Week 2 Supplemental Problem Solutions

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Jan 9, 2024

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19. Excel Solution Xia Corporation is a company whose sole assets are $100,000 in cash and three projects that it will undertake. The projects are risk-free and have the following cash flows. Xia plans to invest any unused cash today at the risk-free interest rate of 10%. In one year, all cash will be paid to investors, and the company will be shut down. What is the NPV of each project? Which projects should Xia undertake, and how much cash should it retain? What is the total value of Xia’s assets (projects and cash) today? What cash flows will the investors in Xia receive? Based on these cash flows, what is the value of Xia today? Suppose Xia pays any unused cash to investors today, rather than investing it. What are the cash flows to the investors in this case? What is the value of Xia now? Explain the relationship in your answers to parts b, c, and d 3-19. a. A 30,000 NPV 20,000 $7, 272.73 1.1    B 25,000 NPV 10,000 $12,727.27 1.1    C 80,000 NPV 60,000 $12,727.27 1.1   

All projects have positive NPV, and Xia has enough cash, so Xia should take all of them. It would retain $10,000 in cash. Total value today = Cash + NPV(projects) = 100,000 + 7,272.73 + 12,727.27 + 12,727,27 = $132,727.27 b. After taking the projects, Xia will have 100,000 – 20,000 – 10,000 – 60,000 = $10,000 in cash left to invest at 10%. Thus, Xia’s cash flows in one year = 30,000 + 25,000 + 80,000 + 10,000 × 1.1 = $146,000. 146,000 Value of Xia today = = $132,727.27 1.1 This is the same as calculated in b. c. Unused cash = 100,000 – 20,000 – 10,000 – 60,000 = $10,000 Cash flows today = $10,000 Cash flows in one year = 30,000 + 25,000 + 80,000 = $135,000 135,000 Value of Xia today = 10,000 + = $132,727.27 1.1 d. Results from b, c, and d are the same because all methods value Xia’s assets today. Whether Xia pays out cash now or invests it at the risk-free rate, investors get the same value today. The point is that a firm cannot increase its value by doing what investors can do by themselves (and is the essence of the second separation principle). A 30,000 NPV 20,000 $7, 272.73 1.1    B 25,000 NPV 10,000 $12,727.27 1.1    C 80,000 NPV 60,000 $12,727.27 1.1    All projects have positive NPV, and Xia has enough cash, so Xia should take all of them. It would retain $10,000 in cash. Total value today = Cash + NPV(projects) = 100,000 + 7,272.73 + 12,727.27 + 12,727,27 = $132,727.27 After taking the projects, Xia will have 100,000 – 20,000 – 10,000 – 60,000 = $10,000 in cash left to invest at 10%. Thus, Xia’s cash flows in one year = 30,000 + 25,000 + 80,000 + 10,000 × 1.1 = $146,000. 146,000 Value of Xia today = = $132,727.27 1.1 This is the same as calculated in b. Unused cash = 100,000 – 20,000 – 10,000 – 60,000 = $10,000 Cash flows today = $10,000 Cash flows in one year = 30,000 + 25,000 + 80,000 = $135,000 135,000 Value of Xia today = 10,000 + = $132,727.27 1.1 e. Results from b, c, and d are the same because all methods value Xia’s assets today. Whether Xia pays out cash now or invests it at the risk-free rate, investors get the same value today. The point is that a firm cannot increase its value by doing what investors can do by themselves (and is the essence of the second separation principle).

3.20. 20. The table here shows the no-arbitrage prices of securities A and B. What are the payoffs of a portfolio of one share of security A and one share of security B? What is the market price of this portfolio? What expected return will you earn from holding this portfolio? What is the risk-free interest rate? a. A B  pays $600 in both cases (i.e., it is risk free). b. Market price 231 346 577    . Expected return is (600 577) 4.0% 577   . c. Since the combination of A and B gives a certain payoff of $600, this is a risk-free security. Thus its expected return calculated in (b) is the risk-free interest rate, so the risk-free interest rate is 4.0%.

