Questions-Risk, Return, and CAPM

Q1) Suppose a stock had an initial price of $58 per share, paid a dividend of $1.90 per share during the year, and had an ending share price of $68. What was the dividend yield and the capital gains yield? The dividend yield is the dividend divided by the price at the beginning of the period, so:  Dividend yield = $1.90 / $58Dividend yield = .0328, or 3.28%   And the capital gains yield is the increase in price divided by the initial price, so:  Capital gains yield = ($68 – 58) / $58Capital gains yield = .1724, or 17.24%

Q2) Using the returns shown above, calculate the arithmetic average return, the variances, and the standard deviations for X and Y Year Return of X Return of Y 1 12% 18% 2 26% 27% 3 -19% -24% 4 10% 5 The average return is the sum of the returns, divided by the number of returns. The average return for each stock was: Rx^Avg=0.082 Ry^Avg=0.098 We calculate the variance of each stock as: Var(X)=0.02722 Var(Y)=0.03932 The standard deviation is the square root of the variance, so the standard deviation of each stock is: σX = (.02722)1/2σX = .1650, or 16.50%   σY = (.03932)1/2σY = .1983, or 19.83%

Q3) You own a portfolio that has $2,400 invested in Stock A and $3,400 invested in Stock B. If the expected returns on these stocks are 9 percent and 12 percent, respectively, what is the expected return on the portfolio? 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:    Portfolio value = $2,400 + 3,400Portfolio value = $5,800   So, the expected return of this portfolio is:   E(RP) = ($2,400 / $5,800)(.09) + ($3,400 / $5,800)(.12)E(RP) = .1076, or 10.76%

Q4) You have $30,000 to invest in a stock portfolio. Your choices are Stock X with an expected return of 11 percent and Stock Y with an expected return of 9 percent. If your goal is to create a portfolio with an expected return of 10.42 percent, how much money will you invest in Stock X? In Stock Y? 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(RP) = .1042 = .11XX + .090(1 − XX)   We can now solve this equation for the weight of Stock X as:   .1042 = .11XX + .090 − .090XX.0142 = .020XXXX = .7100   So, the dollar amount invested in Stock X is the weight of Stock X times the total portfolio value, or:   Investment in X = .7100($30,000)Investment in X = $21,300   And the dollar amount invested in Stock Y is:   Investment in Y = (1 − .7100)($30,000)Investment in Y =  $8,700

Q5) You own a stock portfolio invested 40 percent in Stock Q, 30 percent in Stock R, 20 percent in Stock S, and 10 percent in Stock T. The betas for these four stocks are .87, 1.20, 1.04, and 1.22, respectively. What is the portfolio beta? 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 = .40(.87) + .30(1.20) + .20(1.04) + .10(1.22)βP = 1.04

Q6) You own a portfolio equally invested in a risk-free asset and two stocks. If one of the stocks has a beta of 1.38 and the total portfolio is equally as risky as the market, what must the beta be for the other stock in your portfolio? 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:    βP = 1.0 = 1/3(0) + 1/3(1.38) + 1/3(βX)   Solving for the beta of Stock X, we get:   βX = 1.62

Q7) A stock has a beta of 1.12, the expected return on the market is 10 percent, and the risk-free rate is 3 percent. What must the expected return on this stock be? CAPM states the relationship between the risk of an asset and its expected return. CAPM is:  E(Ri) = Rf + [E(RM) − Rf] × βi  Substituting the values we are given, we find:  E(Ri) = .030 + (.10 − .030)(1.12)E(Ri) = .1084, or 10.84%

Q8) A stock has an expected return of 13.6 percent, the risk-free rate is 6 percent, and the market risk premium is 10 percent. What must the beta of this stock be? 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(Ri) = .136 = .060 + .100βiβi = (.136 − .060) / .100βi = .76

Q9) A stock has an expected return of 10 percent, its beta is 1.10, and the risk-free rate is 4 percent. What must the expected return on the market be? 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(Ri) = .100 = .040 + [E(RM) − .040](1.10)   E(RM) = .0945, or 9.45%