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p.1/10 3/16/2010 6566b409-4023-4661-af36-739f27530abe.doc Instructions: This is the first page of the exam. 1. You have 75 minutes to answer all questions. 2. Show all relevant work: A correct answer with no correct explanation gets at most 10% of total credit. Multiple-choice question is an exception. 3. Be brief and precise. 4. Write legibly. 5. You are not allowed to leave the exam room during the last 15 minutes of exam. When exam is over, please remain seated until proctor collects your exam. 6. You can tear off this page. You do not need to hand this page in. 1. Symbols PV = Present value, what future cash flows bring today FVt = Future value, what cash flows are worth in the future r = Interest rate, rate of return, or discount rate per period t = Number of time periods C = Cash amount 2. PV of a perpetuity of C per period: PV = C/r 3. PV of Growing Perpetuities (payment grows at a rate of g per period): PV = C / (r – g) 4. PV of C per period for t periods discounted at r percent per period: C 1 PV = r 1 - (1+ r )t 5. e lim [ 1 + (1/k) ]k ≈ 2.71828 k Coupon ( Face value price) / maturity 6. Approximate yield to maturity: Yield ( price face value) / 2 2 variance(x) = σx2 = E [ (x - E(x)) ]. cov (X, aY+bZ) = a cov(X, Y) + b cov(X, Z) var (aX + bY) = a2 var (X) + b2 var (Y) + 2 ab cov (X, Y) Covariance: cov (x, y) = E [ (x - E(x)) (y - E(y)) ] = i [ Pi * (xi - E(x))* (yi - E(y)) ] Correlation coefficient: corr (x, y) = cov (x, y) / (σx*σy) Mean and Variance of a portfolio with x1 in asset 1 and 1 - x1 in asset 2 E(Rp) = x1 E(R1) + (1 - x1) E(R2) p 2 = x12 12 + (1 - x1) 2 22 + 2x1(1 - x1) 12 1 2 where 12 = Corr (R1, R2) = cov (R1, R2)/( 1, 2). Beta: i = cov (Ri, Rm) / var(Rm) CAPM: E(R ) = R + [E(Rm) - Rf ] i f i p.2/10 3/16/2010 6566b409-4023-4661-af36-739f27530abe.doc Question 1. Suppose that you just signed a $106,020 mortgage at 12% p.a., and that you always renew your mortgage at 12% p.a. in the future. Note that mortgage rates are by convention quoted with semi-annual compounding. You have two choices of payment: (a) annual payment of $14,560 and (b) weekly payment of $280 (i.e., $14,560/52 weeks). Treat 52 weeks as one year. (a) How long will it take to repay your loan if you make annual payments? (i) Find out the EAR (effective annual rate): (1+EAR) = (1+12%/2)2 EAR = 12.36% p.a. (ii) Use PV of annuity: 106,200 = (14,560/12.36%)*[ 1 – 1/(1+12.36%)t ] (iii) Solve for t (either by using a financial calculator, or by algebraic calculation) t = 19.891 years (b) How long will it take to repay your loan if you make weekly payments? (b) Follow the steps below: (i) We know that the EAR is 12.36%. We need to figure out the weekly rate: (1+EAR) = (1+r/52)52 r = 11.66685%, or r/52 = .224363% p.a. (ii) (ii) Use PV of annuity: 106,200 = [280/(r/52)]*{1 – 1/[1+(r/52)]t} (iii) Solve for t (either by using a financial calculator, or by algebraic calculation) t = 849.42 weeks or 16.33 years (c) Suppose you decide to make annual payments of $14,560. At the end of five years, you want to pay up your mortgage. How much do you pay? At the end of five years, there are fifteen annual payments of $14,560 to make at an interest rate of 12% p.a., or an EAR of 12.36% p.a. At the end of five years, you owe to your bank the PV of fifteen annual payments: PV = (14,560/12.36%)*[ 1 – 1/(1+12.36%)15 ] = 97,289.29 Question 2 On day 1, you bought a two-year bond and a stock in your portfolio. The bond is a zero-coupon bond with face value of $1,000. You paid $1,009.09 for your bond. On day 365, the yield to maturity (YTM) of the bond is 10%. p.3/10 3/16/2010 6566b409-4023-4661-af36-739f27530abe.doc You paid $1,000 for your stock. On day 365, you just received a dividend of $50. The dividend is expected to grow at a steady rate of 5% p.a. Your relevant discount rate is 10% p.a. Right after the receipt of dividend, you sold both your bond and stock. What is rate of return on your portfolio? Answer: You need to find out payoff from