APPLICATION: Mutual Fund Investing. Consider each fund’sperformance relative to holding a balanced market basket. Start off byassumingeach fund has a 50% chance of outperforming the market ineach year. SoPfundkoutperforms market in yeart)1/2.In addition,assumethat a fund’s relative performance is independentfrom one year to the next. (In other words, we assume that fundmanagers are no better than holding a market portfolio, and that relativeperformance in one year is independent of past performance.)82
∙Now, consider a group of mutual funds over a 10-year period. LetBkbe the event that fundkoutperforms the market in all 10 years. Theprobability that any fundkoutperforms the market in all 10 years isPBk121011,024for allk.∙But what is the probability that out of a population of, say, 1,500mutual funds, at least one fund outperforms the market in all 10 years?83
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∙Assume that the eventsB1,B2,...,B1,500are mutually independent.We wantPB1B2…B1,5001−PB1B2…B1,500c1−PB1c∩B2c∩…∩B1,500c1−PB1cPB2cPB1,500c1−1,0231,0241,500≈1−.231.769In other words, there is a better than 75% chance that at least one fundoutperforms the market in every year.∙A special case of calculations with the binomial distribution.84