Option Trading Blog

In this post, I’ll try to explain the different methods to assess the performance of fund managers. How do you choose a fund? If you, the investor, is risk indifferent, you would probably choose the fund that has the biggest return expectancy. How do you calculate the expected return?

Let’s assume you are to choose between two fund managers, and you collect the historical returns of the last four years:

2002 2003 2004 2005

10% -20% 15% 5% : A

-5% 10% 25% 0% : B

Which fund would you choose? If we assume that past performance implies future performance (which will be discussed later), it depends on how we calculate the mean. We could choose to calculate with either arithmetic mean or geometric mean.

For the arithmetic mean:

For the second fund:

According to the arithmetic mean, both funds achieve the same mean, hence we should be indifferent in choosing any of which.

Let’s examine now the geometric mean:

The geometric mean for the second fund is:

According to the geometric mean, the second fund achieved higher mean. The question is now, which method is better then?

Here is a rule of thumb: The geometric mean is always lower then the arithmetic mean. Moreover, the bigger the volatility of returns, the lower the geometric mean would be. Now we know why the first fund had lower returns than the second fund. Since both funds achieved the same arithmetic mean, we expect that both geometric means would be lower than the arithmetic mean. But since the first fund had more volatile returns, it has a lower mean.

We still haven’t answered which way is more “correct”. Let’s take a practical view. Suppose we had invested 100,000$ in both funds four years ago. What would be our compounded returns?

For the first fund, the way to calculate it would be

For the second fund:

Notice that this method is essentially the same as computing a geometric mean. This means that if you had invested in the second fund four years ago, and cashed out after four years, you would receive a bigger return than in the first fund!

It is tempting to conclude now that the geometric mean is the way to go isn’t it? But let’s assume that we change our investment policy. Suppose you invest an initial sum of 100,00$ but now if after one year the fund achieved a positive return, you cash out the return, and if it is a negative return, you add enough money to update the sum to 100,000$ again. At the end of 4 years, you cash out. What is the return in this case?

This means we have to the IRR of such an investment.

For the first fund, the calculation yield a IRR of 2.44%, while for the second, a IRR of 2.35%. Going back to the original question, which fund manager would you choose for a future investment? If we assume that there is no correlation between returns in each year, meaning that the order of returns doesn’t matter, using the arithmetic mean is a valid method. This means that both funds achieved 2.5% and hence we would be indifferent between them.

If you ask though the question who got better returns in the past, it depends in which method. In the first one, when the investor put the money for four years and cashed it out only at the end? or the second one who either cashed out each year or updated the sum to match the initial investment.

If you notice, we haven’t discussed the topic of risk. That is to come in the second part.