Practice Set #2 and Solutions.

Bo Sjö 2011-04-19 Practice Set #2 and Solutions. What to do with this practice set? Practice sets are handed out to help students master the material of the course and prepare for the final exam. These sets contain worked-out problems in corporate finance and, investments and portfolio management, with a bias toward quantitative problems. These sets are not graded, and there is no need to hand-in the solutions. Students are strongly encouraged to solve them, discuss the solutions with other course participants, and discuss any problems with their teacher. Some questions in the final exam might resemble the problems given here. Question 1 ToT Malt (an all-equity firm) is reinvesting 65% of its earnings in projects that provide a ROE of 9%. The expected return on similar risky projects is 14% on the stock market. Given the present policy of the firm its year-end dividend is now 3 per share. a) At what price will the stock sell? b) What is the present value of the growth opportunities for ToT Malt? Question 2 In early 1999 an investor bought 1000 shares of Skandia for 127 SKr per share. During the year dividends were received at 1.15 SKr per share, and finally the shares were sold for 235 SKr per share. What is the investor s a. Total return in SKr? b. Capital gain, dividend yield, and total return in percentage terms? Question 3 HGUS Inc. has a book value of 21,500,000 (including expected retained earnings), expected earnings of 2,640,000, the pay-out ratio is.65. The opportunity cost of capital for the company is 7.5%. There are 1,100,000 outstanding shares. a) What are the excepted growth rate, the price and the P/E ratio of this firm? b) If the plow-back rate were to change to.45 what would be the expected dividend per share, the growth rate, price and P/E ratio, given that all other values remain unchanged? 1

Question 4 McManus Teaching Company is expected to pay a dividend of $2 in the upcoming year. The risk-free rate of return is 4% and the expected return on the market portfolio is 14%. The beta of McManus Teaching Company 1.25. Analysts expect the price of McManus Teaching Company shares to be $22 a year from now. a) What is the market s required rate of return on McManus Teaching's stock? b) What is the intrinsic value of McManus Teaching's stock today? c) If McManus Teaching Company s intrinsic value is $21.00 today, what must be its growth rate? Question 5 The following is given, there are two risky assets A and B with the following characteristics E(r A ) = 12%, E(r B ) = 7%, σ A = 20% σ B = 10%, cov(r A, r B ) = 50 The risk-free rate is r f = 4%, The weights of the optimal risky portfolio are w A = 0.81 and w B = 0.19 a) What is the expected return on the risky portfolio? b) What is the risk (in per cent) of the risky portfolio? c) What is the Capital Allocation Line associated with this risky portfolio? What is the slope of the line? The slop is called reward to variability ratio. (Show a graph and try to calculate numerically). Explain briefly why the slope can be called reward to variability ratio? Question 6 This year, AGL paid its shareholders an annual dividend of $3 a share. A major brokerage firm recently put out a report on AGL stating that, in its opinion, the company s annual dividends should grow at the rate of 10 per cent per year for each of the next 5 years and then level off the grow at the rate of 6 per cent a year thereafter. a) Use the variable-growth dividends technique and a required rate or return of 12 per cent to find the maximum price you should be willing to pay for this share. b) Redo the AGL problem in part a), except this time assume that after year 5, dividends stop growing altogether (for year 6 and beyond, g = 0). Use all the other information given to find the share s intrinsic value. c) Contrast your two answers and comments on your findings. How important is growth to this valuation model? Question 7 MacKLine Bank recently reported net profits after tax of $1,000 million. It has 2.5 million shares outstanding and pays dividends on preference shares equal to $1 million per year. a. Calculate the firm s earnings per share (EPS). 2

b. Assuming that the share currently trades at $75.80 per share, determine what the firm s dividend yield would be if it paid $10 per share to ordinary shareholders. c. What would the firm s dividend payout ratio be if it paid $10 a share in dividends Question 8 You are looking at an asset with a beta value of 1.25. The risk premium on the market is stable around 6% and the risk free rate is 4%. Over the past years the return on this stock as been 10%. a) Use CAPM to discuss if you buy or short sell the stock. b) Next, use an appropriate figure to illustrate the situation, explain the concepts in the figure. In the context of the question explain what is meant by excess return, risk premium and abnormal return. 3

