Question 4. Much is made of the fact that certain mutual...

Question 4. Much is made of the fact that certain mutual funds outperform the market year after year. For concreteness, consider a 10-year period and let the population be the 4,170 mutual funds reported in The Wall Street Journal on Jan 1, 1995. By saying that performance relative to the market is random, we mean that each fund has a 50-50 chance of outperforming the market in any year and that performances is independent from year to year. 4.1. If performance relative to the market is truly random, what is the probability that any particular fund outperformans the market in all 10 years? 4.2 Find the probability that at least one fund out of 4,170 funds outperforms the market in all 10 years.

Transcript

00:01 In this question we need to work out what is the probability that a fund outperforms the market in 10 years in a row.

00:09 So if the probability of outperforming the market is truly random, then that probability is a half.

00:16 And we have 10 years or 10 independent trials of this outcome.

00:25 So n equals 10 trials.

00:27 Then the number of years on which we outperform the market is going to be binomially distributed with n equals 10 and the probability of outperforming the market is a half, which means that the probability that the number of years a fund outperforms the market is equal to some number x.

00:50 Actually, let's call this x, and this is the number of years any fund outperforms the market.

01:07 Then the probability that this random variable x takes some value x is 10 choose x times one half to the power of 10.

01:16 Now what's the, so question 4 .1 is what's the probability that any particular fund outperforms the market in all 10 years? so what's the probability that x is equal to 10? well that's going to be 10 choose 10, which is 1 times 1 half to the power of 10, which is 1 over 1024, or approximately 9 .766 times 10 to the minus 4.

01:50 Okay, so but what if we have, if we have the number of funds is 4 ,170, then we want to find out, what's the probability that any fund outperforms the market in all 10 years? well, let's say why is the number of funds outperforming the market in all 10 years? then y will have a binomial distribution with 4 ,170 trials and 1 over 1 ,070 trials and 1 over 1 over over 1 ,024 chance of success in each trial.

02:44 What's the probability that at least one fund outperforms the market? well, what's the probability y is greater than or equal to 1? well, that's 1 minus the probability that y is equal to 0, which is 1 minus 4 ,172 0 times 1 over 102 to the power of 0 times 1 02 3 over 1024 to the power of 0 to the power of 0, 4 ,170...

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