bond and stock. On day 365, price of the bond is equal to $1,000/(1+10%)=$909.09. So you have a loss of $1,009.09$909.09=$100. For stock, you need to find out price as of day 365: price = D1/(rg) = $50(1+5%)/(10%5%) =$1,050. Therefore, payoff from stock is: (selling pricebuying price)+dividend = ($1,050$1,000)+50=$100 Total payoff is the sum of payoff from bond and stock: $100 + $100 = 0. Therefore rate of return from your portfolio is 0%. Questions 3. You just started an internet company Bubble.com. You plan to pay the first dividend of $2 per share three years from today. You expect that the dividend will grow indefinitely at 6% p.a. Suppose investors require a 10% rate of return. (a) What is current price of your company’s stock? (b) Suppose Joan, one of the shareholders, plans to sell her share right after the first dividend payment. What is the expected rate of return on her investment, expressed as % per annum? (a) Let P0 and Pt denote current price and price t-years later. First dividend will be paid three years later, and dividend will grow at 6% p.a. So, if you apply the growing perpetuity formula, you get PV as of year 2, i.e., the PV is equal to P2. Once you obtain P2, you are almost done. In order to obtain P0, all you need to do is to discount P2 for two years. P2 = $2/ (10% 6%) = $50. P0 = $50/(1+10%)2 = $41.32. (b) Rate of return is defined as rate of return = [ (selling price buying price) + dividend ] / buying price. Joan buys the share today at $41.32 and receives $2 dividend at the end of three years, and sells the share right after the dividend payment. So, the selling price is equal to P3. To find out P3, apply again the growing perpetunity formula: P3 = D4 / (r g) = $2*(1+6%) / (10%6%) = $53. Rate of return = [ (5341.32) + 2 ] / 41.32 = 33.11% for three years. Since this rate of return is for three years, you find out equivalent annual return of return as: (1+x)3 = 1+33.11% x = 10% p.a p.4/10 3/16/2010 6566b409-4023-4661-af36-739f27530abe.doc Questions 4. You are going to form today a portfolio of a bond and some shares of a new internet company Pricecom. You have not yet decided exactly how many shares of the stock you will buy today. In any case, you plan to sell all bond and stocks in your porfolio one year later. The bond is a two-year zero coupon bond with face value of $1,000, currently selling at $800. You expect that the yield to maturity on your bond is 8.69% one year later. The standard deviation of rate of return on the bond investment is 0.10. Pricecom's first dividend will be $2 per share one year from today and that the dividend will grow indefinitely at 6% p.a. If there were zero growth in dividend (i.e., a constant dividend for an indefinite period), stock price will be $20. The standard deviation of rate of return on the stock investment is 0.20. The correlation between the rates of return on bond and stock is 0.5. (a) Suppose you want expected return of 12% p.a. on your portfolio. (a1) How many shares of stock do you need to buy? (a2) What is standard deviation of your portfolio's return? (b) Suppose that the correlation between the rates of return on bond and stock is 0.1, instead of 0.5. You still want expected return of 12% p.a. on your portfolio. Then, (b1) You need to buy [ more, less, same number of, insufficient info is provided ] shares than (as) in (a). (b2) Standard deviation of your portfolio's return is [ greater, smaller, same, insufficient info ] than (as) the one in (a). (c) All the possible combinations of two stocks A, B for a given level of correlation is called an investment opportunity set. Suppose expected return of both A and B is equal to 10% p.a., standard deviation of return for stock A is 0.4, and standard deviation of return for Stock B is 0.1. Assume that correlation between returns of Stock A and B is - 1 (i.e., minus one). Draw the investment opportunity set of a portfolio of stocks A and B. Indicate clearly the points corresponding to A and B. Vertical axis represents the expected return