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After reading this chapter, students should be able to: • Define dollar return and rate of return. • Define risk and calculate the expected rate of return, standard deviation, and coefficient of variation for a probability distribution. • Specify how risk aversion influences required rates of return. • Graph diversifiable risk and market risk; explain which of these is relevant to a welldiversified investor. • State the basic proposition of the Capital Asset Pricing Model (CAPM) and explain how and why a portfolio’s risk may be reduced. • Explain the significance of a stock’s beta coefficient, and use the Security Market Line to calculate a stock’s required rate of return. • List changes in the market or within a firm that would cause the required rate of return on a firm’s stock to change. • Identify concerns about beta and the CAPM. • Explain how stock price volatility is more likely to imply risk than earnings volatility.
After reading this chapter, students should be able to: • Define dollar return and rate of return. • Define risk and calculate the expected rate of return, standard deviation, and coefficient of variation for a probability distribution. • Specify how risk aversion influences required rates of return. • Graph diversifiable risk and market risk; explain which of these is relevant to a well-diversified investor. • State the basic proposition of the Capital Asset Pricing Model (CAPM) and explain how and why a portfolio’s risk may be reduced. • Explain the significance of a stock’s beta coefficient, and use the Security Market Line to calculate a stock’s required rate of return. • List changes in the market or within a firm that would cause the required rate of return on a firm’s stock to change. • Identify concerns about beta and the CAPM. • Explain how stock price volatility is more likely to imply risk than earnings volatility. Learning Objectives: 5 - 1 Chapter 5 Risk and Rates of Return LEARNING OBJECTIVES Risk analysis is an important topic, but it is difficult to teach at the introductory level. We just try to give students an intuitive overview of how risk can be defined and measured, and leave a technical treatment to advanced courses. Our primary goals are to be sure students understand (1) that investment risk is the uncertainty about returns on an asset, (2) the concept of portfolio risk, and (3) the effects of risk on required rates of return. What we cover, and the way we cover it, can be seen by scanning Blueprints, Chapter 5. For other suggestions about the lecture, please see the “Lecture Suggestions” in Chapter 2, where we describe how we conduct our classes. DAYS ON CHAPTER: 3 OF 58 DAYS (50-minute periods) Lecture Suggestions: 5 - 2 LECTURE SUGGESTIONS 5-1 a. The probability distribution for complete certainty is a vertical line. b. The probability distribution for total uncertainty is the X-axis from -∞ to +∞. 5-2 Security A is less risky if held in a diversified portfolio because of its negative correlation with other stocks. In a single-asset portfolio, Security A would be more risky because σ A > σ B and CV A > CV B . 