Midterm Exam Suggested Solutions

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1 JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Date: 7/11/2017 Midterm Exam Suggested Solutions Problem 1. 4 points) Which of the following statements about the relationship between interest rates and bond prices are true? a) There is an inverse relationship between bond prices and interest rates. b) There is a direct relationship between bond prices and interest rates. c) The price of short term bonds fluctuates more than the price of long term bonds for a given change in interest rates assuming that coupon rate is the same for both). d) The price of long term bonds fluctuates more than the price of short term bonds for a given change in interest rates assuming that the coupon rate is the same for both). e) None of the given statements are true. a) and d) only. Problem 2. 2 points) Which of the following statements is true? a) The spot interest rate is a weighted average of yields to maturity. b) Yield to maturity is the weighted average of spot interest rates and estimated forward rates. c) The yield to maturity is always higher than the spot rates. d) None of the given statements are true. b) only. Problem 3. 3 points) Petroleum Inc. owns a lease to extract crude oil from sea. It is considering the construction of a deep sea oil rig at a cost of $50 million C 0 ). The price of oil P is $40/bbl and the extraction and transportation costs are $25/bbl. The quantity of oil Q = 300, 000 bbl per year forever. The risk free rate is 6% per year and that is also the cost of capital. Calculate the NPV of the project. Ignore taxes and any potential changes in prices or costs in the future. NP V = $50 million + $40/bbl $25/bbl) 300, 000 bbl 0.06 = $25 million. Problem points) Use the same assumptions as in the previous problem, but suppose the oil price is uncertain and it will be revealed next year whether it will be high $50/bbl) or low $30/bbl) forever, with equal probability of each outcome. If you have the 1

2 option to postpone the project/investment decision by one year, what is the expected NPV of the project? NP V 1 = $50 million + $50/bbl $25/bbl) 300, 000 bbl 0.06 = $75 million, $30/bbl $25/bbl) 300, 000 bbl NP V 2 = $50 million + = $25 million 0.06 reject the project NP V 2 = 0, Expected NP V = 1 /2NP V /2NP V = $35.4 million. Problem 5. 2 points) Using your answers in the two questions above, what is the value of the option to postpone the project by one year? $35.4 million $25 million = $10.4 million. Problem 6. 5 points) Calculator Company proposes to invest $5 million in a new calculator making plant. Fixed costs are $2 million a year. A calculator costs $5/unit to manufacture and can be sold for $20/unit. If the plant lasts for 3 years and the cost of capital is 12%, what is the approximate break even level i.e. NP V = 0) of annual sales? Assume no taxes. NP V = $5 million + $20/unit $5/unit) sales $2 million 0.12 sales = 272, 117 units. 1 1 ) = Problem 7. 4 points) Summer Co. is expected to pay a dividend or $4 per share out of earnings of $7.5 per share. If the required rate of return on the stock is 15% and dividends are expected to grow at 10% per year forever, calculate the present value of the growth opportunity for the stock PVGO). P 0 = DIV 1 r g & P 0 = EP S 1 r P V GO = DIV 1 r g EP S 1 r = + P V GO $ $ = $30. 2

3 Problem 8. 5 points) Given the following cash flows for project Z: C 0 = $1, 000, C 1 = $600, C 2 = $720, and C 3 = $2, 000, calculate the discounted payback period for the project at a discount rate of 20%. The present values of the cash inflows for the project are shown in the third row of the table below, and the cumulative net present values are shown in the fourth row: C 0 C 1 C 2 C 3 $1, 000 $600 $720 $2,000 $1, 000 $500 = $600 ) $500 = $720 ) ) $2, 000 $1,157.4 = $ 500 $0 $1,157.4 Since the cumulative NPV turns positive in year two, the discounted payback period is 2 years. Problem 9. 4 points) The efficient portfolios: a) have only unique risk b) provide the highest returns for a given level of risk c) provide the least risk for a given level of returns d) have no risk at all b) and c) only. Problem points) The capital asset pricing model CAPM) states that: a) The expected risk premium on an investment is proportional to its beta b) The expected rate of return on an investment is proportional to its beta c) The expected rate of return on an investment depends on the risk free rate and the market rate of return only d) The expected rate of return on an investment is proportional to the risk free rate a) only. Problem points) Given the following data for a stock: beta = 1.5; risk free rate = 4%; market rate of return = 12%; and expected rate of return on the stock = 15% true one!). Then the stock is: a) overpriced b) underpriced c) correctly priced d) cannot be determined r = r f + βr m r f ) = ) = 16% > 15% the stock is overpriced. 3

