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
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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
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7 . Counting and Probability 7 . 1 . Rules for Counting For computing complicated probabilities in equally likely scenarios, one often needs to count various kinds of outcomes given a group of objects. There are two important issues to resolve in most counting arguments. First, if an object is used once, can it be used again? In other words, is selection with or without replacement? A second issue is whether the order that the objects are selected makes any difference. This is the ordered versus unordered distinction. 85
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EXAMPLE : (i) Suppose license plates in a state are determined as three letters (A through Z) followed by three digits (0 through 9). In this case, if a letter is chosen for the first slot the same letter is allowed for the second and third slots. So selection is with replacement because, for example, AAB 235 is a valid license plate. Also, the order is important in the sense that BAA 235 is a different license plate than AAB 235. 86
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(ii) As a different example, suppose that a lottery requires a choice of six different numbers from 0 to 9. So there are 10 choices for the first digit, but only 9 for the second, 8 for the third, and so on. This is selection without replacement. Whether order is relevant depends on the lottery. If the winner must chose the same order that the numbers are drawn, then order is important. For example, 6,3,2,0,1,7 is considered different from 3,2,1,7,0,6 . Whether order matters affects the probability that one wins.
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81 APPLICATION Mutual Fund Investing Consider each funds...

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