5-3 a. No, it is not riskless. The portfolio would be free of default risk and liquidity risk, but inflation could erode the portfolio’s purchasing power. If the actual inflation rate is greater than that expected, interest rates in general will rise to incorporate a larger inflation premium (IP) and as we shall see in Chapter 7 the value of the portfolio would decline. b. No, you would be subject to reinvestment rate risk. You might expect to “roll over” the Treasury bills at a constant (or even increasing) rate of interest, but if interest rates fall, your investment income will decrease. c. A U.S. government-backed bond that provided interest with constant purchasing power (that is, an indexed bond) would be close to riskless. The U.S. Treasury currently issues indexed bonds. 5-4 a. The expected return on a life insurance policy is calculated just as for a common stock. Each outcome is multiplied by its probability of occurrence, and then these products are summed. For example, suppose a 1-year term policy pays $10,000 at death, and the probability of the policyholder’s death in that year is 2 percent. Then, there is a 98 percent probability of zero return and a 2 percent probability of $10,000: Expected return = 0.98($0) + 0.02($10,000) = $200. This expected return could be compared to the premium paid. Generally, the premium will be larger because of sales and administrative costs, and insurance company profits, indicating a negative expected rate of return on the investment in the policy. b. There is a perfect negative correlation between the returns on the life insurance policy and the returns on the policyholder’s human Answers and Solutions: 5 - 3 ANSWERS TO END-OF-CHAPTER QUESTIONS capital. In fact, these events (death and future lifetime earnings capacity) are mutually exclusive. c. People are generally risk averse. Therefore, they are willing to pay a premium to decrease the uncertainty of their future cash flows. A life insurance policy guarantees an income (the face value of the policy) to the policyholder’s beneficiaries when the policyholder’s future earnings capacity drops to zero. 5-5 The risk premium on a high-beta stock would increase more. RP j = Risk Premium for Stock j = (k M - k RF )b j . If risk aversion increases, the slope of the SML will increase, and so will the market risk premium (k M - k RF ). The product (k M - k RF )b j is the risk premium of the jth stock. If b j is low (say, 0.5), then the product will be small; RP j will increase by only half the increase in RP M . However, if b j is large (say, 2.0), then its risk premium will rise by twice the increase in RP M . 5-6 According to the Security Market Line (SML) equation, an increase in beta will increase a company’s expected return by an amount equal to the market risk premium times the change in beta. For example, assume that the risk-free rate is 6 percent, and the market risk premium is 5 percent. If the company’s beta doubles from 0.8 to 1.6 its expected return increases from 10 percent to 14 percent. Therefore, in general, a company’s expected return will not double when its beta doubles. 