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18 suppose that finishing in the top half of all

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18. Suppose that finishing in the top half of all portfolio managers is purely luck, and that the probability of doing so in any year is exactly ½. Then the probability that any particular manager would finish in the top half of the sample five years in a row is (½) 5 = 1/32. We would then expect to find that [350 × (1/32)] = 11 managers finish in the top half for each of the five consecutive years. This is precisely what we found. Thus, we should not conclude that the consistent performance after five years is proof of skill. We would expect to find eleven managers exhibiting precisely this level of "consistency" even if performance is due solely to luck. 4-4
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19. a. After two years, each dollar invested in a fund with a 4% load and a portfolio return equal to r will grow to: $0.96 × (1 + r – 0.005) 2 Each dollar invested in the bank CD will grow to: $1 × 1.06 2 If the mutual fund is to be the better investment, then the portfolio return (r) must satisfy: 0.96 × (1 + r – 0.005) 2 > 1.06 2 0.96 × (1 + r – 0.005) 2 > 1.1236 (1 + r – 0.005) 2 > 1.1704 1 + r – 0.005 > 1.0819 1 + r > 1.0869 Therefore: r > 0.0869 = 8.69% b. If you invest for six years, then the portfolio return must satisfy: 0.96 × (1 + r – 0.005) 6 > 1.06 6 = 1.4185 (1 + r – 0.005) 6 > 1.4776 1 + r – 0.005 > 1.0672 1 + r > 1.0722 r > 7.22% The cutoff rate of return is lower for the six-year investment because the “fixed cost” (i.e., the one-time front-end load) is spread out over a greater number of years. c. With a 12b-1 fee instead of a front-end load, the portfolio must earn a rate of return (r) that satisfies: 1 + r – 0.005 – 0.0075 > 1.06 In this case, r must exceed 7.25% regardless of the investment horizon. 20. The turnover rate is 50%. This means that, on average, 50% of the portfolio is sold and replaced with other securities each year. Trading costs on the sell orders are 0.4%; and the buy orders to replace those securities entail another 0.4% in trading costs. Total trading costs will reduce portfolio returns by: 2 × 0.4% × 0.50 = 0.4% 4-5
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21. For the bond fund, the fraction of portfolio income given up to fees is: % 0 . 4 % 6 . 0 = 0.150 = 15.0% For the equity fund, the fraction of investment earnings given up to fees is: % 0 . 12 % 6 . 0 = 0.050 = 5.0% Fees are a much higher fraction of expected earnings for the bond fund, and therefore may be a more important factor in selecting the bond fund. This may help to explain why unmanaged unit investment trusts are concentrated in the fixed income market. The advantages of unit investment trusts are low turnover and low trading costs and management fees. This is a more important concern to bond-market investors. 4-6
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