Sample Quiz 2 with Solutions

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Finance 402 Capital Budgeting and Corporate Objectives Professor Alex Priest Sample Quiz 2 SOLUTIONS 1

1. (a) (30 points) Suppose you expect Tech Inc.’s earnings to grow at 5% per year forever. This firm pays dividends annually, and pays out all of the earnings as dividends. It just paid a dividend of $2 per share. The cost of capital is 15% per year compounded monthly. i. (23 points) What is the value of Tech Inc.’s stock? Next years dividend will be 2(1 + 0 . 05) = 2 . 1 so that the value of the firms stock is The effective annual rate is (1 + . 15 12 ) 12 - 1 = 0 . 160755 . Which implies that the value of the firm’s stock should be 2 . 1 0 . 1608 - 0 . 05 = $18 . 96 ii. (4 points)What would be the value of Tech Inc.’s stock if instead it were able to grow earnings at a rate of 6.5% by reinvesting 25% of earnings into the firm instead of paying dividends? Going forward, the firm would pay 75% of earnings. Earnings in the first year will be unchanged from before. The earnings in the first year will still be $ 2.10 per share. If 75% of earnings are paid out in the first year, then the first dividend payment will be $2 . 10 × 0 . 75 = $1 . 575 . With this reinvestment, the growth rate increases to 6.5%. Thus the value of the stock with reinvestment is 1 . 575 0 . 1608 - 0 . 065 = $16 . 44 iii. (3 points) If you hold shares of Tech Inc, do you want them to reinvest the earnings or continue to pay out 100%? If you are a shareholder, you prefer that they do not reinvest earnings because the stock will be worth less. (b) (10 Points) Suppose you expect XYZ stock’s quarterly dividends to remain constant for the next 10 years and grow at 4% per quarter thereafter. The firm has just paid its quarterly dividend and you expect the next quarterly dividend to be $1 per share. What is the maximum price that you would be willing to pay for XYZ stock if you require an 8% quarterly rate of return? (Hint: Try to break the problem up into pieces.) Solution: We split the evaluation into 2 parts the first is the present value of a ten year annuity, with quarterly dividends. The second is the present value of the stock price at time 10. Step 1: Find the value of the annuity: 2

1 0 . 08 [1 - 1 1 . 08 40 ] = 11 . 92 (1) Step 2: d 41 = 1(1 . 04) = 1 . 04 (2) Then compute the present value 10 years from today of all the future cash flows: P 40 = d 41 r e - g = 1 . 04 0 . 08 - 0 . 04 = 26 (3) Next, find the present value of this figure. PV ( P 40 ) = P 40 (1 . 08) 40 = 1 . 197 (4) The final price is simply the sum of these two components 11 . 92 + 1 . 197 = 13 . 117 3

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2. Assume the following • Google stock has an expected return of 15 percent, and a standard de- viation of 8 percent. It trades at $80 per share with 1 million shares outstanding. • Yahoo stock has an expected return of 20 percent, and a standard deviation of 20 percent. It trades at $40 per share with 4 million shares outstanding. (a) (15 points) What is the expected return of a value weighted portfolio of Google and Yahoo. First we compute the portfolio weight of Google W G = 80 * 1 80 * 1 + 40 * 4 = 1 3 So that E [ r p ] = 1 3 0 . 15 + 2 3 0 . 2 = 0 . 1833 (b) (10 points) Assuming the correlation between the two stocks is 0.38, cal- culate the standard deviation of an equal weighted portfolio of Google and Yahoo. σ P = p 0 . 5 2 (0 . 08 2 ) + 0 . 5 2 (0 . 2 2 ) + 2(0 . 5)(0 . 5)(0 . 38)(0 . 08)(0 . 2) = 0 . 12 (c) (5 points) If the correlation between the two stocks was instead 0 (still with equal weighting of the two assets), would the standard deviation or expected returns change? If so, calculate. The expected return does not change because expected returns do not depend on correlation. The standard deviation changes because the correlation is now 0. In par- ticular, the standard deviation will become σ P = p 0 . 5 2 (0 . 08 2 ) + 0 . 5 2 (0 . 2 2 ) + 2(0 . 5)(0 . 5)(0)(0 . 08)(0 . 2) = 0 . 1077 The standard deviation decreases with lower correlation. (d) (5 points) What weight in Google will give the minimum variance portfolio assuming the correlation between the two stocks is 0? To solve for the minimum variance portfolio weights, we need to use the formula. w G = σ 2 Y - ρ GY σ G σ Y σ 2 G + σ 2 Y - 2 ρGY σ G σ Y 4

Plugging in: w G = . 2 2 - 0 × 0 . 08 × 0 . 20 . 08 2 + . 2 2 - 2 × 0 × 0 . 08 × 0 . 2 = 0 . 862 The minimum variance portfolio thus has a weight of 86.2% in Google and a weight of (1-86.2%)=13.8% in Yahoo. 3. (20 Points) Assume that CAPM holds for the question and • The market beta of Ford stock is 0.75 • The market beta of GM stock is 1.30 • The risk-free rate is 5% • The expected return of the market is 12% (a) (14 Points) Compute the expected returns for Ford and GM. Use the for- mula for expected returns in CAPM E ( R i ) = R f + βE ( R M - R f ) Plugging in first for Ford E ( R FORD ) = 5% + 0 . 75 × (12% - 5%) = 5% + 5 . 25% = 10 . 25% Now for GM, E ( R GM ) = 5% + 1 . 30 × (12% - 5%) = 5% + 9 . 1% = 14 . 1% (b) (3 points) Would expected returns change if the risk-free rate was instead 3% and everything else stayed the same? Yes. They would become E ( R FORD ) = 3% + 0 . 75 × (12% - 3%) = 3% + 6 . 75% = 9 . 75% Now for GM, E ( R GM ) = 3% + 1 . 30 × (12% - 3%) = 3% + 11 . 7% = 14 . 7% The expected return of Ford decreased and the expected return of GM in- creased. (c) (3 points) Can we say whether Ford or GM has the higher standard deviation from the information given? No, we cannot say. It would depend also on the correlation of GM and Ford with the market. 5