Exercise Session #3 Suggested Solutions

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1 JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Date: 17/10/2017 Exercise Session #3 Suggested Solutions Problem 1. (6.20 Marsha Jones has bought a used Mercedes horse transporter for her Connecticut estate. It cost $35,000. The object is to save on horse transporter rentals. Marsha had been renting a transporter every other week for $200 per day plus $1 per mile. Most of the trips are 80 or 100 miles in total. Marsha usually gives the driver a $40 tip. With the new transporter she will only have to pay for diesel fuel and maintenance, at about $0.45 per mile. Insurance costs for Marsha s transporter are $1,200 per year. The transporter will probably be worth $15,000 (in real terms after eight years, when Marsha s horse Nike will be ready to retire. Is the transporter a positive NPV investment? Assume a nominal discount rate of 9% and a 3% forecasted inflation rate. Marsha s transporter is a personal outlay, not a business or financial investment, so taxes can be ignored. Solution: The following table shows the real cash flows: t = 0 t = 1 t = 7 t = 8 Investment $35, 000 $15,000 Savings $8,580 $8,580 $8,580 (= 26 ( ( /2 Insurance $1, 200 $1, 200 $1, 200 Fuel $1, 053 $1, 053 $1, 053 (= 26 ( 0.45 ( /2 Net Cash Flow $35, 000 $6,327 $6,327 $21,327 NPV (at 5.83% = $14,087.9 The NPV is computed using the real discount rate, which is found as follows: (1 + r nominal = (1 + r real (1 + inflation rate, 1.09 = (1 + r real 1.03 r real = or 5.83%. Problem 2. (6.24 As a result of improvements in product engineering, United Automation is able to sell one of its two milling machines. Both machines perform the same function but differ in age. The newer machine could be sold today for $50,000. Its operating costs are $20,000 a year, but in five years the machine will require a $20,000 overhaul. Thereafter operating costs will be $30,000 until the machine is finally sold in year 10 for $5,000. The older machine could be sold today for $25,000. If it is kept, it will need an immediate $20,000 overhaul. Thereafter operating costs will be $30,000 a year until the machine is finally sold in year 5 for $5,000. 1

Both machines are fully depreciated for tax purposes. The company pays tax at 35%. Cash flows have been forecasted in real terms. The real cost of capital is 12%.

2 Both machines are fully depreciated for tax purposes. The company pays tax at 35%. Cash flows have been forecasted in real terms. The real cost of capital is 12%. Which machine should United Automation sell? Explain the assumptions underlying your answer. Solution: In order to solve this problem, we calculate the equivalent annual cost for each of the two alternatives. (All cash flows are in thousands. Alternative 1 Sell the new machine: If we sell the new machine, we receive the cash flow from the sale, pay taxes on the gain, and pay the costs associated with keeping the old machine. The present value of this alternative is: P V 1 = = $93.8. The equivalent annual cost for the five year period is computed as follows: EAC 1 = P V 1 annuity factor, 5 time periods, 12% = $93.8 ( (1 ( /0.12 = $ Alternative 2 Sell the old machine: If we sell the old machine, we receive the cash flow from the sale, pay taxes on the gain, and pay the costs associated with keeping the new machine. The present value of this alternative is: P V 2 = = $ The equivalent annual cost for the ten year period is computed as follows: EAC 2 = P V 2 annuity factor, 10 time periods, 12% = $ ( (1 ( /0.12 = $ Thus, the least expensive alternative is to sell the old machine because this alternative has the lowest equivalent annual cost. One key assumption underlying this result is that, whenever the machines have to be replaced, the replacement will be a machine that is as efficient to operate as the new machine being replaced. Problem 3. (6.32 Look at the cash flows for machines A and B: The present values of purchase and operating costs are (over three years for A and 21 (over two years for B. The real discount rate is 6% and the inflation rate is 5%. (a Calculate the three and two year level nominal annuities which have present values of and 21. Explain why these annuities are not realistic estimates of equivalent annual costs. (Hint: In real life machinery rentals increase with inflation. 2

