Solutions to Chapter 6



Solutions to Chapter 7

NPV and Other Investment Criteria

5. No. Even though project B has the higher IRR, its NPV is lower than that of project A when the discount rate is lower (as in Problem 1) and higher when the discount rate is higher (as in Problem 3). This example shows that the project with the higher IRR is not necessarily better. The IRR of each project is fixed, but as the discount rate increases, project B becomes relatively more attractive compared to project A. This is because B’s cash flows come earlier, so their present values fall less rapidly when the discount rate increases.

6. The profitability indexes are as follows:

Project A 24.10/100 = .2410

Project B 22.19/100 = .2219

In this case, with equal initial investments, both the profitability index and NPV will give projects the same ranking. This is an unusual case, however, since it is rare for initial investments to be equal.

7. Project A has a payback period of 100/40 = 2.5 years. Project B has a payback period of 2 years.

8. Project A

|Year |Cash Flow ($) |Discounted Cash Flow |Cumulative Discounted Cash Flow ($) |

| | |($) @ 11 percent | |

|0 |-100 |-100 |-100 |

|1 |40 |36.04 |-63.96 |

|2 |40 |32.48 |-31.48 |

|3 |40 |29.24 |-2.24 |

|4 |40 |26.36 |+24.12 |

| |NPV= |24.12 | |

Assuming uniform cash flows across time, the fractional year can now be determined. Since the discounted cash flows are negative until year 3 and become positive by Year 4, the project pays back sometime in the fourth year. Note that out of the total discounted cash flow of $26.36 in Year 4, the first $2.24 comes in by 2.24/26.36 = 0.084 year. Therefore, the discounted payback period for Project A is 3.084 years.

Project B

|Year |Cash Flow ($) |Discounted Cash Flow |Cumulative Discounted Cash Flow ($) |

| | |($) @ 11 percent | |

|0 |-100 |-100 |-100 |

|1 |50 |45.05 |-54.95 |

|2 |50 |40.60 |-14.35 |

|3 |50 |36.55 |+22.20 |

| |NPV= |22.20 | |

The discounted payback for Project B is 2 years + 14.35/36.55 = 2.39 years.

11. NPV = (3,000 + 800 ( annuity factor(10%, 6 years) = $484.21

At this discount rate, you should accept the project.

You can solve for IRR by setting the PV of cash flows equal to 3,000 on your calculator and solving for the interest rate: PV = (3000; n = 6; FV = 0; PMT = 800; compute i. The IRR is 15.34%, which is the highest discount rate before project NPV turns negative.

14. Project at 2 percent discount rate

|Year |Cash Flow ($) |Discounted Cash Flow |Cumulative Discounted Cash Flow ($) |

| | |($) @ 2 percent | |

|0 |-3000 |-3000 |-3000 |

|1 |800 |784 |-2216 |

|2 |800 |768.8 |-1447.2 |

|3 |800 |753.6 |-693.6 |

|4 |800 |739.2 |45.6 |

|5 |800 |724.8 |770.4 |

|6 |800 |710.4 |1480.8 |

| |NPV= |1480.8 | |

Since the discounted cash flows become positive by Year 4, the project pays back sometime in the fourth year. Note that out of the total discounted cash flow of $739.20 in Year 4, the first $693.60 comes in by 693.60/739.20 = 0.94 year. Therefore, the discounted payback for the project is 3.94 years, and thus the project should be pursued.

Project at 12 percent discount rate

|Year |Cash Flow ($) |Discounted Cash Flow |Cumulative Discounted Cash Flow ($) |

| | |($) @ 12 percent | |

|0 |-3000 |-3000 |-3000 |

|1 |800 |714.4 |-2285.6 |

|2 |800 |637.6 |-1648.0 |

|3 |800 |569.6 |-1078.4 |

|4 |800 |508.8 |-569.6 |

|5 |800 |453.6 |-116.0 |

|6 |800 |405.6 |289.6 |

| |NPV= |289.6 | |

Since the discounted cash flows become positive by Year 6, the project pays back in 5 years + 116/405.6 = 5.28 years. Therefore, given the firm’s decision criteria of a discounted payback of 5 years or less, the project should not be pursued.

