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Chapter 13 •Return, Risk, and the Security Market Line McGraw-Hill/Irwin Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. Expected Returns • Expected Rate of Return All possible outcomes and probabilities known • Kμ = E(R) = P1K1 + P2K2 + P3K3 Sample taken of past returns • _ • K = ΣK/n 13-1 Example: Expected Returns • Suppose you have predicted the following returns for stocks C and T in three possible states of nature. What are the expected returns? • State Probability C • Boom 0.3 15 • Normal 0.5 10 • Recession ??? 2 • RC = .3(15) + .5(10) + .2(2) = 9.99% • RT = .3(25) + .5(20) + .2(1) = 17.7% T 25 20 1 13-2 Sample of Observations Returns: 2008 8.6% 2007 14.2% 2006 -4.6% 2005 8.8% E(R) = Mean = (8.6+14.2-4.6+8.8)/4 = 6.75 13-3 Variance and Standard Deviation Variance: σ2 = P1(K1 - Kμ)2 + P2(K2 - Kμ)2 + P3(K3 - Kμ)2 _ _ _ Variance: S2 = [(K1 - K)2 + (K2 - K)2 + (K3 - K)2]/n-1 Standard Deviation = Square Root of Variance 13-4 Sample of Observations Returns: 2008 8.6% 2007 14.2% 2006 -4.6% 2005 8.8% E(R) = Mean = (8.6+14.2-4.6+8.8)/4 = 6/75 13-5 Example: Variance and Standard Deviation • Consider the previous example. What are the variance and standard deviation for each stock? • Stock C • 2 = .3(15-9.9)2 + .5(10-9.9)2 + .2(2-9.9)2 = 20.29 • = 4.5 • Stock T • 2 = .3(25-17.7)2 + .5(20-17.7)2 + .2(1-17.7)2 = 74.41 • = 8.63 13-6 Sample Variance Variance = [(8.6 – 6.75)^2 + (14.2 – 6.75)^2 + (-4.6 – 6.75)^2 + (8.8 – 6.75)^2] /3 = [3.4224 + 55,5025 + 128.8225 + 4.2025]/3 = 63.98333 Standard Deviation = 63.98333^(1/2) = 7.998 Average Distance Around Mean [(8.6 – 6.75) + (14.2 – 6.75) + (-4.6 – 6.75) + (8.8 – 6.75)] /4 = 5.675 13-7 Portfolios vs. Individual Stocks • A portfolio is a collection of assets • An asset’s risk and return are important in how they affect the risk and return of the portfolio • The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets 13-8 Individual Stocks’ Mean Return θ 1 2 3 4 5 P .2 .2 .2 .2 .2 K W -10 40 -5 35 15 40 -10 35 -5 15 Kμ = .2(-10) + .2(40) + .2(-5) + .2(35) + .2(15) = 15 Wμ = .2(40) + .2(-10) + .2(35) + .2(-5) + .2(15) = 15 13-9 Individual Stocks’ Variance and Standard Deviation θ P K W 1 2 3 4 5 .2 .2 .2 .2 .2 -10 40 -5 35 15 40 -10 35 -5 15 (K-Kμ)2 (W-Wμ)2 625 625 400 400 0 625 625 400 400 0 σK = [.2(625) + .2(625) + .2(400) +.2(400) + .2(0)]½ = 20.25 σW = [.2(625) + .2(625) + .2(400) +.2(400) + .2(0)]½ =20.25 13-10 Equally Weighted Portfolio Returns θ P K W (K-Kμ)2 1 2 3 4 5 .2 .2 .2 .2 .2 -10 40 -5 35 15 40 -10 35 -5 15 625 625 400 400 0 (W-Wμ)2 625 625 400 400 0 .5K+.5W 15 15 15 15 15 Ex: (1) .5 x -10 + .5 x 40 = 15 Ex: (2) .5 x 40 + .5 x -10 = 15 13-11 Unequally Weighted Portfolio Variance θ P K W (K-Kμ)2 1 2 3 4 5 .2 .2 .2 .2 .2 -10 40 -5 35 15 40 -10 35 -5 15 625 625 400 400 0 (W-Kμ)2 .75K+.25W 625 625 400 400 0 2.5 27.5 5.0 25.0 15.0 Ex: (1) .75 x -10 + .25 x 40 = 2.5 Ex: (2) .75 x 40 + .25 x -10 = 27.5 13-12 Correlation of Security Returns Perfect Positive = +1 Perfect Negative = -1 Uncorrelated = 0 13-13 Diversification Total Risk = Nondiversifiable Risk + Diversifiable Risk Total Risk = Systematic Risk + Unsystematic Risk Total Risk = Market Risk + Firm Risk 13-14 Systematic Risk • Risk factors that affect a large number of assets • Also known as non-diversifiable risk or market risk • Includes such things as changes in GDP, inflation, interest rates, etc. 13-15 Unsystematic Risk • Risk factors that affect a limited number of assets • Also known as unique risk and assetspecific risk • Includes such things as labor strikes, part shortages, etc. 13-16 Diversification • Portfolio diversification is the investment in several different asset classes or sectors • Diversification is not just holding a lot of assets • For example, if you own 50 internet stocks, you are not diversified • However, if you own 50 stocks that span 20 different industries, then you are diversified 13-17 Table 13.7 13-18 The Principle of Diversification • Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns • This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another • However, there is a minimum level of risk that cannot be diversified away and that is the