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# 1. The value of Crungy's portfolio at the end of Year 1 was:

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### 1. The value of Crungy's portfolio at the end of Year 1 was:

1. 1. 1. The value of Crungy's portfolio at the end of Year 1 was: Value= (100 × \$10) + (300 × \$5) + (250 × \$12) = \$5,500 The value of Crungy's portfolio at the end of Year 2 was: Value= (100 × \$15) + (300 × \$4) + (250 × \$14) = \$6,200 The rate of return on a portfolio with no cash flows is: ROR = (Ending Value - Beginning Value)/Beginning Value In Crungy's case: ROR = (\$6,200 - \$5,500)/\$5,500 = .127 = 12.7% 2. Without intervening cash flows, the performance of a portfolio is simply the percentage change in the portfolio's market value over the measurement period. However, when intervening cash flows occur this calculation is no longer correct. One must account for the fact that the amounts added or withdrawn from the portfolio are not producing investment returns for the full measurement period. Both the dollar-weighted and time-weighted rate of return measures adjust for intervening cash flows. 3. For a portfolio that receives a contribution at the end of the measurement period, the rate of return is: ROR = (End Value - Beg Value - Cont)/Beg Value In Lave's case: ROR = (\$42,000 - \$39,000 - \$4,000)/\$39,000 = -.026 = -2.6% 4. For a portfolio that receives a contribution at the beginning of the measurement period, the rate of return is: Chapter 24 Page 1
2. 2. ROR = (End Value - Beg Value - Cont)/(Beg Value + Cont) In New Lisbon's case: ROR = (\$38 mil - \$30 mil - \$2 mil)/(\$30 mil + \$2 mil) = .188 = 18.8% 5. Calculating the time-weighted return for a portfolio involves valuing the portfolio whenever a cash flow occurs, calculating the rate of return in that subperiod, and then linking the subperiod returns together. In Con's case: Sub-Period Rate of Return 1 (\$9,800 - \$9,000 - \$500)/\$9,000 = .033 = 3.3% 2 (\$10,800 - \$9,800 - \$500)/\$9,800 = .051 = 5.1% 3 (\$11,200 - \$10,800 - \$500)/\$10,800 = -.009 = -0.9% 4 (\$12,000 - \$11,200 - \$500)/\$11,200 = .027 = 2.7% For the year, Con's time-weighted return was: [(1.033 × 1.051 × .991 × 1.027) - 1] = .105 = 10.5% 6. The dollar-weighted rate of return is the interest rate that equates the beginning value of a portfolio to the discounted value of the cash flows and ending value of the portfolio. In Dell's case: \$12,000 = -\$800/(1 + rD)10 + (\$13,977.71)/(1 + rD)30 (Note that a minus sign is attached to the \$800 because it is a contribution) The dollar-weighted rate of return must be solved for iteratively. A value of .30% for rD gives: \$12,000 = -\$800/(1.003)10 + \$13,977.71/(1.003)30 = -\$776.39 + \$12,776.39 = \$12,000 This .30% figure is a daily dollar-weighted return. It can be Chapter 24 Page 2
3. 3. converted to a monthly return as follows: rDM = (1.003)30 - 1 = .094 = 9.4% 7. a. For a three-period example, the dollar-weighted rate of return, rD, is given by: C1 C2 C + Ve Vb = + + 3 (1 + rD )1 (1 + rD ) 2 (1 + rD ) 3 In Ginger's case: Vb = \$10,000 C1 = +\$956 C2 = -\$659 C3 = 0 Ve = \$13,000 Thus: − \$956 + \$659 \$0 + \$13,000 \$10,000 = + + (1 + rD )1 (1 + rD ) 2 (1 + rD ) 3 (Note that a minus sign is attached to C1 because it is a contribution, while C2 is given a plus sign because it is a withdrawal.) The dollar-weighted rate of return must be solved for iteratively. A value of 8% for rD gives: − \$956 + \$659 \$0 + \$13,000 \$10,000 = 1 + + (1 + rD ) (1 + rD ) 2 (1 + rD ) 3 = -\$885.19 + \$564.99 + \$10,319.82 = \$9,999.62 This 8% figure is the monthly dollar-weighted rate of return. It can be converted to a quarterly return as follows: rDQ = (1 + .08)3 - 1 = 1.260 - 1 = 0.260 = 26.0% b. The time-weighted return is found by calculating the change in the portfolio's market value at each date on Chapter 24 Page 3
