Chapter 5, Problem 24QP

Calculating IRR The Utah Mining Corporation is set to open a gold mine near Provo, Utah. According to the treasurer, Monty Goldstein, “This is a golden opportunity.” The mine will cost $2,700,000 to open and will have an econo1nic life of 11 years. It will generate a cash inflow of $435,000 at the end of the first year, and the cash inflows are projected to grow at 8 percent per year for the next 10 years. After 11 years, the mine will be abandoned. Abandonment costs will be $400,000 at the end of Year 11.

1. a. What is the IRR for the gold mine?
2. b. The Utah Mining Corporation requires a return of 10 percent on such undertakings. Should the mine be opened?

Summary Introduction

To find: The IRR for the gold mine.

Internal Rate of Return (IRR):

The internal rate of return is that discounting rate of capital budgeting in which the NPV of cash flows of the project comes equal to zero. IRR is the discounting rate at which present value of cash inflow is equal to present value of cash outflow.

Explanation of Solution

Calculation of IRR of project with the help of excel function IRR.

Table (1)

The IRR for the gold mine is 19%.

Working notes:

Given,

Cash inflow at the end of first year is $435,000.

Growth rate of cash inflow is 8% per year or 0.08 per year.

Calculation of the present value of cash inflow of at the end of 2^nd year,

$\begin{array}{c}\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{inflow}\\ \text{at}\text{the}\text{end}\text{of}2\text{nd}\text{year}\end{array}\right]=\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{at}\text{the}\text{end}\text{of}1\text{st}\text{year}\\ \times \left(1+\text{Growth}\text{rate}\right)\end{array}\right]\\ =\$435,000\times \left(1+0.08\right)\\ =\$435,000\times 1.08\\ =\$469,800\end{array}$

Given,

Cash inflow at the end of 2nd year is $469,800.

Growth rate of cash inflow is 8% per year or 0.08 per year.

Calculation of the present value of cash inflow of at the end of 3rd year

$\begin{array}{c}\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{inflow}\\ \text{at}\text{the}\text{end}\text{of}3\text{rd}\text{year}\end{array}\right]=\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{at}\text{the}\text{end}\text{of}2\text{nd}\text{year}\\ \times \left(1+\text{Growth}\text{rate}\right)\end{array}\right]\\ =\$469,800\times \left(1+0.08\right)\\ =\$469,800\times 1.08\\ =\$507,384\end{array}$

Given,

Cash inflow at the end of 3rd year is $507,384.

Growth rate of cash inflow is 8% per year or 0.08 per year.

Calculation of the present value of cash inflow of at the end of 4th year

$\begin{array}{c}\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{inflow}\\ \text{at}\text{the}\text{end}\text{of}4\text{th}\text{year}\end{array}\right]=\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{at}\text{the}\text{end}\text{of}3\text{rd}\text{year}\\ \times \left(1+\text{Growth}\text{rate}\right)\end{array}\right]\\ =\$507,384\times \left(1+0.08\right)\\ =\$507,384\times 1.08\\ =\$547,974.72\end{array}$

Given,

Cash inflow at the end of 4th year is $547,947.72.

Growth rate of cash inflow is 8% per year or 0.08 per year.

Calculation of the present value of cash inflow of at the end of 5th year

$\begin{array}{c}\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{inflow}\\ \text{at}\text{the}\text{end}\text{of}5\text{th}\text{year}\end{array}\right]=\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{at}\text{the}\text{end}\text{of}4\text{th}\text{year}\\ \times \left(1+\text{Growth}\text{rate}\right)\end{array}\right]\\ =\$547,974.72\times \left(1+0.08\right)\\ =\$547,974.72\times 1.08\\ =\$591,812.69\end{array}$

Given,

Cash inflow at the end of 5th year is $591,812.69.

Growth rate of cash inflow is 8% per year or 0.08 per year.

