Capital Budgeting: Basic Problems

A practice guide for problems 1-19 from the textbook.

Basic Questions & Problems (1-19)

1. Calculating Payback: What is the payback period for the following set of cash flows? [Table of cash flows].

Given: Initial Cost = \$7,700. Cash Flows: Year 1 = \$1,900, Year 2 = \$3,000, Year 3 = \$2,300, Year 4 = \$1,700.

Step 1: Calculate cumulative cash flows.

End of Year 1: \$1,900

End of Year 2: \$1,900 + \$3,000 = \$4,900

End of Year 3: \$4,900 + \$2,300 = \$7,200

At the end of Year 3, we have recovered \$7,200, which is less than the initial cost of \$7,700. We still need to recover $\$7,700 - \$7,200 = \$500$. The cash flow in Year 4 is \$1,700.

Step 2: Calculate the fractional payback period.

$$ \text{Fraction} = \frac{\text{Amount needed}}{\text{Cash flow in Year 4}} = \frac{\$500}{\$1,700} \approx 0.29 \text{ years} $$

Step 3: Sum the whole years and the fraction.

$$ \text{Payback Period} = 3 \text{ years} + 0.29 \text{ years} = \textbf{3.29 years} $$

Tricky Area:

The payback period ignores the time value of money, so you simply add the cash flows sequentially until they equal the initial investment. Don't discount the cash flows for this problem!

2. Calculating Payback: An investment project provides cash inflows of \$835 per year for eight years. What is the project payback period if the initial cost is \$1,900? What if the initial cost is \$3,600? What if it is \$7,400?

Given: Annual Cash Flow = \$835. Project life = 8 years.

Case 1: Initial Cost = \$1,900

The payback period is the initial cost divided by the annual cash flow.

$$ \text{Payback Period} = \frac{\$1,900}{\$835} \approx \textbf{2.275 years} $$

Case 2: Initial Cost = \$3,600

$$ \text{Payback Period} = \frac{\$3,600}{\$835} \approx \textbf{4.311 years} $$

Case 3: Initial Cost = \$7,400

$$ \text{Payback Period} = \frac{\$7,400}{\$835} \approx \textbf{8.862 years} $$

Tricky Area:

The project life is 8 years. In this last case, the payback period is longer than the project life, which means the project never pays back. This highlights a key limitation of the payback rule.

3. Calculating Payback: Kara, Inc., imposes a payback cutoff of three years for its international investment projects. If the company has the following two projects available, should it accept either of them? [Table of cash flows].

Given: Payback Cutoff = 3 years.

Project A: Initial Cost = \$40,000

End of Year 1: \$14,000

End of Year 2: \$14,000 + \$18,000 = \$32,000

Amount needed in Year 3: $\$40,000 - \$32,000 = \$8,000$.

$$ \text{Fraction} = \frac{\$8,000}{\$17,000} \approx 0.47 \text{ years} $$ $$ \text{Payback Period} = 2 + 0.47 = \textbf{2.47 years} $$

Since 2.47 years is less than the 3-year cutoff, **Project A should be accepted**.

Project B: Initial Cost = \$55,000

End of Year 1: \$11,000

End of Year 2: \$11,000 + \$13,000 = \$24,000

End of Year 3: \$24,000 + \$16,000 = \$40,000

At the end of Year 3, the cumulative cash flow is only \$40,000, which is less than the initial cost of \$55,000. Therefore, the payback period is greater than 3 years. **Project B should be rejected**.

4. Calculating Discounted Payback: An investment project has annual cash inflows of \$2,800, \$3,700, \$5,100, and \$4,300, for the next four years, respectively. The discount rate is 9 percent. What is the discounted payback period for these cash flows if the initial cost is \$5,200? What if the initial cost is \$6,400? What if it is \$10,400?

Given: Discount Rate = 9%.

Step 1: Calculate the discounted cash flows (DCF).

Year 1: $\frac{\$2,800}{1.09} = \$2,568.81$

Year 2: $\frac{\$3,700}{1.09^2} = \$3,113.16$

Year 3: $\frac{\$5,100}{1.09^3} = \$3,934.19$

Year 4: $\frac{\$4,300}{1.09^4} = \$3,043.95$

Step 2: Calculate the cumulative DCF.

