slides_1_probability

# 81 application mutual fund investing consider each

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APPLICATION : Mutual Fund Investing. Consider each fund’s performance relative to holding a balanced market basket. Start off by assuming each fund has a 50% chance of outperforming the market in each year. So P fund k outperforms market in year t ) 1/2. In addition, assume that a fund’s relative performance is independent from one year to the next. (In other words, we assume that fund managers are no better than holding a market portfolio, and that relative performance in one year is independent of past performance.) 82
Now, consider a group of mutual funds over a 10-year period. Let B k be the event that fund k outperforms the market in all 10 years. The probability that any fund k outperforms the market in all 10 years is P B k 1 2 10 1 1,024 for all k . But what is the probability that out of a population of, say, 1,500 mutual funds, at least one fund outperforms the market in all 10 years? 83

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Assume that the events B 1 , B 2 ,..., B 1,500 are mutually independent. We want P B 1 B 2 B 1,500 1 P  B 1 B 2 B 1,500 c 1 P B 1 c B 2 c B 1,500 c 1 P B 1 c P B 2 c  P B 1,500 c 1 1,023 1,024 1,500 1 .231 .769 In other words, there is a better than 75% chance that at least one fund outperforms the market in every year. A special case of calculations with the binomial distribution. 84