and horizontal axis represents the standard deviation of returns. [Hint: First, figure out the investment opportunity set when expected return of stock A is 6%. Second, figure out the investment opportunity set when expected return of stock A is 7%. Got the drift?] Answer (A) You want expected return of 12% on your portfolio. Let x denote the proportion of your money that you invest in bond, and let B and S denote the rate of return on bond and stock. Expected return on a portfolio is given by (1) x*E(B) + (1-x)*E(S) = 12 % p.5/10 3/16/2010 6566b409-4023-4661-af36-739f27530abe.doc This equation (1) is the starting point. These are the steps to take. Step 1: find out E(B) and E(S). Step 2: Once you know E(B) and E(S), then you can find out x from equation (1). Step 3: Find out dollar amount of your investment in stock. You buy a bond at $800. The $800 accounts for x of your portfolio. Suppose x=40%, for instance. Since you invest 60% of your money in stock, the dollar amount in stock is equal to $1,200. Step 4: find out current price of a share of stock, using the constant growth model. Dividing $1,200 by the current price gives the number of shares of stock you will buy today. Step 1: Since YTM at the end of year 1 is 8.69%, price at the end of year 1 is equal to $1,000/(1+8.69%), or $920. You buy a bond at $800 and sell it one year later at $920. Expected rate of return on bond, therefore, is 12%. Stock price with constant dividend is given by P = Div/r, or $20 = $2/r, which implies r=10%. Expected rate of return on stock is 10%. Step 2: Given E(B) = 15%, and E(S) = 10%, it is easy to see that x = 40%. Step 3: You spend $800 and buy a bond, which account for 40% of your portfolio. Therefore, 60% of portfolio should be equal to $1,200. Step 4: With a constant growth rate g=6%, stock price is given by P = Div/(r-g) = $2/ (10% - 6%) = $50. You need to buy $1,200/50 = 24 shares of stock. (A2) Standard deviation of the portfolio is given by st dev = [ (0.4)2 (0.1)2 + (0.6)2 (0.2)2 + 2 (0.4)(0.6) (0.5) (0.1) (0.2) ] ½ = 0.1442 (B1) A change in correlation does not affect expected returns of bond and stock. Therefore, you invest the same amount of money in bond and stock. You buy same number of shares of stock as in (A). (B2) The lower the correlation, standard deviation of your portfolio will get smaller. This is the benefit from diversification. Therefore, standard deviation in this case will be smaller than that in (A). (C) The investment opportunity set in this case will be a horizontal line connecting B and 10% at the vertical axis. This line passes through point A. p.6/10 3/16/2010 6566b409-4023-4661-af36-739f27530abe.doc Question 5 Consider the following information on stocks 1 and 2. Stock 1 Stock 2 expected return 16% 10% standard deviation 0.10 0.20 correlation 0.50 Part 1: (1) Suppose you invest $40 in stock 1 and $60 in stock 2. Find out expected rate of return and standard deviation of your portfolio. Answer: E(Rp) = 0.4*16% + 0.6*10% = 12.4%. p = [ (0.4)2*(0.1)2+ (0.6)2*(0.2)2 + 2*0.4*0.6*0.5*0.1*0.2 ]1/2 = 0.1442 Part 2: Assume that expected rate of return on market portfolio is 14% and risk-rate is 6%. (2) Which stock has more systematic risk? Answer: Using CAPM, stock 1: 16% = 6% + 1 (14% 6%) 1 = 1.25 stock 2: 10% = 6% + 2 (14% 6%) 2 = 0.50 Stock 1 has a higher , therefore Stock 1 has more systematic risk. (3) Which stock has more unsystematic risk? Stock 2 has a higher total risk, but lower systematic risk than Stock 1, which implies that Stock 2 has more unsystematic risk than Stock 1. Part 3: This Part 3 is independent of Part 2. Use the information in the table. Assume that the beta of stock 1 is 0.8 and beta of stock 2 is 0.2. (4) Based on CAPM, check if stock 1 and 2 are correctly priced, assuming a risk-free rate of 6%? Based on your answer, what kind of tradings on Stock 1 and Stock 2 would you do to take advantage of the mispricing, if any? Answer: Reward-risk ratio for Stock 1: (16% 6%)/0.8 = 12.5% Reward-risk ratio for Stock 2: (10% 