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Ross5eChap11sm - Chapter 11 Risk and Return The Capital...

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Chapter 11: Risk and Return: The Capital Asset Pricing Model (CAPM) 11.2 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 ) = 0.50(.11) + 0.30(0.17) + 0.20(0.14) = 0.1340 or 13.40% 11.4 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 ) = 0.10(0.06) + 0.60(0.07) + 0.30(0.11) = 0.0810 or 8.10% E(R B ) = 0.10(–0.2) + 0.60(0.13) + 0.30(0.33) = 0.1570 or 15.70% 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, and then add all of these up. The result is the variance. So, the variance and standard deviation of each stock are: σ A 2 = 0.10(0.06 – 0.0810) 2 + 0.60(0.07–0.0810) 2 + 0.30(0.11 – 0.0810) 2 = 0.00037 σ A = (0.00037) 1/2 = 0.0192 or 1.92% σ B 2 = 0.10(–0.2 – 0.1570) 2 + 0.60(0.13–0.1570) 2 + 0.30(0.33 – 0.1570) 2 = 0.02216 σ B = (0.022216) 1/2 = 0.1489 or 14.89% 11.6 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 stock is: E(R A ) = 0.10(–0.045) + 0.20(0.044) + 0.50(0.12) + 0.20(0.207) = 0.1057 or 10.57% 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, and then add all of these up. The result is the variance. So, the variance and standard deviation are: σ 2 = 0.10(–0.045 – 0.1057) 2 + 0.20(0.044 – 0.1057) 2 + 0.50(0.12 – 0.1057) 2 + 0.20(0.207 – 0.1057) 2 = –0.005187 σ = (0.005187) 1/2 = 0.0720 or 7.20% 11.8 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: β p = 1.0 = 1/3(0) + 1/3(1.9) + 1/3( β X ) Solving for the beta of Stock X, we get: β X = 1.10 Answers to End–of–Chapter Problems B– 129
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11.10 a. We 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 ) = 0.4(0.20) + 0.4(0.35) + 0.2(0.60) = 0.3400 or 34.00% Normal: E(R p ) = 0.4(0.15) + 0.4(0.12) + 0.2(0.05) = 0.1180 or 11.80% Bust: E(R p ) = 0.4(0.01) + 0.4(–0.25) + 0.2(–0.50) = –0.1960 or –19.60% And the expected return of the portfolio is: E(R p ) = 0.4(0.34) + 0.4(0.118) + 0.2(–0.196) = 0.1440 or 14.40% 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 2 p = 0.4(0.34 – 0.1440) 2 + 0.4(0.118 – 0.1440) 2 + 0.2(–0.196 – 0.1440) 2 σ 2 p = 0.03876 σ p = (0.03876) 1/2 = 0.1969 or 19.69% b . The risk premium is the return of a risky asset, minus the risk–free rate. T–bills are often
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