4. 4. which a cash flow to or from the portfolio occurs, after adjusting for the cash flow. That is: rT = (Ve - Vb - C)/Vb In the case of Ginger's portfolio: rT1 = (\$9,000 - \$10,000 - \$956)/\$10,000 = -0.196 = -19.6% rT2 = (\$12,000 - \$9,000 + \$659)/\$9,000 = 0.407 = 40.7% rT3 = (\$13,000 - \$12,000 + \$0)/\$12,000 = 0.083 = 8.3% The monthly time-weighted return can be converted to a quarterly return by linking the three monthly returns as follows: rTQ = (1 + rT1) × (1 + rT2) × (1 + rT3) - 1 = (1 - .196) × (1 + .407) × (1 + .083) - 1 = 1.225 - 1 = 0.225 = 22.5% c. The dollar-weighted return is affected by the timing of cash flows to and from the portfolio, while the time- weighted return removes their influence. In this problem, Ginger made a contribution prior to a very strong month of performance (that is, February), which caused the dollar- weighted return to exceed the time-weighted return. 8. The time-weighted rate of return measures the performance of each dollar invested in the portfolio regardless of the size or timing of the cash flow. The dollar-weighted rate of return, on the other hand, measures the growth rate of all funds invested in portfolio. As such, it is sensitive to the size and timing of cash flows. The dollar-weighted rate of return might be a desirable performance measure in situations where the portfolio manager has full discretion over contributions and withdrawals to and Chapter 24 Page 4
5. 5. from the portfolio. In this case the dollar-weighted rate of return's sensitivity to the size and timing of cash flows will reflect not only the manager's skill at investing each dollar in the portfolio, but also the manager's ability to add or subtract funds from the portfolio at appropriate times. 9. The dollar-weighted rate of return is the interest rate that equates the beginning value of a portfolio to the discounted value of the cash flows and ending value of the portfolio. In Oats' case: \$1,500 − \$600 \$21,769.60 \$22,000 = 12 + + (1 + rD ) (1 + rD ) 21 (1 + rD ) 31 \$1,500 − \$600 \$21,769.60 \$22,000 = 12 + 21 + (1 + .001) (1 + .001) (1 + .001) 31 = \$1,482.12 - \$587.54 + \$21,105.42 = \$22,000.00 This 0.10% figure is a daily dollar-weighted return. It can be converted to a monthly return as follows: rDM = (1 + .001)31 - 1 = .031 = 3.1% 10. Calculating the time-weighted return for a portfolio involves valuing the portfolio whenever a cash flow occurs, calculating the rate of return in the subperiod, and then linking the subperiod returns together. In Buttercup's case: Subperiod 1: (\$7,300 - \$5,000 - \$2,000)/\$5,000 = .060 = 6.0% Subperiod 2: (\$9,690.18 - \$7,300)/\$7,300 = .327 = 32.7% For the month, Buttercup's time-weighted return is: (1.060 × 1.327) - 1 = .407 = 40.7% The dollar-weighted rate of return is the interest rate that equates the beginning value of a portfolio to the discounted value of the cash flows and ending value of the portfolio. In Buttercup's case: \$5,000 = -\$2,000/(1 + rD)10 + \$9,690.18/(1 + rD)30 Chapter 24 Page 5
6. 6. The dollar-weighted rate of return must be solved for iteratively. A value of 1.2% for rD gives: \$5,000 = -\$2,000/(1.012)10 + \$9,690.18/(1.012)30 = -\$1,775.11 + \$6,775.11 = \$5,000 This 1.2% figure is a daily dollar-weighted return. It can be converted to a monthly return as follows: rDM = (1 + .012)30 - 1 = .430 = 43.0% The two returns differ by 2.3% because the contribution to Buttercup's portfolio was made prior to a period of relatively high returns for the portfolio. Because the dollar-weighted rate of return is sensitive to the size and timing of cash flows, it shows a higher return than does the time-weighted rate of return. 11. It is impossible to interpret investment performance without some point of reference. Rarely are there unqualified "good" or "bad" investment results. In an environment with a vast array of possible investment opportunities, the selection of a particular portfolio implies foregoing other opportunities. Investment results should therefore be evaluated relative to an valid measure of these opportunities. An appropriate benchmark should be relevant and feasible in the sense that the benchmark should represent an opportunity that could have been chosen instead of the portfolio being evaluated. An appropriate benchmark should also be "fair" in the sense that it should exhibit risk similar to that of the portfolio being evaluated. 12. It is quite possible that the other common stock portfolios are inappropriate benchmarks. They may not satisfy the criteria of being relevant and feasible. That is, they may not represent investment alternatives for the portfolio manager being evaluated. The manager may have no practical opportunity to invest in any of the sample portfolios, particularly because they are not identified in advance. Further, there is no reason to expect that the portfolios comprising the sample have risk similar to that of the portfolio being evaluated. While they might, usually there is Chapter 24 Page 6
