Business 2019 Assignment 3 Suggested Answers Each problem is worth 5 marks. 1. A firm has just paid the moment before valuation a dividend of 55 cents and is expected to exhibit a growth rate of 10% into the indefinite future. If the appropriate discount rate is 14%, what is the value of the stock? Answer: The next dividend, D 1, is 1.1 0.55 = 0.605, and thus the value of the stock is P 0 = 0.605.14.10 = $15.13. 2. The analyst who supplied you with the information in Problem 1 has just revised her forecast. She now realizes that the growth rate of 10% can continue for only five years, after which the company will have a long-term growth rate of 6%. Furthermore, at the end of the five years, she expects the company s dividend to increase from its present 55 cents per share up to 75 cents per share. What value would you assign to the stock of this company? Answer: For each year t, the dividend paid is.551.1 t if t 5, D t =.751.06 t 6 if t 6. Since the dividend grows at a constant rate from time t = 6 on, we can calculate P 5 using the constant dividend growth equation, i.e. P 5 = D 6.14.06 =.75.08 = 9.375. 1
Thus the value of the stock is P 0 = = 5 t=1 D t 1.14 t + P 5 1.14 5 5.605 1.1 1.14.1 1.14 + 9.375 1.14 5 = 7.34. 3. Assume that the forecast for the company in Problem 1 was such that at the end of the fifth year its growth rate was to decline linearly for four years 1% per year to reach the steady-state 6% growth rate. Assume also that the dividend that will be paid at the end of the ninth year is 75 cents. What is the value of this stock? Answer: In this case, we have.551.1 t if t 5,.551.091.1 5 if t = 6, D t =.551.081.091.1 5 if t = 7,.551.071.081.091.1 5 if t = 8,.751.06 t 9 if t 9, and thus P 0 = where This gives us 5 t=1 D t 1.14 t + D 6 1.14 6 + D 7 1.14 7 + D 8 1.14 8 + P 8 1.14 8, P 8 = D 9.14.06 =.75.08 = 9.375. P 0 = 2.47 +.44 +.42 +.39 + 3.29 = 7.01. 4. Consider a bond with semiannual coupon payments of $50, a principal payment of $1,000 in 5 years, and a price of $1,000. Assume that the yield curve is a flat 10%. 2
What is the duration of the bond? Answer: Let i denote the annual coupon rate, let F denote the face amount, let T denote the number of years until maturity, let m denote the number of coupon payments per year, let y denote the yield to maturity and let P denote the bond s price. A bond makes T m payments, where we represent the order of a payment by n i.e. n = 1 for the first payment, n = 2 for the second payment, etc.. Duration, D, is then given by D = 1 P if/m 1 + y m 1 m + if/m 1 + y 2 2 m +... m... + if/m 1 + y n n m +... + if/m + F m 1 + y T m T m m m. In the present case, we have if m gives us D = = 50, m = 2, T = 5, y = 10% and P = 1, 000. This 1 50 1, 000 1.05.5 + 50 1.05 1 + 50 1, 050 1.5 +... + 2 1.05 3 1.05 5 10 = 4.05 years. 5. Consider a bond with annual coupon payments of $100, a principal payment of $1,000 in 10 years, and a cost of $1,000. Assume a flat yield curve with a 10% yield to maturity. What is the duration of the bond? If the yield curve remains unchanged, what is the bond s duration in three years? In five years? In eight years? Answer: This bond is always priced at par since the annual coupon rate, 10%, is always equal to the annualized yield. This makes annual payments, so its duration as time 0 is D = 1 100 1, 000 1.1 1 + 100 1.1 2 + 100 2 1.1 = 6.76 years. 1, 100 3 +... + 3 1.1 10 10 3
More generally, the duration of this bond at time t is given by 10 t 1 100 D t = 1, 000 1.1 n + 1000 10 t. n 1.1 10 t n=1 That is, the duration of this 10-year bond after 3 years is in fact the duration of a 7-year bond with the same features. Hence the duration of a 10-year bond after 5 years is equal to the duration of a 5-year bond with the same features and the duration of a 10-year bond after 8 years is the duration of a 2-year bond with the same features. Therefore, D 3 = 1 100 1, 000 1.1 1 + 100 1, 100 2 +... + 1.1 2 1.1 7 7 = 5.36 years, D 5 = 1 100 1, 000 1.1 1 + 100 1, 100 2 +... + 1.1 2 1.1 5 5 = 4.17 years, D 8 = 1 100 1, 100 1 + 1, 000 1.1 1.1 2 2 = 1.91 years. 6. Your company has an outstanding perpetual bond issue with a face value of $50 million and a coupon rate of 9 percent. the bonds are callable at par plus a $150 call premium per bond; in addition, any new bond issues of you firm will incur fixed costs of $9 million. The bonds must be called now or never. What would the current interest rate have to be for you to be indifferent to a refunding operation? Answer: Assuming these are $1,000 bonds, this bond issue consists of 50,000 bonds. The total cost of calling the bonds is then 50, 000 150 + 9, 000, 000 = $16, 500, 000. 4
