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CHAPTER 6 PORTFOLIO RISK AND RETURN: PART II Presenter Venue Date FORMULAS FOR PORTFOLIO RISK AND RETURN N R E p wi E Ri i 1 N 2 p wi w j C o v (i , j ) i 1, j 1 N wi 1 i 1 G iv e n : C o v ( i , j ) ij i N T hen: 2 p i 1 p 2 p a n d C o v ( i, i ) i 2 j N wi i 2 2 i , j 1, i j w i w j ij i j EXHIBIT 6-1 PORTFOLIO RISK AND RETURN Portfolio X Y Z Return Standard deviation Correlation between Assets 1 and 2 X Weight in Asset 1 25.0% 50.0 75.0 Weight in Asset 2 75.0% 50.0 25.0 10.0% 20.0% 5.0% 10.0% 0.0 .25 .20 .75 .10 2 2 2 2 Portfolio Return 6.25% 7.50 8.75 Portfolio Standard Deviation 9.01% 11.18 15.21 (. 25 )( 0 )(. 20 )(. 10 ) (. 75 )( 0 )(. 10 )(. 20 ) 9 . 01 % PORTFOLIO OF RISK-FREE AND RISKY ASSETS Combine risk-free asset and risky asset Capital allocation line (CAL) Superimpose utility curves on the CAL Optimal Risky Portfolio EXHIBIT 6-2 RISK-FREE ASSET AND PORTFOLIO OF RISKY ASSETS DOES A UNIQUE OPTIMAL RISKY PORTFOLIO EXIST? Single Optimal Portfolio Identical Expectations Different Expectations Different Optimal Portfolios CAPITAL MARKET LINE (CML) Expected Portfolio Return E (Rp) EXHIBIT 6-3 CAPITAL MARKET LINE CML Points above the CML are not achievable Efficient frontier E(Rm) M Individual Securities Rf Standard Deviation of Portfolio p CML: RISK AND RETURN E R p w1 R f 1 w1 E R m p 1 w1 m By substitution, E(Rp) can be expressed in terms of σp, and this yields the equation for the CML: E Rp E Rm R f Rf p m EXAMPLE 6-1 RISK AND RETURN ON THE CML Mr. Miles is a first time investor and wants to build a portfolio using only U.S. T-bills and an index fund that closely tracks the S&P 500 Index. The T-bills have a return of 5%. The S&P 500 has a standard deviation of 20% and an expected return of 15%. 1. Draw the CML and mark the points where the investment in the market is 0%, 25%, 75%, and 100%. 2. Mr. Miles is also interested in determining the exact risk and return at each point. EXAMPLE 6-2 RISK AND RETURN OF A LEVERAGED PORTFOLIO WITH EQUAL LENDING AND BORROWING RATES Mr. Miles decides to set aside a small part of his wealth for investment in a portfolio that has greater risk than his previous investments because he anticipates that the overall market will generate attractive returns in the future. He assumes that he can borrow money at 5% and achieve the same return on the S&P 500 as before: an expected return of 15% with a standard deviation of 20%. Calculate his expected risk and return if he borrows 25%, 50%, and 100% of his initial investment amount. SYSTEMATIC AND NONSYSTEMATIC RISK Can be eliminated by diversification Nonsystematic Risk Total Risk Systematic Risk RETURN-GENERATING MODELS ReturnGenerating Model Different Factors Estimate of Expected Return GENERAL FORMULA FOR RETURNGENERATING MODELS All models contain return on the market portfolio as a key factor Factor weights or factor loadings E Ri R f k E F ij j 1 j i1 E Rm R f Risk factors k E F ij j2 j THE MARKET MODEL E R i R f i E R m R f R i R f i R m R f ei R i i i R m ei Single-index model The difference between expected returns and realized returns is attributable to an error term, ei. The market model: the intercept, αi, and slope coefficient, βi, can be estimated by using historical security and market returns. Note αi = Rf(1 – βi). CALCULATION AND INTERPRETATION OF BETA i i C ov R i , R m 2 m 0.026250 0.02250 i ,m i m 2 m i ,m i m 0.70 0.25 0.15 0.02250 0.70 0.25 0.15 1.17 EXHIBIT 