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11-2 Expected Returns Expected returns are based on the probabilities of possible states of the economy –To simplify, assume states of the economy: Boom Recession Return to Quick Quiz

11-3 Expected Returns Expected Return on Stock A E(R A ) = ∑ (Probability for state of economy X expected return for state of economy) Return to Quick Quiz

11-4 Expected risk premium Expected risk premium = Expected return on risky investment - Return on risk free investment Return to Quick Quiz

11-5 Variance and Standard Deviation Variance and standard deviation measure the volatility of returns Return to Quick Quiz

11-6 Variance and Standard Deviation Variance = Weighted average of squared deviations Ơ 2 = ∑ [(Return for state of economy – Expected return for stock) 2 x Probability for state of economy] Standard deviation = square root of variance

11-7 Variance and Standard Deviation Variance and Standard deviation calculated differently than Chapter 10 Chapter 10: historical / actual events Chapter 11: projections

11-8 Portfolios Portfolio = collection of assets Stock weights in a portfolio: stock value divided by total portfolio value (% of portfolio invested in each asset) The risk-return trade-off for a portfolio is measured by: portfolio expected return standard deviation

11-9 Portfolio Expected Returns The expected return of a portfolio is the weighted average of the expected returns for each asset in the portfolio ∑ (Portfolio weight X expected return for individual stock) Return to Quick Quiz

11-10 Portfolio Risk Variance & Standard Deviation Portfolio standard deviation is NOT a weighted average of the standard deviation of the component securities’ risk –If it were, there would be no benefit to diversification.

11-11 Portfolio Variance Compute portfolio return for each state of economy: ∑ (portfolio weight x expected return for individual stock for that state of economy) Compute the overall expected portfolio ∑ (portfolio weight X expected return for individual stock) Compute the portfolio variance and standard deviation Ơ 2 = ∑ (Portfolio Return for state of economy – Expected return for portfolio) 2 x Probability for state of economy Standard deviation = square root of variance

11-12 Announcements, News and Efficient markets Announcements and news contain both expected and surprise components The surprise component affects stock prices Efficient markets result from investors trading on unexpected news –The easier it is to trade on surprises, the more efficient markets should be Efficient markets involve random price changes because we cannot predict surprises

11-13 Returns Total Return = Expected return + unexpected return R = E(R) + U Unexpected return (U) = Systematic portion (m) + Unsystematic portion (ε) Total Return = Expected return E(R) + Systematic portion m + Unsystematic portion ε = E(R) + m + ε

11-14 Systematic Risk Factors that affect a large number of assets “Non-diversifiable risk” “Market risk” Examples: changes in GDP, inflation, interest rates, etc. Return to Quick Quiz

11-15 Unsystematic Risk = Diversifiable risk Risk factors that affect a limited number of assets Risk that can be eliminated by combining assets into portfolios “Unique risk” “Asset-specific risk” Examples: labor strikes, part shortages, etc. Return to Quick Quiz

11-16 The Principle of Diversification Diversification can substantially reduce risk without an equivalent reduction in expected returns –Reduces the variability of returns –Caused by the offset of worse-than- expected returns from one asset by better- than-expected returns from another Minimum level of risk that cannot be diversified away = systematic portion

11-17 Standard Deviations of Annual Portfolio Returns Table 11.7

11-18 Portfolio Conclusions As more stocks are added, each new stock has a smaller risk-reducing impact on the portfolio  p falls very slowly after about 40 stocks are included –The lower limit for  p ≈ 20% =  M.  Forming well-diversified portfolios can eliminate about half the risk of owning a single stock.

