3 7.1 The Capital Asset Pricing Model AssumptionsMarkets are competitive, equally profitableNo investor is wealthy enough to individually affect pricesAll information publicly available; all securities publicNo taxes on returns, no transaction costsUnlimited borrowing/lending at risk-free rateInvestors are alike except for initial wealth, risk aversionInvestors plan for single-period horizon; they are rational, mean-variance optimizersUse same inputs, consider identical portfolio opportunity sets
4 7.1 The Capital Asset Pricing Model Hypothetical EquilibriumAll investors choose to hold market portfolioMarket portfolio is on efficient frontier, optimal risky portfolio
5 7.1 The Capital Asset Pricing Model Hypothetical EquilibriumRisk premium on market portfolio is proportional to variance of market portfolio and investor’s risk aversionRisk premium on individual assetsProportional to risk premium on market portfolioProportional to beta coefficient of security on market portfolio
6 Figure 7.1 Efficient Frontier and Capital Market Line
7 7.1 The Capital Asset Pricing Model Passive Strategy is EfficientMutual fund theorem: All investors desire same portfolio of risky assets, can be satisfied by single mutual fund composed of that portfolioIf passive strategy is costless and efficient, why follow active strategy?If no one does security analysis, what brings about efficiency of market portfolio?
8 7.1 The Capital Asset Pricing Model Risk Premium of Market PortfolioDemand drives prices, lowers expected rate of return/risk premiumsWhen premiums fall, investors move funds into risk-free assetEquilibrium risk premium of market portfolio proportional toRisk of marketRisk aversion of average investor
10 7.1 The Capital Asset Pricing Model The Security Market Line (SML)Represents expected return-beta relationship of CAPMGraphs individual asset risk premiums as function of asset riskAlphaAbnormal rate of return on security in excess of that predicted by equilibrium model (CAPM)
15 Table 7.1 Monthly Return Statistics 01/06 - 12/10 T-BillsS&P 500GoogleAverage rate of return0.1840.2391.125Average excess return-0.0550.941Standard deviation*0.1775.1110.40Geometric average0.1800.1070.600Cumulative total 5-year return11.656.6043.17Gain Jan 2006-Oct 20079.0427.4570.42Gain Nov 2007-May 20092.29-38.87-40.99Gain June 2009-Dec 20100.1036.8342.36* The rate on T-bills is known in advance, SD does not reflect risk.
18 Figure 7.4 Scatter Diagram/SCL: Google vs. S&P 500, 01/06-12/10
19 Table 7.2 SCL for Google (S&P 500), 01/06-12/10 Linear RegressionRegression StatisticsR0.5914R-square0.3497Adjusted R-square0.3385SE of regression8.4585Total number of observations60Regression equation: Google (excess return) = × S&P 500 (excess return)ANOVAdfSSMSFp-levelRegression131.190.0000Residual5871.55Total59CoefficientsStandard Errort-Statisticp-valueLCLUCLIntercept0.87511.09200.80130.42623.4877S&P 5001.20310.21545.58480.68771.7185t-Statistic (2%)2.3924LCL - Lower confidence interval (95%)UCL - Upper confidence interval (95%)
20 7.2 CAPM and Index Models Estimation results Security Characteristic Line (SCL)Plot of security’s expected excess return over risk-free rate as function of excess return on marketRequired rate = Risk-free rate + β x Expected excess return of index
21 7.2 CAPM and Index Models Predicting Betas Mean reversion Betas move towards mean over timeTo predict future betas, adjust estimates from historical data to account for regression towards 1.0
22 7.3 CAPM and the Real WorldCAPM is false based on validity of its assumptionsUseful predictor of expected returnsUntestable as a theoryPrinciples still validInvestors should diversifySystematic risk is the risk that mattersWell-diversified risky portfolio can be suitable for wide range of investors
25 Table 7.3 Monthly Rates of Return, 01/06-12/10 Monthly Excess Return % *Total ReturnSecurityAverageStandard DeviationGeometric AverageCumulative ReturnT-bill0.1811.65Market index **0.265.440.3019.51SMB0.342.460.3120.70HML0.012.97-0.03-2.06Google0.9410.400.6043.17*Total return for SMB and HML** Includes all NYSE, NASDAQ, and AMEX stocks.