Calling the bonds, on the other hand, would save money to your company in coupon payments if the market interest rate is lower than the coupon rate on the bonds. Let r denote the market interest rate. If the bonds are called, new perpetual bonds will be issued with a coupon rate equal to the market interest rate and thus the present value of the coupon payments would be $50, 000, 000r/r. If the bonds are not called, the coupon rate of 9% will prevail forever, which means a present value of coupon payments of $50, 000, 000.09/r. The present value of the savings from calling the bonds is then 50, 000, 000.09 r r = 4, 500, 000 r 50, 000, 000. If the benefits outweigh the costs, then it is worth calling the bonds. That, the bonds will be called if 4, 500, 000 r 50, 000, 000 > 16, 500, 000 r < 6.77%. 7. An investment under consideration has a payback of six years and a cost of $200,000. If the required return is 12 percent, what is the worst-case NPV? The best-case NPV? Explain. Answer: Since the project has a payback of six years, we know that i the sum of the cash flows from the project is at least $200,000, and ii the last penny of the first $200,000 coming from this project will be received in six years. Note, however, that the NPV calculation takes into account cash flows occurring once the project is paid back, which can be arbitrarily large. Thus the best-case NPV is infinite. The worst-case NPV, on the other hand, results from i the project having a sum of cash flows equal to exactly $200,000, and ii the whole $200,000 is received as late as possible within these six years, that is after six years exactly. This gives a NPV of 200, 000 + 200, 000 1.12 6 = $98, 673.78. 5
8. Fundamentals of Corporate Finance, Chapter 10, Problem 47 Project Evaluation. Pavarotti-in-You PIY, Inc., projects unit sales for a new opera tenor emulation implant as follows: Year Unit Sales 1 100,000 2 105,000 3 110,000 4 114,000 5 75,000 Production of the implants will require $600,000 in net working capital to start and additional net working capital investments each year equal to 40 percent of the projected sales increase for the following year. Because sales are expected to fall in Year 5, there is no NWC cash flow occurring for Year 4. Total fixed costs are $200,000 per year, variable production costs are $200 per unit, and the units are priced at $325 each. The equipment needed to begin production has an installed cost of $13,250,000 and falls into class 8 for tax purposes. In five years, this equipment can be sold for about 30 percent of its acquisition cost. PIY is in the 35 percent marginal tax bracket and has a required return on all its projects of 30 percent. Based on these preliminary project estimates, what is the NPV of the project? What is the IRR? Answer: Let s start with the present value of the CCA tax shield CCATS, assuming that the class 8 asset pool is continued once the project is over. The depreciation rate for tax purposes being 20% for assets in class 8, we have PVCCATS = 13, 250, 000.2.35.2 +.3 1.15 1.3.3 13, 250, 000.2.35.2 +.31.3 5 = 1, 640, 962 149, 882 = 1, 491, 080. 6
Net capital spending is $13,250,000 in Year 0 and $3, 975, 000 at the end of Year 5, for an overall present value of 13, 250, 000 3, 975, 000 1.3 5 = $12, 179, 417. Let s now look at NWC. The latter increases by $600,000 at time 0, by.4 5, 000 325 = $650, 000 in Year 1, by.4 5, 000 325 = $650, 000 in Year 2, by.4 4, 000 325 = $520, 000 in Year 3, and by $0 in Year 4. The NWC recovered at the end of Year 5 is then $2,420,000. The overall present value of additions to NWC is then 600, 000 + 650, 000 1.3 + 650, 000 1.3 2 + 520, 000 1.3 3 2, 420, 000 1.3 5 = $1, 069, 525. The after-tax cash flow from operations is.65325 200 100, 000 200, 000 = 7, 995, 000 in Year 1,.65325 200 105, 000 200, 000 = 8, 401, 250 in Year 2,.65325 200 110, 000 200, 000 = 8, 807, 500 in Year 3,.65325 200 114, 000 200, 000 = 9, 132, 500 in Year 4,.65325 200 75, 000 200, 000 = 5, 963, 750 in Year 5, for an overall present value of 7, 995, 000 1.3 + 8, 401, 250 8, 807, 500 9, 132, 500 5, 963, 750 + + + = $19, 933, 783. 1.3 2 1.3 3 1.3 4 1.3 5 The net present value of the project is then 19, 933, 783 + 1, 491, 080 12, 179, 417 1, 069, 525 = $8, 175, 921. 7