End of Year 1: \$2,568.81

End of Year 2: \$2,568.81 + \$3,113.16 = \$5,681.97

End of Year 3: \$5,681.97 + \$3,934.19 = \$9,616.16

End of Year 4: \$9,616.16 + \$3,043.95 = \$12,660.11

Case 1: Initial Cost = \$5,200

At the end of Year 1, the cumulative DCF is \$2,568.81. We need to recover $\$5,200 - \$2,568.81 = \$2,631.19$ in Year 2. The DCF for Year 2 is \$3,113.16.

$$ \text{Fraction} = \frac{\$2,631.19}{\$3,113.16} \approx 0.845 \text{ years} $$ $$ \text{Discounted Payback} = 1 + 0.845 = \textbf{1.845 years} $$

Case 2: Initial Cost = \$6,400

At the end of Year 2, the cumulative DCF is \$5,681.97. We need to recover $\$6,400 - \$5,681.97 = \$718.03$ in Year 3. The DCF for Year 3 is \$3,934.19.

$$ \text{Fraction} = \frac{\$718.03}{\$3,934.19} \approx 0.1825 \text{ years} $$ $$ \text{Discounted Payback} = 2 + 0.1825 = \textbf{2.1825 years} $$

Case 3: Initial Cost = \$10,400

At the end of Year 3, the cumulative DCF is \$9,616.16. We need to recover $\$10,400 - \$9,616.16 = \$783.84$ in Year 4. The DCF for Year 4 is \$3,043.95.

$$ \text{Fraction} = \frac{\$783.84}{\$3,043.95} \approx 0.2575 \text{ years} $$ $$ \text{Discounted Payback} = 3 + 0.2575 = \textbf{3.2575 years} $$
5. Calculating Discounted Payback: An investment project costs \$19,000 and has annual cash flows of \$5,100 for six years. What is the discounted payback period if the discount rate is zero percent? What if the discount rate is 5 percent? If it is 19 percent?

Given: Initial Cost = \$19,000, Annual Cash Flow = \$5,100 for 6 years.

Case 1: Discount Rate = 0%

When the discount rate is 0%, the discounted payback is the same as the regular payback.

$$ \text{Payback Period} = \frac{\$19,000}{\$5,100} \approx \textbf{3.725 years} $$

Case 2: Discount Rate = 5%

We need to find the present value of the cash flows until they equal \$19,000.

PV of 3 years' cash flows = $\$5,100 \times \left[ \frac{1 - 1/1.05^3}{0.05} \right] = \$13,858.91$

PV of 4 years' cash flows = $\$5,100 \times \left[ \frac{1 - 1/1.05^4}{0.05} \right] = \$18,103.73$

PV of 5 years' cash flows = $\$5,100 \times \left[ \frac{1 - 1/1.05^5}{0.05} \right] = \$22,042.86$

The payback is between Year 4 and Year 5. We need to recover $\$19,000 - \$18,103.73 = \$896.27$ in Year 5. The discounted cash flow for Year 5 is $\frac{\$5,100}{1.05^5} = \$4,000.41$.

$$ \text{Fraction} = \frac{\$896.27}{\$4,000.41} \approx 0.224 \text{ years} $$ $$ \text{Discounted Payback} = 4 + 0.224 = \textbf{4.224 years} $$

Case 3: Discount Rate = 19%

PV of 6 years' cash flows = $\$5,100 \times \left[ \frac{1 - 1/1.19^6}{0.19} \right] = \$17,994.46$. This is less than the initial cost of \$19,000. Therefore, the project **never pays back** on a discounted basis at 19%.

Tricky Area:

A project can have a positive payback period but never pay back on a discounted basis if the discount rate is high enough to make the present value of all cash flows less than the initial cost.

6. Calculating AAR: You're trying to determine whether to expand your business by building a new manufacturing plant. The plant has an installation cost of \$12.6 million, which will be depreciated straight-line to zero over its four-year life. If the plant has projected net income of \$1,430,000, \$1,523,460, \$1,716,300, and \$1,097,400 over these four years, respectively, what is the project's average accounting return (AAR)?

Given: Initial Cost = \$12.6M, life = 4 years. Net Incomes for each year are provided.