5-7 Yes, if the portfolio’s beta is equal to zero. In practice, however, it may be impossible to find individual stocks that have a nonpositive beta. In this case it would also be impossible to have a stock portfolio with a zero beta. Even if such a portfolio could be constructed, investors would probably be better off just purchasing Treasury bills, or other zero beta investments. 5-8 No. For a stock to have a negative beta, its returns would have to logically be expected to go up in the future when other stocks’ returns were falling. Just because in one year the stock’s return increases when the market declined doesn’t mean the stock has a negative beta. A stock in a given year may move counter to the overall market, even though the stock’s beta is positive. Answers and Solutions: 5 - 4 5-1 k ˆ = (0.1)(-50%) + (0.2)(-5%) + (0.4)(16%) + (0.2)(25%) + (0.1)(60%) = 11.40%. σ 2 = (-50% - 11.40%) 2 (0.1) + (-5% - 11.40%) 2 (0.2) + (16% - 11.40%) 2 (0.4) + (25% - 11.40%) 2 (0.2) + (60% - 11.40%) 2 (0.1) σ 2 = 712.44; σ = 26.69%. CV = 11.40% 26.69% = 2.34. 5-2 Investment Beta $35,000 0.8 40,000 1.4 Total $75,000 b p = ($35,000/$75,000)(0.8) + ($40,000/$75,000)(1.4) = 1.12. 5-3 k RF = 5%; RP M = 6%; k M = ? k M = 5% + (6%)1 = 11%. k when b = 1.2 = ? k = 5% + 6%(1.2) = 12.2%. 5-4 k RF = 6%; k M = 13%; b = 0.7; k = ? k = k RF + (k M - k RF )b = 6% + (13% - 6%)0.7 = 10.9%. 5-5 a. k = 11%; k RF = 7%; RP M = 4%. k = k RF + (k M – k RF )b 11% = 7% + 4%b 4% = 4%b b = 1. Answers and Solutions: 5 - 5 SOLUTIONS TO END-OF-CHAPTER PROBLEMS b. k RF = 7%; RP M = 6%; b = 1. k = k RF + (k M – k RF )b k = 7% + (6%)1 k = 13%. 5-6 a. ∑ = = n 1i ii kPk ˆ . Y k ˆ = 0.1(-35%) + 0.2(0%) + 0.4(20%) + 0.2(25%) + 0.1(45%) = 14% versus 12% for X. b. σ = ∑ = − n 1i i 2 i P)k ˆ k( . 2 X σ = (-10% - 12%) 2 (0.1) + (2% - 12%) 2 (0.2) + (12% - 12%) 2 (0.4) + (20% - 12%) 2 (0.2) + (38% - 12%) 2 (0.1) = 148.8%. σ X = 12.20% versus 20.35% for Y. CV X = σ X / k ˆ X = 12.20%/12% = 1.02, while CV Y = 20.35%/14% = 1.45. If Stock Y is less highly correlated with the market than X, then it might have a lower beta than Stock X, and hence be less risky in a portfolio sense. 5-7 a. k i = k RF + (k M - k RF )b i = 9% + (14% - 9%)1.3 = 15.5%. b. 1. k RF increases to 10%: k M increases by 1 percentage point, from 14% to 15%. k i = k RF + (k M - k RF )b i = 10% + (15% - 10%)1.3 = 16.5%. 2. k RF decreases to 8%: k M decreases by 1%, from 14% to 13%. k i = k RF + (k M - k RF )b i = 8% + (13% - 8%)1.3 = 14.5%. c. 1. k M increases to 16%: k i = k RF + (k M - k RF )b i = 9% + (16% - 9%)1.3 = 18.1%. 2. k M decreases to 13%: k i = k RF + (k M - k RF )b i = 9% + (13% - 9%)1.3 = 14.2%. 5-8 Old portfolio beta = $150,000 $142,500 (b) + $150,000 $7,500 (1.00) 1.12 = 0.95b + 0.05 1.07 = 0.95b 1.1263 = b. New portfolio beta = 0.95(1.1263) + 0.05(1.75) = 1.1575 ≈ 1.16. Alternative Solutions: 1. Old portfolio beta = 1.12 = (0.05)b 1 + (0.05)b 2 + + (0.05)b 20 1.12 = ∑ )b( i (0.05) ∑ i b = 1.12/0.05 = 22.4. New portfolio beta = (22.4 - 1.0 + 1.75)(0.05) = 1.1575 ≈ 1.16. 