6-6 BETA ESTIMATION USING A PLOT OF SECURITY AND MARKET RETURNS Ri Slope = β [Beta] Rm Market Return CAPITAL ASSET PRICING MODEL (CAPM) Beta is the primary determinant of expected return E R i R f i E R m R f E R i 3% 1.5 9% 3% 12.0% E R i 3% 1.0 9% 3% 9.0% E R i 3% 0.5 9% 3% 6.0% E R i 3% 0.0 9% 3% 3.0% ASSUMPTIONS OF THE CAPM Investors are risk-averse, utility-maximizing, rational individuals. Markets are frictionless, including no transaction costs or taxes. Investors plan for the same single holding period. Investors have homogeneous expectations or beliefs. All investments are infinitely divisible. Investors are price takers. EXHIBIT 6-7 THE SECURITY MARKET LINE (SML) E(Ri) Expected Return SML E(Rm) M βi = βm Slope = Rm – Rf Rf 1.0 Beta The SML is a graphical representation of the CAPM. PORTFOLIO BETA Portfolio beta is the weighted sum of the betas of the component securities: N p w i i (0 .4 0 1 .5 0 ) (0 .6 0 1 .2 0 ) 1 .3 2 i 1 The portfolio’s expected return given by the CAPM is: E R p R f p E R m R f E R p 3 % 1 .3 2 9 % 3 % 1 0 .9 2 % APPLICATIONS OF THE CAPM Estimates of Expected Return CAPM Applications Security Selection Performance Appraisal PERFORMANCE EVALUATION: SHARPE RATIO AND TREYNOR RATIO S h arp e ratio T reyn o r ratio Rp Rf p Rp Rf p PERFORMANCE EVALUATION: MSQUARED (M2) Sharpe Ratio •Identical rankings •Expressed in percentage terms M 2 Rp Rf m p Rm R f PERFORMANCE EVALUATION: JENSEN’S ALPHA p R p R f p Rm R f EXHIBIT 6-8 MEASURES OF PORTFOLIO PERFORMANCE EVALUATION Manager Ri σi βi E(Ri) X Y Z M Rf 10.0% 11.0 12.0 9.0 3.0 20.0% 10.0 25.0 19.0 0.0 1.10 0.70 0.60 1.00 0.00 9.6% 7.2 6.6 9.0 3.0 Sharpe Treynor Ratio Ratio 0.35 0.064 0.80 0.114 0.36 0.150 0.32 0.060 – – M2 αi 0.65% 9.20 0.84 0.00 – 0.40% 3.80 5.40 0.00 0.00 EXHIBIT 6-11 THE SECURITY CHARACTERISTIC LINE (SCL) Ri – Rf Excess Security Return Excess Returns Jensen’s Alpha [Beta] Rm – Rf Excess Market Return EXHIBIT 6-12 SECURITY SELECTION USING SML SML 15 % A ( = 11%, β = 0.5) B ( = 12%, β = 1.0) Rf = 5% = 1.0 Beta Overvalued Return on Investment C ( = 20%, β = 1.2) Undervalued Ri DECOMPOSITION OF TOTAL RISK FOR A SINGLE-INDEX MODEL R i R f i R m R f ei T o ta l va ria n ce = S yste m a tic va ria n ce + N o n syste m a tic va ria n ce i i m e i 2C o v R m , e i 2 2 2 2 Zero 2 2 2 i m ei EXHIBIT 6-13 DIVERSIFICATION WITH NUMBER OF STOCKS Non-Systematic Variance Variance Total Variance Variance of Market Portfolio Systematic Variance 1 5 10 20 Number of Stocks 30 WHAT SHOULD THE RELATIVE WEIGHT OF SECURITIES IN THE PORTFOLIO BE? Higher Alpha →Higher Weight Inform ation ratio Greater NonSystematic Risk →Lower Weight i ei 2 LIMITATIONS OF THE CAPM Theoretical Practical • Single-factor model • Single-period model • • • • • Market portfolio Proxy for a market portfolio Estimation of beta Poor predictor of returns Homogeneity in investor expectations EXTENSIONS TO THE CAPM: ARBITRAGE PRICING THEORY (APT) Risk Premium for Factor 1 E R p R F 1 p ,1 Sensitivity of the Portfolio to Factor 1 K p ,K FOUR-FACTOR MODEL Value Anomaly E R it i i , M K T M K Tt i , SM B SM B t i , H M L H M L t i ,U M D U M D t Size Anomaly SUMMARY • Portfolio risk and return • Optimal risky portfolio and the capital market line (CML) • Return-generating models and the market model • Systematic and non-systematic risk • Capital asset pricing model (CAPM) and the security market line (SML) • Performance measures • Arbitrage pricing theory (APT) and factor models