7. 7. no way to verify that fact. The performance measurement services typically do not supply detailed risk information on the individual portfolios comprising their samples. 13. Students may have different answers, depending on whether they choose different dates. Using a July 1, 1998 The Wall Street Journal, the DJIA's component security closing prices (as of June 30) were: Allied Signal 44.37500 Hew Packard 59.87500 Alcoa 65.93750 IBM 114.81250 Am Express 113.75000 Intl Paper 43.00000 AT & T 57.12500 J and J 74.00000 Boeing 44.56250 McDonalds 69.00000 Caterpillar 52.90625 Merck 133.75000 Chevron 83.75000 3M 82.81250 CocaCola 85.50000 JP Morgan 82.81250 Disney 105.06250 Phill Morris 39.37500 Dupont 74.68750 Proc Gamble 91.06250 Eastman Kodak 73.06250 Sears 61.06250 Exxon 71.37500 Travelers 60.75000 Gen Electric 90.87500 Union Carbide 53.37500 Gen Motors 66.81250 Untd Tech 92.50000 Goodyear 64.43750 Walmart 60.75000 Also, on June 30, 1998 the closing value of the DJIA was 8,592.02. The value of the DJIA is given by: DJIA = Sum of prices/Divisor The sum of the prices of the component securities as of the June 30 close was 2,247.28125. Solving the DJIA value equation for the divisor gives: 8,592.02 = 2,247.28125/Divisor Divisor = 2,247.28125/8,592.02 = 0.251036 14. a. A 5% stock dividend should reduce the price of stock A by a factor of 1.05. Thus after the stock dividend the price of stock A should be: Chapter 24 Page 7
8. 8. \$16/1.05 = \$15.250 (rounded to nearest sixteenth) Because the value of the price-weighted index before the stock dividend was: Index = (16 + 30)/2 = 23 After the stock dividend the divisor will adjust to maintain the same index value. Thus: 23 = (15.250 + 30)/Divisor Divisor = 1.967 b. After its 3-1 stock split, the price of stock B should be reduced by a factor of 3. Thus it should be worth \$10 per share. Again, the value of the divisor must adjust to keep the index's value constant after the split. Thus: 23 = (16 + 10)/Divisor Divisor = 1.130 c. After its 4-1 split, the price of stock A should be reduced by a factor of 4. Thus it should be worth \$4 per share. Again, adjusting the value of the divisor to maintain a constant index value gives: 23 = (4 + 30)/Divisor Divisor = 1.478 15. The S&P 500 is composed of considerably more stocks than the DJIA (that is, 500 versus 30). Thus by sheer force of numbers it is a better statistical representation of the entire market. More importantly, however, the S&P 500 is composed of stocks from many different sectors of the stock market, whereas the DJIA is composed of only large, mature, and primarily industrial companies (although the non-industrial weighting of the DJIA has been increased in recent years). As a result the S&P 500 provides a significantly more complete representation of the stock market's industry composition than does the DJIA. Chapter 24 Page 8
9. 9. In addition, the S&P 500 is a value-weighted index while the DJIA is a price-weighted index. Because the stock market's performance is reflective of the market capitalization of all stocks, the S&P 500 more accurately represents the aggregate impact of stock price changes on the value of the entire stock market. 16. a. The aggregate value of the market equals the price of each security in the market times the number of shares of each respective security. Therefore: Value of market = \$20 × 20,000 + \$35 × 40,000 + \$30 × 40,000 \$3,000,000 b. A 20% increase in security C's price boosts its value to \$36 per share and the market value of its shares outstanding to \$1,440,000 (\$36 × 40,000). The aggregate value of the market will then be \$3,240,00, an 8.0% increase. c. A 2-1 split by security B will reduce the value of its price by 50% to \$17.50 per share and double the number of shares outstanding to 80,000. The net result is no change in the total value of security B's shares and therefore no change in the aggregate value of the market. 17. a. On Date 1, the price-weighted index's value is given by: Ipw = (PX + PY + PZ)/3 = (16 + 5 + 24)/3 = 15.0 b. On Date 2: Ipw = (22 + 4 + 30)/3 = 18.7 c. The 4-for-1 split of stock X causes its price to fall to \$5.50. But the split should not be permitted to change the index's value. Therefore the divisor of the price- weighted index must be adjusted to maintain an index value of 18.7. Ipw = (5.5 + 4 + 30)/D = 18.7 Chapter 24 Page 9
