An investment’s return is the profit it earns as a percentage of the initial investment value. An excess return is the return an investment earns in excess of the return of a benchmark, which is another investment with which you compare the performance of your investment. There are two types of excess returns. The arithmetic excess return is the difference between the investment’s return and the benchmark’s return. The geometric excess return measures an investment’s excess return as a percentage of the benchmark’s final value.

Determine the values of an investment and the values of a benchmark at the beginning and end of a period over which you want to calculate the investment’s excess returns. Use a benchmark that represents a broad measure of returns, such as a stock or bond index, for the asset class in which you are calculating excess returns. For example, assume an investment’s beginning and ending values were $1,000 and $1,200, respectively. Also assume a benchmark’s beginning and ending values were $1,150 and $1,250, respectively.

Subtract the investment’s and the benchmark’s beginning values from their respective ending values to determine the change in value for each. In the example from the previous step, subtract $1,000 from $1,200 to get a $200 change in value of the investment. Subtract $1,150 from $1,250 to get a $100 change in value of the benchmark.

Divide the investment’s and the benchmark’s changes in value by their respective beginning values to determine the return of each. In this example, divide $200 by $1,000 to get 0.2, or a 20 percent return for the investment. Divide $100 by $1,150 to get 0.087, or an 8.7 percent return for the benchmark.

Subtract the benchmark’s return from the investment’s return to calculate the arithmetic excess return of the investment. In this example, subtract 0.087 from 0.2 to get an arithmetic excess return of 0.113, or 11.3 percent. This means that the investment’s return was 11.3 percent greater than the benchmark’s.

Substitute the return values into the geometric return formula, [(1 + R)/(1 + B)] - 1, in which R represents the investment’s return and B represents the benchmark’s return. In this example round .087 up to .09 and substitute the values to get [(1 + 0.2)/(1 + 0.09)] - 1.

Solve the formula to determine the geometric excess return of the investment. Continuing with the example, calculate the numbers in parentheses to get (1.2/1.09). Divide 1.2 by 1.09 to get 1.1. Subtract 1 from 1.1 to get a geometric excess return of 0.1, or 10 percent. This means that the investment grew 10 percent more than it would have if it were invested in the benchmark.

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