Step 1: Calculate the average net income.

$$ \text{Average Net Income} = \frac{\$1,430,000 + \$1,523,460 + \$1,716,300 + \$1,097,400}{4} = \frac{\$5,767,160}{4} = \$1,441,790 $$

Step 2: Calculate the average book value.

$$ \text{Average Book Value} = \frac{\text{Initial Cost} + \text{Final Value}}{2} = \frac{\$12,600,000 + 0}{2} = \$6,300,000 $$

Step 3: Calculate the AAR.

$$ \text{AAR} = \frac{\text{Average Net Income}}{\text{Average Book Value}} = \frac{\$1,441,790}{\$6,300,000} \approx 0.2288 \text{ or } \textbf{22.88\%} $$

Tricky Area:

The AAR is an accounting measure, not an economic one. It does not consider the time value of money, and the "rate of return" it provides is not comparable to market-based returns like a discount rate.

7. Calculating IRR: A firm evaluates all of its projects by applying the IRR rule. If the required return is 14 percent, should the firm accept the following project? [Table of cash flows].

Given: Initial Cost = \$41,000. Cash Flows: Year 1 = \$20,000, Year 2 = \$23,000, Year 3 = \$14,000. Required Return = 14%.

Step 1: Set up the NPV equation and solve for IRR.

$$ 0 = -\$41,000 + \frac{\$20,000}{(1+IRR)^1} + \frac{\$23,000}{(1+IRR)^2} + \frac{\$14,000}{(1+IRR)^3} $$

Using a financial calculator or spreadsheet, we find that IRR is approximately **15.75%**.

Step 2: Apply the IRR rule.

Since the IRR of 15.75% is greater than the required return of 14%, the firm **should accept** the project.

8. Calculating NPV: For the cash flows in the previous problem, suppose the firm uses the NPV decision rule. At a required return of 11 percent, should the firm accept this project? What if the required return is 24 percent?

Given: Cash flows from Problem 7. Initial Cost = \$41,000. Cash Flows: Year 1 = \$20,000, Year 2 = \$23,000, Year 3 = \$14,000.

Case 1: Required Return = 11%

$$ \text{NPV} = -\$41,000 + \frac{\$20,000}{1.11} + \frac{\$23,000}{1.11^2} + \frac{\$14,000}{1.11^3} $$ $$ \text{NPV} = -\$41,000 + \$18,018.02 + \$18,639.73 + \$10,230.12 = \textbf{\$5,887.87} $$

Since the NPV is positive, the firm **should accept** the project.

Case 2: Required Return = 24%

$$ \text{NPV} = -\$41,000 + \frac{\$20,000}{1.24} + \frac{\$23,000}{1.24^2} + \frac{\$14,000}{1.24^3} $$ $$ \text{NPV} = -\$41,000 + \$16,129.03 + \$14,942.50 + \$7,322.25 = \textbf{-\$2,606.22} $$

Since the NPV is negative, the firm **should reject** the project.

9. Calculating NPV and IRR: A project that provides annual cash flows of \$15,300 for nine years costs \$74,000 today. Is this a good project if the required return is 8 percent? What if it's 20 percent? At what discount rate would you be indifferent between accepting the project and rejecting it?

Given: Initial Cost = \$74,000. Annual Cash Flow = \$15,300 for 9 years.

Case 1: Required Return = 8%

$$ \text{NPV} = -\$74,000 + \$15,300 \times \left[ \frac{1 - \frac{1}{(1.08)^9}}{0.08} \right] $$ $$ \text{NPV} = -\$74,000 + \$15,300 \times 6.24689 = -\$74,000 + \$95,584.02 = \textbf{\$21,584.02} $$

Since the NPV is positive, this is a **good project** at 8%.

Case 2: Required Return = 20%

$$ \text{NPV} = -\$74,000 + \$15,300 \times \left[ \frac{1 - \frac{1}{(1.20)^9}}{0.20} \right] $$ $$ \text{NPV} = -\$74,000 + \$15,300 \times 4.30755 = -\$74,000 + \$65,995.52 = \textbf{-\$8,004.48} $$

Since the NPV is negative, this is **not a good project** at 20%.

Indifferent Discount Rate (IRR):

You would be indifferent when the NPV is zero, which is the IRR. We need to find the rate that makes the present value of a 9-year, \$15,300 annuity equal to \$74,000.

$$ \$74,000 = \$15,300 \times \text{PVAIF}(IRR, 9 \text{ years}) $$ $$ \text{PVAIF} = \frac{\$74,000}{\$15,300} = 4.8366 $$

Using a financial calculator or spreadsheet, the rate that corresponds to this annuity factor is approximately **14.39%**. This is the IRR.