10. 10. Solving for the divisor D gives: D = (5.5 + 4 + 30)/18.7 = 2.11 d. The aggregate market value of the value-weighted index on Date 1 is: \$16/shr × 100 shrs + \$5/shr × 200 shrs + \$24/shr × 100 shrs = \$5,000 The aggregate market value of the value-weighted index on Date 2 is: \$22/shr × 100 shrs (pre-split) + \$4/shr × 200 shrs + \$30/shr × 100 shrs = \$6,000 The ratio of the Date 2 index market value to the Date 1 index market value is: \$6,000/\$5,000 = 1.20 Multiplying by 100 to scale the Date 2 value-weighted index value to the initial value of 100 on Date 1 gives: Ivw = 1.20 × 100 = 120.00 (Note: the split of stock X has no impact on the value of the value-weighted index. Either the pre-split or post- split price and shares of stock X can be used.) 18. Stock market indices composed of a small number of stocks relative to the entire market are not designed simply to measure the performance of the component securities. Rather, the indices are designed to measure the performance of the entire market. For example, the performance of the thirty stocks comprising the DJIA is not of great importance in and of itself. Instead, it is the fact that changes in the value of those thirty stocks are reasonably good indicators of changes in the Chapter 24 Page 10
11. 11. aggregate value of the many thousands of stocks not included in the DJIA that makes the DJIA a useful representation of market performance. 19. a. The return on the three securities from Date 1 to Date 2 is: Security Return A \$55/\$50 = 1.100 = 10.0% B \$28/\$30 = 0.933 = -6.7% C \$75/\$70 = 1.071 = 7.1% The equal-weighted index will assign each of the three security returns an equal weight. Summing the three securities' returns and dividing by 3 gives a return of 3.5% [(10.0 - 6.7 + 7.1)/3]. b. Similarly, from Date 2 to Date 3, the securities' returns are: Security Return A \$60/\$55 = 1.091 = 9.1% B \$30/\$28 = 1.071 = 7.1% C \$73/\$75 = .973 = -2.7% Thus, the equal-weighted index return is: [(9.1 + 7.1 - 2.7)/3] = 4.5% 20. The returns of the three securities from Date 1 to Date 2 and from Date 2 to Date 3 are: Date 1-2 Date 2-3 Security Return Return L \$23/\$20 = 15.0% \$30/\$23 = 30.4% M \$30/\$27 = 11.1% \$31/\$30 = 3.3% N \$35/\$40 = -12.5% \$29/\$35 = -17.1% From Date 1 to Date 2, the geometric return on the index is: RG = (1.150 × 1.111 × .875)1/3 = 3.8% From Date 2 to Date 3, the geometric return on the index is: Chapter 24 Page 11
12. 12. RG = (1.304 × 1.033 × .829)1/3 = 3.7% Thus, if the value of the index was 200 on Date 1, it will be 207.6 (200 × 1.038) on Date 2 and 215.3 (207.6 × 1.037) on Date 3. 21. Based on the ex post SML, the risk-adjusted return for a portfolio is: αp = arp - [arf + (arm - arf)ßp] In Pickles' case: αp = 16.8% - [7.4% + (15.2% - 7.4%) × 1.10] = 16.8% - 16.0% = 0.8% 22. Applying the three measures of risk-adjusted performance to the Venus Fund: Ex post alpha = (arp - arf) - (arm - arf)ßp For the Venus Fund: Ex post alpha = 0.60 - (0.50 × 1.10) = +0.05 Reward-to-volatility = (arp - arf)/ßp For the Venus Fund: Reward-to-volatility = 0.60/1.10 = +0.55 For the S&P 500: Reward-to-volatility = 0.50/1.00 = +0.50 Sharpe ratio = (arp - arf)/σp For the Venus Fund: Sharpe ratio = 0.60/9.90 = +0.06 For the S&P 500: Chapter 24 Page 12
13. 13. Sharpe ratio = 0.50/6.60 = +0.08 Two of the performance measures (ex post alpha and reward-to- volatility ratio) show that the Venus Fund outperformed the market index on a risk-adjusted basis. Conversely, the Sharpe ratio indicates that the Venus Fund underperformed the market index. As a result, using only historical performance data, the answer regarding which fund you should recommend to Dazzy is indeterminate. However, if the portfolio happened to represent all of Dazzy's wealth, you should place more emphasis on the results of the Sharpe ratio and recommend that Dazzy invest in the index fund. 23. If the portfolio represents the entire wealth of its owner, then the total risk of the portfolio is the relevant risk measure for performance evaluation purposes. The variability of the portfolio's return is likely to be of more concern to the investor than is beta, which measures only the market- related portion of the portfolio's total risk. 24. The four risk-adjusted performance measures can give conflicting answers to the question of whether a portfolio outperformed the market index. Specifically, the