2. ∑ i b excluding the stock with the beta equal to 1.0 is 22.4 - 1.0 = 21.4, so the beta of the portfolio excluding this stock is b = 21.4/19 = 1.1263. The beta of the new portfolio is: 1.1263(0.95) + 1.75(0.05) = 1.1575 ≈ 1.16. 5-9 Portfolio beta = $4,000,000 $400,000 (1.50) + $4,000,000 $600,000 (-0.50) + $4,000,000 $1,000,000 (1.25) + $4,000,000 $2,000,000 (0.75) b p = (0.1)(1.5) + (0.15)(-0.50) + (0.25)(1.25) + (0.5)(0.75) = 0.15 - 0.075 + 0.3125 + 0.375 = 0.7625. k p = k RF + (k M - k RF )(b p ) = 6% + (14% - 6%)(0.7625) = 12.1%. Alternative solution: First, calculate the return for each stock using the CAPM equation [k RF + (k M - k RF )b], and then calculate the weighted average of these returns. k RF = 6% and (k M - k RF ) = 8%. Answers and Solutions: 5 - 7 Stock Investment Beta k = k RF + (k M - k RF )b Weight A $ 400,000 1.50 18% 0.10 B 600,000 (0.50) 2 0.15 C 1,000,000 1.25 16 0.25 D 2,000,000 0.75 12 0.50 Total $4,000,000 1.00 k p = 18%(0.10) + 2%(0.15) + 16%(0.25) + 12%(0.50) = 12.1%. 5-10 We know that b R = 1.50, b S = 0.75, k M = 13%, k RF = 7%. k i = k RF + (k M - k RF )b i = 7% + (13% - 7%)b i . k R = 7% + 6%(1.50) = 16.0% k S = 7% + 6%(0.75) = 11.5 4.5 % 5-11 X k ˆ = 10%; b X = 0.9; σ X = 35%. Y k ˆ = 12.5%; b Y = 1.2; σ Y = 25%. k RF = 6%; RP M = 5%. a. CV X = 35%/10% = 3.5. CV Y = 25%/12.5% = 2.0. b. For diversified investors the relevant risk is measured by beta. Therefore, the stock with the higher beta is more risky. Stock Y has the higher beta so it is more risky than Stock X. c. k X = 6% + 5%(0.9) k X = 10.5%. k Y = 6% + 5%(1.2) k Y = 12%. d. k X = 10.5%; X k ˆ = 10%. k Y = 12%; Y k ˆ = 12.5%. Stock Y would be most attractive to a diversified investor since its expected return of 12.5% is greater than its required return of 12%. e. b p = ($7,500/$10,000)0.9 + ($2,500/$10,000)1.2 = 0.6750 + 0.30 = 0.9750. k p = 6% + 5%(0.975) k p = 10.875%. f. If RP M increases from 5% to 6%, the stock with the highest beta will have the largest increase in its required return. Therefore, Stock Y will have the greatest increase. Check: k X = 6% + 6%(0.9) = 11.4%. Increase 10.5% to 11.4%. k Y = 6% + 6%(1.2) = 13.2%. Increase 12% to 13.2%. 5-12 k RF = k* + IP = 2.5% + 3.5% = 6%. k s = 6% + (6.5%)1.7 = 17.05%. 5-13 Using Stock X (or any stock): 9% = k RF + (k M – k RF )b X 9% = 5.5% + (k M – k RF )0.8 (k M – k RF ) = 4.375%. 5-14 In equilibrium: k J = J k ˆ = 12.5%. k J = k RF + (k M - k RF )b 12.5% = 4.5% + (10.5% - 4.5%)b b = 1.33. 5-15 b HRI = 1.8; b LRI = 0.6. No changes occur. k RF = 6%. Decreases by 1.5% to 4.5%. k M = 13%. Falls to 10.5%. Now SML: k i = k RF + (k M - k RF )b i . k HRI = 4.5% + (10.5% - 4.5%)1.8 = 4.5% + 6%(1.8) = 15.3% k LRI = 4.5% + (10.5% - 4.5%)0.6 = 4.5% + 6%(0.6) = 8.1% Difference 7.2% 5-16 An index fund will have a beta of 1.0. If k M is 12.5 percent (given in the problem) and the risk-free rate is 5 percent, you can calculate the market risk premium (RP M ) calculated as k M - k RF as follows: k = k RF + (RP M )b 12.5% = 5% + (RP M )1.0 7.5% = RP M . Now, you can use the RP M , the k RF , and the two stocks’ betas to calculate their required returns. Answers and Solutions: 5 - 9 Bradford: k B = k RF + (RP M )b = 5% + (7.5%)1.45 = 5% + 10.875% = 15.875%. Farley: k F = k RF + (RP M )b = 5% + (7.5%)0.85 = 5% + 6.375% = 11.375%. The difference in their required returns is: 15.875% - 11.375% = 4.5%. 