Chapter 9, Problem 17QP

Comparing Investment Criteria [LO1, 2, 3, 5, 7] Consider the following two mutually exclusive projects:

Year	Cash Flow (A)	Cash Flow (B)
0	−$455,000	−$65,000
1	58,000	31,000
2	85,000	28,000
3	85,000	25,500
4	572,000	19,000

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 (a) through (e), which project will you finally choose? Why?

Summary Introduction

To compute: The payback period.

Introduction:

Capital budgeting is a process, where the business identifies and assesses the potential investments or expenses that are larger in general.

Answer to Problem 17QP

Here, Project B must be accepted as it pays back sooner than Project A.

Explanation of Solution

Given information:

The cash flows for two mutually exclusive projects are $58,000, $85,000, $85,000, $572,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $455,000. The cash flows of Project B are $31,000, $28,000, $25,500, and $19,000 respectively for year 1, year 2, year 3, and year 4. The initial investment for Project B is $65,000. The rate of return is 11%.

Formula to compute the payback period of a project:

$\text{Payback period}=\left[\begin{array}{l}\text{Maximum number of years to }\\ \text{recover the amount}+\left(\frac{\text{Amount remaining to be recovered}}{4^{\text{th}}\text{ year cash flow}}\right)\end{array}\right]$

Compute the payback period of a project for Project A:

$\begin{array}{c}\text{Payback period}=\left[\begin{array}{l}\text{Maximum number of years to }\\ \text{recover the amount}+\left(\frac{\text{Amount remaining to be recovered}}{4^{\text{th}}\text{ year cash flow}}\right)\end{array}\right]\\ =3+\left(\frac{\$227,000}{\$572,000}\right)\\ =3.40\text{ years}\end{array}$

Hence, the payback period is 3.40 years for Project A.

Compute the payback period of a project for Project B:

$\begin{array}{c}\text{Payback period}=\left[\begin{array}{l}\text{Maximum number of years to }\\ \text{recover the amount}+\left(\frac{\text{Amount remaining to be recovered}}{3^{\text{rd}}\text{ year cash flow}}\right)\end{array}\right]\\ =2+\left(\frac{\$6,000}{\$25,500}\right)\\ =2.24\text{ years}\end{array}$

Hence, the payback period is 2.24 years for Project B.

Summary Introduction

To compute: The discounted payback period.

Introduction:

Capital budgeting is a process, where the business identifies and assesses the potential investments or expenses that are larger in general.

Answer to Problem 17QP

Here, Project B must be accepted as it pays back sooner than Project A.

Explanation of Solution

Given information:

The cash flows for two mutually exclusive projects are $58,000, $85,000, $85,000, $572,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $455,000. The cash flows of Project B are $31,000, $28,000, $25,500, and $19,000 respectively for year 1, year 2, year 3, and year 4. The initial investment for Project B is $65,000. The rate of return is 11%.

Table showing the calculation of discounted payback period for Project A:

Year	Net Cash Inflows
	Annual	Today’s value (discounted) at 11%	Accumulated
1	$58,000	$52,252.25 $\left(\frac{\$58,000}{1.11}\right)$	$52,252.25
2	$85,000	$68,987.91 $\left(\frac{\$85,000}{{1.11}^2}\right)$	$121,240.16
3	$85,000	$62,151.27 $\left(\frac{\$85,000}{{1.11}^3}\right)$	$183,391.43
4	$572,000	$376,794.12 $\left(\frac{\$572,000}{{1.11}^4}\right)$	$560,185.55

Formula to compute the discounted payback period:

By the end of the year 3, the recovery amount is shorter than the initial investment. In the year 4, the recovery amount is higher than initial investment. It means the discounted payback period will be in between the year 3 and 4.

$\text{Discounted payback period}=3\text{ years}+\left(\frac{\text{Initial cost}-\left(\begin{array}{l}\text{Accumulated discounted }\\ \text{cash inflows in year3}\end{array}\right)}{\text{Discounted cash inflows in year 4}}\right)$

Calculate discounted payback period, if initial cost is $455,000:

$\begin{array}{c}\text{Discounted payback period}=3\text{ years}+\left(\frac{\text{Initial cost}-\left(\begin{array}{l}\text{Accumulated discounted }\\ \text{cash inflows in year3}\end{array}\right)}{\text{Discounted cash inflows in year 4}}\right)\\ =3\text{years}+\frac{\$455,000-\$183,391.43}{\$376,794.12}\\ =3\text{years}+\text{0}\text{.72 year}\\ =3.\text{72 years}\end{array}$

Hence, the discounted payback period for Project A is 3.72 years.