ex post alpha and reward-to-volatility ratio will always give the same answer, as will the Sharpe ratio and M-squared, but the two pairs of ratios may give differing answers. The reason for this possible disparity is that the ex post alpha and the reward-to-volatility ratio use market risk (beta) as the relevant measure of portfolio risk, while the Sharpe ratio and M-squared use total risk (standard deviation) as the relevant measure of portfolio risk. If the portfolio has a material amount of unique risk, then the market risk of the portfolio will differ significantly from its total risk and therefore could lead to conflicting answers as to whether the portfolio outperformed the market index on a risk-adjusted basis. 25. The portfolio's ex post alpha will measure gains due to both security selection and market timing. If the portfolio manager is able to position the portfolio so that it is more (less) sensitive to market index moves when the market goes up (down), then the relationship between the portfolio's excess return and that of the market index will be a convex curve. Assuming that security selection is neutral, when a straight Chapter 24 Page 13
14. 14. line is fitted to this curved relationship the intercept on the vertical axis (alpha) will be positive, reflecting successful market timing. On the other hand, perverse market timing will lead to a negative alpha value when security selection is neutral. If security selection is not neutral, the net value of alpha will incorporate results due to both security selection and market timing. 26. If broad stock market indices are poor proxies for the market portfolio, then the conclusions drawn from the ex post alpha performance measure are potentially meaningless. The ex post alpha based on the CAPM requires the identification of the market portfolio. Presuming that the market portfolio is misspecified by the stock market indices, then it is possible that the rankings produced by the ex post alpha are arbitrary and simply a function of the particular stock market index used. Nevertheless, even if the stock market indices are poor market proxies it is possible that they may still lead to valid rankings by the ex post alpha. However, because this is not necessarily the case, the uncertainty surrounding this issue has been discomforting to users of this performance measure. 27. a. The average beta is calculated as: Beta = (.90 + .95 + .95 + 1.00 + 1.00 + .90 + .80 + .75 + .80 + .85)/10 = .89 Treasury bills have a beta of 0.0 and the S&P 500 has a beta of 1.0. Further, the beta of any portfolio is the weighted average beta of the component securities. Therefore, for a portfolio composed of the S&P 500 and Treasury bills, the percentage investment in the S&P 500 needed to produce a given beta is simply the desired beta. In this problem, to produce a beta of .89, 89% of the portfolio should be invested in the S&P 500. b. For a portfolio 89% invested in the S&P 500 and 11% invested in Treasury bills over the ten-year period, the year-by-year returns are found as follows: 1 (.89 × -8.50%) + (.11 × 6.58%) = -6.84% Chapter 24 Page 14
15. 15. 2 (.89 × 4.01%) + (.11 × 6.53%) = 4.29% 3 (.89 × 14.31%) + (.11 × 4.39%) = 13.22% 4 (.89 × 18.98%) + (.11 × 3.84%) = 17.31% 5 (.89 × -14.66%) + (.11 × 6.93%) = -12.29% 6 (.89 × -26.47%) + (.11 × 8.00%) = -22.68% 7 (.89 × 37.20%) + (.11 × 5.80%) = 33.75% 8 (.89 × 23.84%) + (.11 × 5.08%) = 21.78% 9 (.89 × -7.18%) + (.11 × 5.12%) = -5.83% 10 (.89 × 6.56%) + (.11 × 7.18%) = 6.63% c. As discussed, for a portfolio composed of the S&P 500 and Treasury bills to achieve a desired beta value, the necessary weighting in the S&P 500 component of the portfolio is equal to the beta. For the various beta values exhibited by Jupiter Fund over the ten-year period, the resulting returns would have been earned on an appropriately weighted S&P 500/Treasury bills portfolio: 1 (0.90 × -8.50%) + (.10 × 6.58%) = -6.99% 2 (0.95 × 4.01%) + (.05 × 6.53%) = 4.14% 3 (0.95 × 14.31%) + (.05 × 4.39%) = 13.81% 4 (1.00 × 18.98%) + (.00 × 3.84%) = 18.98% 5 (1.00 × -14.66%) + (.00 × 6.93%) = -14.66% 6 (0.90 × -26.47%) + (.10 × 8.00%) = -23.02% 7 (0.80 × 37.20%) + (.20 × 5.80%) = 30.92% 8 (0.75 × 23.84%) + (.25 × 5.08%) = 19.15% Chapter 24 Page 15
