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Lecture 6

ADMS 4501 Lecture 6: Investments8Ce_ISM_Ch21

14 Pages
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Department
Administrative Studies
Course Code
ADMS 4501
Professor
Lois King

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CHAPTER 21 ACTIVE MANAGEMENT AND PERFORMANCE MEASUREMENT 1. The dollar-weighted average return or IRR for the cash flow given, based on daily compounding, is .05553 percent or an annual rate of 22.46 percent. (Solved through spreadsheet by using XIRR function, and assuming 365 days in a year.) 2. As established in the following result from the text, the Sharpe ratio depends on both alpha for the portfoli () and the correlation between the portfolio and the P market index (ρ): E(r P r f   P  S   M P P Specifically, this result demonstrates that a lower correlation with the market index reduces the Sharpe ratio. Hence, if alpha is not sufficiently large, the portfolio is inferior to the index. Another way to think about this conclusion is to note that, even for a portfolio with a positive alpha, if its diversifiable risk is sufficiently large, thereby reducing the correlation with the market index, this can result in a lower Sharpe ratio. 3. The IRR (i.e., the dollar-weighted return) cannot be ranked relative to either the geometric average return (i.e., the time-weighted return) or the arithmetic average return. Under some conditions, the IRR is greater than each of the other two averages, and similarly, under other conditions, the IRR can also be less than each of the other averages. A number of scenarios can be developed to illustrate this conclusion. For example, consider a scenario where the rate of return each period consistently increases over several time periods. If the amount invested also increases each period, and then all of the proceeds are withdrawn at the end of several periods, the IRR is greater than either the geometric or the arithmetic average because more money is invested at the higher rates than at the lower rates. On the other hand, if withdrawals gradually reduce the amount invested as the rate of return increases, then the IRR is less than each of the other averages. (Similar scenarios are illustrated with numerical examples in the text, where the IRR is shown to be less than the geometric average, and in Concept Check 1, where the IRR is greater than the geometric average.) 4. It is not necessarily wise to shift resources to timing at the expense of security selection. There is also tremendous potential value in security analysis. The decision on whether to shift resources has to be made on the basis of the macro, compared to the micro, forecasting ability of the portfolio management team. – – 5. a. Arithmetic average: r ABC = 10% r XYZ = 10% 21-1 b. Dispersion: ABC = 7.07%, XYZ = 13.91%. XYZ has greater dispersion. (We used 5 degrees of freedom to calculate standard deviations.) c. Geometric average: rABC = (1.2  1.12  1.14  1.03  1.01)5– 1 = .0977 = 9.77% rXYZ = (1.3  1.12  1.18  1.0  .90)– 1 = .0911 = 9.11% Despite the equal arithmetic averages, XYZ has a lower geometric average. The reason is 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 return. Therefore, if the data reflect the probabilities of future returns, 10 percent is the expected return of both stocks. 6. a. Time-weighted average returns are based on year-by-year rates of return. Year Return [(Capital gains + Dividend)/Price] 2010–2011 [($120 – $100) + $4]/$100 = 24.00% 2011–2012 [($90 – $120) + $4]/$120 = –21.67% 2012–2013 [($100 – $90) + $4]/$90 = 15.56% Arithmetic mean: (24% – 21.67% + 15.56%)/3 = 5.96% Geometric mean: (1.24  .7833  1.1556) 1/3– 1 = .0392 = 3.92% b. Date Cash Flow Explanation 1/1/2010 –300 Purchase of three shares at $100 each. 1/1/2011 –228 Purchase of two shares at $120 less dividend income on three shares held. 1/1/2012 110 Dividends on five shares plus sale of one share at $90. 1/1/2013 416 Dividends on four shares plus sale of four shares at $100 each. 21-2 Dollar-weighted return = Internal rate of return = –.1606%. 7. Time Cash Flow ($) 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 = .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%. (You can find this using a financial calculator by setting n = 3, PV = (–)270, FV = 0, PMT = 100, and solving for the interest rate.) The IRR exceeds the other averages because the investment fund was the largest when the highest return occurred. 21-3 8. a. E(r) (%)   Portfolio A 12 12 .7 Portfolio B 16 31 1.4 Market index 13 18 1.0 Risk-free asset 5 0 0.0 The alphas for the two portfolios are: A= 12% – [5% + 0.7(13% – 5%)] = 1.4% B= 16% – [5% + 1.4(13% – 5%)] = – 0.2% Ideally, you would want to take a long position in A and a short position in B. b. If you will hold only one of the two portfolios, then the Sharpe measure is the appropriate criterion: 12 – 5 SA= 12 = .583 16 – 5 SB= 31 = .355 Portfolio A is preferred using the Sharpe criterion. 9. a. Stock A Stock B i. Alpha is the intercept of the regression 1% 2% ii. Appraisal ratio =  /(e ) .0971 .1047 p p iii. Sharpe ratio* = (r – r )/ .4907 .3373 p f p iv. Treynor measure** = (r – r )/ 8.833% 10.500% p f p *To compute the Sharpe ratio, note that for each stockp r f r can be computed from the right-hand side of the regression equation, using the assumed parameters r M 14% and r = f%. The standard deviation of each stock’s returns is given in the problem. **The beta to use for the Treynor measure is the slope coefficient of the regression equation presented in the problem. b. i. If this is the only risky asset, then Sharpe’s measure is the one to use. A’s is higher, so it is preferred. ii. If the stock is mixed with the index fund, the contribution to the overall Sharpe measure is determined by the appraisal ratio; therefore, B is preferred. 