Table showing the calculation of discounted payback period for Project B:

Year	Net Cash Inflows
	Annual	Today’s value (discounted) at 11%	Accumulated
1	$31,000	$27,927.93 $\left(\frac{\$31,000}{1.11}\right)$	$27,927.93
2	$28,000	$22,725.43 $\left(\frac{\$28,000}{{1.11}^2}\right)$	$50,653.36
3	$25,500	$18,645.38 $\left(\frac{\$25,500}{{1.11}^3}\right)$	$69,298.74
4	$19,000	$12,515.89 $\left(\frac{\$19,000}{{1.11}^4}\right)$	$81,814.63

Formula to calculate the discounted payback period for Project B, if initial cost is $65,000:

By the end of the year 2, the recovery amount is shorter than the initial investment. In the year 3, the recovery amount is higher than initial investment. It means the discounted payback period will be in between the year 2 and 3.

$\text{Discounted payback period}=2\text{ years}+\left(\frac{\text{Initial cost}-\left(\begin{array}{l}\text{Accumulated discounted }\\ \text{cash inflows in year 2}\end{array}\right)}{\text{Discounted cash inflows in year 3}}\right)$

Calculate the discounted payback period for Project B, if initial cost is $65,000:

$\begin{array}{c}\text{Discounted payback period}=2\text{ years}+\left(\frac{\text{Initial cost}-\left(\begin{array}{l}\text{Accumulated discounted }\\ \text{cash inflows in year 2}\end{array}\right)}{\text{Discounted cash inflows in year 3}}\right)\\ =2\text{years}+\frac{\$65,000-\$50,653.36}{\$18,645.38}\\ =2\text{years}+\text{0}\text{.77 year}\\ =2.\text{77 years}\end{array}$

Hence, the discounted payback period for Project B is 2.77 years.

Summary Introduction

To compute: The NPV (Net present value).

Introduction:

Capital budgeting is a process, where the business identifies and assesses the potential investments or expenses that are larger in general.

Answer to Problem 17QP

Here, Project A must be accepted as the NPV is higher in Project A.

Explanation of Solution

Given information:

The cash flows for two mutually exclusive projects are $58,000, $85,000, $85,000, $572,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $455,000. The cash flows of Project B are $31,000, $28,000, $25,500, and $19,000 respectively for year 1, year 2, year 3, and year 4. The initial investment for Project B is $65,000. The rate of return is 11%.

Formula to calculate the NPV:

$\text{NPV}=\left[\begin{array}{l}\frac{\text{Cash flow of Year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow of Year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}4}{1+\text{Rate of discount}}-\\ \text{Initial investment}\end{array}\right]$

Calculate the NPV for Project A:

$\begin{array}{c}\text{NPV for Project A}=\left[\begin{array}{l}\frac{\text{Cash flow of Year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow of Year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}4}{1+\text{Rate of discount}}-\\ \text{Initial investment}\end{array}\right]\\ =\frac{\text{\$58,000}}{1.11}+\frac{\$85,000}{{\left(\text{1}\text{.11}\right)}^{\text{2}}}+\frac{\$85,000}{{\left(\text{1}\text{.11}\right)}^3}+\frac{\$572,000}{{\left(\text{1}\text{.11}\right)}^4}-\$455,000\\ =\$52,252.25+\$68,987.91+\$62,151.27+\$376,794.12-\$455,000\\ =105,185.54\end{array}$

Hence, the NPV for Project A is $105,185.54.

Calculate the NPV for Project B:

$\begin{array}{c}\text{NPV for Project B}=\left[\begin{array}{l}\frac{\text{Cash flow of Year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow of Year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}4}{1+\text{Rate of discount}}-\\ \text{Initial investment}\end{array}\right]\\ =\frac{\text{\$31,000}}{1.11}+\frac{\$28,000}{{\left(\text{1}\text{.11}\right)}^{\text{2}}}+\frac{\$25,500}{{\left(\text{1}\text{.11}\right)}^3}+\frac{\$19,000}{{\left(\text{1}\text{.11}\right)}^4}-\$65,000\\ =\$27,927.93+\$22,725.43+\$18,645.38+\$12,515.89-\$65,000\\ =\$16,814.63\end{array}$

Hence, the NPV for Project B is $16,814.63.

Summary Introduction

To compute: The IRR for the project.

Introduction:

Capital budgeting is a process, where the business identifies and assesses the potential investments or expenses that are larger in general.

Answer to Problem 17QP

Here, Project B must be accepted as the IRR is higher in Project B.

Explanation of Solution

Given information:

The cash flows for two mutually exclusive projects are $58,000, $85,000, $85,000, $572,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $455,000. The cash flows of Project B are $31,000, $28,000, $25,500, and $19,000 respectively for year 1, year 2, year 3, and year 4. The initial investment for Project B is $65,000. The rate of return is 11%.

Equation of IRR of Project A:

$0=-\$455,000+\frac{\$58,000}{\left(1+\text{IRR}\right)}+\frac{\$85,000}{{\left(1+\text{IRR}\right)}^2}+\frac{\$85,000}{{\left(1+\text{IRR}\right)}^3}+\frac{\$572,000}{{\left(1+\text{IRR}\right)}^4}$

Compute IRR for Project A using a spreadsheet:

Step 1:

• Type the equation of NPV in H6 in the spreadsheet and consider the IRR value as H7

Step 2:

• Assume the IRR value as 10%

Step 3:

• In the spreadsheet go to data and select the what-if analysis
• In what-if analysis select goal seek
• In set cell select H6 (the Formula)
• The “To value” is considered as 0 (the assumption value for NPV)
• The H7 cell is selected for the by changing cell

Step 4:

• Following the previous step click OK in the goal seek. The goal seek status appears with the IRR value

Step 5:

• Thevalue appears to be 18.1542216674247%

Hence, the IRRvalue is 18.15%.