16. 16. 9 (0.80 × -7.18%) + (.20 × 5.12%) = -4.72% 10 (0.85 × 6.56%) + (.15 × 7.18%) = 6.65% d. Subtracting the part (b) calculations from the part (c) calculations is a reasonable measure of the fund's market timing ability because all stock selection effects have been removed from the analysis. Only the effect of adjusting the fund's beta around its ten-year average value is considered. Performing the subtraction on a year-by-year basis gives: 1 -6.99% - (-6.84%) = -0.15% 2 4.14% - 4.29% = -0.15% 3 13.81% - 13.22% = 0.59% 4 18.98% - 17.31% = 1.67% 5 -14.66% - (-12.29%) = -2.37% 6 -23.02% - (-22.68%) = -0.34% 7 30.92% - 33.75% = -2.83% 8 19.15% - 21.78% = -2.63% 9 -4.72% - (- 5.83%) = 1.11% 10 6.65% - 6.63% = 0.02% The average difference over the ten-year period is -0.51% per year. The evidence does not indicate that Jupiter Fund was a superior market timer. e. Subtracting the calculations in part (c) from Jupiter Fund's actual returns is a reasonable measure of security selection ability because it represents the residual return after the fund's market timing actions have been removed from the fund's returns. Performing the subtraction on a year-by-year basis gives: 1 2.99% - (-6.99%) = 4.00% 2 0.63% - 4.14% = -3.51% Chapter 24 Page 16
17. 17. 3 22.01% - 13.81% = 8.20% 4 24.08% - 18.98% = 5.10% 5 -22.46% - (-14.66%) = -7.80% 6 -25.12% - (-23.02%) = -2.10% 7 29.72% - 30.92% = -1.20% 8 22.15% - 19.15% = 3.00% 9 0.48% - (-4.72%) = 5.20% 10 6.85% - 6.65% = 0.20% The average difference is 1.11% per year, indicating that Jupiter Fund had superior security selection abilities. 28. The three risk-adjusted performance measures are calculated as follows: Ex post alpha = (arp - arf) - (arm - arf)ßp Reward-to-volatility = (arp - arf)/ßp Sharpe ratio = (arp - arf)/σp A portfolio's beta is defined as: Beta = [(T × ΣXY) - (ΣY × ΣX)]/[(T × ΣX²) - (ΣX)²] Using the Minifund's returns as reported in the question, as well as the returns on the market and the riskfree return as given in Table 1-1 in the text: ßp = (20 ×9727.09)-(321.05 × 249.71)/(20 × 8633.97) - (249.71)² = 1.037 Thus for Minifund: Ex post alpha = (16.05% - 7.69%) - (12.48% - 7.69%) × 1.037 Chapter 24 Page 17
18. 18. = 3.38 Reward-to-volatility = (16.05% - 7.69%)/1.037 = 8.06 Sharpe ratio = (16.05% - 7.69%)/24.79 = 0.34 In contrast, the same performance measures for the market index are: Ex post alpha = 0.00 (by definition) Reward-to-volatility = (12.48% - 7.69%)/1.00 = 4.79 Sharpe ratio = (12.48% - 7.69%)/16.02 = 0.30 By all three measures, the Minifund has produced superior returns over the twenty-year period. Note however that the benchmark used for performance evaluation purposes is inappropriate. The large capitalization nature of the S&P 500 is not relevant to the investment strategy of the Minifund, which is to invest in only very small companies. In fact none of the holdings of the Minifund would ever be found in the S&P 500. 29. While details are not shown here, using the regression analysis available in a computer spreadsheet, the following calculations were made for Keynes's portfolio performance: Ex Post Alpha Alpha = 14.5 Std Error Alpha = 19.8 Beta = 1.8 Std Error Beta = 0.4 These calculations are consistent with the Chua-Woodward study. Chapter 24 Page 18
19. 19. Quadratic Equation Alpha = 15.8 Std Error Alpha = 20.3 Beta = 1.7 Std Error Beta = 0.4 Quadratic Coefficient = -.009 Std Error Quad Coeff = 0.02 Dummy Variable Equation Alpha = 15.3 Std Error Alpha = 20.4 Beta = 1.8 Std Error Beta = 0.6 Dummy Variable Coefficient = -.20 Std Error Dummy Coeff = 1.30 The negative values for the market timing coefficients in both the quadratic and dummy variable equations indicate the absence of market timing skill on Keynes' part. Interestingly, Chua and Woodward mention that Keynes eschewed market timing, believing it to be an unproductive strategy. 30. The primary drawback of using the bond market line approach to evaluating bond portfolio performance is that it is may be an inadequate explanation of the relationship between bond risk and return. Although duration is certainly a significant factor affecting bond returns, there may be other systematic factors such as the quality of the issuer or the convexity of the bond. Further, even if these other factors were not systematically priced, duration may still be an unreliable factor in explaining bond returns. Depending on yield curve movements, portfolios with the same durations may experience significantly different returns depending on the maturity and cash flow characteristics of the portfolios' underlying securities. 31. Performance attribution attempts to explain why a particular Chapter 24 Page 19