systematic portion 13-19 Figure 13.1 13-20 Diversifiable Risk • The risk that can be eliminated by combining assets into a portfolio • Often considered the same as unsystematic, unique or asset-specific risk • If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away 13-21 Total Risk • Total risk = systematic risk + unsystematic risk • The standard deviation of returns is a measure of total risk • For well-diversified portfolios, unsystematic risk is very small • Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk 13-22 Systematic Risk Principle • There is a reward for bearing risk • There is not a reward for bearing risk unnecessarily • The expected return on a risky asset depends only on that asset’s systematic risk since unsystematic risk can be diversified away 13-23 Measuring Systematic Risk • How do we measure systematic risk? • We use the beta coefficient to measure systematic risk • What does beta tell us? • A beta of 1 implies the asset has the same systematic risk as the overall market • A beta < 1 implies the asset has less systematic risk than the overall market • A beta > 1 implies the asset has more systematic risk than the overall market 13-24 Total versus Systematic Risk • Consider the following information: • Security C • Security K Standard Deviation 20% 30% Beta 1.25 0.95 • Which security has more total risk? • Which security has more systematic risk? • Which security should have the higher expected return? 13-25 Examples of Betas Edison Electricity Coor’s Brewing Sony General Motors Intel Dell .55 .75 .85 1.25 1.40 1.55 13-26 Example: Portfolio Betas • Consider the following four securities • • • • • Security DCLK KO INTC KEI Weight .133 .2 .267 .4 Beta 2.685 0.195 2.161 2.434 • What is the portfolio beta? • .133(2.685) + .2(.195) + .267(2.161) + .4(2.434) = 1.9467 13-27 Beta and the Risk Premium • Remember that the risk premium = expected return – risk-free rate • The higher the beta, the greater the risk premium should be • Can we define the relationship between the risk premium and beta so that we can estimate the expected return? • YES! 13-28 Example: Portfolio Expected Returns and Betas 30% Expected Return 25% E(RA) 20% 15% 10% Rf 5% 0% 0 0.5 1 1.5A 2 2.5 3 Beta 13-29 Reward-to-Risk Ratio: Definition and Example • The reward-to-risk ratio is the slope of the line illustrated in the previous example • Slope = (E(RA) – Rf) / (A – 0) • Reward-to-risk ratio for previous example = (20 – 8) / (1.6 – 0) = 7.5 • What if an asset has a reward-to-risk ratio of 8 (implying that the asset plots above the line)? • What if an asset has a reward-to-risk ratio of 7 (implying that the asset plots below the line)? 13-30 Market Equilibrium • In equilibrium, all assets and portfolios must have the same reward-to-risk ratio and they all must equal the reward-to-risk ratio for the market E ( RA ) R f A E ( RM R f ) M 13-31 Security Market Line • The security market line (SML) is the representation of market equilibrium • The slope of the SML is the reward-to-risk ratio: (E(RM) – Rf) / M • But since the beta for the market is ALWAYS equal to one, the slope can be rewritten • Slope = E(RM) – Rf = market risk premium 13-32 The Capital Asset Pricing Model (CAPM) • The capital asset pricing model defines the relationship between risk and return • E(RA) = Rf + A(E(RM) – Rf) • If we know an asset’s systematic risk, we can use the CAPM to determine its expected return • This is true whether we are talking about financial assets or physical assets 13-33 Factors Affecting Expected Return • Pure time value of money – measured by the risk-free rate • Reward for bearing systematic risk – measured by the market risk premium • Amount of systematic risk – measured by beta 13-34 Example - CAPM • Consider the betas for each of the assets given earlier. If the risk-free rate is 2.13% and the market risk premium is 8.6%, what is the expected return for each? Security DCLK KO INTC KEI Beta 2.685 0.195 2.161 2.434 Expected Return 2.13 + 2.685(8.6) = 25.22% 2.13 + 0.195(8.6) = 3.81% 2.13 + 2.161(8.6) = 20.71% 2.13 + 2.434(8.6) = 23.06% 13-35 Figure 13.4 13-36 Chapter 12 •End of Chapter McGraw-Hill/Irwin Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.