26 Table 7.4 Regression Statistics: Alternative Specifications Regression statistics for:1.A Single index with S&P 500 as market proxy1.B Single index with broad market index (NYSE+NASDAQ+AMEX)2. Fama French three-factor model (Broad Market+SMB+HML)Monthly returns January December 2010Single Index SpecificationFF 3-Factor SpecificationEstimateS&P 500Broad Market Indexwith Broad Market IndexCorrelation coefficient0.590.610.70Adjusted R-Square0.340.360.47Residual SD = Regression SE (%)8.468.337.61Alpha = Intercept (%)0.88 (1.09)0.64 (1.08)0.62 (0.99)Market beta1.20 (0.21)1.16 (0.20)1.51 (0.21)SMB (size) beta--0.20 (0.44)HML (book to market) beta-1.33 (0.37)Standard errors in parenthesis
27 7.5 Arbitrage Pricing Theory Relative mispricing creates riskless profitArbitrage Pricing Theory (APT)Risk-return relationships from no-arbitrage considerations in large capital marketsWell-diversified portfolioNonsystematic risk is negligibleArbitrage portfolioPositive return, zero-net-investment, risk-free portfolio
28 7.5 Arbitrage Pricing Theory Calculating APTReturns on well-diversified portfolio
29 Table 7.5 Portfolio Conversion Steps to convert a well-diversified portfolio into an arbitrage portfolio*When alpha is negative, you would reverse the signs of each portfolio weight to achieve a portfolio A with positive alpha and no net investment.
30 Table 7.6 Largest Capitalization Stocks in S&P 500 WeightStockWeight
31 Table 7.7 Regression Statistics of S&P 500 Portfolio on Benchmark Portfolio, 01/06-12/10 Linear RegressionRegression StatisticsR0.9933R-square0.9866Adjusted R-square0.9864AnnualizedRegression SE0.59682.067Total number of observations60S&P 500 = × BenchmarkCoefficientsStandard Errort-statp-levelIntercept0.07710.0163Benchmark0.93370.01430.0000
34 7.5 Arbitrage Pricing Theory Multifactor Generalization of APT and CAPMFactor portfolioWell-diversified portfolio constructed to have beta of 1.0 on one factor and beta of zero on any other factorTwo-Factor Model for APT
35 Table 7.9 Constructing an Arbitrage Portfolio Constructing an arbitrage portfolio with two systemic factors
38 Problem 1 Which stock? Which stock? X = 1.8%Y = -1.5%Which stock?Well diversified: Relevant Risk Measure? Best Choice?Which stock?Held alone: Relevant Risk Measure? Best Choice?β: CAPM ModelStock X with the positive alphaCalculate Sharpe ratios7-38
39 Problem 1 (continued) Sharpe Ratios Held Alone: Sharpe Ratio X = Sharpe Ratio Y =Sharpe Ratio Index =(0.14 – 0.05)/0.36 = 0.25(0.17 – 0.05)/0.25 = 0.48Better(0.14 – 0.05)/0.15 = 0.607-39
41 Problem 3 E(rP) = rf + b[E(rM) – rf] E(rp) when double the beta: If the stock pays a constant dividend in perpetuity, then we know from the original data that the dividend (D) must satisfy the equation for a perpetuity:Price = Dividend / E(r)$40 = Dividend / 0.13At the new discount rate of 19%, the stock would be worth:$5.20 / 0.19 = $27.3713% = 7% + β(8%) orβ = 0.75E(rP) = 7% + 1.5(8%) orE(rP) = 19%so the Dividend = $40 x 0.13 = $5.20The increase in stock risk has reduced the value of the stock by($ $40) / $40 = %.7-41
42 Problem 4 False. b = 0 implies E(r) = rf , not zero. Depends on what one means by ‘volatility.’ If one means the then this statement is false. Investors require a risk premium for bearing systematic (i.e., market or undiversifiable) risk.False. You should invest 0.75 of your portfolio in the market portfolio, which has β = 1, and the remainder in T-bills. Then:bP = (0.75 x 1) + (0.25 x 0) = 0.757-42
43 Problems 5 & 65.6.Not possible. Portfolio A has a higher beta than Portfolio B, but the expected return for Portfolio A is lower.Possible. Portfolio A's lower expected rate of return can be paired with a higher standard deviation, as long as Portfolio A's beta is lower than that of Portfolio B.If the CAPM is valid, the expected rate of return compensates only for systematic (market) risk as measured by beta, rather than the standard deviation, which includes nonsystematic risk. Thus, Portfolio A's lower expected rate of return can be paired with a higher standard deviation, as long as Portfolio A's beta is lower than that of Portfolio B.7-43
