Dde 321 - Solutions Exercise 2

Advanced Corporate Finance

Leonidas Rompolis

EXERCISES -2 (SOLUTIONS) Chapter 6, Practice Questions 1. See the table below. We begin with the cash flows given in the text, Table 6.6, line 8, and utilize the following relationship: Real cash flow = nominal cash flow/(1 + inflation rate)t Here, the nominal rate is 20 percent, the expected inflation rate is 10 percent, and the real rate is given by the following: (1 + rnominal) 1.20 rreal

= (1 + rreal) × (1 + inflation rate) = (1 + rreal) × (1.10) = 0.0909 = 9.09%

As can be seen in the table, the NPV is unchanged (to within a rounding error).

Net Cash Flows Net Cash Flows (Real)

Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 -12,600 -1,484 2,947 6,323 10,534 9,985 5,757 3,269 -12,600 -1,349 2,436 4,751 7,195 6,200 3,250 1,678 NPV of Real Cash Flows (at 9.09%) = $3,804

13. 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: 30 30 30 30 30 PV1 = 50 -[0.35(50 - 0)] - 20 2 3 4 1.12 1.12 1.12 1.12 1.125 5 0.35 (5 - 0) + =-$93.80 5 1.12 1.125 The equivalent annual cost for the five-year period is computed as follows: PV1 = EAC1 × [annuity factor, 5 time periods, 12%] –93.80 = EAC1 × [3.605] EAC1 = –26.02, or an equivalent annual cost of $26,020 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: 20 20 20 20 20 PV2 = 25 - [0.35(25 - 0)] 2 3 4 1.12 1.12 1.12 1.12 1.125 20 30 30 30 30 30 5 6 7 8 9 1.12 1.12 1.12 1.12 1.12 1.1210

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Advanced Corporate Finance

Leonidas Rompolis

5 0 .35 (5 - 0) =-$127.51 10 1.12 1.1210 The equivalent annual cost for the ten-year period is computed as follows: PV2 = EAC2 × [annuity factor, 10 time periods, 12%] –127.51 = EAC2 × [5.650] EAC2 = –22.57, or an equivalent annual cost of $22,570 Thus, the least expensive alternative is to sell the old machine because this alternative has the lowest equivalent annual cost. +

15. The table presents the NFV for the five following years. The present values of these NFV are calculated by: NFVt NPVt = (1 + OCC) t For example, if the investment starts the first year the NPV1 would be: NFV1 1.64 NPV1 = = = 1.43 1 + OCC 1.14 when the OCC = 14% and NFV1 1.64 NPV1 = = = 1.36 1 + OCC 1.2 when the OCC = 20%. Following the same procedure for the other years we construct the following table:

OCC = 14% OCC = 20%

0 1 1

1 1.43 1.36

Year of investment 2 3 1.76 1.85 1.5902 1.5914

4 1.84 1.50

5 1.77 1.37

In both cases the optimal time to invest is year 3. 17. a. PVA = $66,730 (Note that this is a cost.) PVB = 50,000 +

8,000 8,000 8,000 8,000 + + + 1.06 1.062 1.063 1.064

PVB = $77,721 (Note that this is a cost.) Equivalent annual cost (EAC) is found by: PVA = EACA × [annuity factor, 6%, 3 time periods] 66,730 = EACA × 2.673 EACA = $24,964 per year rental PVB = EACB × [annuity factor, 6%, 4 time periods]

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Advanced Corporate Finance

Leonidas Rompolis

77,721 = EACB × 3.465 EACB = $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 percent per year. For example, for Machine A, rent for the first year would be $24,964; rent for the second year would be ($24,964 × 1.08) = $26,961; etc. 1. a. The following table presents the cash flows for capital budgeting: CI COUT CI – COUT (1 – τc) τcDep CF

1 90,000 75,000 15,000 9,000 3,200 12,200

2 93,600 79,500 14,100 8,460 3,200 11,660

3 97,344 84,270 13,074 7,844 3,200 11,044

4 101,238 89,263 11,912 7,147 3,200 10,347

5 105,287 94,686 10,601 6,361 3,200 9,561

The NPV is: 12, 200 11, 660 11, 044 10,347 9,561 + + + + = $4,172 1.08 1.082 1.083 1.084 1.085 and the project should be accepted. b. We must discount nominal cash flows to nominal rates. Therefore, we must calculate the nominal OCC. This is equal to: rnom = (1 + 0.08)(1 + 0.06) − 1 = 14.48% The NPV of the project is now: NPV = - $1,779, and the project should be rejected. NPV = −40, 000 +