As illustrated by the two scenarios above, the firm’s decision will change as the discount rate changes. As the discount rate increases, the discounted payback period gets extended.

19. NPV = 10,000 + [pic]+ [pic]= $-2,029.08

which is negative. So the project is not attractive.

However, you can note that the IRR of the project is 37.03 %. Since the IRR of the project is greater than the required rate of return of 12%, the project should be accepted using this rule. On balance, we would use the NPV rule and reject the project.

20. NPV9% = –20,000 + 4,000 ( annuity factor(9%, 8 periods)

= $2139.28

NPV14% = –20,000 + 4,000 ( annuity factor(14%, 8 periods)

= –$1,444.54

The IRR is 11.81%. To confirm this on your calculator, set PV = (()20,000; PMT = 4000; FV = 0; n = 8, and compute i. The project will be rejected for any discount rate above this rate.

21. a. The present value of the savings is 100/r

If r = .08, PV = 1,250 and NPV = –1,000 + 1,250 = $250

If r = .10, PV = 1,000 and NPV = –1,000 + 1,000 = $0

b. IRR = .10 or 10%. At this discount rate, NPV = $0.

c. Payback = 10 years.

d. Discounted Payback

|Year |Cash Flow ($) |Discounted Cash Flow |Cumulative Discounted Cash Flow ($) |

| | |($) @ 8 percent | |

|0 |-1000 |-1000 |-1000 |

|1 |100 |92.6 |-907.4 |

|2 |100 |85.7 |-821.7 |

|3 |100 |79.4 |-742.3 |

|4 |100 |73.5 |-668.8 |

|5 |100 |68.1 |-600.7 |

|6 |100 |63.0 |-537.7 |

|7 |100 |58.3 |-479.4 |

|8 |100 |54.0 |-425.4 |

|9 |100 |50.0 |-375.4 |

|10 |100 |46.3 |-329.1 |

|11 |100 |42.9 |-286.2 |

|12 |100 |39.7 |-246.5 |

|13 |100 |36.8 |-209.7 |

|14 |100 |34.0 |-175.7 |

|15 |100 |31.5 |-144.2 |

|16 |100 |29.2 |-115.0 |

|17 |100 |27.0 |-88.0 |

|18 |100 |25.0 |-63.0 |

|19 |100 |23.2 |-39.8 |

|20 |100 |21.5 |-18.3 |

|21 |100 |19.9 |1.6 |

Discounted payback when cost of capital is 8 percent = 20 years +18.3/19.9 = 20.95 years.

The NPV=0 when the cost of capital =10%. The savings are supposed to last forever. Therefore, there is no finite discounted payback period when cost of capital is 10%.

25. a. Project A

|Year |Cash Flow ($) |Discounted Cash Flow |Cumulative Discounted Cash Flow ($) |

| | |($) @ 10 percent | |

|0 |-5000 |-5000.00 |-5000.00 |

|1 |1000 |909.09 |-4090.91 |

|2 |1000 |826.45 |-3264.46 |

|3 |3000 |2253.94 |-1010.52 |

|4 |0.00 |0.00 |-1010.52 |

| |NPV= |-1010.52 | |

The payback period for Project A is 3 years.

Project A does not pay back on a discounted basis since cumulative discounted cash flows remain negative until the end of Year 4.

Project B

|Year |Cash Flow ($) |Discounted Cash Flow |Cumulative Discounted Cash Flow ($) |

| | |($) @ 10 percent | |

|0 |-1000 |-1000.00 |-1000 |

|1 |0 |0 |-1000 |

|2 |1000 |826.45 |-173.55 |

|3 |2000 |1502.63 |1329.08 |

|4 |3000 |2049.04 |3378.12 |

| |NPV= |3378.12 | |

The payback period for Project B is 2 years.

