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Return, Risk, and the Security Market Line

13. Return, Risk, and the Security Market Line. Expected Returns. Expected Rate of Return All possible outcomes and probabilities known Kμ = E(R) = P 1 K 1 + P 2 K 2 + P 3 K 3 Sample taken of past returns _ K = ΣK/n. Example: Expected Returns.

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Return, Risk, and the Security Market Line

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  1. 13 Return, Risk, and the Security Market Line

  2. 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

  3. 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 T • Boom 0.3 15 25 • Normal 0.5 10 20 • Recession ??? 2 1 • RC = .3(15) + .5(10) + .2(2) = 9.99% • RT = .3(25) + .5(20) + .2(1) = 17.7%

  4. Sample of Observations Returns: • 8.6% 2007 14.2% • -4.6% • 8.8% E(R) = Mean = (8.6+14.2-4.6+8.8)/4 = 6.75

  5. 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

  6. Sample of Observations Returns: • 8.6% 2007 14.2% • -4.6% • 8.8% E(R) = Mean = (8.6+14.2-4.6+8.8)/4 = 6/75

  7. Example: Variance andStandard 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

  8. 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

  9. 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

  10. Individual Stocks’ Mean Return θ P K W 1 .2 -10 40 2 .2 40 -10 3 .2 -5 35 4 .2 35 -5 5 .2 15 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

  11. Individual Stocks’ Varianceand Standard Deviation θ P K W (K-Kμ)2 (W-Wμ)2 1 .2 -10 40 625 625 2 .2 40 -10 625 625 3 .2 -5 35 400 400 4 .2 35 -5 400 400 5 .2 15 15 0 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

  12. Equally WeightedPortfolio Returns θ P K W (K-Kμ)2 (W-Wμ)2 .5K+.5W 1 .2 -10 40 625 625 15 2 .2 40 -10 625 625 15 3 .2 -5 35 400 400 15 4 .2 35 -5 400 400 15 5 .2 15 15 0 0 15 Ex: (1) .5 x -10 + .5 x 40 = 15 Ex: (2) .5 x 40 + .5 x -10 = 15

  13. Unequally Weighted Portfolio Variance θ P K W (K-Kμ)2 (W-Kμ)2 .75K+.25W 1 .2 -10 40 625 625 2.5 2 .2 40 -10 625 625 27.5 3 .2 -5 35 400 400 5.0 4 .2 35 -5 400 400 25.0 5 .2 15 15 0 0 15.0 Ex: (1) .75 x -10 + .25 x 40 = 2.5 Ex: (2) .75 x 40 + .25 x -10 = 27.5

  14. Correlation of Security Returns Perfect Positive = +1 Perfect Negative = -1 Uncorrelated = 0

  15. Diversification Total Risk = Nondiversifiable Risk + Diversifiable Risk Total Risk = Systematic Risk + Unsystematic Risk Total Risk = Market Risk + Firm Risk

  16. 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.

  17. Unsystematic Risk • Risk factors that affect a limited number of assets • Also known as unique risk and asset-specific risk • Includes such things as labor strikes, part shortages, etc.

  18. 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

  19. Table 13.7

  20. 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

  21. Figure 13.1

  22. 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

  23. 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

  24. 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

  25. 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

  26. Total versus Systematic Risk • Consider the following information: Standard Deviation Beta • Security C 20% 1.25 • Security K 30% 0.95 • Which security has more total risk? • Which security has more systematic risk? • Which security should have the higher expected return?

  27. Examples of Betas Edison Electricity .55 Coor’s Brewing .75 Sony .85 General Motors 1.25 Intel 1.40 Dell 1.55

  28. Example: Portfolio Betas • Consider the following four securities • Security Weight Beta • DCLK .133 2.685 • KO .2 0.195 • INTC .267 2.161 • KEI .4 2.434 • What is the portfolio beta? • .133(2.685) + .2(.195) + .267(2.161) + .4(2.434) = 1.9467

  29. 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!

  30. Example: Portfolio Expected Returns and Betas E(RA) Rf A

  31. 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)?

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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?

  37. Figure 13.4

  38. 12 End of Chapter

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