10. Calculating IRR: What is the IRR of the following set of cash flows? [Table of cash flows].

Given: Initial Cost = \$18,700. Cash Flows: Year 1 = \$9,400, Year 2 = \$10,400, Year 3 = \$6,500.

Step 1: Set up the NPV equation and solve for IRR.

$$ 0 = -\$18,700 + \frac{\$9,400}{(1+IRR)^1} + \frac{\$10,400}{(1+IRR)^2} + \frac{\$6,500}{(1+IRR)^3} $$

Using a financial calculator or spreadsheet, the IRR is approximately **19.34%**.

11. Calculating NPV: For the cash flows in the previous problem, what is the NPV at a discount rate of zero percent? What if the discount rate is 10 percent? If it is 20 percent? If it is 30 percent?

Given: Cash flows from Problem 10. Initial Cost = \$18,700. Cash Flows: Year 1 = \$9,400, Year 2 = \$10,400, Year 3 = \$6,500.

Case 1: Discount Rate = 0%

$$ \text{NPV} = -\$18,700 + \$9,400 + \$10,400 + \$6,500 = \textbf{\$7,600} $$

Case 2: Discount Rate = 10%

$$ \text{NPV} = -\$18,700 + \frac{\$9,400}{1.10} + \frac{\$10,400}{1.10^2} + \frac{\$6,500}{1.10^3} $$ $$ \text{NPV} = -\$18,700 + \$8,545.45 + \$8,595.04 + \$4,883.31 = \textbf{\$3,323.80} $$

Case 3: Discount Rate = 20%

$$ \text{NPV} = -\$18,700 + \frac{\$9,400}{1.20} + \frac{\$10,400}{1.20^2} + \frac{\$6,500}{1.20^3} $$ $$ \text{NPV} = -\$18,700 + \$7,833.33 + \$7,222.22 + \$3,761.57 = \textbf{\$10.92} $$

Case 4: Discount Rate = 30%

$$ \text{NPV} = -\$18,700 + \frac{\$9,400}{1.30} + \frac{\$10,400}{1.30^2} + \frac{\$6,500}{1.30^3} $$ $$ \text{NPV} = -\$18,700 + \$7,230.77 + \$6,170.83 + \$2,933.28 = \textbf{-\$2,365.12} $$
12. NPV versus IRR: Bruin, Inc., has identified the following two mutually exclusive projects: [Table of cash flows]. a. What is the IRR for each of these projects? Using the IRR decision rule, which project should the company accept? Is this decision necessarily correct? b. If the required return is 11 percent, what is the NPV for each of these projects? Which project will the company choose if it applies the NPV decision rule? c. Over what range of discount rates would the company choose Project A? Project B? At what discount rate would the company be indifferent between these two projects? Explain.

Given: Initial Cost = \$41,300 for both projects.

a. IRR and IRR Rule:

Project A IRR: The IRR is approximately **24.57%**.

Project B IRR: The IRR is approximately **22.58%**.

Using the IRR decision rule, the company should accept Project A because its IRR is higher. However, this decision is not necessarily correct because these are mutually exclusive projects, and the IRR rule can lead to incorrect rankings.

b. NPV at 11%:

$$ \text{NPV(A)} = -\$41,300 + \frac{\$19,100}{1.11} + \frac{\$17,800}{1.11^2} + \frac{\$15,200}{1.11^3} + \frac{\$8,400}{1.11^4} $$ $$ \text{NPV(A)} = -\$41,300 + \$17,207.21 + \$14,401.59 + \$11,189.60 + \$5,532.74 = \textbf{\$7,031.14} $$
$$ \text{NPV(B)} = -\$41,300 + \frac{\$6,300}{1.11} + \frac{\$14,200}{1.11^2} + \frac{\$17,900}{1.11^3} + \frac{\$30,300}{1.11^4} $$ $$ \text{NPV(B)} = -\$41,300 + \$5,675.68 + \$11,489.19 + \$13,172.93 + \$19,957.94 = \textbf{\$9,095.74} $$

Using the NPV rule, the company should choose Project B because it has a higher NPV.

Tricky Area:

This problem demonstrates the conflict between the NPV and IRR rules for mutually exclusive projects. The IRR rule says to pick A, but the NPV rule, which is the correct one, says to pick B.

c. Crossover Rate:

To find the crossover rate, we take the difference in cash flows (e.g., A - B) and find the IRR of that new project. The difference in cash flows is [0, \$12,800, \$3,600, -\$2,700, -\$21,900].