Solutions #2 Solution 1 Given the policy of the management the growth rate can be estimated as, g = ROE x b = 9% x.65 = 5.4% The earning of the firm can solved from the figures given in the question. Dividends (D), Earnings (E) and b is the plow-back ratio (reinvestment ratio). Thus D = E x (1-b). What is not reinvested is paid out as dividends. Thus, earnings per share is, E = D/(1-b) = 3/.35 = 8.57. Suppose the price per share should is estimated according to the Gordon s formula P 0 = 3 / (.14 -.054) = 34.88, where.14 is the discount rate on this type of cash flow. If the firm was to pay out its earnings as dividends its value would be, E/r = 8.57/.14 = 61.22. The PV of growth opportunities is, PVGO = Price per share - No-growth value per share = 34.88 - E 1 / r 0 = 34.88-8.57/.14 = 34.88-61.22 = - 26.34 Thus, PVGO is negative, since the firm has no growth opportunities that yield a return over or at least in par with the market return for similar projects. The rate of return on ToT Malt is less than the opportunity cost of capital. The reason for a takeover is that the firm can be bought today for a price of 34.88 per share, lower than its price during a different management and investment policy. A new management can change the dividend policy and pay out all earnings as dividends. The price of the firm per share will then rise to 61.22. And 61.22-34.88 (-transaction costs) is profit for the new owner. Solution 2 a. The total return is SEK1,000(235-127+1.15) = EK109,150 b. The capital gain is (235-127)/127=0.85 and dividend yield is 1.15/127=0.009 and the total return is 0.85+0.009=0.859. Solution 3 ROE = 2,640,000/ 21,500,000 =.123 or 12.3% Expected dividend per share Div = ( Earnings/share) x pay-out ratio = 2.4 x.65 = 1.56 4

Retention ratio = b = 1- pay-out ratio = 1 -.65 =.35 Expected growth rate g = b x ROE =.35 x 12.3% = 4.3% Expected Price P 0 = D 1 / (k-g) = 1.56 / (.075 -.043) = 48.75 Expected P/E P 0 /E 1 = 48.75/ 2.4 = 20.31 Change b to.45, g =.45 x 12.3% = 5.5% Div 1 = 2.4 x (1-.45) = 1.32 P 0 = 1.32 / (.075 -.043) = 41.25 P 0 / E 1 = 41.25 / 2.4 = 17.19 Solution 4 a) Apply CAPM o get the markets required rate of return McManus's stocks: E( r i ) = r f + B i E(r m - r f ) 4% + 1.25(14% - 4%) = 16.5%. b) Again apply CAPM, solve for required, expected equilibrium return k = 0.04 + 1.25 (0.14-0.04) ; k = 0.165 or 16.5% The required rate should be the HPR in equilibrium so, 0.165 = (22 P + 2) / P Then solve for the price (=the intrinsic price) 0.165P = 24 P 1.165P = 24 P = 20.60 c) a) Start with CAPM to get equilibrium expected return on this stock, k = 0.04 + 1.25 (0.14-0.04) k = 0.165 b) Next apply dividend growth formula and solve for the growth rate, 5