6%)/0.2 = 20% Stock 2 has a higher reward-risk ratio, so you want to buy Stock 2 and sell Stock 1. (5) This question is independent of (4). Suppose these securities are correctly priced according to CAPM. What is the expected rate of return on the market portfolio? Answer: Reward-risk ratio of two stocks should be identical: (16% Rf)/0.8 = (10% Rf)/0.2 Solving for Rf yields Rf=8%. With Rf=8%, the reward-risk ratio is (16% 8%)/0.8=10%. The reward-risk ratio for the market portfolio is also 10%, i.e., (E(Rm) Rf)/m = 10%. Since Rf=8%, and m=1 (by definition of ), E(Rm) = 18%. p.7/10 3/16/2010 6566b409-4023-4661-af36-739f27530abe.doc (6) Suppose you want to form a portfolio of stock 1 and 2 so that the beta of the portfolio is equal to 0.62. How much do you need to invest in stock 1 out of your $100? Answer The of a portfolio is a weighted average of ’s of individual assets in the portfolio. x1(0.8) + (1 x1)(0.2) = 0.62 x1 = 70%. So you invest $70 in stock 1. Question 6 (Finding the WACC) 5 million shares of common shares at $40, E = 1.2. 750,000 shares of 7% preferred stock selling at $75. 250,000 units of 11% semi-annual bonds, par value = $1,000, 15 years-to-maturity selling at 93.5% of the par. Find out the after tax WACC. E(Rm) – Rf = 6%, Rf = 4%, TC = 34% Step 1: find out market value of common, preferred stocks and bonds. See column 1 of table. Step 2: find out fraction in total firm value. See column 2. Step 3: find out after tax cost of common, preferred stocks and bonds. See column 3. (a) RE = Rf + [E(Rm) – Rf] = 4% + 1.2 * 6%= 11.2% (b) To obtain RD, first find out the bond yield using the formula (or using financial calculator) Yield ≈ [Coupon + (Face value – Price)/maturity ] / [ (Price + Face value)/2 ] = [110 + (1,000 – 935)/15 ] / [ (1,000 + 935)/2 ] = 11.8% Yield (from a financial calculator) = 11.94% After-tax: 11.94% (1 – TC) = 11.94% * (66%) = 7.88% Step 4: calculate the weighted average of cost. See column 4. Capital structure fraction of (after-tax) return WACC (1) firm value (2) (3) (4) Common $40 * 5 m = $200 m 40.8% 11.2% Preferred $75 * 0.75m = 56.25 11.5% 7.00% Bond $935*0.25m = 233.75 47.7% 7.88% value of the firm $490 9.13% Question 7: (Using the WACC) Suppose WallStore. is considering opening another store. The expansion will cost $50,000 and is expected to generate after-tax cash flows of $10,000 per year in perpetuity. The firm has a target debt/equity ratio of .50. New equity has a required return of 15%, while new debt has a required return of 10%. Cost of capital: WACC = (E/V) × RE + (D/V) × RD × (1 – TC) WACC = 2/3 × 15% + 1/3 × 10% × (1 – .34) = 12.2%. p.8/10 3/16/2010 6566b409-4023-4661-af36-739f27530abe.doc NPV: Investment = $50,000 = $50,000 PV of cash flow = $10,000/.122 = $81,967 NPV = – $50,000 + $81,967 = $31,967 Key Point: WACC is the appropriate discount rate for the proposed project, only when the proposed project is in the same risk class as the firm. Question 8: SML and WACC An all-equity firm with 14% cost of capital considers the projects: Project Beta Expected return market data X 0.85 13% T-bill = 5% Y 1.15 15% E(Rm) = 14% (1) If the firm is using 14% WACC (without considering the different level of risk associated with projects), which project does it accept? Project Y. (2) When the firm considers risk of the projects, which project should it accept? Project X. E(X) = 5% + 0.85*(14% 5%) = 12.65% . E(X) based on CAPM < expected return of 13%. E(Y) = 5% + 1.15*(14% 5%) = 15.35% . E(Y) based on CAPM > expected return of 15%. ■ Short Answer Questions Qu 1. True or false: The most important characteristic in determining the expected return of a well diversified portfolio is the variances of the individual assets in the portfolio. Explain. Qu 2. If a portfolio has a positive investment in every asset, can the standard deviation on the portfolio be less than that on every asset in the portfolio) What about the portfolio beta? Qu 3. Is it possible that a risky asset could have a beta of zero? Explain. Based on the