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Risk, Return, and the S ecurity M arket L ine ( SML ) Chapter 13

Risk, Return, and the S ecurity M arket L ine ( SML ) Chapter 13. Expected returns are based on the probabilities of possible outcomes In this context, “expected” means average if the process is repeated many times. Flip a coin for $1 if heads.

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Risk, Return, and the S ecurity M arket L ine ( SML ) Chapter 13

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  1. Risk, Return, and the Security Market Line (SML)Chapter 13 • Expected returns are based on the probabilities of possible outcomes • In this context, “expected” means average if the process is repeated many times. Flip a coin for $1 if heads. • The “expected” return does not even have to be a possible return

  2. Expected Returns for Two StocksCitibank (C) and AT&T (T) • Suppose you have predicted the following returns for stocks Cand 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) = 9.9% RT = .3(25) + .5(20) + .2(1) = 17.7%

  3. Risk Premium • What’s the probability of a recession? • The risk-free return doesn’t change in each state of the world. • If the risk-free rate is 6.15%, what is the risk premium? • Stock C: 9.9 – 6.15 = 3.75% • Stock T: 17.7 – 6.15 = 11.55%

  4. Variance and Standard Deviation • Variance and standard deviation still measure the volatility of returns • Population Variance vs. Sample Variance • Using unequal probabilities for the entire range of possibilities • Weighted average of squared deviations

  5. 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

  6. Portfolios • 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

  7. 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 AAPL • $3000 of KO • $4000 of INTC • $6000 of HOG AAPL: 2/15 = .133 KO: 3/15 = .2 INTC: 4/15 = .267 HOG: 6/15 = .4 Apple, Coke, Intel Corp, and Harley Davidson

  8. Portfolio Expected Returns • The expected return of a portfolio is the weighted average of the expected returns of the respective assets in the portfolio • You can also find the expected return by finding the portfolio return in each possible state and computing the expected value as we did with individual securities

  9. 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? • AAPL: 12.21% • KO: 7.31% • INTC: 8.41% • HOG: 12.55% • E(RP) = .133(12.21) + .2(7.31) + .267(8.41) + .4(12.55) = 10.35%

  10. Portfolio Variance • Compute the portfolio return for each state:RP = w1R1 + w2R2 + … + wmRm • 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

  11. Portfolio Variance for Two Securities Consider the following information • Invest 50:50 in Assets A & B State Probability A B • Boom .4 30% -5% • Bust .6 -10% 25% • What are the expected return and standard deviation for each asset? • R(A) = 6% Std Dev (A) = 19.59% • R(B) = 13% & Std Dev (B) = 14.7% • What are the expected return and standard deviation for the portfolio? • R(P) = 9.5% & Std Dev (P) = 2.45% < either one alone Portfolio R 12.5% 7.5%

  12. Calculations Individual Stocks: • R(A) = .4 (30%) + .6(-10%) = 6% • R(B) = .4 (-5%) + .6 (25%) =13% • sA = √[.4(30-6)2 + .6(-10-6)2] = 19.59% • sB = √[.4(-5-13)2 + .6(25-13)2] = 14.7% Portfolio: • RP = .5 (6%) + .5 (13%) = 9.5% sP = √[.4(12.5-9.5)2 + .6(7.5-9.5)2] = 2.45%

  13. Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X,Y) • Cov(X,Y) = XY · x· y • XY= Cov (X,Y)/(x·y ) • Rho is the correlation coefficient, which ranges from -1 up to +1. • Var(a•X + b•Y) = a2·Var(X) + b2·Var(Y) + 2·a·b·Cov(X,Y) PROBLEM 2-Assets:Find the expected value, variance, and standard deviation of variation an equally weighted portfoliowhere E(X) = .12, E(Y) = .16, with standard deviations of .05, and .08, respectively. The correlation coefficient is 0.1.

  14. Answer • Expected value is 14% • Variance = .52(.0025) + .52(.0064) + 2(.5)(.5)(.05)(.08)(0.1) = .00403 • Standard deviation = SQRT(.00403) = .0634 • Notice 14% ROR with just • 6.34% portfolio standard deviation

  15. Risk - Return Tradeoff RETURNS Security Y .16 .14 .12 Portfolio Security X standard deviation as the measure of risk

  16. Expected versusUnexpected Returns • Realized returns are generally not equal to expected returns • There is the expected component and the unexpected component • At any point in time, the unexpected return can be either positive or negative • Over time, the average of the unexpected component is zero

  17. Announcements and News • Announcements and news contain both an expected component and a surprise component • It is the surprise component that affects a stock’s price and therefore its return • This is very obvious when we watch how stock prices move when an unexpected announcement is made or earnings are different than anticipated

  18. Efficient Markets • Efficient markets are a result of investors trading on the unexpected portion of announcements • Some also trade for liquidity purposes • 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

  19. Systematic Risk • Risk factors that affect a large number of assets • Also known as non-diversifiable risk or just market risk • Includes such things as changes in GDP, inflation, interest rates, corporate tax changes, etc.

