Geometric Mean and the Nth root

[BCF00]

When calculating the Geometric mean for a portfolio and the period to be calculated is not in full years, for example, 2.5 years or 3 1/4 years. Is it correct to use 2.5 as the Nth power of the radical? for example,

You want to find the annualized return of a portfolio.

you multiply all the returns and find a product of 2.137
the time between the start date and the end date is 5.25 years
then to calculate the geometric return, 2.137 ^(1/5.25)-1 expressed as a %
equals 15.56% annualized rate of return.

What I am trying to understand is if it is correct to use a number that has a decimal or a fraction of a year in the Nth power..

 

[blamocur]

When calculating the Geometric mean for a portfolio and the period to be calculated is not in full years, for example, 2.5 years or 3 1/4 years. Is it correct to use 2.5 as the Nth power of the radical? for example,

You want to find the annualized return of a portfolio.

you multiply all the returns and find a product of 2.137
the time between the start date and the end date is 5.25 years
then to calculate the geometric return, 2.137 ^(1/5.25)-1 expressed as a %
equals 15.56% annualized rate of return.

What I am trying to understand is if it is correct to use a number that has a decimal or a fraction of a year in the Nth power..

This looks right to me, although I am not familiar with the conventions in finance. One way to look at it to find monthly returns, then use them to compute annual returns -- the result should be the same.

 

[JeffM]

Yes, there is nothing wrong mathematically with using a number that is not an integer as an exponent. Your 5.25 years is just a different way to say 21 quarters. And obviously you can calculate the average return per quarter. which in your example would be [imath]\sqrt[21]{2.137} - 1 \approx 0.03862...[/imath]. You can then turn that into an average annual return of [imath](1 + 0.03862...)^4 - 1 \approx 0.15563.[/imath]

When you use 5.25 as the exponent, you are just combining steps.

What I do not understand is why you go to the effort of multiplying twenty-one quarterly returns. Divide the final portfolio value by the initial portfolio value and take the root of that quotient.