Chapter 9, Problem 17QP

a)

Summary Introduction

To compute: The payback period.

Introduction:

Capital budgeting is a process, where the business identifies and assesses 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 considered. The cash flows if Project A are $45,000, $65,000, $65,000, $440,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $350,000. The cash flows of Project B are $24,000, $22,000, $19,500, and $14,600 respectively for year 1, year 2, year 3, and year 4. The initial investment of Project B is $50,000. The rate of return is 15%.

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:

$\begin{array}{c}\text{Payback period}=\left[\begin{array}{l}\text{Maximum number of years to }\\ \text{recover the amount}+\left(\frac{\text{Amount remaining to be recovered}}{4^{\text{th}}\text{ year cash flow}}\right)\end{array}\right]\\ =3+\left(\frac{\$175,000}{\$440,000}\right)\\ =3.40\text{ years}\end{array}$

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

Compute the payback period of a project for Project B:

$\begin{array}{c}\text{Payback period}=\left[\begin{array}{l}\text{Maximum number of years to }\\ \text{recover the amount}+\left(\frac{\text{Amount remaining to be recovered}}{3^{\text{rd}}\text{ year cash flow}}\right)\end{array}\right]\\ =2+\left(\frac{\$4,000}{\$19,500}\right)\\ =2.21\text{ years}\end{array}$

Hence, the payback period for Project B is 2.21 years.

b)

Summary Introduction

To compute: The discounted payback period.

Introduction:

Capital budgeting is a process, where the business identifies and assesses 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 considered. The cash flows if Project A are $45,000, $65,000, $65,000, $440,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $350,000. The cash flows of Project B are $24,000, $22,000, $19,500, and $14,600 respectively for year 1, year 2, year 3, and year 4. The initial investment of Project B is $50,000. The rate of return is 15%.

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

Year	Net Cash Inflows
	Annual	Today’s value (discounted) at 15%	Accumulated
1	$45,000	$39,130.43 $\left(\frac{\$45,000}{1.15}\right)$	$39,130.43
2	$65,000	$49,149.34 $\left(\frac{\$65,000}{{1.15}^2}\right)$	$88,279.77
3	$65,000	$42,738.56 $\left(\frac{\$65,000}{{1.15}^3}\right)$	$131,018.33
4	$440,000	$251,571.43 $\left(\frac{\$440,000}{{1.15}^4}\right)$	$382,589.76

Formula to compute the discounted payback period:

By the end of year 3, the recovery amount will be shorter than the initial investment. In year 4, the recovery amount will be higher than initial investment. It means the discounted payback period will be in between 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 the initial cost is $455,000:

$\begin{array}{c}\text{Discounted payback period}=3\text{ years}+\left(\frac{\text{Initial cost}-\left(\begin{array}{l}\text{Accumulated discounted }\\ \text{cash inflows in year 3}\end{array}\right)}{\text{Discounted cash inflows in year 4}}\right)\\ =3\text{years}+\frac{\$350,000-\$131,018.33}{\$382,589.76}\\ =3\text{years}+\text{0}\text{.57 year}\\ =3.\text{57 years}\end{array}$

Hence, the discounted payback period for Project A is 3.57 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	$24,000	$20,869.57 $\left(\frac{\$24,000}{1.15}\right)$	$20,869.57
2	$22,000	$16,635.16 $\left(\frac{\$22,000}{{1.15}^2}\right)$	$37,504.73
3	$19,500	$12,821.57 $\left(\frac{\$19,500}{{1.15}^3}\right)$	$50,326.30
4	$14,600	$8,347.60 $\left(\frac{\$14,600}{{1.15}^4}\right)$	$58,673.90

Formula to calculate the discounted payback period for Project B, if the initial cost is $50,000:

By the end of year 2, the recovery amount will be shorter than the initial investment. In year 3, the recovery amount will be higher than initial investment. It means the discounted payback period will be in between 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 $50,000:

$\begin{array}{c}\text{Discounted payback period}=2\text{ years}+\left(\frac{\text{Initial cost}-\left(\begin{array}{l}\text{Accumulated discounted }\\ \text{cash inflows in year 2}\end{array}\right)}{\text{Discounted cash inflows in year 3}}\right)\\ =2\text{years}+\frac{\$50,000-\$37,504.73}{\$50,326.30}\\ =2\text{years}+\text{0}\text{.25 year}\\ =2.\text{25 years}\end{array}$

Hence, the discounted payback period for Project B is 2.25 years.

c)

Summary Introduction

To compute: The NPV (Net present value).

Introduction:

Capital budgeting is a process, where the business identifies and assesses 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 considered. The cash flows if Project A are $45,000, $65,000, $65,000, $440,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $350,000. The cash flows of Project B are $24,000, $22,000, $19,500, and $14,600 respectively for year 1, year 2, year 3, and year 4. The initial investment of Project B is $50,000. The rate of return is 15%.

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:

$\begin{array}{c}\text{NPV for Project A}=\left[\begin{array}{l}\frac{\text{Cash flow of Year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow of Year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}4}{1+\text{Rate of discount}}-\\ \text{Initial investment}\end{array}\right]\\ =\frac{\text{\$45,000}}{1.15}+\frac{\$65,000}{{\left(\text{1}\text{.15}\right)}^{\text{2}}}+\frac{\$65,000}{{\left(\text{1}\text{.15}\right)}^3}+\frac{\$440,000}{{\left(\text{1}\text{.15}\right)}^4}-\$350,000\\ =\$39,130.43+\$49,149.34+\$42,738.56+\$251,571.43-\$350,000\\ =\$32,589.76\end{array}$

Hence, the NPV for Project A is $32,589.76.

