Topic F Exercises

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Seton Hall University *

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May 1, 2024

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Exercises Topic F CAPM, Single-Index Model and Multi-Factor Models Exercises #1 (Slide 9) CML (a special case of CAL, see Topic E Slides 21,22,31) SML (slide 7 of Topic F) Relevant Risk Measures total risk - Standard dev. Measures systematic risk - beta Horizontal Axis Standard dev. - Complete portfolio Beta – individual securities Vertical Axis Expected return – complete portfolio Expected return – individual securities Intercept Rf (risk free rate) Rf (risk free rate) Slope Sharpe ratio – market portfolio Expected market of risk premium Exercise #2 (Slide 11) You invest $800 in security A with a Beta of 1.4 and $500 in security B with a beta of 0.7. What is the weighted average Beta of the portfolio? A weight: $800 / ($800 + $500) = $800 / $1,300 = 0.6153 = 61.50% B weight: $500 / $1,300 = 0.3846 = 38.46% Weighted average Beta of the Portfolio: A weight * A beta + B weight * B beta 0.6153 * 1.4 + 0.3846 * 0.7 0.862 + 0.269 = 1.31 1

Exercise #3 (Slide 17) Within the context of the CAPM, assume:  Expected Return on the market=12%  Risk-free rate=2%  Expected Return on the ABC stock=16%  Beta on the ABC stock=1.5 (a) What is the ABC stock’s fair expected rate of return according to CAPM? 2% + 1.5 (12% - 2%) 2% + 1.5 * 10% 2% + 15% = 17% (b) Is the stock underpriced, overpriced or fairly priced? How much is the alpha? The stock is overpriced because the ABC stock expected return is 16% which is more than 17%. Alpha: 16% - 17% = -1% Exercise #4 (Slide 28) Suppose that the index model for stocks A and B is estimated from R A = 3% + .7 R M + e A R B =− 2% + 1.2 R M + e B σ M = 20%; RSquared A = .20 ; RSquared B = .12 excess returns with the following results: 2

(a) What is the standard deviation of each stock? A: .7^2 * .2^2 / .2 = 31.3% B: 1.2^2 * 0.2^2 / .12 = 69.28% (b) Break down the variance of each stock to the systematic and firm-specific components. A Systematic: .7^2 * .2^2 = 0.0196 Unsystemtaic= .0980 - .0196 = 0.0784 B Systemtaic: 1.2^2 * .2^2 = 0.0576 Unsytematic: .4800 – 0.0576 = 0.4224 (c) What are the covariance and correlation between the two stocks? 0.0336 / 31.30% * 69.28% = 0.155 (d) What is the covariance between each stock and the market index? 0.70 * 0.0400 = 0.028 1.20 * 0.0400 = 0.048 Exercise #5 (Slide 34) (a) Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 3,000 stocks in order to construct a mean-variance efficient portfolio constrained by 3,000 investments. They will need to calculate (N) 3,000____ expected returns, (N)_3,000__________ variances of returns and __4,498,500______ covariances. (b) (b)Assume that stock market returns do follow a single-index structure. An investment fund analyzes 3,000 stocks in order to construct a mean-variance efficient portfolio constrained by 3,000 investments. They will need to calculate 3

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