5-17 Step 1: Determine the market risk premium from the CAPM: 0.12 = 0.0525 + (k M - k RF )1.25 (k M - k RF ) = 0.054. Step 2: Calculate the beta of the new portfolio: The beta of the new portfolio is ($500,000/$5,500,000)(0.75) + ($5,000,000/$5,500,000)(1.25) = 1.2045. Step 3: Calculate the required return on the new portfolio: The required return on the new portfolio is: 5.25% + (5.4%)(1.2045) = 11.75%. 5-18 After additional investments are made, for the entire fund to have an expected return of 13%, the portfolio must have a beta of 1.5455 as shown below: 13% = 4.5% + (5.5%)b b = 1.5455. Since the fund’s beta is a weighted average of the betas of all the individual investments, we can calculate the required beta on the additional investment as follows: 1.5455 = 0$25,000,00 00)(1.5)($20,000,0 + 0$25,000,00 X$5,000,000 1.5455 = 1.2 + 0.2X 0.3455 = 0.2X X = 1.7275. 5-19 a. ($1 million)(0.5) + ($0)(0.5) = $0.5 million. b. You would probably take the sure $0.5 million. c. Risk averter. [...]... 15.3%/15.0% = 1.0 WHEN WE MEASURE RISK PER UNIT OF RETURN, COLLECTIONS, WITH ITS LOW EXPECTED RETURN, BECOMES THE MOST RISKY STOCK THE CV IS A BETTER MEASURE OF AN ASSET’S STAND-ALONE RISK THAN σ BECAUSE CV CONSIDERS BOTH THE EXPECTED VALUE AND THE DISPERSION OF A DISTRIBUTION A SECURITY WITH A LOW EXPECTED RETURN AND A LOW STANDARD DEVIATION COULD HAVE A HIGHER CHANCE OF A LOSS THAN ONE WITH A HIGH σ... INSURANCE POLICY I 4 WHAT WOULD BE THE MARKET RISK AND THE REQUIRED RETURN OF A 50-50 PORTFOLIO OF HIGH TECH AND COLLECTIONS? OF HIGH TECH AND U.S RUBBER? ANSWER: [SHOW S5-45 AND S5-46 HERE.] NOTE THAT THE BETA OF A PORTFOLIO IS SIMPLY OF THE PORTFOLIO WEIGHTED AVERAGE THE BETAS OF THE STOCKS IN THE THUS, THE BETA OF A PORTFOLIO WITH 50 PERCENT HIGH TECH AND 50 PERCENT COLLECTIONS IS: bp = n ∑ wb i... RUBBER, AND T-BILLS Integrated Case: 5 - 19 ANSWER: Probability of Occurrence T-Bills High Tech U.S Rubber -60 -45 -30 -15 0 15 30 45 60 Rate of Return (%) ON THE BASIS OF THESE DATA, HIGH TECH IS THE MOST RISKY INVESTMENT, T-BILLS THE LEAST RISKY D SUPPOSE YOU SUDDENLY REMEMBERED THAT THE COEFFICIENT OF VARIATION (CV) IS GENERALLY REGARDED AS BEING A BETTER MEASURE OF STAND-ALONE RISK THAN THE STANDARD... B CALCULATE THE EXPECTED RATE OF RETURN ON EACH ALTERNATIVE AND FILL IN ˆ THE BLANKS ON THE ROW FOR k IN THE TABLE ABOVE ANSWER: [SHOW S5-9 AND S5-10 HERE.] ˆ THE EXPECTED RATE OF RETURN, k , IS EXPRESSED AS FOLLOWS: ˆ k = n ∑ Pk i i i=1 HERE Pi IS THE PROBABILITY OF OCCURRENCE OF THE iTH STATE, k i IS THE ESTIMATED RATE OF RETURN FOR THAT STATE, AND n IS THE NUMBER OF STATES HERE IS THE CALCULATION... 