3 (b Suppose the inflation rate increases to 25%. The real interest rate stays at 6%. Recalculate the level nominal annuities. Note that the ranking of machines A and B appears to change. Why? Solution: (a With a real rate of 6% and an inflation rate of 5%, the nominal rate, r, is determined as follows: (1 + r = ( ( r = or 11.3%. 1 ( For a three year annuity at 11.3%, the annuity factor is: ( For a two year annuity, the annuity factor is: For a three year annuity with a present value of $28.37, the nominal annuity is: $ = $ For a two year annuity with a present value of $21 the nominal annuity is: $ = $ These nominal annuities are not realistic estimates of equivalent annual costs because the appropriate rental cost (i.e., the equivalent annual cost must take into account the effects of inflation. (b With a real rate of 6% and an inflation rate of 25%, the nominal rate, r, is determined as follows: (1 + r = ( ( r = or 32.5%. 1 ( For a three year annuity at 32.5%, the annuity factor is: ( For a two year annuity, the annuity factor is: For a three year annuity with a present value of $28.37, the nominal annuity is: $ = $ For a two year annuity with a present value of $21, the nominal annuity is: $ = $ With an inflation rate of 5%, Machine A has the lower nominal annual cost ($11.67 compared to $ With inflation at 25%, Machine B has the lower nominal annual cost ($15.86 compared to $ Thus, it is clear that inflation has a significant impact on the calculation of equivalent annual cost, and hence, the warning in the text to do these calculations in real terms. The rankings change because, at the higher inflation rate, the machine with the longer life (here, Machine A is affected more. Problem 4. (10.21 Magna Charter is a new corporation formed by Agnes Magna to provide an executive flying service for the southeastern United States. The founder thinks there will be a ready demand from businesses that cannot justify a full time company plane 3

4 but nevertheless need one from time to time. However, the venture is not a sure thing. There is a 40% chance that demand in the first year will be low. If it is low, there is a 60% chance that it will remain low in subsequent years. On the other hand, if the initial demand is high, there is an 80% chance that it will stay high. The immediate problem is to decide what kind of plane to buy. A turboprop costs $550,000. A piston engine plane costs only $250,000 but has less capacity. Moreover, the piston engine plane is an old design and likely to depreciate rapidly. Ms. Magna thinks that next year second hand piston aircraft will be available for only $150,000. The table below shows how the payoffs in years 1 and 2 from both planes depend on the pattern of demand. You can see, for example, that if demand is high in both years 1 and 2, the turbo will provide a payoff of $960,000 in year 2. If demand is high in year 1 but low in year 2, the turbo s payoff in the second year is only $220,000. Think of the payoffs in the second year as the cash flow that year plus the year 2 value of any subsequent cash flows. Also think of these cash flows as certainty equivalents, which can therefore be discounted at the risk free interest rate of 10%. Ms. Magna now has an idea: Why not start out with one piston plane. If demand is low in the first year, Magna Charter can sit tight with this one relatively inexpensive aircraft. On the other hand, if demand is high in the first year she can buy a second piston engine plane for only $150,000. In this case, if demand continues to be high, the payoff in year 2 from the two piston planes will be $800,000. However, if demand in year 2 were to decline, the payoff would be only $100,000. (a Draw a decision tree setting out Magna Charter s choices. (b If Magna Charter buys a piston plane, should it expand if demand turns out to be high in the first year? (c Given your answer to (b, would you recommend that Ms. Magna buys the turboprop or the piston engine plane today? (d What would be the NPV of an investment in a piston plane if there were no option to expand? How much extra value is contributed by the option to expand? Table 1: The possible payoffs from Ms. Magna s flying service. (All figures are in thousands. Probabilities are in parentheses. Payoffs from the Turboprop Year 1 demand High (0.6 Low (0.4 Year 1 payoff $150 $30 Year 2 demand High (0.8 Low (0.2 High (0.4 Low (0.6 Year 2 payoff $960 $220 $930 $140 Payoffs from the Piston Engine Year 1 demand High (0.6 Low (0.4 Year 1 payoff $100 $50 Year 2 demand High (0.8 Low (0.2 High (0.4 Low (0.6 Year 2 payoff $410 $180 $220 $100 Solution: (a Decision tree is the following: 4

5 (b Analyze the decision tree by working backwards. If we purchase the piston plane and demand is high: The NPV at t = 1 of the Expand branch is: The NPV at t = 1 of the Continue branch is: = $331. = $450. Thus, if we purchase the piston plane and demand is high, we should expand further at t = 1. This branch has the highest NPV. (c Continuing the analysis, if we purchase the piston plane and demand is low: The NPV of the Continue branch is: = $135. We can now use these results to calculate the NPV of the Piston Engine branch 5

6 at t = 0: ( ( = $117. Similarly for the Turboprop branch, if demand is high, the expected cash flow at t = 1 is: = $812. If demand is low, the expected cash flow is: = $456. So, for the Turboprop branch, the combined NPV is: NP V = = $96. Therefore, the company should buy the Piston Engine plane today. (d To determine the value of the option to expand, we first compute the NPV without the option to expand: NP V = ( ( = $52. 2 Therefore, the value of the option to expand is: $117 $52 = $

Homework #3 Suggested Solutions

JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Homework #3 Suggested Solutions Problem 1. (6.6) When appraising mutually exclusive investments in plant and equipment, financial