Capital Budgeting: Intermediate & Challenge Problems

A practice guide for problems 20-28 from the textbook.

Intermediate Questions & Problems (20-22)

20. MIRR: Suppose the company in the previous problem uses a discount rate of 11 percent and a reinvestment rate of 8 percent on all of its projects. Calculate the MIRR of the project using all three methods.

Given: Cash flows from Problem 19. Required return = 11%. Reinvestment rate = 8%. Nonconventional cash flows.

Tricky Area:

The problem provides both a discount rate and a reinvestment rate. You must use the appropriate rate for each method, as the problem specifies different rates for discounting and compounding.

Method 1: Discounting Approach

Discount all negative cash flows back to the present at the **required return** of 11%. The positive cash flows are reinvested to the end of the project at the **reinvestment rate** of 8%.

PV of Negative Cash Flows (at 11%):

$$ \text{PV of negative CF} = \$53,000 + \frac{\$8,600}{(1.11)^5} = \$53,000 + \$5,110.15 = \$58,110.15 $$

FV of Positive Cash Flows (at 8%):

$$ \text{FV of positive CF} = \$16,700(1.08)^4 + \$21,900(1.08)^3 + \$27,300(1.08)^2 + \$20,400(1.08)^1 = \$92,853.51 $$ $$ \text{MIRR} = \left( \frac{\text{FV of positive CF}}{\text{PV of negative CF}} \right)^{1/5} - 1 = \left( \frac{\$92,853.51}{\$58,110.15} \right)^{1/5} - 1 = \textbf{9.80\%} $$

Method 2: Reinvestment Approach

Compound all cash flows (except the initial cost) to the end of the project's life at the **reinvestment rate** of 8%. The initial cost is already at the beginning of the project.

FV of all cash flows (at 8%):

$$ \text{FV} = \$16,700(1.08)^4 + \$21,900(1.08)^3 + \$27,300(1.08)^2 + \$20,400(1.08)^1 - \$8,600 = \$84,251.27 $$ $$ \text{MIRR} = \left( \frac{\text{FV of total CF}}{\text{Initial Cost}} \right)^{1/5} - 1 = \left( \frac{\$84,251.27}{\$53,000} \right)^{1/5} - 1 = \textbf{9.68\%} $$

Method 3: Combination Approach

Discount negative cash flows to the present at the **required return** (11%) and compound positive cash flows to the future at the **reinvestment rate** (8%).

PV of Negative CF (at 11%):

$$ \text{PV of negative CF} = \$53,000 + \frac{\$8,600}{(1.11)^5} = \$58,110.15 $$

FV of Positive CF (at 8%):

$$ \text{FV of positive CF} = \$16,700(1.08)^4 + \$21,900(1.08)^3 + \$27,300(1.08)^2 + \$20,400(1.08)^1 = \$92,853.51 $$ $$ \text{MIRR} = \left( \frac{\text{FV of positive CF}}{\text{PV of negative CF}} \right)^{1/5} - 1 = \left( \frac{\$92,853.51}{\$58,110.15} \right)^{1/5} - 1 = \textbf{9.80\%} $$
21. NPV and the Profitability Index: If we define the NPV index as the ratio of NPV to cost, what is the relationship between this index and the profitability index?

Theoretical Foundations:

The Profitability Index (PI) is defined as: $$ PI = \frac{\text{PV of Future Cash Flows}}{\text{Initial Cost}} $$

The NPV is defined as: $$ NPV = \text{PV of Future Cash Flows} - \text{Initial Cost} $$

The new NPV Index is defined as: $$ \text{NPV Index} = \frac{NPV}{\text{Initial Cost}} = \frac{\text{PV of Future Cash Flows} - \text{Initial Cost}}{\text{Initial Cost}} $$

The Relationship:

$$ \text{NPV Index} = \frac{\text{PV of Future Cash Flows}}{\text{Initial Cost}} - \frac{\text{Initial Cost}}{\text{Initial Cost}} $$ $$ \text{NPV Index} = PI - 1 $$

The relationship is that the NPV index is simply the profitability index minus 1. This means a positive NPV (and thus a positive NPV index) corresponds to a PI greater than 1.

22. Cash Flow Intuition: A project has an initial cost of I, a required return of R, and pays C annually for N years. a. Find C in terms of I and N such that the project has a payback period just equal to its life. b. Find C in terms of I, N, and R such that this is a profitable project according to the NPV decision rule. c. Find C in terms of I, N, and R such that the project has a benefit-cost ratio of 2.