Solution 5 0.165 = 2/21 + g g = 0.07 or 7% a) E(r p ) = w A E(r A ) + w B E(r B ) = 0.81 x 12 + 0.19 x 7 = 9.72 + 1.33 = 11.05 b) He formula for the variance of a combination of two stochastic returns c) σ 2 p = w 2 A σ 2 A + w 2 Bσ 2 B + 2 w A w B cov(r A, r B ) = (0.81) 2 20 2 + (0.19) 2 10 2 + 2 (0.81) (0.19) 50 = 281.44 σ p = 16.78% The reward to variability ratio is the slope of the capital allocation line. The straight line that combines the risk-free rate with the points [E(r p ), σ p ] = [11.05, 16.78]. The slope of this CAL is given by [E(r p ) - r f ] / σ p = (11.05-4) / 16.78 = 0.42 A graph that illustrates the relationship is necessary. Reward to variability comes from the assumption of risk-averse investors. The higher the risk the higher must the expected return be for the investor to be happy. To get max points you have to discuss the meaning of the slope [E(r p ) - r f ] / σ p, and not state the ratio. Solution 6 a. Projected annual dividends: Year Dividends 0 $3.00 1 3.30 (g = 10%) 2 3.63 (g = 10%) 3 3.99 (g = 10%) 4 4.39 (g = 10%) 5 4.83 (g = 10%) 6 5.31 (g = 6%) Estimated annual growth rate for year 6 and beyond: 6% Step 1: Present value of dividends using a required rate of return of 12%: Year Dividends x PVIF, 12% Present Value 1 3.30.893 $2.95 2 3.63.797 2.89 3 3.99.712 2.84 4 4.39.636 2.79 5 4.83.567 2.74 Total $14.21 6

Step 2: Price of share at the end of year 5: P 5 = D 6 = $5.12 = $5.12 = $85.33 k - g.12 -.06.06 Step 3: Present value of the share price: PV = $85.33 x PVIF 12%, 5 yrs. = $85.33 x.567 = $48.38 Step 4: Value of SWL&G share = $14.21 (Step 1) + $48.38 (Step 3) = $62.59 Therefore, $62.59 is the maximum price you should be willing to pay for this share. b. Since g = 0 for year 6 and beyond, dividends for year 6 will be the same as the dividend for year 5; i.e., $4.83. We just need to redo steps 2 and 3 to find the intrinsic value of the share: Step 2: Price of share at the end of year 5: P 5 = D 6 = $4.83 = $4.83=$40.25 k -g.12-0.12 Step 3: Present value of the share price: PV (P 5 ) = $40.25 x PVIF 12%, 5 yrs. = $40.25 x.567 = $22.82 Since the present value of the first five years of dividends is the same as in (a), intrinsic value (=valuation based on estimated cash flows) of the share is: above, the Intrinsic value = $14.21 + $22.82 = $37.03 c. The intrinsic value of the share in (a) is much higher than that calculated in (b). In (a), dividends are growing at 6% per year beyond year 5, while in (b), the dividends do not grow after year 5. The dividend valuation model is very sensitive to the growth rate in dividends; the higher the rate of growth in dividends, the higher the intrinsic value of the share. Solution 7 a. Earnings per share (EPS) = Net profits after taxes - Preference dividends Number of ordinary shares outstanding For Mackline Bank: EPS = $1,000,000,000 - $1,000,000 = $399.6 2,500,000 b. Dividend yield = Cash dividends per share Market price per share 7

For MacKline Bank: Dividend yield = $10 / $75.80 = 13.19% c. Dividend payout ratio = Dividends per share EPS For MacKLine Bank: Dividend payout ratio = $10 / $399.6 = 2.5% Solution 8 a) Answer: Do CAPM calculation of the required return on this stock. E(r i ) = 4% + 1.25(6) =11.5.5% CAPM predicts that the present price is too high. If the stocks pay 10% and CAPM predicts 11.5% it means that the return is lower than compensation for risk. Present holders are making abnormal negative returns of 10-11.5 = -1.5%. If the price goes down the return will go up. The model predicts that market forces will drive down the price, and bring up the return to the equilibrium return of 11.5%. Thus, you should sell the stock before the price goes down in order to make a profit. b) Answer: Set up the security market line (SML) and discuss of the stock is over priced or under priced. 8