The IRR of this differential cash flow is approximately **19.86%**. This is the crossover rate. Below this rate, Project B has a higher NPV. Above this rate, Project A has a higher NPV.

The company would choose Project B for required returns between 0% and 19.86% and Project A for required returns between 19.86% and the IRR of Project A (24.57%).

13. NPV versus IRR: Consider the following two mutually exclusive projects: [Table of cash flows]. Sketch the NPV profiles for X and Y over a range of discount rates from zero to 25 percent. What is the crossover rate for these two projects?

Given: Initial Cost = \$30,000 for both projects.

NPV Profiles:

The crossover rate is the discount rate at which the NPV of both projects is equal. We can find this by taking the difference in cash flows (X - Y) and finding the IRR of that new project. The difference in cash flows is [0, -\$1,900, \$2,000, \$100].

$$ 0 = \frac{-\$1,900}{(1+IRR)^1} + \frac{\$2,000}{(1+IRR)^2} + \frac{\$100}{(1+IRR)^3} $$

Using a financial calculator or spreadsheet, the IRR of the differential cash flows is approximately **3.40%**. This is the crossover rate.

14. Problems with IRR: Howell Petroleum, Inc., is trying to evaluate a generation project with the following cash flows: [Table of cash flows]. a. If the company requires a return of 12 percent on its investments, should it accept this project? Why? b. Compute the IRR for this project. How many IRRs are there? Using the IRR decision rule, should the company accept the project? What's going on here?

Given: Initial Cost = -\$52M. Cash Flows: Year 1 = \$74M, Year 2 = -\$12M.

a. NPV at 12%:

$$ \text{NPV} = -\$52M + \frac{\$74M}{1.12} + \frac{-\$12M}{1.12^2} $$ $$ \text{NPV} = -\$52M + \$66.07M - \$9.57M = \textbf{\$4.5M} $$

Since the NPV is positive, the company **should accept** the project.

b. IRR and Nonconventional Cash Flows:

The cash flows are nonconventional (negative, positive, negative). According to Descartes' Rule of Signs, there can be up to two IRRs. We can find them by setting NPV to zero.

$$ 0 = -\$52M + \frac{\$74M}{(1+IRR)^1} + \frac{-\$12M}{(1+IRR)^2} $$

This is a quadratic equation. Solving for IRR, we find two possible values: **13.43%** and **14.28%**. Since both IRRs are greater than the 12% required return, the IRR rule would suggest accepting the project. This is consistent with the positive NPV we found in part (a).

This illustrates the multiple rates of return problem. The IRR rule is not reliable because there are two IRRs. The NPV rule, however, gives a clear answer.

15. Calculating Profitability Index: What is the profitability index for the following set of cash flows if the relevant discount rate is 10 percent? What if the discount rate is 15 percent? If it is 22 percent?

Given: Initial Cost = \$14,800. Cash Flows: Year 1 = \$8,400, Year 2 = \$7,600, Year 3 = \$4,300.

Step 1: Calculate the present value (PV) of future cash flows for each rate.

10%: PV = $\frac{\$8,400}{1.10} + \frac{\$7,600}{1.10^2} + \frac{\$4,300}{1.10^3} = \$7,636.36 + \$6,280.99 + \$3,230.82 = \$17,148.17$

15%: PV = $\frac{\$8,400}{1.15} + \frac{\$7,600}{1.15^2} + \frac{\$4,300}{1.15^3} = \$7,304.35 + \$5,749.50 + \$2,827.67 = \$15,881.52$

22%: PV = $\frac{\$8,400}{1.22} + \frac{\$7,600}{1.22^2} + \frac{\$4,300}{1.22^3} = \$6,885.25 + \$5,108.62 + \$2,357.73 = \$14,351.60$

Step 2: Calculate the Profitability Index (PI).

10%: PI = $\frac{\$17,148.17}{\$14,800} = \textbf{1.1587}$

15%: PI = $\frac{\$15,881.52}{\$14,800} = \textbf{1.0731}$

22%: PI = $\frac{\$14,351.60}{\$14,800} = \textbf{0.9697}$

16. Problems with Profitability Index: The Michner Corporation is trying to choose between the following two mutually exclusive design projects: [Table of cash flows]. a. If the required return is 10 percent and the company applies the profitability index decision rule, which project should the firm accept? b. If the company applies the NPV decision rule, which project should it take? c. Explain why your answers in parts (a) and (b) are different.

