Chapter 8, Problem 15QP

a)

Summary Introduction

To compute: The payback period and the investment that must be selected.

Introduction:

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

Answer to Problem 15QP

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 $38,000, $47,000, $62,000, $455,000 for Project A for year 1, year 2, year 3, and year 4respectively. Project A has an initial investment of -$365,000. The cash flows of Project B are $20,300, $15,200, $14,100, and $11,200 for year 1, year 2, year 3, and year 4respectively. The initial investment for Project B is -$40,000. The rate of return is 13%.

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 Project A:

$\begin{array}{c}\text{Payback period}=\left[\begin{array}{l}\text{Maximum number of years to }\\ \text{recover the amount}+\left(\frac{\text{Amount remaining to be recovered}}{4^{\text{th}}\text{ year cash flow}}\right)\end{array}\right]\\ =3+\left(\frac{\$218,000}{\$455,000}\right)\\ =3.47\text{ years}\end{array}$

Hence, the payback period is 3.47 years for Project A.

Compute the payback period of Project B:

$\begin{array}{c}\text{Payback period}=\left[\begin{array}{l}\text{Maximum number of years to }\\ \text{recover the amount}+\left(\frac{\text{Amount remaining to be recovered}}{2^{\text{nd}}\text{ year cash flow}}\right)\end{array}\right]\\ =1+\left(\frac{\$19,700}{\$15,200}\right)\\ =2.29\text{years}\end{array}$

Hence, the payback period is 2.29 years for Project B.

b)

Summary Introduction

To compute: The NPV (Net Present Value) and the investment that must be selected.

Introduction:

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

Answer to Problem 15QP

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 $38,000, $47,000, $62,000, $455,000 for Project A for year 1, year 2, year 3, and year 4respectively. Project A has an initial investment of -$365,000. The cash flows of Project B are $20,300, $15,200, $14,100, and $11,200 for year 1, year 2, year 3, and year 4respectively. The initial investment for Project B is -$40,000. The rate of return is 13%.

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:

$\begin{array}{c}\text{NPV for Project A}=\left[\begin{array}{l}\frac{\text{Cash flow of Year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow of Year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}4}{1+\text{Rate of discount}}-\\ \text{Initial investment}\end{array}\right]\\ =\frac{\text{\$38,000}}{1.13}+\frac{\$47,000}{{\left(\text{1}\text{.13}\right)}^{\text{2}}}+\frac{\$62,000}{{\left(\text{1}\text{.13}\right)}^3}+\frac{\$455,000}{{\left(\text{1}\text{.13}\right)}^4}-\$365,000\\ =\$33,628.31858+\$36,807.89412+\$42,969.11006+\\ \$279,060.0211-\$365,000\\ =\$27,465.34385\end{array}$

Hence, the NPV for Project A is $27,465.34385.

Calculate the NPV for Project B:

$\begin{array}{c}\text{NPV for Project A}=\left[\begin{array}{l}\frac{\text{Cash flow of Year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow of Year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow of Year}4}{1+\text{Rate of discount}}-\\ \text{Initial investment}\end{array}\right]\\ =\frac{\text{\$20,300}}{1.13}+\frac{\$15,200}{{\left(\text{1}\text{.13}\right)}^{\text{2}}}+\frac{\$14,100}{{\left(\text{1}\text{.13}\right)}^3}+\frac{\$11,200}{{\left(\text{1}\text{.13}\right)}^4}-\$40,000\\ =\$17,964.60177+\$11,903.2959+\$9,772.007288+\\ \$6,869.16975-\$40,000\\ =\$6,509.074\end{array}$

Hence, the NPV for Project B is $6,509.074.

c)

Summary Introduction

To compute: The IRR for the project and the investment that must be selected.

Introduction:

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

Answer to Problem 15QP

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 $38,000, $47,000, $62,000, $455,000 for Project A for year 1, year 2, year 3, and year 4respectively. Project A has an initial investment of -$365,000. The cash flows of Project B are $20,300, $15,200, $14,100, and $11,200 for year 1, year 2, year 3, and year 4respectively. The initial investment for Project B is -$40,000. The rate of return is 13%.

