Individual investors face a daunting choice of thousands of stock and bond funds and fund managers. The following (stylized) spreadsheet example tries to distinguish superior fund managers from the throng of average managers based on their past track records of performance. Suppose there are three types of mutual fund managers: Superior (20 percent of the population), Average (60 percent), and Inferior (20 percent). In any six-month period, superior managers earn positive returns 70 percent of the time, average managers 60 percent of the time, and inferior managers just 50 percent of the time. At the risk of oversimplifying, we assume an average six-month gain of 25 percent (at an annual rate) versus a possible loss of 15 percent. Thus, the superior manager's expected return is calculated as: (.7)(25) + (.3)(-15) = 13 percent. Similarly, an average fund manager's expected return is 9 percent, and an inferior manager's expected return is only 5 percent. These rates of return mimic real-life fund performance and are listed in the top of the spreadsheet.

Consider the strong 10-year track record of manager G. This fund has had positive returns in 15 (six-month) periods and negative returns in only 5 periods. This long and strong track record suggests that G is a superior manager. Using the spreadsheet, our task is to compute the revised prob- ability: Pr(Sƒ15 of 20). To do this, we first compute the converse probabil- ity: Pr(15 of 20ƒS), the likelihood of such a track record if Manager G is truly superior (i.e. has a .7 chance of earning a positive return in any par- ticular six-month period). This likelihood follows a binomial probability and is shown to be about .18 in cell D15. This probability is computed using the following

Excel formula:= BINOMDIST(C15, B15, D9, 0).

Here, the first argument (cell C15) is the number of binomial successes, the second argument is the total number of trials, the third argument is the probability of success on each trial, and the last argument is always zero. By similar formulas, we compute Pr(15 of 20 ƒ A) and Pr(15 of 20ƒI) in cells E15 and F15. (Notice, it is much less likely that such a strong track record would be recorded by an average fund or especially by an inferior fund.)

a. Complete the joint probability table by computing the missing entries in cells E20 and F20. (Cell D20 has been calculated for you.) In turn, compute the revised probabilities (i.e., Pr(S ƒ15 of 20) and so on) in row 24.

b. Even with an impressive past performance record, how confident are you that manager G is of the superior type? If you invest your money with manager G, what return would you predict on average? Compute the expected return in cell J24 using the revised probabilities found in part (a).

c. Experiment with different performance records (cells B15 and C15) and greater differences between types of fund managers (cells D9 and F9). What effect does each factor have on one's ability to identify superior managers?