Question

Which is not right?

When rates of return are not the same for all years, the arithmetic mean will always be higher than the geometric mean.

When rates of return are the same for all years, the arithmetic mean and the geometric mean will be equal.

Annual geometric mean is not the same as annual holding period yield

While the arithmetic mean is best used as an “expected value” for an individual year, while the geometric mean is the best measure of an asset’s long-term performance

Answer 1

Consider example of rates of return being 1%,3% and 5%.

Now lets evaluate the first statement:

1. AM = (1.01+1.03+1.05)/3 = 3 which is the addition of all records divided by the number of records and GM = (1*3*5)^1/3 = 2.46 which is the nth root of the product of n records.

However, considering the rates of return are 1%, -3% and 5%.

AM = (1-3+5)/3 = 1 while the GM = (1*-3*5)^1/3 = -2.44

The above evaluation shows that the first statement is right.

Let us evaluate the second statement.

2. Consider example of rates of return being 3,3 and 3 for last three years.

AM = (3+3+3)/3 = 3 and GM = (3*3*3)^1/3 = 3

The above evaluation shows that the second statement is right.

Let us evaluate the third statement.

Annual Geometric mean is the is the average rate of return of a set of values calculated using the products of the terms and taking the nth root of the product of all values. It is calculated as below:

Annual Geometric mean = [(1.1*1.3*1.5)^1/3] - 1= 28.64% when the annual rate of return on an asset/ portfolio are 10%,30% and 50% .

Annual Holding period return is calculated on the basis of total returns and is particularly useful for comparing returns between investments held for different periods of time. It is calculated as below using the above example:

Annual Holding period return = [(1.1*1.3*1.5)^1/3] - 1= 28.64% when the holding period rate of return on an asset/ portfolio are 10%,30% and 50%.

The Annual geometric mean is the same as annual holding period yield when the annual rate of return and holding period returns are the same. Else, the annual geometric mean and annual holding period return will not be equal.

Hence, the third statement is right.

The fourth statement is right as the arithmetic mean is best used as an “expected value” for an individual year, while the geometric mean is the best measure of an asset’s long-term performance