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3.21. 21. Suppose security C has a payoff of $600 when the economy is weak and $1800 when the economy is strong. Suppose security D has a payoff of $1800 when the economy is weak and $600 when the economy is strong. Security C has the same payoffs as what portfolio of the securities A and B in Problem 20? Security D has the same payoffs as what portfolio of the securities A and B in Problem 20? What is the no-arbitrage price of security C? What is the no-arbitrage price of security D? What is the expected return of security C if both states are equally likely? What is its risk premium? (Refer to the answer to Problem 20c for the risk-free interest rate.) What is the expected return of security D if both states are equally likely? What is its risk premium? (Refer to the answer to Problem 20c for the risk-free interest rate.) What is the difference between the return of security C when the economy is strong and when it is weak? If security C had a risk premium of 10%, what arbitrage opportunity would be available? What is the difference between the return of security D when the economy is strong and when it is weak? If security D had a risk premium of 10%, what arbitrage opportunity would be available? a. C 3A B   b. D = 3B + A c. Price of C 3 231 346 1039     d. Price of D 3 346 231 1269     e. Expected payoff of C is 600 1800 1200 2 2   . Expected return  1200  1039 1039  15.5% . Risk premium 15.5 4 11.5%    . f. Expected payoff of D is 1800 600 1200 2 2   . Expected return 1200 1269 5.4% 1269    . Risk premium 5.4 4 9.4%    . g. Return when strong 1800 1039 73% 1039    , return when weak 600 1039 42% 1039    . Difference   73 42 115%     . h. Price of C given 10% risk premium = expected cash flow discounted using risk-adjusted rate 1200 $1053 1.14   .

Buy 3A B  for 1039, sell C for 1053, and earn a profit of 1053 1039 $14   . i. Return when strong 600 1269 53% 1269    , return when weak 1800 1269 42% 1269    . Difference 53 42 95%    j. Price of D given 10% risk premium = expected cash flow discounted using risk-adjusted rate 1200 $1053 1.14   . Buy D for 1053 and sell   3B A  for 1269, and earn a profit of 1269 1053 $216   . 3-24. *24. Suppose Hewlett-Packard (HPQ) stock is currently trading on the NYSE with a bid price of $28 and an ask price of $28.10. At the same time, a NASDAQ dealer posts a bid price for HPQ of $27.85 and an ask price of $27.95. Is there an arbitrage opportunity in this case? If so, how would you exploit it? Suppose the NASDAQ dealer revises the quotes to a bid price of $27.95 and an ask price of $28.05. Is there an arbitrage opportunity now? If so, how would you exploit it? What must be true of the highest bid price and the lowest ask price for no arbitrage opportunity to exist? a There is an arbitrage opportunity. One would buy from the NASDAQ dealer at $27.95 and sell to the NYSE dealer at $28.00, making a profit of $0.05 per share. b. There is no arbitrage opportunity. c. To eliminate any arbitrage opportunity, the highest bid price should be lower than or equal to the lowest ask price. 4-7. 7. Suppose you invest $1000 in an account paying 8% interest per year. What is the balance in the account after three years? How much of this balance corresponds to “interest on interest”? What is the balance in the account after 25 years? How much of this balance corresponds to interest on interest? a. The balance after three years is $1259.71; interest on interest is $19.71. b. The balance after 25 years is $6848.48; interest on interest is $3848.38. 4-8. 8. Your daughter is currently eight years old. You anticipate that she will be going to university in 10 years. You would like to have $100,000 in a registered education savings plan (RESP) to fund her education at that time. If the account promises to pay a fixed interest rate of 3% per year, how much money do you need

to put into the account today (ignoring government grants) to ensure that you will have $100,000 in 10 years? Timeline: 0 1 2 3 10 PV=? 100,000 10 100, 000 PV= 74, 409.39 1.03  4-27. 27. Excel Solution Your oldest daughter is about to start kindergarten at a private school. Tuition is $10,000 per year, payable at the beginning of the school year. You expect to keep your daughter in private school through high school. You expect tuition to increase at a rate of 5% per year over the 13 years of her schooling. What is the present value of the tuition payments if the interest rate is 5% per year? How much would you need to have in the bank now to fund all 13 years of tuition? Timeline: 0 1 2 3 12 13 10,000 10,000(1.05) 10,000(1.05) 2 10,000(1.05) 3 10,000(1.05) 12 0 This problem consists of two parts: today’s tuition payment of $10,000 and a 12-year growing annuity with first payment of 10,000(1.05). However we cannot use the growing annuity formula because in this case r = g. We can just calculate the present values of the payments and add them up:                 2 3 12 GA 2 3 12 10, 000 1.05 10,000 1.05 10, 000 1.05 10, 000 1.05 PV 1.05 1.05 1.05 1.05 10, 000 10,000 10, 000 10, 000 10, 000 12 120, 000                Adding the initial tuition payment gives 120, 000 10,000 130, 000  

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