11-19 Total Risk = Stand-alone 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  Total risk for a diversified portfolio is essentially equivalent to the systematic risk

11-20 Systematic Risk Principle There is a reward for bearing risk There is no reward for bearing risk unnecessarily The expected return (market required return) on an asset depends only on that asset’s systematic or market risk. Return to Quick Quiz

11-21 Market Risk for Individual Securities The contribution of a security to the overall riskiness of a portfolio Relevant for stocks held in well-diversified portfolios Measured by a stock’s beta coefficient Measures the stock’s volatility relative to the market

11-22 Interpretation of beta If  = 1.0, stock has average risk If  > 1.0, stock is riskier than average If  < 1.0, stock is less risky than average Most stocks have betas in the range of 0.5 to 1.5 Beta of the market = 1.0 Beta of a T-Bill = 0

11-23 Beta Coefficients for Selected Companies Table 11.8

11-24 Example: Work the Web Many sites provide betas for companies Yahoo! Finance provides beta, plus a lot of other information under its profile link Click on the Web surfer to go to Yahoo! Finance –Enter a ticker symbol and get a basic quote –Click on key statistics –Beta is reported under stock price history

11-25 Quick Quiz: Total vs. Systematic Risk Consider the following information: Standard DeviationBeta Security C20%1.25 Security K30%0.95 Which security has more total risk? Which security has more systematic risk? Which security should have the higher expected return?

11-26 Beta and the Risk Premium Risk premium = E(R ) – R f 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!

11-27 SML and Equilibrium Figure 11.4

11-28 Reward-to-Risk Ratio Reward-to-Risk Ratio: = Slope of line on graph In equilibrium, ratio should be the same for all assets When E(R) is plotted against β for all assets, the result should be a straight line

11-29 Market Equilibrium In equilibrium, all assets and portfolios must have the same reward-to-risk ratio Each ratio must equal the reward-to-risk ratio for the market

11-30 Security Market Line The security market line (SML) is the representation of market equilibrium The slope of the SML = reward-to-risk ratio: (E(R M ) – R f ) /  M Slope = E(R M ) – R f = market risk premium –Since  of the market is always 1.0

11-31 The SML and Required Return The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM) R f = Risk-free rate (T-Bill or T-Bond) R M = Market return ≈ S&P 500 RP M = Market risk premium = E(R M ) – R f E(R i ) = “Required Return”

11-32 Capital Asset Pricing Model The capital asset pricing model (CAPM) defines the relationship between risk and return E(R A ) = R f + (E(R M ) – R f )β A If an asset’s systematic risk (  ) is known, CAPM can be used to determine its expected return

11-33 SML example

11-34 Factors Affecting Required Return R f measures the pure time value of money RP M = (E(R M )-R f ) measures the reward for bearing systematic risk  i measures the amount of systematic risk

11-35 Portfolio Beta β p = Weighted average of the Betas of the assets in the portfolio Weights (w i ) = % of portfolio invested in asset i

11-36 Covariance of Returns Measures how much the returns on two risky assets move together.

11-37 Covariance vs. Variance of Returns

11-38 Correlation Coefficient Correlation Coefficient = ρ (rho) Scales covariance to [-1,+1] –-1 = Perfectly negatively correlated – 0 = Uncorrelated; not related –+1 = Perfectly positively correlated

11-39 Two-Stock Portfolios If  = -1.0 –Two stocks can be combined to form a riskless portfolio If  = +1.0 –No risk reduction at all In general, stocks have  ≈ 0.65 –Risk is lowered but not eliminated Investors typically hold many stocks

11-40 Covariance & Correlation Coefficient

11-41  of n-Stock Portfolio  Subscripts denote stocks i and j  i,j = Correlation between stocks i and j  σ i and σ j =Standard deviations of stocks i and j  σ ij = Covariance of stocks i and j

11-42 Portfolio Risk-n Risky Assets i jfor n=2 11w 1 w 1  11 = w 1 2  1 2 12w 1 w 2  12 21w 2 w 1  21 22w 2 w 2  22 = w 2 2  2 2  p 2 = w 1 2  1 2 + w 2 2  2 2 + 2w 1 w 2  12

11-43  p 2 = w 1 2  1 2 + w 2 2  2 2 + 2w 1 w 2  12 Portfolio Risk-2 Risky Assets i jfor n=2 11 w 1 w 1  11 = w 1 2  1 2 = (.50)(.50)(45)(45) 12 w 1 w 2  12 = (.50)(.50)(-.045) 21 w 2 w 1  21 = (.50)(.50)(-.045) 22 w 2 w 2  22 = w 2 2  2 2 = (.50)(.50)(10)(10)  p 2 = w 1 2  1 2 + w 2 2  2 2 + 2w 1 w 2  12 = 0.030625