Given: Required return = 10%. Projects I and II are mutually exclusive.

Step 1: Calculate the present value (PV) of cash flows for both projects.

Project I: PV = $\frac{\$37,600}{1.10} + \frac{\$37,600}{1.10^2} + \frac{\$37,600}{1.10^3} = \$34,181.82 + \$31,074.38 + \$28,249.44 = \$93,505.64$

Project II: PV = $\frac{\$11,200}{1.10} + \frac{\$11,200}{1.10^2} + \frac{\$11,200}{1.10^3} = \$10,181.82 + \$9,256.20 + \$8,414.73 = \$27,852.75$

a. PI Rule:

Project I: PI = $\frac{\$93,505.64}{\$82,000} = \textbf{1.1403}$

Project II: PI = $\frac{\$27,852.75}{\$21,700} = \textbf{1.2835}$

Based on the PI rule, the company should accept Project **II** because it has a higher PI.

b. NPV Rule:

Project I: NPV = $\$93,505.64 - \$82,000 = \textbf{\$11,505.64}$

Project II: NPV = $\$27,852.75 - \$21,700 = \textbf{\$6,152.75}$

Based on the NPV rule, the company should take Project **I** because it has a higher NPV.

c. Explanation:

The answers are different because the two projects are mutually exclusive. The PI measures the value created per dollar invested, while the NPV measures the total value created. Project II has a higher "rate of return" per dollar invested, but Project I creates more total value for the firm. The NPV rule is the correct one to use when choosing between mutually exclusive projects because the goal is to maximize total shareholder wealth.

17. Comparing Investment Criteria: Consider the following two mutually exclusive projects: [Table of cash flows]. Whichever project you choose, if any, you require a return of 11 percent on your investment. a. If you apply the payback criterion, which investment will you choose? Why? b. If you apply the discounted payback criterion, which investment will you choose? Why? c. If you apply the NPV criterion, which investment will you choose? Why? d. If you apply the IRR criterion, which investment will you choose? Why? e. If you apply the profitability index criterion, which investment will you choose? Why? f. Based on your answers in parts (a) through (e), which project will you finally choose? Why?

Given: Required return = 11%. Projects A and B are mutually exclusive.

a. Payback:

Project A: Payback is between Year 3 and Year 4. Amount needed in Year 4: $\$291,000 - (\$37,000 + \$55,000 + \$55,000) = \$144,000$. Payback = $3 + \frac{\$144,000}{\$366,000} = \textbf{3.39 years}$.

Project B: Payback is between Year 2 and Year 3. Amount needed in Year 3: $\$41,600 - (\$20,000 + \$17,600) = \$4,000$. Payback = $2 + \frac{\$4,000}{\$17,200} = \textbf{2.23 years}$.

Based on the payback rule, choose Project **B** because it has a shorter payback period.

b. Discounted Payback:

The discounted cash flows for Project A never recover the initial investment, as the NPV is negative. Therefore, Project A has no discounted payback. Project B has a discounted payback of **2.97 years**. Based on the discounted payback rule, choose Project **B**.

c. NPV:

Project A: NPV = $-\$291,000 + \frac{\$37,000}{1.11} + \frac{\$55,000}{1.11^2} + \frac{\$55,000}{1.11^3} + \frac{\$366,000}{1.11^4} = \textbf{-\$21,079.91}$

Project B: NPV = $-\$41,600 + \frac{\$20,000}{1.11} + \frac{\$17,600}{1.11^2} + \frac{\$17,200}{1.11^3} + \frac{\$14,000}{1.11^4} = \textbf{\$6,039.46}$

Based on the NPV rule, choose Project **B** because it has a positive NPV, while Project A has a negative NPV.

d. IRR:

Project A: IRR is approximately **7.21%**.

Project B: IRR is approximately **17.84%**.

Based on the IRR rule, choose Project **B** because its IRR is greater than the required return of 11%, while Project A's IRR is less than 11%.

e. PI:

PV of cash flows for Project A = \$269,920.09. PV of cash flows for Project B = \$47,639.46.

Project A: PI = $\frac{\$269,920.09}{\$291,000} = \textbf{0.9276}$

Project B: PI = $\frac{\$47,639.46}{\$41,600} = \textbf{1.1452}$

Based on the PI rule, choose Project **B** because its PI is greater than 1, while Project A's PI is less than 1.

f. Final Choice:

The best choice is Project **B**. In this case, all criteria point to the same decision. This is because Project A has a negative NPV, making it a bad investment. The primary goal is to accept only positive NPV projects, and Project B is the only one that meets that condition.