CAPM, what is the expected return on such an asset? Is it possible that a risky asset could have a negative beta? What does the CAPM predict about the expected return on such an asset? Can you give an explanation for your answer? Qu 4. If a portfolio has a positive investment in every asset, can the expected return on the portfolio be greater than that on every asset in the portfolio? Can it be less thal that on every asset in the portfolio? If you answer yes to one or both of these questions, give an example to support your answer. Qu 5. What are the advantages of using the DCF model for determining the cost of equity capital? What are the disadvantages? What specific piece of information do you need to find the cost of equity using this model? What are some of the ways in which you could get this estimate? Qu 6. What are the advantages of using the SML approach to finding the cost of equity capital? What are the disadvantages? What are the specific pieces of information needed to use p.9/10 3/16/2010 6566b409-4023-4661-af36-739f27530abe.doc this method? Are all of these variables observable, or do they need to be estimated? What are some of the some of the ways in which you could get these estimates? Qu 7. How do you determine the appropriate cost of debt for a company? Does it make a difference if the company's debt is privately placed as opposed to being publicly traded? How would you estimate the cost of debt for a firm whose only debt issues are privately held by institutional investors? Qu 8. Suppose Tom O'Bedlam, president of Bedlam Products, Inc., has hired you to determine the firm's cost of debt and cost of equity capital. a. The stock currently sells for $50 per share, and the dividend per share will probably be about $5. Tom argues, "It will cost us $5 per share to use the stockholders' money this year, so the cost of equity is equal to 10 percent ($5/50)." What's wrong with this conclusion? b. Based on the most recent financial statements, Bedlam Products' total liabilities are $8 million. Total interest expense for the coming year will be about $1 million. Tom therefore reasons, "We owe $8 million, and we will pay $1 million interest. Therefore, our cost of debt is obviously $1 million/8 million = 12.5%." What's wrong with this conclusion? c. Based on his own analysis, Tom is recommending that the company increase its use of equity financing, because "debt costs 12.5 percent, but equity only costs 10 percent; thus equity is cheaper." Ignoring all the other issues, what do you think about the conclusion that the cost of equity is less than the cost of debt? Qu 9. Both Dow Chemical Company, a large natural gas user, and Superior Oil, a major natural gas producer, are thinking of investing in natural gas wells near Edmonton. Both are all- equity-financed companies. Dow and Superior are looking at identical projects. They've analyzed their respective investments, which would involve a negative cash flow now and positive expected cash flows in the future. These cash flows Would be the same for both firms. No debt would be used to finance the projects. Both companies estimate that their project would have a net present value of $1 million at an 18 percent discount rate and a -$ 1.1 million NPV at a 22% discount rate. Dow has a beta of 1.25, whereas Superior has a beta of .75. The expected risk premium on the market is 8 percent, and risk-free bonds are yielding 12 percent. Should either company proceed? Should both? Explain. Answer: 1. False. The variance of the individual assets is a measure of the total risk. The variance on a well-diversified portfolio is a function of systematic risk only. 2. Yes, the standard deviation can be less than that of every asset in the portfolio. However, cannot be less than the smallest beta because p is a weighted average of the individual asset betas. 3. Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be equal to the risk-free rate. It is also possible to have a negative beta; the return would be less than the risk-free rate. A negative beta asset would carry a negative risk premium because of its value as a diversification instrument. p.10/10 3/16/2010 6566b409-4023-4661-af36-739f27530abe.doc 4. No to both questions. The portfolio expected return is a weighted average of the asset returns, so it must be less than the largest asset return and greater than the smallest asset return. 5. The primary advantage of the DCF model is its simplicity. The method is disadvantaged in that (1) the model is applicable only to firms that actually pay dividends; many do not; (2) even if a firm does pay dividends, the DCF model requires a constant dividend growth rate forever; (3) the estimated cost of equity from this method is very sensitive to changes in g, which is a very uncertain parameter; and (4) the model does not explicitly consider risk, although risk is implicitly considered to the extent that the market has impounded the relevant risk of the stock into its market price. While the share price and most recent dividend can be observed in the market, the dividend growth rate must be estimated. Two common methods of estimating g are to use analysts’ earnings and payout forecasts, or to determine some appropriate average historical g from the firm’s available data. 6. Two primary advantages of the SML approach are that the model explicitly incorporates the relevant risk of the stock, and the method is more widely applicable than is the DCF model, since the SML doesn’t make any assumptions about the firm’s dividends. The primary disadvantages of the SML method are (1) three parameters, the risk-free rate, the expected return on the market, and beta must be estimated, and (2) the method essentially uses historical information to estimate these parameters. The risk-free rate is usually estimated to be the yield on very short maturity T-bills and is hence observable; the market risk premium is usually estimated from historical risk premiums and hence is not observable. The stock beta, which is unobservable, is usually estimated either by determining some average historical beta from the firm and the market’s return data, or using beta estimates provided by analysts and investment firms. 7. The appropriate aftertax cost of debt to the company is the interest rate it would have to pay if it were to issue new debt today. Hence, if the YTM on outstanding bonds of the company is observed, the company has an accurate estimate of its cost of debt. If the debt is privately- placed, the firm could still estimate its cost of debt by (1) looking at the cost of debt for similar firms in similar risk classes, (2) looking at the average debt cost for firms with the same credit rating (assuming the firm’s private debt is rated), or (3) consulting analysts and investment bankers. Even if the debt is publicly traded, an additional complication occurs when the firm has more than one issue outstanding; these issues rarely have the same yield because no two issues are ever completely homogeneous. 8. a. This only considers the dividend yield component and ignores the capital gain yield component of the required return on equity. b. This is the current yield only, not the promised yield to maturity. In addition, it is based on the book value of the liability, and it ignores taxes. c. Equity is inherently more risky than debt (except, perhaps, in the unusual case where a firm’s assets have a negative beta). For this reason, the cost of equity exceeds the cost of debt. Additionally, he is basing his decision on incorrect calculations for the cost of debt and the cost of equity. 9. Rsup = .12 + .75(.08) = 18% Both should proceed. The appropriate discount rate does not depend on which company is investing; it depends on the risk of the project. Since Superior is in the business, it is closer to a pure play. Therefore, its cost of capital should be used. With an 18% cost of capital, the project has an NPV of $1 million regardless of who takes it.

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