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**Unformatted text preview: **Finance 100: Problem Set 5 Alternative Solutions Note: Where appropriate, the “final answer” for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas This section contains the formulas that you will need for this homework set: 1. Present Value of an Annuity Formula: A = a (1 + i ) + a (1 + i ) 2 + ... + a (1 + i ) N- 1 + a (1 + i ) N = a · 1- (1 + i )- N i (1) where a is the amount of the annuity payment, i is the periodic interest rate and N is the total number of compounding periods. 2. Present Value of a Perpetuity Formula: A = a (1 + i ) + a (1 + i ) 2 + a (1 + i ) 3 + ... (2) = a i (3) 1 II. Problems 1. 1.a The present value of the cash flows (in $thousands) to both machines for one operating cycle are: NPV Machine A =- 80 + 50 1 . 1 + 50 (1 . 1) 2 + 50 (1 . 1) 3 + 25 (1 . 1) 4 = 61 . 42 NPV Machine B =- 100 + 60 1 . 1 + 60 (1 . 1) 2 + 60 (1 . 1) 3 = 49 . 21 In order to compare these investments, we must now compute their equivalent annuity payments. From the annuity formula (equation (1)) we have a = PV × i 1- (1 + i )- N . For machine A, a = 61 . 42 × . 10 1- (1 + 0 . 10)- 4 = 19 . 376 . For machine B a = 49 . 21 × . 10 1- (1 + 0 . 10)- 3 = 19 . 788 . Therefore, machine B is the better choice . 1.b Using the results above, we can consider the value of each machine used in perpetuity. NPV MachineA = 19 . 376 . 1 = 193 . 76 . NPV MachineB = 19 . 788 . 1 = 197 . 88 . Thus, over the entire lifetime, machine B will provide the company with $197 . 88- $193 . 76 = $4 . 12 additional thousands of dollars. The up front cost of $10,000 for machine B negates this advantage, implying that we should choose machine A under this scenario . 2 1.c They should replace the existing machine as soon as it’s annual cash flows fall below the annual equivalent of its replacement. This occurs in year 3. 1.d Comparing one-year annual equivalent cash flows produces the correct an- swer regardless of the time-horizon. Thus, without the re-tooling fee of $10,000 on machine B, we should choose machine B . The aggregate NPV of both machines is now (using the perpetuity formula in equation (3)): NPV MachineA = 19 . 376 × 1- (1 + 0 . 10)- 24 ) . 1 = 174 . 09 NPV MachineB = 19 . 788 × 1- (1 + 0 . 10)- 24 ) . 1 = 177 . 79 Therefore, with the re-tooling fee for machine B, we should choose machine A . 2. The first step is to determine the cash flows and their timing, which are outlined in Table 1 Table 1: Cash Flows ($) Year 1 2 ... 8 EBITDA 11,000 11,000 ... 11,000 Depreciation-6,000-6,000 ...-6,000 EBIT 5,000 5,000 ... 5,000 Tax-1,700-1,700 ...-1,700 Salvage Value ... 12,000 Cost-60,000 ... Total-60,000 9,300 9,300 ... 21,300 Depreciation is computed using a straight-line (interpolated) method, which amounts to computing the average value loss per year ((60 , 000- 12 , 000) / 8 = 6 , 000). The tax figures are computed assuming a 34% mar- ginal corporate tax rate (5 , 000 × . 34 = 1 , 700). The final cash flows are 3 obtained by subtracting the tax from the EBITDA (depreciation is not a...

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