Question

A fund manager has claimed that their portfolio outperforms the market average return in 90% of cases. To test this claim, you have checked the monthly performance of their portfolio over the last three years. Assuming that the performance of each month is independent, you have found that only 86% of the months showed a higher return from the fund manager's portfolio compared to the market return. Given the fund manager's claim, the following questions arise: a. What is the probability that, according to the fund manager's claim, only 86% or less of the months in the past three years his portfolio beat the market? Answer: b. If you were to check the monthly performance of the fund manager's portfolio over the past fifteen years and find that it still outperforms the market in only 86% of the months, what is the probability that the fund manager's claim is true?

Answer 1

Using a binomial distribution, we can calculate the probability of the fund manager's portfolio outperforming the market in only 86% of the months over a three-year period and similarly over fifteen years. These calculations typically require statistical software. The longer the underperformance at 86% continues, the more unlikely the claim of 90% success becomes.

Step-by-step explanation:

To address the question of how to calculate the probability that the fund manager's portfolio outperforms the market average return in only 86% or less of the months in the past three years, we need to use statistical methods. The fund manager's claim that the portfolio beats the market 90% of the time suggests a binomial distribution of outcomes, with a 'success' being outperforming the market in any given month.

For a three-year period, with 12 months per year, there are a total of 36 months. If each month is an independent event, and given the portfolio's actual performance of 86% over 36 months, we would calculate the probability using the binomial formula or binomial distribution table. However, this calculation can be complex and typically requires the use of statistical software or a calculator that can handle binomial probabilities.

As for a 15-year period, the calculations would be similar, but the number of months would increase to 180. Again, the binomial distribution would be applied. However, over a longer period, if the fund manager still only outperforms the market 86% of the time, it becomes increasingly unlikely that the manager's claim of 90% success is true. The likelihood can be assessed using the same binomial distribution approach.

It's important to note that this approach assumes the performances of each month are independent and identically distributed, which might not always be the case in real-world scenarios. Moreover, it's well-understood in the field of finance that beating the market consistently is difficult, and many professional investors do not manage to do so over the long term.