18. NPV and Discount Rates: An investment has an installed cost of \$574,380. The cash flows over the four-year life of the investment are projected to be \$216,700, \$259,300, \$214,600, and \$167,410, respectively. If the discount rate is zero, what is the NPV? If the discount rate is infinite, what is the NPV? At what discount rate is the NPV just equal to zero? Sketch the NPV profile for this investment based on these three points.

Given: Initial Cost = -\$574,380. Cash Flows: Year 1 = \$216,700, Year 2 = \$259,300, Year 3 = \$214,600, Year 4 = \$167,410.

Case 1: Discount Rate = 0%

At a 0% discount rate, the NPV is simply the sum of all cash flows (undiscounted) minus the initial cost.

$$ \text{NPV} = (\$216,700 + \$259,300 + \$214,600 + \$167,410) - \$574,380 = \$858,010 - \$574,380 = \textbf{\$283,630} $$

Case 2: Discount Rate = $\infty$

As the discount rate approaches infinity, the present value of all future cash flows approaches zero. Therefore, the NPV approaches the negative of the initial cost.

$$ \text{NPV} = 0 - \$574,380 = \textbf{-\$574,380} $$

Case 3: NPV = 0 (IRR)

The discount rate that makes NPV zero is the IRR. Using a financial calculator or spreadsheet, the IRR is approximately **20.44%**.

19. MIRR: Duo Corp. is evaluating a project with the following cash flows: [Table of cash flows]. The company uses an interest rate of 10 percent on all of its projects. Calculate the MIRR of the project using all three methods.

Given: Cash flows: Year 0 = -\$53,000, Year 1 = \$16,700, Year 2 = \$21,900, Year 3 = \$27,300, Year 4 = \$20,400, Year 5 = -\$8,600. Required return = 10%.

Tricky Area:

This is a nonconventional cash flow problem (negative cash flows at the beginning and end), which is a perfect scenario for using the MIRR to avoid multiple IRRs.

Method 1: Discounting Approach

Discount all negative cash flows to the present. The negative cash flow at Year 5 is discounted back to Year 0.

$$ \text{PV of negative cash flows} = \$53,000 + \frac{\$8,600}{(1.10)^5} = \$53,000 + \$5,340.54 = \$58,340.54 $$ $$ \text{Future Value of positive cash flows} = \$16,700(1.10)^4 + \$21,900(1.10)^3 + \$27,300(1.10)^2 + \$20,400(1.10)^1 = \$100,530.40 $$ $$ \text{MIRR} = \left( \frac{\text{FV of positive CF}}{\text{PV of negative CF}} \right)^{1/5} - 1 = \left( \frac{\$100,530.40}{\$58,340.54} \right)^{1/5} - 1 = \textbf{11.53\%} $$

Method 2: Reinvestment Approach

Compound all cash flows to the end of the project's life (Year 5). The initial cost is not compounded, and the Year 5 negative cash flow is already at the end of the project.

$$ \text{FV of cash flows} = (\$16,700(1.10)^4 + \$21,900(1.10)^3 + \$27,300(1.10)^2 + \$20,400(1.10)^1) - \$8,600 = \$92,854.40 $$ $$ \text{MIRR} = \left( \frac{\text{FV of total CF}}{\text{Initial Cost}} \right)^{1/5} - 1 = \left( \frac{\$92,854.40}{\$53,000} \right)^{1/5} - 1 = \textbf{11.83\%} $$

Method 3: Combination Approach

This method combines the previous two. Discount negative cash flows to the present and compound positive cash flows to the future.

$$ \text{PV of negative CF} = \$53,000 + \frac{\$8,600}{(1.10)^5} = \$58,340.54 $$ $$ \text{FV of positive CF} = \$16,700(1.10)^4 + \$21,900(1.10)^3 + \$27,300(1.10)^2 + \$20,400(1.10)^1 = \$100,530.40 $$ $$ \text{MIRR} = \left( \frac{\text{FV of positive CF}}{\text{PV of negative CF}} \right)^{1/5} - 1 = \left( \frac{\$100,530.40}{\$58,340.54} \right)^{1/5} - 1 = \textbf{11.53\%} $$