Given: Initial Cost = $I$, Annual Cash Flow = $C$, Life = $N$ years, Required Return = $R$.

a. Payback Period = Life:

For the payback period to equal the project's life, the sum of the undiscounted cash flows must equal the initial cost. $$ C \times N = I $$

$$ C = \frac{I}{N} $$

b. Profitable Project (NPV > 0):

A project is profitable if its NPV is greater than zero. The present value of the cash flows (an annuity) must be greater than the initial cost.

$$ \text{PV of Annuity} > I $$
$$ C \times \left[ \frac{1 - \frac{1}{(1+R)^N}}{R} \right] > I $$ $$ C > \frac{I}{\left[ \frac{1 - \frac{1}{(1+R)^N}}{R} \right]} $$ $$ C > I \times \frac{R}{1 - (1+R)^{-N}} $$

c. Benefit-Cost Ratio (PI) = 2:

The benefit-cost ratio, or profitability index, is the ratio of the present value of cash flows to the initial cost. We set this ratio equal to 2 and solve for $C$.

$$ PI = \frac{\text{PV of Annuity}}{I} = 2 $$
$$ C \times \left[ \frac{1 - \frac{1}{(1+R)^N}}{R} \right] = 2I $$ $$ C = \frac{2I}{\left[ \frac{1 - \frac{1}{(1+R)^N}}{R} \right]} = 2I \times \frac{R}{1 - (1+R)^{-N}} $$

Challenge Questions & Problems (23-28)

23. Payback and NPV: An investment under consideration has a payback of seven years and a cost of \$745,000. If the required return is 11 percent, what is the worst-case NPV? The best-case NPV? Explain. Assume the cash flows are conventional.

Given: Initial Cost = \$745,000, Payback Period = 7 years, Required Return = 11%. Conventional cash flows.

Tricky Area:

This is a conceptual problem that tests your understanding of the relationship between payback and NPV. The cash flows are conventional, which simplifies the analysis.

Worst-Case NPV:

The worst-case NPV occurs when all the cash flows arrive as late as possible while still achieving a 7-year payback. This would mean that the cash flows are zero for the first six years, and then a single cash flow of \$745,000 arrives in Year 7.

$$ \text{Worst-Case NPV} = -\$745,000 + \frac{\$745,000}{(1.11)^7} = -\$745,000 + \$358,079.52 = \textbf{-\$386,920.48} $$

Best-Case NPV:

The best-case NPV occurs when all the cash flows arrive as early as possible while still achieving a 7-year payback. This would mean a cash flow of C for seven years, where $7C = \$745,000$, so $C = \$106,428.57$. The cash flows from year 8 to infinity would be positive, as it is a conventional project.

$$ \text{Best-Case NPV} = -\$745,000 + \sum_{t=1}^{7} \frac{\$106,428.57}{(1.11)^t} + \frac{\text{CF}_8}{(1.11)^8} + ... $$

In this scenario, the NPV is positive. The worst case NPV is definitely negative and the best case NPV is definitely positive. The payback period alone is not sufficient to determine the sign of the NPV.

24. Multiple IRRs: This problem is useful for testing the ability of financial calculators and spreadsheets. Consider the following cash flows. How many different IRRs are there? (Hint: Search between 20 percent and 70 percent.) When should we take this project? [Table of cash flows].

Given: Cash flows: Year 0 = -\$3,024, Year 1 = \$17,172, Year 2 = -\$36,420, Year 3 = \$34,200, Year 4 = -\$12,000.

Tricky Area:

This is a classic nonconventional cash flow problem. The cash flows change sign four times (negative to positive, positive to negative, negative to positive, positive to negative). According to Descartes' Rule of Signs, there can be up to four IRRs. The hint suggests searching between 20% and 70%.

By plotting the NPV profile or using a financial calculator with multiple starting points, you will find four IRRs: **25%**, **33.33%**, **50%**, and **60%**.

Based on the NPV profile, the project should be accepted when the required return is between 25% and 33.33% or between 50% and 60%. The NPV is positive only in these ranges.

25. NPV Valuation: The Yurdone Corporation wants to set up a private cemetery business. According to the CFO, Barry M. Deep, business is "looking up." As a result, the cemetery project will provide a net cash inflow of \$180,000 for the firm during the first year, and the cash flows are projected to grow at a rate of 4 percent per year forever. The project requires an initial investment of \$2.2 million. a. If the company requires an 11 percent return on such undertakings, should the cemetery business be started? b. The company is somewhat unsure about the assumption of a growth rate of 4 percent in its cash flows. At what constant growth rate would the company just break even if it still required a return of 11 percent on investment?

Given: Initial Investment = \$2.2M, First Cash Flow ($CF_1$) = \$180,000, Growth Rate ($g$) = 4%, Required Return ($R$) = 11%.

a. Should the business be started?

This is a growing perpetuity problem. We use the Gordon Growth Model formula to find the present value of the cash flows and then calculate the NPV.

$$ \text{PV of Cash Flows} = \frac{CF_1}{R-g} = \frac{\$180,000}{0.11 - 0.04} = \frac{\$180,000}{0.07} = \$2,571,428.57 $$ $$ \text{NPV} = \text{PV of Cash Flows} - \text{Initial Cost} = \$2,571,428.57 - \$2,200,000 = \textbf{\$371,428.57} $$

Since the NPV is positive, the cemetery business **should be started**.

b. Break-even growth rate:

The company breaks even when NPV = 0. This means the PV of cash flows must equal the initial investment of \$2.2M.

$$ \text{PV of Cash Flows} = \frac{CF_1}{R-g} = \text{Initial Cost} $$ $$ \$2,200,000 = \frac{\$180,000}{0.11 - g} $$ $$ 0.11 - g = \frac{\$180,000}{\$2,200,000} \approx 0.0818 $$ $$ g = 0.11 - 0.0818 = 0.0282 \text{ or } \textbf{2.82\%} $$

The company would break even at a growth rate of 2.82%.