9.6)2(0.1)]½ = 3.3% CVp = 3.3%/9.6% = 0.34 E 2 HOW DOES THE RISKINESS OF THIS 2-STOCK PORTFOLIO COMPARE WITH THE RISKINESS OF THE INDIVIDUAL STOCKS IF THEY WERE HELD IN ISOLATION? ANSWER: [SHOW S5-24 THROUGH S5-27 HERE.] ALONE RISK MEASURE, SIGNIFICANTLY LESS THE THAN USING EITHER σ OR CV AS OUR STAND- STAND-ALONE THE RISK STAND-ALONE OF RISK THE OF PORTFOLIO THE IS INDIVIDUAL STOCKS THIS IS BECAUSE THE... SHOULD RECOGNIZE THAT BASING A DECISION SOLELY ON RETURNS IS ONLY APPROPRIATE FOR RISK- NEUTRAL INDIVIDUALS EXPECTED SINCE YOUR CLIENT, LIKE VIRTUALLY EVERYONE, IS RISK AVERSE, THE RISKINESS OF EACH ALTERNATIVE IS AN IMPORTANT ASPECT OF THE DECISION ONE POSSIBLE MEASURE OF RISK IS THE STANDARD DEVIATION OF RETURNS 1 CALCULATE THIS VALUE FOR EACH ALTERNATIVE, AND FILL IN THE BLANK ON THE ROW FOR σ IN THE... ANSWER: [SHOW S5-14 AND S5-15 HERE.] THE STANDARD DEVIATION IS A MEASURE OF A SECURITY’S (OR A PORTFOLIO’S) STAND-ALONE RISK THE LARGER THE STANDARD DEVIATION, THE HIGHER THE PROBABILITY THAT ACTUAL REALIZED RETURNS WILL FALL FAR BELOW THE EXPECTED RETURN, AND THAT LOSSES RATHER THAN PROFITS WILL BE INCURRED Integrated Case: 5 - 18 C 3 DRAW A GRAPH THAT SHOWS ROUGHLY THE SHAPE OF THE PROBABILITY DISTRIBUTIONS... HIGH TECH AND $50,000 IN COLLECTIONS ˆ 1 CALCULATE THE EXPECTED RETURN ( k p ), THE STANDARD DEVIATION (σ p), AND THE COEFFICIENT OF VARIATION (CVp) FOR THIS PORTFOLIO AND FILL IN THE APPROPRIATE BLANKS IN THE TABLE ABOVE ANSWER: [SHOW S5-20 THROUGH S5-23 HERE.] TO FIND THE EXPECTED RATE OF RETURN ON THE TWO-STOCK PORTFOLIO, WE FIRST CALCULATE THE RATE OF RETURN ON THE PORTFOLIO IN EACH STATE OF THE ECONOMY... TWO-STOCK PORTFOLIO’S RETURNS THE PORTFOLIO’S σ DEPENDS JOINTLY ON (1) EACH SECURITY’S σ AND (2) THE CORRELATION BETWEEN THE SECURITIES’ RETURNS THE BEST WAY TO APPROACH THE PROBLEM IS TO ESTIMATE THE PORTFOLIO’S RISK AND RETURN IN EACH STATE OF THE ECONOMY, AND THEN TO ESTIMATE σp WITH THE σ FORMULA GIVEN THE DISTRIBUTION OF RETURNS FOR THE PORTFOLIO, WE CAN CALCULATE THE PORTFOLIO’S σ AND CV AS SHOWN BELOW:... (5%)2.0 = 16% An expected return of 15 percent on the new stock is below the 16 percent required rate of return on an investment with a risk of b = ˆ 2.0 Since kN = 16% > k N = 15%, the new stock should not be purchased The expected rate of return that would make the fund indifferent to purchasing the stock is 16 percent Answers and Solutions: 5 - 11 5-21 The answers to a, b, c, and d are given below: . (RP M )b = 5% + (7 .5% )1. 45 = 5% + 10.8 75% = 15. 8 75% . Farley: k F = k RF + (RP M )b = 5% + (7 .5% )0. 85 = 5% + 6.3 75% = 11.3 75% . The difference in their required returns is: 15. 8 75% - 11.3 75% = 4 .5% . 5- 17. investment as follows: 1 .54 55 = 0$ 25, 000,00 00)(1 .5) ($20,000,0 + 0$ 25, 000,00 X $5, 000,000 1 .54 55 = 1.2 + 0.2X 0.3 455 = 0.2X X = 1.72 75. 5- 19 a. ($1 million)(0 .5) + ($0)(0 .5) = $0 .5 million. b. You. 14.2%. 5- 8 Old portfolio beta = $ 150 ,000 $142 ,50 0 (b) + $ 150 ,000 $7 ,50 0 (1.00) 1.12 = 0.95b + 0. 05 1.07 = 0.95b 1.1263 = b. New portfolio beta = 0. 95( 1.1263) + 0. 05( 1. 75) = 1. 157 5 ≈ 1.16. Alternative