Equation of IRR of Project B:

$0=-\$65,000+\frac{\$31,000}{\left(1+\text{IRR}\right)}+\frac{\$28,000}{{\left(1+\text{IRR}\right)}^2}+\frac{\$25,500}{{\left(1+\text{IRR}\right)}^3}+\frac{\$19,000}{{\left(1+\text{IRR}\right)}^4}$

Compute IRR for Project B using a spreadsheet:

Step 1:

• Type the equation of NPV in H6 in the spreadsheet and consider the IRR value as H7

Step 2:

• Assume the IRR value as 10%

Step 3:

• In the spreadsheet go to data and select the what-if analysis
• In what-if analysis select goal seek
• In set cell select H6 (the Formula)
• The “To value” is considered as 0 (the assumption value for NPV)
• The H7 cell is selected for the by changing cell

Step 4:

• Following the previous step click OK in the goal seek. The goal seek status appears with the IRR value

Step 5:

• Thevalue appears to be 23.6503617037489%

Hence, the IRRvalue is 23.65%.

Summary Introduction

To compute: The profitability index.

Introduction:

Capital budgeting is a process, where the business identifies and assesses the potential investments or expenses that are larger in general.

Answer to Problem 17QP

Here, Project B must be accepted as the PI is higher in Project B.

Explanation of Solution

Given information:

The cash flows for two mutually exclusive projects are $58,000, $85,000, $85,000, $572,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $455,000. The cash flows of Project B are $31,000, $28,000, $25,500, and $19,000 respectively for year 1, year 2, year 3, and year 4. The initial investment for Project B is $65,000. The rate of return is 11%.

Formula to compute the profitability index:

$\text{PI}\left(\text{Profitability Index}\right)=\frac{\left(\begin{array}{l}\frac{\text{Cash flow for year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow for year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year4}}{1+\text{Rate of discount}}\end{array}\right)}{\text{Initial investment}}$

Compute the profitability index for Project A:

$\begin{array}{c}\text{PI}\left(\text{Profitability Index}\right)=\left[\frac{\left(\begin{array}{l}\frac{\text{Cash flow for year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow for year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year4}}{1+\text{Rate of discount}}\end{array}\right)}{\text{Initial investment}}\right]\\ =\left[\frac{\frac{\text{\$58,000}}{1.11}+\frac{\$85,000}{{\left(\text{1}\text{.11}\right)}^{\text{2}}}+\frac{\$85,000}{{\left(\text{1}\text{.11}\right)}^3}+\frac{\$572,000}{{\left(\text{1}\text{.11}\right)}^4}}{\$455,000}\right]\\ =\frac{\$52,252.25+\$68,987.91+\$62,151.27+\$376,794.12}{\$455,000}\\ =1.23\end{array}$

Hence, the profitability index for Project A is $1.23.

Compute the profitability index for Project B:

$\begin{array}{c}\text{PI}\left(\text{Profitability Index}\right)=\left[\frac{\left(\begin{array}{l}\frac{\text{Cash flow for year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow for year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year4}}{1+\text{Rate of discount}}\end{array}\right)}{\text{Initial investment}}\right]\\ \text{=}\frac{\frac{\text{\$31,000}}{1.11}+\frac{\$28,000}{{\left(\text{1}\text{.11}\right)}^{\text{2}}}+\frac{\$25,500}{{\left(\text{1}\text{.11}\right)}^3}+\frac{\$19,000}{{\left(\text{1}\text{.11}\right)}^4}}{\$65,000}\\ =\frac{\$27,927.93+\$22,725.43+\$18,645.38+\$12,515.89}{\$65,000}\\ =1.26\end{array}$

Hence, the profitability index for Project B is $1.26.

Summary Introduction

To discuss: The project that Person X will select with a reason.

Introduction:

Capital budgeting is a process, where the business identifies and assesses the potential investments or expenses that are larger in general.

Explanation of Solution

Given information:

The cash flows for two mutually exclusive projects are $58,000, $85,000, $85,000, $572,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $455,000. The cash flows of Project B are $31,000, $28,000, $25,500, and $19,000 respectively for year 1, year 2, year 3, and year 4. The initial investment for Project B is $65,000. The rate of return is 11%.

Explanation:

In this case, the criteria of NPV denote that Project A must be accepted, while payback period, discounted payback, profitability index, and IRR denote that Project B must be accepted. The overall decision must be based on the NPV as it does not have the ranking problem when compared with the other techniques of capital budgeting. Hence, Project A must be accepted.