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Department
Sociology
Course
SOC 202
Professor
Louis Pike
Semester
Fall

Description
CHAPTER13 RETURN, RISK, AND THE SECURITY MARKET LINE Learning Objectives LO1 The calculation for expected returns and standard deviation for individual securities and portfolios. LO2 The principle of diversification and the role of correlation. LO3 Systematic and unsystematic risk. LO4 Beta as a measure of risk and the security market line. Answers to Concepts Review and Critical Thinking Questions 1. (LO3) Some of the risk in holding any asset is unique to the asset in question. By investing in a variety of assets, this unique portion of the total risk can be eliminated at little cost. On the other hand, there are some risks that affect all investments. This portion of the total risk of an asset cannot be costlessly eliminated. In other words, systematic risk can be controlled, but only by a costly reduction in expected returns. 2. (LO3) If the market expected the growth rate in the coming year to be 2 percent, then there would be no change in security prices if this expectation had been fully anticipated and priced. However, if the market had been expecting a growth rate different than 2 percent and the expectation was incorporated into security prices, then the government’s announcement would most likely cause security prices in general to change; prices would drop if the anticipated growth rate had been more than 2 percent, and prices would rise if the anticipated growth rate had been less than 2 percent. 3. (LO3) a. systematic b. unsystematic c. both; probably mostly systematic d. unsystematic e. unsystematic f. systematic 4. (LO3) a. a change in systematic risk has occurred; market prices in general will most likely decline. b. no change in unsystematic risk; company price will most likely stay constant. c. no change in systematic risk; market prices in general will most likely stay constant. d. a change in unsystematic risk has occurred; company price will most likely decline. e. no change in systematic risk; market prices in general will most likely stay constant. 5. (LO1) No to both questions. The portfolio expected return is a weighted average of the asset returns, so it must be less than the largest asset return and greater than the smallest asset return. 6. (LO2) False. The variance of the individual assets is a measure of the total risk. The variance on a well- diversified portfolio is a function of systematic risk only. 7. (LO2) Yes, the standard deviation can be less than that of every asset in the portfolio. However, β pannot be less than the smallest beta because β is a weighted average of the individual asset betas. p 8. (LO4) Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be equal to the risk-free rate. It is also possible to have a negative beta; the return would be less than the risk-free rate. A negative beta asset would carry a negative risk premium because of its value as a diversification instrument. S13-1 9. (LO1) Such layoffs generally occur in the context of corporate restructurings. To the extent that the market views a restructuring as value-creating, stock prices will rise. So, it’s not layoffs per se that are being cheered on. Nonetheless, Bay Street does encourage corporations to takes actions to create value, even if such actions involve layoffs. 10. (LO1) Earnings contain information about recent sales and costs. This information is useful for projecting future growth rates and cash flows. Thus, unexpectedly low earnings often lead market participants to reduce estimates of future growth rates and cash flows; price drops are the result. The reverse is often true for unexpectedly high earnings. Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. (LO1) The portfolio weight of an asset is total investment in that asset divided by the total portfolio value. First, we will find the portfolio value, which is: Total value = 180($45) + 140($27) = $11,880 The portfolio weight for each stock is: Weight A 180($45)/$11,880 = .6818 Weight B 140($27)/$11,880 = .3182 2. (LO1) The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. The total value of the portfolio is: Total value = $2,950 + 3,700 = $6,650 So, the expected return of this portfolio is: E(R p = ($2,950/$6,650)(0.11) + ($3,700/$6,650)(0.15) = .1323 or 13.23% 3. (LO1) The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. So, the expected return of the portfolio is: E(R p = .60(.09) + .25(.17) + .15(.13) = .1160 or 11.60% 4. (LO1) Here we are given the expected return of the portfolio and the expected return of each asset in the portfolio, and are asked to find the weight of each asset. We can use the equation for the expected return of a portfolio to solve this problem. Since the total weight of a portfolio must equal 1 (100%), the weight of Stock Y must be one minus the weight of Stock X. Mathematically speaking, this means: E(R p = .124 = .14w X .105(1 – w ) X We can now solve this equation for the weight of Stock X as: .124 = .14w X .105 – .105w X .019 = .035w X w = 0.542857 X S13-2 So, the dollar amount invested in Stock X is the weight of Stock X times the total portfolio value, or: Investment in X = 0.542857($10,000) = $5,428.57 And the dollar amount invested in Stock Y is: Investment in Y = (1 – 0.542857)($10,000) = $4,571.43 5. (LO1) The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of the asset is: E(R) = .25(–.08) + .75(.21) = .1375 or 13.75% 6. (LO1) The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of the asset is: E(R) = .20(–.05) + .50(.12) + .30(.25) = .1250 or 12.50% 7. (LO1) The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock asset is: E(R A = .15(.05) + .65(.08) + .20(.13) = .0855 or 8.55% E(R B = .15(–.17) + .65(.12) + .20(.29) = .1105 or 11.05% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, then add all of these up. The result is the variance. So, the variance and standard deviation of each stock is: σA=.15(.05 – .0855) + .65(.08 – .0855) + .20(.13 – .0855) = .00060 σA= (.00060) = .0246 or 2.46% σB=.15(–.17 – .1105) + .65(.12 – .1105) + .20(.29 – .1105) = .01830 σB= (.01830) = .1353 or 13.53% 8. (LO1) The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. So, the expected return of the portfolio is: E(R p = .25(.08) + .55(.15) + .20(.24) = .1505 or 15.05% If we own this portfolio, we would expect to get a return of 15.05 percent. 