20. 20. portfolio produced a given return over a performance evaluation period. In general, performance attribution decomposes portfolio returns into various sources of returns. These sources are associated with important investment management decisions on the part of the portfolio manager (for example, market timing, security selection, or industry or factor emphases). Many of the same types of problems that hinder measures of how well a portfolio performed can also cause difficulties in determining why a portfolio produced a given return. For example, misspecification of the factor model generating security returns will lead to erroneous conclusions as to the sources of portfolio returns. Further, considerably more data than is usually available is required to establish statistical significance. Additionally, performance attribution assumes that certain portfolio characteristics remain constant over the period of analysis. This assumption may be incorrect. 32. The following table presents the difference in performance between Portfolio A and the market index due to the different sector/factor attributes of the two portfolios. The table shows the sector/factor attributes of Portfolio A and the market, the difference between those two sets of attributes, the return to each sector/factor as stated in the problem, and the resulting return associated with the different sector/factor attributes of Portfolio A as opposed to those of the market index. (a) (b) (c=a-b) (d) (e=c×d) Factor Port A Market Diff Fact Ret Diff Ret Beta 1.10 1.00 0.10 -0.50 -0.05% Size 1.30 6.00 -4.70 -0.60 2.82 Indust 0.40 0.80 -0.40 8.00 -3.20 Non- Indust 0.60 0.20 0.40 16.00 6.40 Of the 7.0 percentage point difference in performance between Portfolio A and the market index, 5.97 percentage points is due to the sector/factor differences between the two portfolios (that is, -.05 + 2.82 - 3.20 + 6.40). The remaining amount (1.03 percentage points) is, by definition, due to non-sector/factor differences between the two portfolios. The table below breaks down the returns of the two portfolios into sector/factor returns and non-sector/factor returns. Chapter 24 Page 20
21. 21. Port A Market Difference Sec/Fac Returns 11.47% 5.50% 5.97% Non-Sec/Fac Returns 1.03 0.00 1.03 Total Return 12.50 5.50 7.00 That is, the 11.47% sector/factor return of Portfolio A is found by multiplying the respective sector/factor values of the Portfolio A times the associated returns on the sector/factors and summing over all four sector/factors. The difference between the Portfolio A's sector/factor return and its total return is the return due to non-sector/factors. 33. (From The CFA Candidate Study and Examination Program Review, 1992.) a. Factors which can affect past investment results are so varied and pervasive that past investment performance alone provides very little insight into future investment performance. Pitfalls exist in interpreting any performance results, whether those results are monitored by consultants, audited by public accountants, reported in compliance with legislated or accepted standards, or otherwise. Hiring an investment manager is an investment in that manager's future performance, just as a security purchase is an investment in that security's future performance. A fund administrator must look beyond past or recent investment performance, just as a security analyst must look beyond past or recent earnings per share. b. Manager A - Simulated Performance An important requirement for the validity of simulated investment performance is that realistic assumptions be used in the simulation. Back-tests and simulations attempt to reproduce the decision process, trading and record of an investment strategy by making assumptions about how the process could have been applied in the past. The validity of assumptions with respect to executable market prices, transaction costs, timely availability of required data, etc., is critical. Simulations often rely on the analysis and manipulation of huge amounts of data, but the final results are often represented by summary statistics. It is important that the analytic techniques used are proper and the Chapter 24 Page 21
