested for the second year. This means the arithmetic average must be checked.

The "brute force" way of calculating average annual returns, if we assume that compounding takes place annually, of initial sum V₀ growing to V_n over n years is:

(1)  R_a = (V_n / V₀)^1/n − 1

We may also make a calculation of the precise level of V₂ in two years as we are aware that V₀ = 500. Specifically,

(2)  V₂ = 500 (1.06) (1.14) = 604.2

Thus we know in this instance:

(3)  R_a = (604.2 / 500)^1/2 − 1 = .0992 = 9.92%

So here the arithmetic mean is larger than the real annual average return, as we are aware that .0992 is accurate as it was derived from a calculation using an accurate definition of the annual return.

We could create a solution for average annual returns for a two-year time span employing the actual annual rates, 6% and 14%, or r₁ for year 1 and r₂ for year 2.

We know that this is the case:

(4)  (V₂ / V₀) = (V₂ / V₁) (V₁ / V₀)

Additionally we are aware that the annual return can be defined by (V₁ / V₀) = 1 + r₁ and (V₂ / V₁) = 1 + r₂ and so using substitution we can change the expression above to:

(5)  V₂ / V₀ = (1 + r₂) (1 + r₁)

In the light of equation (1), we can arrive at:

(6)  R_a = [(V₂ / V₁) (V₁ / V₀)]^1/2 − 1

(7)  R_a = [(1 + r₁) (1 + r₂)]^1/2 − 1

The final expression represents the geometric average of r₁ and r₂. If we substitute .06 for r₁ and .14 for r₂, equation (7) gives us .0992, i.e., the correct answer.

In more general terms, if r₁ represents the return for year 1, r₂ represents the return for year 2 and r_n represents the return for year n, then an accurate formula for calculation of average annual returns, making an assumption that profits are continuously reinvested year on year, is the geometric average of r₁, r₂, ..., r_n, which we find with this formula:

R_a = [(1 + r₁) (1 + r₂) ... (1 + r_n)]^1/n − 1

Recognizing the Difference

The gap between geometric average and arithmetic average may appear negligible in this example (but 8 basis points can sometimes be very significant). Actually, the two outcomes would be identical if r₁ = r₂ = ... r_n. Nevertheless, should r₁ and r_n be substantially different, we can get substantial variations in the results produced by the two methods.