9. (LO1, 2) a. To find the expected return of the portfolio, we need to find the return of the portfolio in each state of the economy. This portfolio is a special case since all three assets have the same weight. To find the expected return in an equally weighted portfolio, we can sum the returns of each asset and divide by the number of assets, so the expected return of the portfolio in each state of the economy is: Boom: E(R ) = p.07 + .15 + .33)/3 = .1833 or 18.33% Bust: E(R ) = (.13 + .03 −.06)/3 = .0333 or 3.33% p S13-3 To find the expected return of the portfolio, we multiply the return in each state of the economy by the probability of that state occurring, and then sum. Doing this, we find: E(R )p= .35(.1833) + .65(.0333) = .0858 or 8.58% b. This portfolio does not have an equal weight in each asset. We still need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: E(R ) = p20(.07) +.20(.15) + .60(.33) =.2420 or 24.20% Bust: E(R p = .20(.13) +.20(.03) + .60(−.06) = –.0040 or –0.40% And the expected return of the portfolio is: E(R )p= .35(.2420) + .65(−.004) = .0821 or 8.21% To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, than add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio is: σ p .35(.2420 – .0821) + .65(−.0040 – .0821) = .013767 10. (LO1, 2) a. This portfolio does not have an equal weight in each asset. We first need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: E(R )p= .30(.3) + .40(.45) + .30(.33) = .3690 or 36.90% Good: E(R p = .30(.12) + .40(.10) + .30(.15) = .1210 or 12.10% Poor: E(R )p= .30(.01) + .40(–.15) + .30(–.05) = –.0720 or –7.20% Bust: E(R )p= .30(–.06) + .40(–.30) + .30(–.09) = –.1650 or –16.50% And the expected return of the portfolio is: E(R ) = .15(.3690) + .45(.1210) + .35(–.0720) + .05(–.1650) = .0764 or 7.64% p b. To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, than add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio is: σ p .15(.3690 – .0764) + .45(.1210 – .0764) + .35(–.0720 – .0764) + .05(–.1650 – .0764) 2 2 σ p .02436 σ p (.02436) = .1561 or 15.61% 11. (LO4) The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. So, the beta of the portfolio is: βp= .25(.84) + .20(1.17) + .15(1.11) + .40(1.36) = 1.15 S13-4 12. (LO4) The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. If the portfolio is as risky as the market it must have the same beta as the market. Since the beta of the market is one, we know the beta of our portfolio is one. We also need to remember that the beta of the risk-free asset is zero. It has to be zero since the asset has no risk. Setting up the equation for the beta of our portfolio, we get: 1 1 1 β p 1.0 = / (03 + / (1.33) + / (β ) 3 X Solving for the beta of Stock X, we get: β X 1.62 13. (LO1, 4) CAPM states the relationship between the risk of an asset and its expected return. CAPM is: E(R) = R + [E(R ) – R] × β i f M f i Substituting the values we are given, we find: E(R) i .052 + (.11 – .052)(1.05) = .1129 or 11.29% 14. (LO1, 4) We are given the values for the CAPM except for the β of the stock. We need to substitute these values into the CAPM, and solve for the β of the stock. One important thing we need to realize is that we are given the market risk premium. The market risk premium is the expected return of the market minus the risk- free rate. We must be careful not to use this value as the expected return of the market. Using the CAPM, we find: E(R) i .102 = .045+ .085β i β i 0.67 15. (LO1, 4) Here we need to find the expected return of the market using the CAPM. Substituting the values given, and solving for the expected return of the market, we find: E(R) i .135 = .055 + [E(R ) – .M55](1.17) E(R )M= .1234 or 12.34% 16. (LO4) Here we need to find the risk-free rate using the CAPM. Substituting the values given, and solving for the risk-free rate, we find: E(R) i .14 = R + (f115 – R)(1.45f .14 = R + .16675 – 1.45R f f R f .0594 or 5.94% 17. (LO1, 4) a. Again we have a special case where the portfolio is equally weighted, so we can sum the returns of each asset and divide by the number of assets. The expected return of the portfolio is: E(R p = (.16 + .048)/2 = .1040 or 10.40% S13-5 b. We need to find the portfolio weights that result in a portfolio with a β of 0.95. We know the β of the risk-free asset is zero. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So: β p 0.95 = w (1S35) + (1 – w )(0S 0.95 = 1.35w +S0 – 0w S w S 0.95/1.35 w = .7037 S And, the weight of the risk-free asset is: w Rf1 – .7037 = .2963 c. We need to find the portfolio weights that result in a portfolio with an expected return of 8 percent. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So: E(R p = .08 = .16w +S.048(1 – w ) S .08 = .16w +S.048 – .048w S .032 = .112w S w S .2857 So, the β of the portfolio will be: β p .2857(1.35) + (1 – .2857)(0) = 0.386 d. Solving for the β of the portfolio as we did in part a, we find: β p 2.70 = w (1S35) + (1 – w )(0S w S 2.70/1.35 = 2 w Rf1 – 2 = –1
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