bkmsolch249e corrected 8910.docx

1、bkmsolch249e corrected 8910CHAPTER 24: PORTFOLIO PERFORMANCE EVALUATIONPROBLEM SETS1. As established in the following result from the text, the Sharpe ratio depends on both alpha for the portfolio () and the correlation between the portfolio and the market index ():Specifically, this result demonstr

2、ates 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 suffic

3、iently large, thereby reducing the correlation with the market index, this can result in a lower Sharpe ratio.2. The IRR (i.e., the dollar-weighted return) can not be ranked relative to either the geometric average return (i.e., the time-weighted return) or the arithmetic average return. Under some

4、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 pe

5、riod 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 If the amount invested also increases each period, and then all of the proceeds are withdrawn at the end of several periods, the IRR is greater than either the geometric or the arithmetic average because more money is invested at the higher rate

6、s 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

7、geometric average, and in Concept Check 1, where the IRR is greater than the geometric average.)3. 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 as to whether to shift resource

8、s has to be made on the basis of the macro, compared to the micro, forecasting ability of the portfolio management team.4. a. Arithmetic average:; b. Dispersion: ABC = 7.07%; XYZ = 13.91%Stock XYZ has greater dispersion.(Note: We used 5 degrees of freedom in calculating standard deviations.)c. Geome

9、tric average:rABC = (1.20 1.12 1.14 1.03 1.01)1/5 1 = 0.0977 = 9.77%rXYZ = (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 th

10、e 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

11、the expected rate of return for both stocks.5. a. Time-weighted average returns are based on year-by-year rates of return:YearReturn = (capital gains + dividend)/price2007 2008($120 $100) + $4/$100 = 24.00%2008 2009($90 $120) + $4/$120 = 21.67%2009 2010($100 $90) + $4/$90 = 15.56%Arithmetic mean: (2

12、4% 21.67% + 15.56%)/3 = 5.96%Geometric mean: (1.24 0.7833 1.1556)1/3 1 = 0.0392 = 3.92%b.DateCashFlowExplanation1/1/07$300Purchase of three shares at $100 each1/1/08$228Purchase of two shares at $120 less dividend income on three shares held1/1/09$110Dividends on five shares plus sale of one share a

13、t $901/1/10$416Dividends on four shares plus sale of four shares at $100 each416110Date: 1/1/07 1/1/08 1/1/09 1/1/10228300Dollar-weighted return = Internal rate of return = 0.1607%6.TimeCash flowHolding period return03($90) = $2701$100(10090)/90 = 11.11%2$1000%3$1000%a. Time-weighted geometric avera

14、ge 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

15、rate of return = IRR = 5.46%Using a financial calculator, enter: n = 3, PV = 270, FV = 0, PMT = 100. Then compute the interest rate, or use the CF0=300, CF1=100, F1=3, then compute IRR. The IRR exceeds the other averages because the investment fund was the largest when the highest return occurred.7.

16、 a. 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 Portfolio A and a short position in Portfolio B.b. If you will hold only one of the two portfolios, then the Sharpe measure is the appropriate

17、criterion:Using the Sharpe criterion, Portfolio A is the preferred portfolio.8.a.Stock AStock B (i)Alpha = regression intercept1.0%2.0% (ii)Information ratio = 0.09710.1047 (iii)*Sharpe measure = 0.49070.3373 (iv)*Treynor measure = 8.83310.500* To compute the Sharpe measure, note that for each stock

18、, (rP rf ) can be computed from the right-hand side of the regression equation, using the assumed parameters rM = 14% and rf = 6%. The standard deviation of each stocks returns is given in the problem.* The beta to use for the Treynor measure is the slope coefficient of the regression equation prese

19、nted in the problem.b. (i) If this is the only risky asset held by the investor, then Sharpes measure is the appropriate measure. Since the Sharpe measure is higher for Stock A, then A is the best choice.(ii) If the stock is mixed with the market index fund, then the contribution to the overall Shar

20、pe measure is determined by the appraisal ratio; therefore, Stock B is preferred.(iii) If the stock is one of many stocks, then Treynors measure is the appropriate measure, and Stock B is preferred.9. We need to distinguish between market timing and security selection abilities. The intercept of the

21、 scatter diagram is a measure of stock selection ability. If the manager tends to have a positive excess return even when the markets 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 so

22、urce 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 along the horizontal axis show good timing ability. The steeper slope shows that the manager maintained higher portfolio sensitivity to market swin

23、gs (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 mark

24、et 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 AbilityTiming AbilityA.BadGoodB.GoodGoodC.GoodBadD.BadBad10. a. Bogey: (0.60 2.5%) + (0.30 1.2%) + (0.10 0.5%) = 1.91

25、%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)MarketDifferential returnwithin market(Manager index)Managersportfolio weightContribution toperformanceEquity0.5%0.700.35%Bonds0.2%0.200.04%Cash0.0%0.100.00%Contribution of securit

26、y selection:0.39%c. Asset Allocation:(1)(2)(3) = (1) (2)MarketExcess weight(Manager benchmark)IndexReturnContribution toperformanceEquity0.10%2.5%0.25%Bonds0.10%1.2%0.12%Cash0.00%0.5%0.00%Contribution of asset allocation:0.13%Summary:Security selection 0.39%Asset allocation 0.13%Excess performance 0

27、.26%11. a. Manager: (0.30 20%) + (0.10 15%) + (0.40 10%) + (0.20 5%) = 12.50%Bogey: (0.15 12%) + (0.30 15%) + (0.45 14%) + (0.10 12%) = 13.80%Added value: 1.30%b. Added value from country allocation:(1)(2)(3) = (1) (2)CountryExcess weight(Manager benchmark)Index Returnminus bogeyContribution toperfo

28、rmanceU.K.0.151.8%0.27%Japan0.201.2%0.24%U.S.0.050.2%0.01%Germany0.101.8%0.18%Contribution of country allocation:0.70%c. Added value from stock selection:(1)(2)(3) = (1) (2)CountryDifferential returnwithin country(Manager Index)Managerscountry weightContribution toperformanceU.K.0.080.30%2.4%Japan0.

29、000.10%0.0%U.S.0.040.40%1.6%Germany0.070.20%1.4%Contribution of stock selection:0.6%Summary:Country allocation 0.70%Stock selection 0.60%Excess performance 1.30%12. Support: A manager could be a better performer in one type of circumstance than in another. For example, a manager who does no timing,

30、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