The discounted payback period for Project B is 2 years + 173.55/1502.63 = 2.12 years.

Project C

|Year |Cash Flow ($) |Discounted Cash Flow |Cumulative Discounted Cash Flow ($) |

| | |($) @ 10 percent | |

|0 |-5000 |-5000.00 |-5000.00 |

|1 |1000 |909.09 |-4090.91 |

|2 |1000 |826.45 |-3264.46 |

|3 |3000 |2253.94 |-1010.52 |

|4 |5000 |3415.07 |2404.55 |

| |NPV= |2404.55 | |

The payback period for Project C is 3 years.

The discounted payback period for Project C is 3 years + 1010.52/3415.07 = 3.3 years.

b. Only B satisfies the 2-year payback criterion.

c. You would accept Project B

d. Projects B and C

|Project |NPV |

|A |-1010.52 |

|B |3378.12 |

|C |2404.55 |

e. False. Payback gives no weight to cash flows after the cutoff date.

29. a. The present values of the project cash flows (net of the initial investments) are:

NPVA = –2100 + + = $400

NPVB = –2100 + + = $300

The initial investment for each project is 2100.

Profitability index (A) = 400/2100 = 0.1905

Profitability index (B) = 300/2100 = 0.1429

b. If you can choose only one project, choose A for its higher profitability index. If you can take both projects, you should: Both have positive profitability index.

30. a. The less–risky projects should have lower discount rates.

b. First, find the profitability index of each project.

PV of Profitability

Project Cash flow Investment NPV Index

A 3.79 3 0.79 0.26

B 4.97 4 0.97 0.24

C 6.62 5 1.62 0.32

D 3.87 3 0.87 0.29

E 4.11 3 1.11 0.37

Then select projects with the highest profitability index until the $8 million budget is exhausted. Choose, therefore, projects E and C.

c. All the projects have positive NPV. All will be chosen if there is no rationing.

33. a. Assuming an opportunity cost of capital of 6%

(i) Payback Period

|Project |Cash Flows, Dollars |Payback Period, |

| | |years |

| |C0 |C1 |C2 |C3 |C4 |C5 | |

|I |-250000 |12000 |18000 |18000 |30000 |250000 |5.69 |

|II |-25000 |15000 |8000 |6000 |6000 |500 |3.33 |

Project II provides the lowest payback period, and thus is the project of choice under this decision criterion.

(ii) Discounted Payback Period

Project I

|Year |Cash Flow ($) |Discounted Cash Flow |Cumulative Discounted Cash Flow ($) |

| | |($) @ 6 percent | |

|0 |-250000.00 |-250000.00 |-250000.00 |

|1 |12000.00 |11320.75 |-238679.25 |

|2 |18000.00 |16019.94 |-222659.31 |

|3 |18000.00 |15113.15 |-207546.16 |

|4 |30000.00 |23762.81 |-183783.35 |

|5 |250000.00 |186814.54 |3031.19 |

| |NPV= |3031.19 | |

Discounted payback period is 4 years + 183783.35/186814.54 = 4.98 years.

Project II

|Year |Cash Flow ($) |Discounted Cash Flow |Cumulative Discounted Cash Flow ($) |

| | |($) @ 6 percent | |

|0 |-25000.00 |-25000.00 |-25000.00 |

|1 |15000.00 |14150.94 |-10849.06 |

|2 |8000.00 |7119.97 |-3729.09 |

|3 |6000.00 |5037.72 |1308.63 |

|4 |6000.00 |4752.56 |6061.19 |

|5 |500.00 |373.63 |6434.82 |

| |NPV= |6434.82 | |

Discounted payback period is 2 years + 3729.09/5037.72 = 2.74 years.

Project II has the lower discounted payback period at 2.74 years, and thus it is the best choice under this decision criterion.

(iii) NPV at 6% for: Project I = $3031.19

Project II = $6434.82

Since Project II offers a higher NPV, it is the best choice under this decision criterion.

