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bkmsol_ch24 - CHAPTER 24 PORTFOLIO PERFORMANCE EVALUATION 1...

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CHAPTER 24: PORTFOLIO PERFORMANCE EVALUATION 1. a. Arithmetic average: % 10 r ABC = ; % 10 r XYZ = b. Dispersion: σ ABC = 7.07%; σ XYZ = 13.91% Stock XYZ has greater dispersion. (Note: We used 5 degrees of freedom in calculating standard deviations.) c. Geometric average: r ABC = (1.20 × 1.12 × 1.14 × 1.03 × 1.01) 1/5 – 1 = 0.0977 = 9.77% r XYZ = (1.30 × 1.12 × 1.18 × 1.00 × 0.90) 1/5 – 1 = 0.0911 = 9.11% Despite the fact that the two stocks have the same arithmetic average, the geometric average for XYZ is less than the geometric average for ABC. The reason for this result is the fact that the greater variance of XYZ drives the geometric average further below the arithmetic average. d. In terms of “forward looking” statistics, the arithmetic average is the better estimate of expected rate of return. Therefore, if the data reflect the probabilities of future returns, 10% is the expected rate of return for both stocks. 2. a.Time-weighted average returns are based on year-by-year rates of return: Year Return = [(capital gains + dividend)/price 1998 1999 [(\$120 – \$100) + \$4]/\$100 = 24.00% 1999 2000 [(\$90 – \$120) + \$4]/\$120 = –21.67% 2000 2001 [(\$100 – \$90) + \$4]/\$90 = 15.56% Arithmetic mean: (24% – 21.67% + 15.56%)/3 = 5.96% Geometric mean: (1.24 × 0.7833 × 1.1556) 1/3 – 1 = 0.0392 = 3.92% b. Date Cash Flow Explanation 1/1/98 –\$300 Purchase of three shares at \$100 each 1/1/99 –\$228 Purchase of two shares at \$120 less dividend income on three shares held 1/1/00 \$110 Dividends on five shares plus sale of one share at \$90 1/1/01 \$416 Dividends on four shares plus sale of four shares at \$100 each 24-1

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416 110 Date: 1/1/98 1/1/99 1/1/00 1/1/01 - 228 - 300 Dollar-weighted return = Internal rate of return = –0.1607% 3. Time Cash flow Holding period return 0 3(–\$90) = –\$270 1 \$100 (100–90)/90 = 11.11% 2 \$100 0% 3 \$100 0% a.Time-weighted geometric average rate of return = (1.1111 × 1.0 × 1.0) 1/3 – 1 = 0.0357 = 3.57% b. Time-weighted arithmetic average rate of return = (11.11% + 0 + 0)/3 = 3.70% The arithmetic average is always greater than or equal to the geometric average; the greater the dispersion, the greater the difference. c.Dollar-weighted average rate of return = IRR = 5.46% [Using a financial calculator, enter: n = 3, PV = –270, FV = 0, PMT = 100. Then compute the interest rate.] The IRR exceeds the other averages because the investment fund was the largest when the highest return occurred. 4. a.The alphas for the two portfolios are: α A = 12% – [5% + 0.7(13% – 5%)] = 1.4% α B = 16% – [5% + 1.4(13% – 5%)] = –0.2% Ideally, you would want to take a long position in Portfolio A and a short position in Portfolio B. 24-2
b. If you will hold only one of the two portfolios, then the Sharpe measure is the appropriate criterion: 583 . 0 12 5 12 S A = - = 355 . 0 31 5 16 S B = - = Using the Sharpe criterion, Portfolio A is the preferred portfolio. 5. a. Stock A Stock B (i) Alpha = regression intercept 1.0% 2.0% (ii) Information ratio = α P / σ (e P ) 0.0971 0.1047 (iii) *Sharpe measure = (r P – r f )/ σ P 0.4907 0.3373 (iv) **Treynor measure = (r P – r f )/ β P 8.833 10.500 * To compute the Sharpe measure, note that for each stock, (r P – r f ) can be computed from the right-hand side of the regression equation, using the assumed parameters r M = 14% and r f = 6%. The standard deviation of each stock’s returns is given in the problem.

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bkmsol_ch24 - CHAPTER 24 PORTFOLIO PERFORMANCE EVALUATION 1...

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