1. Every year, you observe the performance of 100 different mutual funds. Specif
ID: 3062531 • Letter: 1
Question
1. Every year, you observe the performance of 100 different mutual funds. Specif- ically, you observe whether they perform better than the S & P 500 or worse (a) If the mutual fund performance is determined entirely by luck, rather than skill of the manager, each mutual fund has only a 50% probability of beating the S & P 500 each year. If performance is driven by luck, what is the probability that at least one fund will beat the market every year for 10 years? (b) After 15 years? (c) Suppose you are not sure whether skill affects mutual fund performance or not. You notice that five of the 100 funds have beaten the market every year for the last five years. What is the probability of observing at least that many funds beating the market every year for five years, if performance is determined entirely by luck? (d) Suppose you see that one of the 100 funds has beaten the market every year for the last x years. How high does x have to be for this to have a less than 5% chance of occurring if fund performance is determined entirely by luckExplanation / Answer
a. The probability that a fund beats the market for 10 years = 0.510
The probability that a fund doesn't beat the market for 10 years = 1-0.510
The probability that no fund will beat the market every year for 10 years = (1-0.510)100
The probability that at least one fund will beat the market for 10 years = 1 - Probability that no fund will beat the market every year for 10 years = 1 - (1-0.510)100
b. Similarly, for 15 years, we have the required probability = 1 - (1-0.515)100
c. The probability of a fund beating the market for the next 5 years = 5/100 = 0.05
So, the probability of 5 or more funds beating the market for next 5 years = 1 - P(0, 1, 2, 3 or 4 funds beating the market for next 5 years) = 1 - 100C0(0.95)100 - 100C10.05*(0.95)99 - 100C2*(0.05)2(0.95)98 - 100C3(0.05)3(0.95)97 - 100C4(0.05)4(0.95)96
d. The probability of a fund beating the market every year for x years = 0.5x
The probability of only one out of 100 funds beating the market every year for x years = 100C10.5x(1-0.5x)99
So, 100C10.5x(1-0.5x)99 < 0.05
0.5x(1-0.5x)99 < 0.0005
Solving we get max value of x=4
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