26. Problems with IRR: A project has the following cash flows: [Table of cash flows]. What is the IRR for this project? If the required return is 12 percent, should the firm accept the project? What is the NPV of this project? What is the NPV of the project if the required return is 0 percent? 24 percent? What is going on here? Sketch the NPV profile to help you with your answer.

Given: Initial Cash Flow = \$74,000, Year 1 = -\$49,000, Year 2 = -\$41,000.

Tricky Area:

This is a financing-type cash flow problem, which is a key exception to the standard IRR rule. You receive cash up front and pay it out later, which is the opposite of a typical investment project.

IRR:

Setting NPV to zero, we get: $$ 0 = \$74,000 - \frac{\$49,000}{(1+IRR)^1} - \frac{\$41,000}{(1+IRR)^2} $$

Solving for IRR, we find a single IRR of **15.22%**.

NPV at various rates:

0%: NPV = $\$74,000 - \$49,000 - \$41,000 = \textbf{-\$16,000}$

12%: NPV = $\$74,000 - \frac{\$49,000}{1.12} - \frac{\$41,000}{1.12^2} = \$74,000 - \$43,750 - \$32,731.25 = \textbf{-\$2,481.25}$

24%: NPV = $\$74,000 - \frac{\$49,000}{1.24} - \frac{\$41,000}{1.24^2} = \$74,000 - \$39,516.13 - \$26,629.74 = \textbf{\$7,854.13}$

What is going on here?

This is a financing-type cash flow project. The NPV is negative at low discount rates and becomes positive as the rate increases. The IRR rule for this type of project is the opposite of a conventional project: accept if the IRR is **less than** the required return. Since the IRR of 15.22% is greater than the required return of 12%, you should **reject** the project. This is consistent with the negative NPV at 12%.

27. Problems with IRR: Mako Corp. has a project with the following cash flows: [Table of cash flows]. What is the IRR of the project? What is happening here?

Given: Initial Cash Flow = \$25,000, Year 1 = -\$11,000, Year 2 = \$7,000.

IRR:

This is another nonconventional project. You must plot the NPV profile to see what is happening. The IRR is approximately **-31.5%** and **67.6%**.

What is happening here?

The cash flows change sign twice, so there are two IRRs. The IRR rule is not reliable. You should rely on the NPV rule instead. For example, if the required return is 10%, the NPV is positive (\$17,294), and you should accept. If the required return is 80%, the NPV is negative (\$1,911), and you should reject.

28. NPV and IRR: Anderson International Limited is evaluating a project in Erewhon. The project will create the following cash flows: [Table of cash flows]. All cash flows will occur in Erewhon and are expressed in dollars. In an attempt to improve its economy, the Erewhonian government has declared that all cash flows created by a foreign company are "blocked" and must be reinvested with the government for one year. The reinvestment rate for these funds is 4 percent. If Anderson uses a required return of 11 percent on this project, what are the NPV and IRR of the project? Is the IRR you calculated the MIRR of the project? Why or why not?

Given: Initial Cost = -\$1,785,000. Cash Flows: Year 1 = \$610,000, Year 2 = \$707,000, Year 3 = \$580,000, Year 4 = \$483,000. Required Return = 11%. Reinvestment Rate = 4%.

Tricky Area:

This problem is a blend of a standard NPV calculation with the concept of MIRR. You must first adjust the cash flows for the reinvestment rate before calculating the final NPV and IRR.

Step 1: Adjust the cash flows for the reinvestment rule.

Each cash flow is reinvested for one year at 4%. This means the actual cash flow available to the company is the compounded value of the original cash flow after one year.

CF from Year 1 becomes available at Year 2: $\$610,000 \times 1.04 = \$634,400$

CF from Year 2 becomes available at Year 3: $\$707,000 \times 1.04 = \$735,280$

CF from Year 3 becomes available at Year 4: $\$580,000 \times 1.04 = \$603,200$

Step 2: Recalculate NPV with the adjusted cash flows and the 11% required return.

$$ \text{NPV} = -\$1,785,000 + \frac{\$634,400}{1.11^2} + \frac{\$735,280}{1.11^3} + \frac{\$603,200 + \$483,000}{1.11^4} $$ $$ \text{NPV} = -\$1,785,000 + \$513,446.85 + \$537,707.03 + \$716,666.90 = \textbf{-\$117,179.22} $$

Step 3: Recalculate IRR with the adjusted cash flows.

We solve for the rate that makes NPV = 0 with the adjusted cash flows. The IRR is approximately **7.91%**.

Is the IRR the MIRR?

Yes. The IRR you calculated is effectively the MIRR of the project. The problem is set up in a way that forces you to modify the cash flows by a given reinvestment rate (4%) before you can solve for the project's internal rate of return. Since this rate depends on an externally supplied reinvestment rate, it is, by definition, a Modified IRR.