(iv) IRRI = 6.29%

IRRII = 19.04%

Project II has the higher IRR and is, therefore, the better choice.

(v) Profitability Index (PI) = [pic]

PI Project I = [pic]

PI Project II = [pic]

Project II offers the highest ratio of net present value to investment; therefore, we would choose Project II.

b. Of the two projects, Project II is the better choice. It has lower payback and discounted payback periods. In addition, Project II has the higher NPV, IRR and profitability index.

36. a. The following table shows the NPV profile of the project. NPV is zero at an interest rate between 7% and 8% and at an interest rate between 33% and 34%. These are the two IRRs of the project. You can use your calculator to confirm that the two IRR’s are, more precisely, 7.16% and 33.67%.

| |Discount rate |NPV | |Discount rate |NPV |

| |0.00 |–2.00 | |0.21 |0.82 |

| |0.01 |–1.62 | |0.22 |0.79 |

| |0.02 |–1.28 | |0.23 |0.75 |

| |0.03 |–0.97 | |0.24 |0.71 |

| |0.04 |–0.69 | |0.25 |0.66 |

| |0.05 |–0.44 | |0.26 |0.60 |

| |0.06 |–0.22 | |0.27 |0.54 |

| |0.07 |–0.03 | |0.28 |0.47 |

| |0.08 |0.14 | |0.29 |0.39 |

| |0.09 |0.29 | |0.30 |0.32 |

| |0.10 |0.42 | |0.31 |0.24 |

| |0.11 |0.53 | |0.32 |0.15 |

| |0.12 |0.62 | |0.33 |0.06 |

| |0.13 |0.69 | |0.34 |–0.03 |

| |0.14 |0.75 | |0.35 |–0.13 |

| |0.15 |0.79 | |0.36 |–0.22 |

| |0.16 |0.83 | |0.37 |–0.32 |

| |0.17 |0.85 | |0.38 |–0.42 |

| |0.18 |0.85 | |0.39 |–0.53 |

| |0.19 |0.85 | |0.40 |–0.63 |

| |0.20 |0.84 | |0.41 |–0.74 |

b. At 5% the NPV is:

NPV = –22 + + + – = –0.443

Since the NPV is negative the project is not attractive.

c. At 20% the NPV is:

NPV = –22 + + + – = 0.840

At 40% the NPV is:

NPV = –22 + + + – = –0.634

d. At a low discount rate, the positive cash flows ($20 for 3 years) are not discounted much. However, the final negative cash flow of $40 does not get discounted very heavily either. The net effect is a negative NPV.

At very high rates, the positive cash flows are discounted very heavily, resulting in a negative NPV. For mid-range discount rates, the positive cash flows that occur in the middle of the project dominate and project NPV is positive.

46.

a) Payback period:

| |Project A |A’s Cumulative |Project B |B’s cumulative |

|Year |cash flow |cash flow |cash flow |cash flow |

|0 |- 10,000 |-10,000 |-15,000 |-15,000 |

|1 |6,000 |-4,000 |3,000 |-12,000 |

|2 |6,000 |2,000 |5,000 |-7,000 |

|3 |6,000 |8,000 |7,000 |0 |

|4 |6,000 |14,000 |8,000 |8,000 |

Payback for Project A = 1 + 4,000/6,000

= 1.667 years

Payback for Project B = 3 years

According to this method, Project A is preferred since it pays back earlier than Project B.

While the payback method is relatively easy to use, its major disadvantages are: (i) It does not take into consideration the time value of money and (ii) it ignores cash flows beyond the payback period.

Discounted Payback and NPV:

| |Discount |A’s discounted cash |A’s Cumulative |B’s discounted cash |B’s cumulative |

|Year |Factor 10 % |flow |Discount cash flow |flow |Discount cash flow |

|0 |1.000 |-10,000 |-10,000 |-15,000 |-15,000 |

|1 |.909 |5,454 |-4,546 |2,727 |-12,273 |

|2 |.826 |4,956 |410 |4,130 |-8,143 |

|3 |.751 |4,506 |4,916 |5,257 |-2,886 |

|4 |.683 |4,098 |9,014 |5,464 |2,578 |

|NPV |9,014 | |2,578 | |

Based on NPV project A is preferred.

