# Geometric Average Return (GAR) Calculator

This handy geometric average return (GAR) calculator can be used with investments that undergo compounding over a number of timespans to calculate the average rate per period.

The geometric average return (GAR) is 5% per year.

^{ 1/1}− 1 = 5%

## 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 V_{0} growing to V_{n} over n years is:

**(1)** R_{a} = (V_{n} / V_{0})^{1/n} − 1

We may also make a calculation of the precise level of V_{2} in two years as we are aware that V_{0} = 500. Specifically,

**(2)** V_{2} = 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 r_{1} for year 1 and r_{2} for year 2.

We know that this is the case:

**(4)** (V_{2} / V_{0}) = (V_{2} / V_{1}) (V_{1} / V_{0})

Additionally we are aware that the annual return can be defined by **(V _{1} / V_{0}) = 1 + r_{1}** and

**(V**and so using substitution we can change the expression above to:

_{2}/ V_{1}) = 1 + r_{2}**(5)** V_{2} / V_{0} = (1 + r_{2}) (1 + r_{1})

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

**(6)** R_{a} = [(V_{2} / V_{1}) (V_{1} / V_{0})]^{1/2} − 1

**(7)** R_{a} = [(1 + r_{1}) (1 + r_{2})]^{1/2} − 1

The final expression represents the geometric average of r_{1} and r_{2}. If we substitute .06 for r_{1} and .14 for r_{2}, equation (7) gives us .0992, i.e., the correct answer.

In more general terms, if r_{1} represents the return for year 1, r_{2} 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 r_{1}, r_{2}, ..., r_{n}, which we find with this formula:

R_{a} = [(1 + r_{1}) (1 + r_{2}) ... (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 r_{1} = r_{2} = ... r_{n}. Nevertheless, should r_{1} and r_{n} be substantially different, we can get substantial variations in the results produced by the two methods.

## Geometric Average Return (GAR) Calculator Example

Certainly! Here's an example of using a table to calculate the Geometric Average Return (GAR) for a series of investment returns:

Year | Investment Return |
---|---|

2018 | 10% |

2019 | 5% |

2020 | 8% |

In this example, we have three different years with their respective investment returns. We will calculate the Geometric Average Return (GAR) for these returns.

To calculate the Geometric Average Return (GAR), you can use the formula:

GAR = ((1 + R1) * (1 + R2) * ... * (1 + Rn)) ^ (1/n) - 1

where R1, R2, ..., Rn represent the individual investment returns, and n is the number of returns.

For instance, for the given investment returns of 10%, 5%, and 8%, the calculation would be as follows:

GAR = ((1 + 0.10) * (1 + 0.05) * (1 + 0.08)) ^ (1/3) - 1

This equation simplifies to:

GAR ≈ (1.10 * 1.05 * 1.08) ^ (1/3) - 1

After evaluating the expression, the approximate Geometric Average Return (GAR) would be approximately 7.58%.

The Geometric Average Return (GAR) is a measure of the compounded annual growth rate over a series of returns. It takes into account the compounding effect and is useful in evaluating the overall performance of an investment over multiple periods. By calculating the GAR, investors can assess the average annual return achieved and compare it with other investments to make informed investment decisions.