44 Problem 7 7. Calculate Sharpe ratios for both portfolios: Not possible. The reward-to-variability ratio for Portfolio A is better than that of the market, which is not possible according to the CAPM, since the CAPM predicts that the market portfolio is the portfolio with the highest return per unit of risk.7-44
45 Problem 8 8. Need to calculate Sharpe ratios? Not possible. Portfolio A clearly dominates the market portfolio. It has a lower standard deviation with a higher expected return.7-45
46 Problem 9 9. Given the data, the SML is: E(r) = 10% + b(18% – 10%) A portfolio with beta of 1.5 should have an expected return of:E(r) = 10% + 1.5(18% – 10%) = 22%Not Possible: The expected return for Portfolio A is 16% so that Portfolio A plots below the SML (i.e., has an = –6%), and hence is an overpriced portfolio. This is inconsistent with the CAPM.7-46
47 Problem 10 10. E(r) = 10% + b(18% – 10%) The SML is the same as in the prior problem. Here, the required expected return for Portfolio A is:10% + (0.9 8%) = 17.2%Not Possible: The required return is higher than 16%. Portfolio A is overpriced, with = –1.2%.7-47
48 Problem 1111.Sharpe A =Sharpe M =Possible: Portfolio A's ratio of risk premium to standard deviation is less attractive than the market's. This situation is consistent with the CAPM. The market portfolio should provide the highest reward-to-variability ratio.(16% - 10%) / 22% = .27(18% - 10%) / 24% = .337-48
49 Problem 1212Since the stock's beta is equal to 1.0, its expected rate of return should be equal to ______________________.E(r) =0.18 =the market return, or 18%or P1 = $1097-49
50 Problem 13 Part c? r1 = 19%; r2 = 16%; b1 = 1.5; b2 = 1.0 We can’t tell which adviser did the better job selecting stocks because we can’t calculate either the alpha or the return per unit of risk.CAPM: ri = 6% + β(14%-6%)r1 = 19%; r2 = 16%; b1 = 1.5; b2 = 1.0, rf = 6%; rM = 14%a1 =a2 =The second adviser did the better job selecting stocks (bigger + alpha)19% –16% –[6% + 1.5(14% – 6%)] =19% – 18% = 1%[6% + 1.0(14% – 6%)] =16% – 14% = 2%Part c?7-50
51 Problem 13 CAPM: ri = 3% + β(15%-3%) r1 = 19%; r2 = 16%; b1 = 1.5; b2 = 1.0, rf = 3%; rM = 15%a1 =a2 =Here, not only does the second investment adviser appear to be a better stock selector, but the first adviser's selections appear valueless (or worse).19% – [3% + 1.5(15% – 3%)] =16% – [3%+ 1.0(15% – 3%)] =19% – 21% = –2%16% – 15% = 1%7-51
52 Problem 14a. McKay should borrow funds and invest those funds proportionally in Murray’s existing portfolio (i.e., buy more risky assets on margin). In addition to increased expected return, the alternative portfolio on the capital market line (CML) will also have increased variability (risk), which is caused by the higher proportion of risky assets in the total portfolio.b. McKay should substitute low beta stocks for high beta stocks in order to reduce the overall beta of York’s portfolio. Because York does not permit borrowing or lending, McKay cannot reduce risk by selling equities and using the proceeds to buy risk free assets (i.e., by lending part of the portfolio).7-52
53 Problem 15Since the beta for Portfolio F is zero, the expected return for Portfolio F equals the risk-free rate.For Portfolio A, the ratio of risk premium to beta is:The ratio for Portfolio E is:(10% - 4%)/1 = 6%(9% - 4%)/(2/3) = 7.5%Create Portfolio P by buying Portfolio E and shorting F in the proportions to give βp = βA = 1, the same beta as A. βp =Wi βi1 = WE(βE) + (1-WE)(βF);E(rp) =WE = 1 / (2/3) orWE = 1.5 andWF = (1-WE) = -.51.5(9) (4) = 11.5%,Buying Portfolio P and shorting A creates an arbitrage opportunity since both have β = 1p,-A =11.5% - 10% = 1.5%7-53
54 Problem 16 E(IP) = 4% & E(IR) = 6%; E(rstock) = 14% βIP = 1.0 & βIR = 0.4Actual IP = 5%, so unexpected ΔIP = 1%Actual IR = 7%, so unexpected ΔIR = 1%The revised estimate of the expected rate of return of the stock would be the old estimate plus the sum of the unexpected changes in the factors times the sensitivity coefficients, as follows:Revised estimate =Note IP is defined as the growth rate of Industrial Production and of course the inflation rate is a change variableE(rstock) + Δ due to unexpected Δ Factors14% +[(1 1%) + (0.4 1%)] =15.4%7-54
Your consent to our cookies if you continue to use this website.