Discounted payback for Project A = 1 + (4,546/4,956) = 1.92 years

Discounted payback for Project B = 3 + (2,886/5464) = 3.53 years

Once again, according to the discounted payback method, Project A is preferred since it pays back earlier than Project B.

The main advantage of the discounted payback period is that this method considers the time value of money, unlike the payback period.

The main flaw of the discounted payback period is that it does not consider cash flows beyond the payback period and therefore, it may on occasion incorrectly reject positive NPV projects.

(b). We see from the computations above that the NPV for Project A is $9,014 whereas for Project B it is $2,578. In general, notice also that if projects are able meet the cutoff in terms of a discounted payback, they must have a positive NPV.

Based on the NPV approach, Project A is preferred to Project B as it has the higher NPV.

(c). The profitability index for Project A = $9,014/$10,000 = .9014

For Project B, it is $2,578/$15,000 = .172.

Once again, Project A is preferable using this method.

(d). Internal Rate of return-

Trial and Error Approach.: Essentially, with this approach we try to select the discount rate at which the IRR for the project =0. This discount rate is, then, also the project’s IRR.

Project A

Let us try a discount rate of 48 %. At this rate,

NPV = -10,000 + {6,000/ (1+.48)1} + {6,000/ (1+.48)2} + {6,000/ (1+.48)3} + {6,000/ (1+.48)4}

NPV = - 105.28

Since NPV is negative at this rate, the IRR should be lower than 48 percent.

Let us try a discount rate of 46 %. At this rate,

NPV = -10,000 + {6,000/ (1+.46)1} + {6,000/ (1+.46)2} + {6,000/ (1+.46)3} + {6,000/ (1+.46)4}

NPV = 172.82

So, we can assume that the IRR for project A is somewhere between 46% and 48 %.

Notice that this 2 % difference between the two rates has an “NPV distance” of 278.1 (i.e. 105.28 + 172.82).

So, by interpolation, NPV will be 0 at a rate which is (2/278.1 x 105.28) below 48%; or 0.76% below 48% = 47.24%

It is actually 47.23 % (using a financial calculator).

Project B

Let us try an initial discount rate of 17 %. At this rate,

NPV = -15,000 + {3,000/ (1+.17)1} + {5,000/ (1+.17)2} + {7,000/ (1+.17)3} + {8,000/ (1+.17)4}

NPV = - 143.54

Since NPV is negative at 17%, the IRR should be lower than this rate.

Let us try a discount rate of 15 %. At this rate,

NPV = - 15,000 + {3,000/ (1+.15)1} + {5,000/ (1+.15)2} + {7,000/ (1+.15)3} + {8,000/ (1+.15)4}

NPV = 566.04

So, we know that the IRR for project B is some where between 15 % and 17 %.

Once again, notice that this 2 % difference between the two rates has an “NPV distance” of 709.58 (i.e. 143.54 + 566.04).

So, by interpolation, NPV will be 0 at a rate which is (2/709.58 x 143.54) below 17%; or 0.4% below 17% = 16.6%

It is actually 16.58 % (using a financial calculator).

Notice that, for both Projects A and B, we were able to get results for the IRR that were quite close to the actual numbers obtained through a financial calculator.

Using the IRR rule, Project A (with the higher IRR) is preferred over Project B.

e) Independent projects would be evaluated for acceptance or rejection on a “standalone” basis. Mutually exclusive projects would be selected on an “either/or” basis and only those contributing most toward shareholder wealth would be selected.

f) In this question we are able to reach the same decision using all the methods (payback, discounted payback, NPV, profitability index, and IRR) However, if we get conflicting decisions using different methods then the NPV method should be used as it is generally considered to be the most robust.

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