22. 22. computations be accurate. It is necessary that statistical measures be applied and interpreted appropriately, that the investment process is faithfully reproduced (such as income reinvestment or portfolio rebalancing), that the computations be verified for accuracy and internal consistency, etc. It is important that simulations be free of data-mining and any a priori bias. An investment process should be based on sound fundamental investment concepts which are tested by simulation after development of the idea, not discovered by trial-and-error to find what works. Data- mining can also take the form of selecting time periods which may not be representative of the norm. Independent reproduction of back-tests and simulations for the purpose of verification is nearly impossible. Therefore, at some point you must trust the integrity, qualifications, and intellect of the manager. It is important that the manager exhibits high ethical standards and can demonstrate that your trust is deserved. Manager B - Anomalies It is important that there be a sound fundamental reason why the purported market inefficiency or anomaly should be expected to produce superior returns. Without a good explanation of why an anomaly worked in the past, it is difficult to justify why it should continue to work into the future. It is not sufficient to merely know that it has worked over short periods of time with small sums of money invested. Another important factor is the longevity of the market inefficiency or anomaly. Once discovered, true market inefficiencies and anomalies are likely to be exploited to the point that the market is made efficient. The common phrase is to say the anomaly is "arbitraged away." By exploiting an anomaly, underpriced securities are bought (or overpriced sold). The resulting demand (or supply) for the securities eventually raises (or lowers) the price to where the security is no longer systematically mispriced. The time required for this to occur depends on many factors, such as the nature of the anomaly, the uniqueness or "secrecy" of the manager's insight, structural barriers (such as tax laws or regulatory Chapter 24 Page 22
23. 23. limitations), etc. Manager C - Established Process The factors which determine portfolio performance, in decreasing order of importance, are (i) long-term asset allocation policy and asset classes used, (ii) shorter- term asset allocation decisions and market timing, and (iii) security selection. Depending upon the particular investment decision process in question, these basic factors may be further subdivided. For example, sector rotation or industry group selection may fall between market timing and security selection for a particular equity manager, or may include currency for global managers. The key in assessing Manager C's performance is to identify which determinants of returns are being consciously and actively controlled, and the degree of success attained in that activity. Important factors to consider: Asset allocation between countries - The choice of countries and the allocation of assets among various world markets strongly affect portfolio performance, since different countries can have widely varying results in any period of time. For example, a portfolio heavily concentrated in Europe will have different results than one diversified to include Japan. It is important to know if country diversification is actively or passively controlled by the manager, and whether it is his choice or reflects, instead, the preferences of the client. Asset allocation between stocks and bonds within countries -It is important to know if the manager actively varies the allocation within countries and whether he is more successful in some countries than in others. Currency management - Currency exposure in global portfolios is a major factor in determining portfolio performance. It is important to know if the manager's portfolios are always hedged to the U.S. dollar, always unhedged, or actively managed. If passive (either always hedged or unhedged), it is important to know if being passive was his decision or was imposed by client Chapter 24 Page 23
24. 24. preferences. Security selection - Style of investing, such as growth stocks or low P/E stocks, affects investment results. It is important to understand how much of a managers' performance results from his choice of security selection style versus his ability to skillfully execute the style itself. Chapter 24 Page 24
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