Calculate the NPV for Project B:

$\begin{array}{c}\text{NPV for Project B}=\left[\begin{array}{l}\frac{\text{Cash flow of Year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow of Year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}4}{1+\text{Rate of discount}}-\\ \text{Initial investment}\end{array}\right]\\ =\frac{\text{\$24,000}}{1.15}+\frac{\$22,000}{{\left(\text{1}\text{.15}\right)}^{\text{2}}}+\frac{\$19,500}{{\left(\text{1}\text{.15}\right)}^3}+\frac{\$14,600}{{\left(\text{1}\text{.15}\right)}^4}-\$50,000\\ =\$20,869.57+\$16,635.16+\$12,821.57+\$8,347.60-\$50,000\\ =\$8,673.9\end{array}$

Hence, the NPV for Project B is $8,673.9.

d)

Summary Introduction

To compute: The IRR for the project.

Introduction:

Capital budgeting is a process, where the business identifies and assesses 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 considered. The cash flows if Project A are $45,000, $65,000, $65,000, $440,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $350,000. The cash flows of Project B are $24,000, $22,000, $19,500, and $14,600 respectively for year 1, year 2, year 3, and year 4. The initial investment of Project B is $50,000. The rate of return is 15%.

Equation of IRR of Project A:

$0=-\$350,000+\frac{\$45,000}{\left(1+\text{IRR}\right)}+\frac{\$65,000}{{\left(1+\text{IRR}\right)}^2}+\frac{\$65,000}{{\left(1+\text{IRR}\right)}^3}+\frac{\$440,000}{{\left(1+\text{IRR}\right)}^4}$

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 IRRvalue as 10%

Step 3:

• In the spreadsheet, go to data and select 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 status. The goal seek status appears with the IRRvalue

Step 5:

• Thevalue appears to be 18.1354509925898%.

Hence, the IRRvalue is 18.14%.

Equation of IRR of Project B:

$0=-\$50,000+\frac{\$24,000}{\left(1+\text{IRR}\right)}+\frac{\$22,000}{{\left(1+\text{IRR}\right)}^2}+\frac{\$19,500}{{\left(1+\text{IRR}\right)}^3}+\frac{\$14,600}{{\left(1+\text{IRR}\right)}^4}$

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 IRRvalue as 10%

Step 3:

• In the spreadsheet go to data, and select 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 status. The goal seek status appears with the IRRvalue

Step 5:

• Thevalue appears to be 24.0789943942319%.

Hence, the IRRvalue is 24%.

e)

Summary Introduction

To compute: The profitability index.

Introduction:

Capital budgeting is a process, where the business identifies and assesses 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 considered. The cash flows if Project A are $45,000, $65,000, $65,000, $440,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $350,000. The cash flows of Project B are $24,000, $22,000, $19,500, and $14,600 respectively for year 1, year 2, year 3, and year 4. The initial investment of Project B is $50,000. The rate of return is 15%.

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:

$\begin{array}{c}\text{PI}\left(\text{Profitability Index}\right)=\left[\frac{\left(\begin{array}{l}\frac{\text{Cash flow for year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow for year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year4}}{1+\text{Rate of discount}}\end{array}\right)}{\text{Initial investment}}\right]\\ =\left[\frac{\frac{\text{\$45,000}}{1.15}+\frac{\$65,000}{{\left(\text{1}\text{.15}\right)}^{\text{2}}}+\frac{\$65,000}{{\left(\text{1}\text{.15}\right)}^3}+\frac{\$440,000}{{\left(\text{1}\text{.15}\right)}^4}}{\$350,000}\right]\\ =\frac{\$39,130.43+\$49,149.34+\$42,738.56+\$251,571.43}{\$350,000}\\ =1.09\end{array}$

Hence, the profitability index for Project A is $1.09.

Compute the profitability index for Project B:

$\begin{array}{c}\text{PI}\left(\text{Profitability Index}\right)=\left[\frac{\left(\begin{array}{l}\frac{\text{Cash flow for year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow for year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year4}}{1+\text{Rate of discount}}\end{array}\right)}{\text{Initial investment}}\right]\\ \text{=}\frac{\frac{\text{\$24,000}}{1.15}+\frac{\$22,000}{{\left(\text{1}\text{.15}\right)}^{\text{2}}}+\frac{\$19,500}{{\left(\text{1}\text{.15}\right)}^3}+\frac{\$14,600}{{\left(\text{1}\text{.15}\right)}^4}}{\$50,000}\\ =\frac{\$20,869.57+\$16,635.16+\$12,821.57+\$8,347.60}{\$50,000}\\ =1.17\end{array}$

Hence, the profitability index for Project B is $1.17.

f)

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 potential investments or expenses that are larger in general.

Explanation of Solution

Given information:

The cash flows for two mutually exclusive projects are considered. The cash flows if Project A are $45,000, $65,000, $65,000, $440,000 for Project A respectively for year 1, year 2, year 3, and year 4. Project A has an initial investment of $350,000. The cash flows of Project B are $24,000, $22,000, $19,500, and $14,600 respectively for year 1, year 2, year 3, and year 4. The initial investment of Project B is $50,000. The rate of return is 15%.

Explanation:

In this case, the criteria of NPV denote that Project A must be accepted, while payback period, discounted payback, profitability index, and IRR denotes 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.