Homework #3 Suggested Solutions

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1 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 managers calculate the investments equivalent annual costs and rank the investments on this basis. Why is this necessary? Why not just compare the investments NPVs? Explain briefly. Comparing present values can be misleading when projects have different economic lives and the projects are part of an ongoing business. For example, a machine that costs $100,000 per year to buy and lasts 5 years is not necessarily more expensive than a machine that costs $75,000 per year to buy but lasts only 3 years. Calculating the machines equivalent annual costs allows an unbiased comparison. Problem 2. (6.7) Air conditioning for a college dormitory will cost $1.5 million to install and $200,000 per year to operate. The system should last 25 years. The real cost of capital is 5%, and the college pays no taxes. What is the equivalent annual cost? P V (cost) = 1.5 million + 200, Equivalent annual cost = ( 1 1 ) = $4.319 million $4.319 million 1/0.05 (1 1 / ) = $306, 000. Problem 3. (6.8) Machines A and B are mutually exclusive and are expected to produce the following real cash flows: The real opportunity cost of capital is 10%. (a) Calculate the NPV of each machine. (b) Calculate the equivalent annual cash flow from each machine. (c) Which machine should you buy? 1

$110, 000 (a) NP V A = $100, 000 + NP V B = $120, 000 + $110, 000 (b) Equivalent cash flow of A = Equivalent cash flow of B = (c) Machine B.

2 $110, 000 (a) NP V A = $100, NP V B = $120, $110, 000 (b) Equivalent cash flow of A = Equivalent cash flow of B = (c) Machine B. $121, = $100, 000; 2 $121, $100, 000 = $57, 604; 1/0.1 (1 1 / 2 ) $180, 000 1/0.1 (1 1 / 3 ) $133, = $180, 000. = $72, 376. Problem 4. (6.9) Machine C was purchased five years ago for $200,000 and produces an annual real cash flow of $80,000. It has no salvage value but is expected to last another five years. The company can replace machine C with machine B (see the previous problem) either now or at the end of five years. Which should it do? Replace at end of 5 years as $80, 000 (Equivalent cash f low of C) > $72, 376 (Equivalent cash flow of B). Problem 5. (6.25) Low energy lightbulbs cost $3.5, have a life of nine years, and use about $1.6 of electricity a year. Conventional lightbulbs cost only $0.5, but last only about a year and use about $6.6 of energy. If the real discount rate is 5%, what is the equivalent annual cost of the two products? Assuming that the light bulb purchases occur at year 0 (for use during the following year or years), the cost structure and PV of each option is Year PV at 5% Low Energy $3.5 $1.6 $1.6 $1.6 $1.6 $1.6 $1.6 $1.6 $1.6 $1.6 $14.87 Conventional $0.5 $6.6 $6.79 The equivalent annual cost for the low energy bulb is computed as follows: EAC LE = P V LE annuity factor, 9 time periods, 5% = $14.87 ( (1 (1+0.05) 9 )/0.05) = $2.09, which is much cheaper than the $6.79 cost of using a conventional light bulb for the year. Problem 6. (6.28) The Borstal Company has to choose between two machines that do the same job but have different lives. The two machines have the following costs: 2

3 These costs are expressed in real terms. (a) Suppose you are Borstal s financial manager. If you had to buy one or the other machine and rent it to the production manager for that machine s economic life, what annual rental payment would you have to charge? Assume a 6% real discount rate and ignore taxes. (b) Which machine should Borstal buy? (c) Usually the rental payments you derived in part (a) are just hypothetical a way of calculating and interpreting equivalent annual cost. Suppose you actually do buy one of the machines and rent it to the production manager. How much would you actually have to charge in each future year if there is steady 8% per year inflation? (Note: The rental payments calculated in part (a) are real cash flows. You would have to mark up those payments to cover inflation.) 10, , , 000 (a) P V A = 40, = $66, 730 (note that this is a cost) , 000 8, 000 8, 000 8, 000 P V B = 50, = $77, 721 (note that this is a cost) Equivalent annual cost (EAC) is found by: EAC A = EAC B = P V A annuity factor, 6%, 3 time periods = $66, 730 ( (1 (1+0.06) 3 )/0.06) P V B annuity factor, 6%, 4 time periods = $77, 271 ( (1 (1+0.06) 4 )/0.06) = $24, 964 per year rental. = $22, 430 per year rental. (b) Annual rental is $24,964 for Machine A and $22,430 for Machine B. Borstal should buy Machine B. (c) The payments would increase by 8% per year. For example, for Machine A, rent for the first year would be $24,964; rent for the second year would be $24, = $26, 961; etc. Problem 7. (6.29) Look again at your calculations for the previous problem. Suppose that technological change is expected to reduce costs by 10% per year. There will be new machines in year 1 that cost 10% less to buy and operate than A and B. In year 2 there will be a second crop of new machines incorporating a further 10% reduction, and so on. How does this change the equivalent annual costs of machines A and B? Because the cost of a new machine now decreases by 10% per year, the rent on such a machine also decreases by 10% per year. Therefore: P V A = 40, a cost). 10, , P V B = 50, that this is a cost). + 10, , = $61, 820 (note that this is , , , = $71, 614 (note

