# © 2003 The McGraw-Hill Companies, Inc. All rights reserved. Return, Risk, and the Security Market Line Chapter Thirteen.

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© 2003 The McGraw-Hill Companies, Inc. All rights reserved. Return, Risk, and the Security Market Line Chapter Thirteen

13.1 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Key Concepts and Skills Know how to calculate expected returns Understand the impact of diversification Understand the systematic risk principle Understand the security market line Understand the risk-return trade-off Be able to use the Capital Asset Pricing Model (CAPM)

13.2 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Chapter Outline Expected Returns and Variances Portfolios Risk: Systematic and Unsystematic Diversification and Portfolio Risk Systematic Risk and Beta The Security Market Line The SML and the Cost of Capital: A Preview Arbitrage Pricing Theory

13.3 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Expected Returns 13.1 Expected returns are based on the probabilities of possible outcomes In this context, “expected” means average if the process is repeated many times The “expected” return does not even have to be a possible return n = total number of states, p = probability that state i occurs, R = return in state i

13.4 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Expected Returns – Example 1 Suppose you have predicted the following returns for stocks C and T in three possible states of nature. What are the expected returns? –StateProbabilityCT –Boom0.30.150.25 –Normal0.50.100.20 –Recession0.20.020.01 R C =.3(.15) +.5(.10) +.2(.02) =.099 = 9.9% R T =.3(.25) +.5(.20) +.2(.01) =.177 = 17.7%

13.5 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Variance and Standard Deviation Variance (σ²) and standard deviation (σ) still measure the volatility of returns You can use unequal probabilities for the entire range of possibilities Weighted average of squared deviations

13.6 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Variance and Standard Deviation – Example 1 Consider the previous example. What is the variance and standard deviation for each stock? Stock C  2 =.3(.15-.099) 2 +.5(.1-.099) 2 +.2(.02-.099) 2 =.002029  =.045 Stock T  2 =.3(.25-.177) 2 +.5(.2-.177) 2 +.2(.01-.177) 2 =.007441  =.0863

13.7 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Portfolios 13.2 A portfolio is a collection of assets An asset’s risk and return is important in how it affects 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 The sum of risks of individual assets does not equal the risk of the portfolio

13.8 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Example: Portfolio Weights Suppose you have \$15,000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security? –\$2000 of ABC –\$3000 of DEF –\$4000 of GHI –\$6000 of JKL ABC: 2/15 =.133 DEF: 3/15 =.2 GHI: 4/15 =.267 JKL: 6/15 =.4

13.9 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Portfolio Expected Returns The expected return of a portfolio is the weighted average of the expected returns for each asset in the portfolio w = the weight of asset j in the portfolio

13.10 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Example: Expected Portfolio Returns Consider the portfolio weights computed previously. If the individual stocks have the following expected returns, what is the expected return for the portfolio? –ABC: 19.65% –DEF: 8.96% –GHI: 9.67% –JKL: 8.13% E(R P ) =.133(19.65) +.2(8.96) +.267(9.67) +.4(8.13) = 10.24%

13.11 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Portfolio Variance Compute the portfolio return for each state: R P = w 1 R 1 + w 2 R 2 + … + w m R m Compute the expected portfolio return using the same formula as for an individual asset Compute the portfolio variance and standard deviation using the same formulas as for an individual asset

13.12 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Example: Portfolio Variance Consider the following information –Invest 50% of your money in Asset A –StateProbabilityABPortfolio –Boom.570%10% 7.3% –Bust.5-20%30% 12.8% What is the expected return and standard deviation for each asset? What is the expected return and standard deviation for the portfolio?

13.13 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Portfolio Variance Asset A E( R)= 0.5 (0.7) + 0.5 (-0.2) = 0.35 – 0.1 = 0.25 σ²= 0.5 (0.7 - 0.25)² + 0.5 (-0.2 - 0.25)² = 0.2025 σ= 0.45 Asset B E( R)= 0.5 (0.1) + 0.5 (0.3) = 0.05 + 0.15 = 0.2 σ²= 0.5 (0.1 - 0.2)² + 0.5 (0.3 - 0.2)² = 0.01 σ= 0.1 Portfolio E( R)= 0.5 (0.25) + 0.5 (0.2) = 0.125 + 0.1 = 0.225 σ²= 0.5 (0.25 – 0.225)² + 0.5 (0.2 – 0.225)² = 0.000625 σ= 0.025

13.14 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Another Way to Calculate Portfolio Variance Portfolio variance can also be calculated using the following formula:

13.18 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Figure 13.2 – Graphs of Possible Relationships Between Two Stocks

13.19 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Diversification There are benefits to diversification whenever the correlation between two stocks is less than perfect (p < 1.0) Figure 13.4

13.20 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Terminology Feasible set (also called the opportunity set) – the curve that comprises all of the possible portfolio combinations Efficient set – the portion of the feasible set that only includes the efficient portfolio (where the maximum return is achieved for a given level of risk, or where the minimum risk is accepted for a given level of return) Minimum Variance Portfolio – the possible portfolio with the least amount of risk

13.21 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Quick Quiz II Consider the following information –StateProbabilityXZ –Boom.2515%10% –Normal.6010%9% –Recession.155%10% What is the expected return and standard deviation for a portfolio with an investment of \$6000 in asset X and \$4000 in asset Y?

13.22 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Systematic Risk 13.4 Risk that cannot be diversified away through portfolio formation (rewarded with return) 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.

13.23 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Unsystematic Risk Risk that can be diversified away through portfolio formation (no reward of return) Risk factors that affect a limited number of assets Also known as diversifiable risk, unique risk and asset-specific risk Includes such things as labor strikes, shortages, etc.

13.24 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Diversification 13.5 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

13.26 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. 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

13.28 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Diversifiable (Unsystematic) Risk The risk that can be eliminated by combining assets into a portfolio Synonymous with 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 The market will not compensate investors for assuming unnecessary risk

13.29 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. 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

13.30 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Systematic Risk Principle 13.6 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

13.31 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Measuring Systematic Risk How do we measure systematic risk? –We use the beta coefficient to measure systematic risk.  i is the percent change in asset i’s return for a 1% change in the market portfolio’s return. 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

13.32 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Figure 13.7 – High and Low Betas

13.33 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Table 13.10 – Beta Coefficients for Selected Companies

13.34 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Example: Portfolio Betas Consider the previous example with the following four securities –SecurityWeightBeta –ABC.1333.69 –DEF.20.64 –GHI.2671.64 –JKL.41.79 What is the portfolio beta?.133(3.69) +.2(.64) +.267(1.64) +.4(1.79) = 1.773

13.35 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Beta and the Risk Premium 13.7 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!

13.36 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Figure 13.8A – Portfolio Expected Returns and Betas RfRf

13.37 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. 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(R A ) – R f ) / (  A – 0) –Reward-to-risk ratio for previous example = (20 – 8) / (1.6 – 0) = 7.5 –  f = 0 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)?

13.38 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. 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

13.39 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. 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(R M ) – R f ) /  M But since the beta for the market is ALWAYS equal to one, the slope can be rewritten Slope = E(R M ) – R f = market risk premium

13.41 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. The Capital Asset Pricing Model (CAPM) The capital asset pricing model defines the relationship between risk and return E(R A ) = R f +  A (E(R M ) – R f ) 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

13.42 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. 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

13.43 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Example - CAPM Consider the betas for each of the assets given earlier. If the risk-free rate is 4.5% and the market risk premium is 8.5%, what is the expected return for each? SecurityBetaExpected Return ABC3.694.5 + 3.69(8.5) = 35.865% DEF.644.5 +.64(8.5) = 9.940% GHI1.644.5 + 1.64(8.5) = 18.440% JKL1.794.5 + 1.79(8.5) = 19.715%

13.44 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Example Calculate the market risk premium on the asset i, and the expected return on asset i using the following information:  i = 1.5, E(R M )= 15%, R f =5% Market risk premium = 15% - 5% = 10% Risk premium on asset i = 1.5(15% - 5%) = 15% Expected return on asset i = 5% + 1.5(15% - 5%) = 20%

13.45 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Arbitrage Pricing Theory (APT) 13.9 The major advantage of the APT model is that it can handle multiple factors not included in CAPM. Like CAPM, the APT model assumes that stock returns depend on both expected and unexpected returns. Unlike CAPM, the unexpected return in the APT model is related to several market factors. Assuming those factors are unanticipated changes in inflation, GNP, and interest rates, the expected return would be written as:

13.46 Copyright © 2005 McGraw-Hill Ryerson Limited. All rights reserved. Summary 13.10 There is a reward for bearing risk Total risk has two parts: systematic risk and unsystematic risk Unsystematic risk can be eliminated through diversification Systematic risk cannot be eliminated and is rewarded with a risk premium Systematic risk is measured by the beta The equation for the SML is the CAPM, and it determines the equilibrium required return for a given level of risk

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