Equation of the IRR of Project A:

$0=-\$365,000+\frac{\$38,000}{\left(1+\text{IRR}\right)}+\frac{\$47,000}{{\left(1+\text{IRR}\right)}^2}+\frac{\$62,000}{{\left(1+\text{IRR}\right)}^3}+\frac{\$455,000}{{\left(1+\text{IRR}\right)}^4}$

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 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 “By changing cell”.

Step 4:

• Following the previous step, click OK in the Goal Seek Status. Goal Seek Status appears with the IRR value.

Step 5:

• The value appears to be 15.4115219168648%.

Hence, the IRR value is 15.411%.

Equation of IRR of Project B:

$0=-\$40,000+\frac{\$20,300}{\left(1+\text{IRR}\right)}+\frac{\$15,200}{{\left(1+\text{IRR}\right)}^2}+\frac{\$14,100}{{\left(1+\text{IRR}\right)}^3}+\frac{\$11,200}{{\left(1+\text{IRR}\right)}^4}$

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 What-if analysis.
• In the 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 “By changing cell”.

Step 4:

• Following the previous step, click OK in the Goal seek. Goal Seek Status appears with the IRR value.

Step 5:

• The value appears to be 21.5074361897684%.

Hence, the IRR value is 21.50%.

d)

Summary Introduction

To compute: The profitability index.

Introduction:

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

Answer to Problem 15QP

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 $38,000, $47,000, $62,000, $455,000 for Project A for year 1, year 2, year 3, and year 4respectively. Project A has an initial investment of -$365,000. The cash flows of Project B are $20,300, $15,200, $14,100, and $11,200 for year 1, year 2, year 3, and year 4respectively. The initial investment for Project B is -$40,000. The rate of return is 13%.

Formula to compute the profitability index:

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

Compute the profitability index for Project A:

$\begin{array}{c}\text{PI}\left(\text{Profitability Index}\right)=\left[\frac{\left(\begin{array}{l}\frac{\text{Cash flow for year}1}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year}2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow for year}3}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year4}}{1+\text{Rate of discount}}\end{array}\right)}{\text{Initial investment}}\right]\\ =\left[\frac{\frac{\text{\$38,000}}{1.13}+\frac{\$47,000}{{\left(\text{1}\text{.13}\right)}^{\text{2}}}+\frac{\$62,000}{{\left(\text{1}\text{.13}\right)}^3}+\frac{\$455,000}{{\left(\text{1}\text{.13}\right)}^4}}{\$365,000}\right]\\ =\frac{\begin{array}{l}\$33,628.31858+\$36,807.89412+\\ \$42,969.11006+\$279,060.0211\end{array}}{\$365,000}\\ =1.075\end{array}$

Hence, the profitability index for Project A is $1.075.

Compute the profitability index for Project B:

$\begin{array}{c}\text{PI}\left(\text{Profitability Index}\right)=\left[\frac{\left(\begin{array}{l}\frac{\text{Cash flow for year }1}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year }2}{1+\text{Rate of discount}}+\\ \frac{\text{Cash flow for year }3}{1+\text{Rate of discount}}+\frac{\text{Cash flow for year 4}}{1+\text{Rate of discount}}\end{array}\right)}{\text{Initial investment}}\right]\\ \text{=}\frac{\frac{\text{\$20,300}}{1.13}+\frac{\$15,200}{{\left(\text{1}\text{.13}\right)}^{\text{2}}}+\frac{\$14,100}{{\left(\text{1}\text{.13}\right)}^3}+\frac{\$11,200}{{\left(\text{1}\text{.13}\right)}^4}}{\$40,000}\\ =\frac{\$17,964.60177+\$11,903.82959+\$9,772.007288+\$6,869.16975}{\$40,000}\\ =1.162\end{array}$

Hence, the profitability index for Project B is $1.162.

e)

Summary Introduction

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

Introduction:

Capital budgeting is a process where a 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 $38,000, $47,000, $62,000, $455,000 for Project A for year 1, year 2, year 3, and year 4respectively. Project A has an initial investment of -$365,000. The cash flows of Project B are $20,300, $15,200, $14,100, and $11,200 for year 1, year 2, year 3, and year 4respectively. The initial investment for Project B is -$40,000. The rate of return is 13%.

Explanation:

In this case, the criteria of NPV denote that Project A must be accepted while the payback period, 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 ranking problem compared to the other techniques of capital budgeting. Hence, Project A must be accepted.