4 Equivalent annual cost (EAC) is found as follows: EAC A = EAC B = P V A annuity factor, 6%, 3 time periods = $61, 820 ( (1 (1+0.06) 3 )/0.06) P V B annuity factor, 6%, 4 time periods = $71, 614 ( (1 (1+0.06) 4 )/0.06) = $23, 128, a reduction of 7.35%. = $20, 668, a reduction of 7.86%. Problem 8. (10.3) Define and briefly explain each of the following terms or procedures: (a) Sensitivity analysis (b) Scenario analysis (c) Break even analysis (d) Monte Carlo simulation (e) Decision tree (f) Real option (g) Abandonment value (h) Expansion value (a) Analysis of how project profitability and NPV change if different assumptions are made about sales, cost, and other key variables. (b) Project NPV is recalculated by changing several inputs to new, but consistent, values. (c) Determines the level of future sales at which project profitability or NPV equals zero. (d) An extension of sensitivity analysis that explores all possible outcomes and weights each by its probability. (e) A graphical technique for displaying possible future events and decisions taken in response to those events. (f) Option to modify a project at a future date. (g) The additional present value created by the option to bail out of a project, and recover part of the initial investment, if the project performs poorly. (h) The additional present value created by the option to invest more and expand output, if a project performs well. Problem 9. (10.9) Look at the cash flows for projects F and G: Cash Flows ($) Project C 0 C 1 C 2 C 3 C 4 C 5 Etc. F 9, ,000 +5,000 +4, G 9, ,800 +1,800 +1,800 +1,800 +1, The cost of capital is assumed to be 10%. Assume that the forecasted cash flows for projects of this type are overstated by 8% on average. That is, the forecast for each cash flow from each project should be reduced by 8%. But a lazy financial manager, unwilling to take the time to argue with the projects sponsors, instructs them to use a discount rate of 18%. (a) What are the projects true NPVs? 4

5 (b) What are the NPVs at the 18% discount rate? (c) Are there any circumstances in which the 18% discount rate would give the correct NPVs? (Hint: Could upward bias be more severe for more distant cash flows?) $6, 000 (1 0.08) $5, 000 (1 0.08) $4, 000 (1 0.08) (a) NP V F = $9, = 2 3 $2, $1, 800 (1 0.08) NP V G = $9, = $7, $6, 000 $5, 000 $4, 000 (b) NP V F = $9, = $2, $1, 800 NP V G = $9, = $1, (c) The 18% discount rate would give an approximation to the correct NP V s for projects with all (or most) of the inflows in the first year. The present value of $1 to be received one year from now, discounted at 18% is: $ The present value of $1 (1 0.08) (that is, $0.92) to be received one year from now, discounted at 10% is: $ The former calculation overstates the correct answer by approximately 1%. However, for cash flows five or ten years in to the future, discounting by 18% understates the correct present value by approximately 13% (= 0.92 / 5 1 /8 5 ) and 16% (= 0.92 / 10 1 /8 10 ), respectively. The error increases substantially because the incorrect factor (i.e., 8) is compounded, causing the denominator of the present value calculation to be greatly overstated so that the present value is greatly understated. 5

Exercise Session #3 Suggested Solutions

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