2. The first step is to compute the NPVs of the projects: 1 ⎡ 1 ⎤ NPVA = −48, 000 + 20, 000 ⎢ − = $17, 485 5⎥ ⎣ 0.16 0.16 × 1.16 ⎦ Similarly, NPVB = $6,900, NPVC = $17,328 and NPVD = $19,750. The second step is to assume constant scale replication and to calculate: ⎡ (1 + k) N ⎤ NPV(N, ∞) = NPV(N) ⎢ ⎥ N ⎣ (1 + k) − 1 ⎦ The results are: NPVA (N, ∞) = $33,366 (best) NPVB (N, ∞) = $7, 735 (worst) NPVC (N, ∞) = $22, 407 NPVD (N, ∞) = $22,139 3

Advanced Corporate Finance

Leonidas Rompolis

3. a. The NPVs of the project are: NPVA = −10 +

6 6 + = 0.41 1.10 1.102

and NPVB = −10 +

6.55 6.55 6.55 + + = 0.41 1.4 1.42 1.43

b. ⎛ 1.102 ⎞ NPVA ( N, ∞ ) = 0.41⎜ ⎟ = 2.36 2 ⎝ 1.10 − 1 ⎠ ⎛ 1.403 ⎞ NPVB ( N, ∞ ) = 0.41⎜ ⎟ = 0.65 3 ⎝ 1.40 − 1 ⎠

c. The EAVs are

EAVA = 0.1⋅ NPVA ( N, ∞ ) = 0.236 EAVB = 0.4 ⋅ NPVB ( N, ∞ ) = 0.260

d. Project A should be accepted, following the NPV ( N, ∞ ) criterion. The EAV cannot be used, since the two projects have different risk.

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12, 000 = 3, 260 t t =1 1.12 and NPVB = 3,840, NPVC = 5,200 and NPVD = 6,725. Following the NPV rule the best project is D, while the worst is A. ⎡ (1 + k) N ⎤ b. NPVA (N, ∞) = NPVA ⎢ ⎥ =7,535 , where k = 12% N ⎣ (1 + k) − 1 ⎦

4. a. NPVA = −40, 000 + ∑

Similarly we have, NPVB (N, ∞) = 8,876 , NPVC (N, ∞) = 7, 669 and NPVD (N, ∞) = 9,918 . Project D should be selected. c. We start by computing the PVI for each project. These are, PVIA = 1.08, PVIB = 1.15, PVIC = 1.13 and PVID = 1.22. We have a budget constraint of $70,000. Therefore, the only combination that we cannot use is project A and C. Assume the projects A and B. Then we have that: 40, 000 25, 000 5, 000 PVI A,B = 1.08 + 1.15 + 1.0 = 1.099 70, 000 70, 000 70, 000 Similarly we calculate the PVI of all other combinations and we observe that the maximum is for C and D equal to PVIC,D = 1.169. Thus, projects C and D should be selected for investment.

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Advanced Corporate Finance

Leonidas Rompolis

5. We have that DVt ( Vt − C ) k 100, 000 (100, 000 ln t − 50, 000 ) 0.15 = ⇒ = dt 1 − e − kt t 1 − e−0.15t The solution is approximately 3.55 years. 6. The NPVA = 20.41, NPVB = -23.55 and NPVC = 15.86. The problem is set as follows: max ( 20.41X1 − 23.55X 2 + 15.86X3 ) subject to:

30X1 + 50X 2 + 80X 3 ≤ 120 30X1 + 20X 2 + 30X 3 ≤ 50 10X1 + 20X 2 + 50X 3 ≤ 50 0 ≤ X j ≤ 1, j = 1, 2,3

The solution is: X1 = 1, X2 = 0 and X3 = 0.667. The NPV of the selected combination of project is NPV = 30.99. We select project A, which has the maximum NPV, while we reject project B which has a negative NPV.

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