  20. UnsystematicRisk • 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, deaths of chief executives, etc.

  21. Returns • Total Return = expected return + unexpected return = E(R) + U • Unexpected return = U = m + e = systematic portion + unsystematic portion • Therefore, total return can be expressed as follows: • R = E(R) + U = expected return + systematic portion + unsystematic portion

  22. 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, when Jim Cramer screams, “AreYou diversified?” on Mad Money, you can shout back, Boo-Yah! -- yes I am, Jim.

  23. Table 13.7, page 432 I like 30 stocks Jim Cramer 14

  24. 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 Don’t put all your eggs in one basket.

  25. Figure 13.1 Portfolio Diversification

  26. Risk Declines Through Diversification Total Risk • Greater risk reduction with foreign equities. • Standard deviation of US market portfolio is between 15 and 20%. US only .182 .117 International systematic risk N securities

  27. 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

  28. 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

  29. 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

  30. Table13.8 Beta of Companies p. 435 • Systematic risk varies across firms and industries. • Lower betas is drugs and food • Why are autos, tech & media firms higher beta firms do you think? Source: Yahoo Finance, 2017 R Coca-Cola= a + .74 (R Market )

  31. Measuring Systematic Risk • How do we measure systematic risk? • We use the beta coefficient ( b)to measure systematic risk • What does beta tell us? • When beta = 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 Ri = a + b RM + ei

  32. Total versus Systematic RiskCitibank (C) and Ford (F) • Consider the following information: Standard Deviation Beta • Security C 14.94% 1.55 • Security F 35.35% 1.29 • Which security has more total risk? C or F? • Which security has more systematic risk? C or F? • Which security should have the higher expected return? C or F? Circle the correct response.

  33. Portfolio Betas • Consider the previous example with the following four securities Security Weight Beta • AAPL .133 1.44 • KO .2 .74 • JNJ .267 .67 • HOG .4 1.49 • What is the portfolio beta? • ANSWER: P= .133 (1.44) + .2 (.74) + .267 (.67) + .4 (1.44) = 1.1144

  34. 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!

  35. Portfolio Expected Returns and Betas SML A E(RA) B Rf A = 1.6

  36. Reward-to-Risk Ratio • 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 in graph = (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)?

  37. Risk-Reward the Same for All Assets Math: A: (20-8)/1.6 = 7.5% B: (16 – 8)/1.2 = 6.667% HENCE: Asset B cannot persist. Investors will abandon Asset B until the price falls enough and risk-reward is equal.

  38. 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

  39. Security Market Line (SML) • The security market line 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

  40. 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 E(RA) = Rf + A(E(RM) – Rf)

  41. Factors Affecting Expected Return • Pure time value of money is measured by the risk-free rate • Reward for bearing systematic risk is measured by the market risk premium • Amount of systematic risk is measured by beta

  42. 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 7%, what is the expected return for each?

  43. Diversification, Risk & Return • The concept is that RISK involves correlation with the market. • If international equities are less correlated with the US market, their “Betas” will tend to be lower than otherwise similar domestic equities. • A portfolio of stocks from different countries would then be more diversifiedthan an otherwise similar portfolio of US only stocks

  44. Beta as Systematic Risk The beta (ß): A measure of systematic risk with market. kiis return on asset i. Taken from a simple linear regression: ki = i+ i·km + iMarket Model V(ki) = i2·V(km) + V(i) for a portfolio kp = p+ p·km + p p2·V(km) + i2·i2] systematic riskunsystematic risk

  45. Unsystematic Risk Declines Through Diversification  [wi2·i2] • However, the portfolio beta, p, tends to 1, and the variance of the market does not change. Hence, systematic risk remains. Note the squared weights (wi2) become tiny as N rises N securities

  46. Problem • The risk free rate is 4%, and the required return on the market is 12%. What is the required return on an asset with a beta of 1.5? • Is it something close to 16%? • R = .04 + 1.5 x (.12 - .04) = .16 • What is the required return on a portfolio consisting of 40% of the asset above and the rest in an asset with an average amount of systematic risk? • Could it be (.4 x .16) + (.6 x .12) = .136 ?

  47. Quiz • How do you compute the expected return and standard deviation for an individual asset? For a portfolio? • What is the difference between systematic and unsystematic risk? • What type of risk is relevant for determining the expected return? • Consider an asset with a beta of 1.2, a risk-free rate of 5%, and a market return of 13%. • What is the reward-to-risk ratio in equilibrium? A: Reward-to-risk ratio = 13 – 5 = 8% • What is the expected return on the asset? A: Expected return = 5 + 1.2 (8) = 14.6%

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