21-4 iii. If it is one of many stocks, then Treynor’s measure counts, and B is preferred. 10. We need to distinguish between market timing and security selection abilities. The intercept of the scatter diagram is a measure of stock selection ability. If the manager tends to have a positive excess return even when the market’s performance is merely “neutral” (i.e., has zero excess return) then we conclude that the manager has on average made good stock picks. Stock selection must be the source of the positive excess returns. Timing ability is indicated by the curvature of the plotted line. Lines that become steeper as you move to the right of the graph show good timing ability. The steeper slope shows that the manager maintained higher portfolio sensitivity to market swings (i.e., a higher beta) in periods when the market performed well. This ability to choose more market-sensitive securities in anticipation of market upturns is the essence of good timing. In contrast, a declining slope as you move to the right means that the portfolio was more sensitive to the market when the market did poorly and less sensitive when the market did well. This indicates poor timing. We can therefore classify performance for the four managers as follows: Selection Ability Timing Ability A. Bad Good B. Good Good C. Good Bad D. Bad Bad 11. a. Sharpe ratio = (r P r )f P Williamson Capital: Sharpe ratio = (22.1%  5.0%)/16.8% = 1.02 Joyner Asset Management: Sharpe ratio = (24.2%  5.0%)/20.2% = 0.95 Treynor measure = (r P r f/ P Williamson Capital: Treynor measure = (22.1%  5.0%)/1.2 = 14.25 Joyner Asset Management: Treynor measure = (24.2%  5.0%)/0.8 = 24.00 b. The difference in the rankings of Williamson and Joyner results directly from the difference in diversification of the portfolios. Joyner has a higher Treynor measure (24.00) and a lower Sharpe ratio (0.95) than does Williamson (14.25 and 1.02, respectively), so Joyner must be less diversified than Williamson. The Treynor measure indicates that Joyner has a higher return per unit of 21-5 systematic risk than does Williamson, while the Sharpe ratio indicates that Joyner has a lower return per unit of total risk than does Williamson. 12. Support. A manager could be a better performer in one type of circumstance. For example, a manager who does no timing, but simply maintains a high beta, will do better in up markets and worse in down markets. Therefore, we should observe performance over an entire cycle. Also, to the extent that observing a manager over an entire cycle increases the number of observations, it would improve the reliability of the measurement. Contradict. If we adequately control for exposure to the market (i.e., adjust for beta), then market performance should not affect the relative performance of individual managers. It is therefore not necessary to wait for an entire market cycle to pass before you evaluate a manager. 13. It does, to some degree, if those manager groups can be made sufficiently homogeneous with respect to style. 14. a. The manager’s alpha is 10 – [6 + .5(14 – 6)] = 0. b. From Black-Jensen-Scholes and others, we know that, on average, portfolios with low beta have had positive alphas. (The slope of the empirical security market line is shallower than predicted by the CAPM—see Chapter 7.) Therefore, given the manager’s low beta, performance could be subpar despite the estimated alpha of zero. 15. a. Bogey: (0.60 2.5%) + (0.30 1.2%) + (0.10  0.5%) = 1.91% Actual: (0.70 2.0%) + (0.20 1.0%) + (0.10  0.5%) = 1.65% Underperformance: 0.26% b. Security selection: (1) (2) (3) = (1) × (2) Differential Manager’s Contribution Return Within Market Market (Manager Portfolio to – Index) Weight Performance Equity –0.5% 0.70 −0.35% Bonds –0.2% 0.20 –0.04% Cash 0.0% 0.10 0.00% Contribution of security selection: −0.39% c. Asset allocation: 21-6 (1) (2) (3) = (1) × (2) Excess weight Index Contribution Market (Manager – to Benchmark) Return Performance Equity 0.10 2.5% 0.25% Bonds –0.10 1.2% –0.12% Cash 0.00 0.5% 0.00% Contribution of asset allocation: 0.13% Summary: Security selection –0.39% Asset allocation 0.13% Excess performance –0.26% 16. Time-weighted average return = (1.151.1) 1/1 12.47% 15% 10% (The arithmetic mean is: 2 12.5% .) To compute dollar-weighted rate of return, cash flows are: CF 0 −$500,000 CF 1 −$500,000 CF 2 ($500,000 × 1.15 × 1.10) + ($500,000 × 1.10) = $1,182,500 Dollar-weighted rate of return = 11.71% 17. a. The most likely reason for a difference in ranking is due to the absence of diversification in fund A. The Sharpe ratio measures excess return per unit of total risk, while the Treynor ratio measures excess return per unit of systematic risk. Since fund A performed well on the Treynor measure and so poorly on the Sharpe measure, it seems that the fund carries a greater amount of unsystematic risk, meaning it is not well diversified and systematic risk is not the relevant risk measure. 18. The within sector selection calculates the return according to security selection. This is done by summing the weight of the security in the portfolio multiplied by the return of the security in the portfolio minus the return of the security in the benchmark: 21-7 Lar
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