4 Problem points) The market value of XYZ Corporation s common stock is $40 million and the market value of the risk free debt is $60 million. The beta of the company s common stock is 0.8, and the expected market risk premium is 10%. If the Treasury bill rate is 6%, what is the firm s cost of capital? Assume no taxes. r E = r f + βr m r f ) = = 14%, E Cost of capital = r E D + E + r D D D + E = $40 million = 14 $40 million + $60 million + 6 $60 million $40 million + $60 million = 9.2%. Problem 13. Arbitrage Pricing Theory) Imagine that the Arbitrage Pricing Theory holds and that there are only two pervasive macroeconomic factors factor 1 and factor 2). Investments A, B, and C have the following sensitivities to these two factors: Investment b 1 b 2 A B 3 6 C 6 3 We assume that the expected risk premium is 2% on factor 1 and 4% on factor 2. Treasury bills offer zero risk premium. a) 5 points) According to the APT, what is the risk premium on each of these three stocks? b) 10 points) Suppose you buy $400 of A and $100 of B and sell $300 of C. What is the sensitivity of your portfolio to each of the two factors? What is the expected risk premium and why? c) 10 points) Suggest two possible ways that you could construct a portfolio that has a sensitivity of 0.5 to factor 1 we do not care about sensitivity to factor 2). Now compare the risk premiums on each of these two investments and explain any differences. d) 5 points) Suppose that the APT did not hold and that A offered a risk premium of 4%, B offered a premium of 7%, and C offered a premium of 8%. Devise an investment that has zero sensitivity to each factor and has a positive risk premium. a) r A = = 13.5%, r B = = 18%, r C = = 24%. b) This portfolio has the following portfolio weights: X A = 400 / ) = 2, 4

5 X B = 100 / ) = 0.5, X C = 300 / ) = 1.5. The portfolio s sensitivities to the factors are: F actor 1 : ) + 1.5) 6 = 0, F actor 2 : ) 3 = 0. Because the sensitivities are both zero, the expected risk premium is zero. c) The sensitivity requirement can be expressed as: In addition, we know that: F actor 1 : X A X B 3) + X C 6 = 0.5. X A + X B + X C = 1. With two linear equations in three variables, there is an infinite number of solutions. Two of these are: 1. X A = 0, X B = 11 /18, X C = 7 /18, 2. X A = 14 /33, X B = 19 /33, X C = 0. The risk premiums for these two funds are: r 1 = ) + 11 /18 3) ) + 7 / ) = 20.33%, r 2 = 14 / ) + 19 /33 3) ) ) = 16.09%. These risk premiums differ because, while each fund has a sensitivity of 0.5 to factor 1, they differ in their sensitivities to factor 2. d) Because the sensitivities to the two factors are the same as in part b), one portfolio with zero sensitivity to each factor is given by: The risk premium for this portfolio is: X A = 2, X B = 0.5, X C = ) 8 = 0.5%. Because this is an example of a portfolio with zero sensitivity to each factor and a nonzero risk premium, it is clear that the Arbitrage Pricing Theory does not hold in this case. A portfolio with a positive risk premium is: X A = 2, X B = 0.5, X C = 1.5. Problem 14. Real options) You are a telecom company considering an investment into 1st generation GSM analog mobile phone coverage) network. You need to buy the license from the government and build the infrastructure, for total investment of EUR 400 million today and you will receive net cash inflows of EUR 100 million at the end of each of next five years, after which the 5

6 investment will be worthless. The appropriate discount rate for this project is 10%, while the risk free rate is 5%. If and only if) you invest today into the 1st generation network, you will be able but not obliged) to invest into the 2nd generation digital) network in five years. Then, the needed investment will be EUR 800 million and you expect to get net cash inflow of EUR 200 million in each of the five years following the investment. As mobile phones are a new industry in your country, you see great growth potential, but also large uncertainty in the future cash flow development. Your analysts believe, based on comparable projects and industries, that the most appropriate volatility standard deviation) of the cash inflows/value is 25% per year. a) 8 points) Calculate NPVs for both generations of GSM networks i.e. for 1st and 2nd generations at the time of actual and initial investment). b) 10 points) State the problem in terms of real options and identify key parameters needed to quantify option value. Calculate the real option value using the Black Scholes formula. c) 2 points) Make a recommendation whether investing in the 1st generation network makes sense and explain why not). If needed, please use approximate values for Nd): d Nd) d Nd) a) 1st generation: NP V = 400 million million ) = 20.9 million EUR nd generation at the time of actual investment t = 5): 200 million NP V = 800 million ) = million EUR nd generation at the time of initial investment t = 0): NP V = million = million EUR. b) Black Scholes formula: V alue of call option = Nd 1 ) P Nd 2 ) P V EX), 6

7 where d 1 = ) P log P V EX) σ t d 2 = d 1 σ t, + σ t 2, Nd) = cumulative normal probability density function, EX = exercise price of option; P V EX) is calculated by discounting at the risk free interest rate r f, t = number of periods to exercise date, P = price of stock now, σ = standard deviation per period of continuously compounded) rate of return on stock. In our case, t = 5, σ = 0.25, r f = 0.05, EX = 800 million EUR needed investment for 2nd generation network). P V EX) = 800 = million EUR P = million 1 1 ) = million EUR d 1 = 0.23, d 2 = 0.79, Nd 1 ) = 0.42, Nd 2 ) = V alue of call option = = million EUR. c) Yes, makes sense because the option value is greater than the 1st generation NPV loss. 7

Midterm Review. P resent value = P V =

Midterm Review. P resent value = P V = JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Midterm Review F uture value of $100 = $100 (1 + r) t Suppose that you will receive a cash flow of C t dollars at the end of