Calculation of the present value of cash inflow of at the end of 6th year,

$\begin{array}{c}\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{inflow}\\ \text{at}\text{the}\text{end}\text{of}6\text{th}\text{year}\end{array}\right]=\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{at}\text{the}\text{end}\text{of}5\text{th}\text{year}\\ \times \left(1+\text{Growth}\text{rate}\right)\end{array}\right]\\ =\$591,812.69\times \left(1+0.08\right)\\ =\$591,812.69\times 1.08\\ =\$639,157.71\end{array}$

Given,

Cash inflow at the end of 6th year is $639,157.71.

Growth rate of cash inflow is 8% per year or 0.08 per year.

Calculation of the present value of cash inflow of at the end of 7th year,

$\begin{array}{c}\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{inflow}\\ \text{at}\text{the}\text{end}\text{of}7\text{th}\text{year}\end{array}\right]=\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{at}\text{the}\text{end}\text{of}6\text{th}\text{year}\\ \times \left(1+\text{Growth}\text{rate}\right)\end{array}\right]\\ =\$639,157.71\times \left(1+0.08\right)\\ =\$639,157.71\times 1.08\\ =\$690,290.33\end{array}$

Given,

Cash inflow at the end of 7th year is $690,290.33.

Growth rate of cash inflow is 8% per year or 0.08 per year.

Calculation of the present value of cash inflow of at the end of 8th year,

$\begin{array}{c}\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{inflow}\\ \text{at}\text{the}\text{end}\text{of}8\text{th}\text{year}\end{array}\right]=\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{at}\text{the}\text{end}\text{of}7\text{th}\text{year}\\ \times \left(1+\text{Growth}\text{rate}\right)\end{array}\right]\\ =\$690,290.33\times \left(1+0.08\right)\\ =\$690,290.33\times 1.08\\ =\$745,513.55\end{array}$

Given,

Cash inflow at the end of 8th year is $745,513.55.

Growth rate of cash inflow is 8% per year or 0.08 per year.

Calculation of the present value of cash inflow of at the end of 9th year,

$\begin{array}{c}\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{inflow}\\ \text{at}\text{the}\text{end}\text{of}9\text{th}\text{year}\end{array}\right]=\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{at}\text{the}\text{end}\text{of}8\text{th}\text{year}\\ \times \left(1+\text{Growth}\text{rate}\right)\end{array}\right]\\ =\$745,513.55\times \left(1+0.08\right)\\ =\$745,513.55\times 1.08\\ =\$805,154.64\end{array}$

Given,

Cash inflow at the end of 9th year is $805,154.64.

Growth rate of cash inflow is 8% per year or 0.08 per year.

Calculation of the present value of cash inflow of at the end of 10th year,

$\begin{array}{c}\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{inflow}\\ \text{at}\text{the}\text{end}\text{of}10\text{th}\text{year}\end{array}\right]=\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{at}\text{the}\text{end}\text{of}9\text{th}\text{year}\\ \times \left(1+\text{Growth}\text{rate}\right)\end{array}\right]\\ =\$805,154.64\times \left(1+0.08\right)\\ =\$805,154.64\times 1.08\\ =\$869,567.01\end{array}$

Given,

Cash inflow at the end of 10th year is $869,567.01.

Growth rate of cash inflow is 8% per year or 0.08 per year.

Abandonment cost of mine at the end of 11^th year is $400,000.

Calculation of the present value of cash inflow of at the end of 11th year,

$\begin{array}{c}\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{inflow}\\ \text{at}\text{the}\text{end}\text{of}11\text{th}\text{year}\end{array}\right]=\left[\begin{array}{l}\text{Present}\text{value}\text{of}\text{cash}\text{at}\\ \text{the}\text{end}\text{of}9\text{th}\text{year}\\ \times \left(1+\text{Growth}\text{rate}\right)\end{array}\right]+\left[\begin{array}{l}\text{Abandonment}\\ \text{cost}\text{ofmine}\end{array}\right]\\ =\left[\$869,567.01\times \left(1+0.08\right)\right]+\$400,000\\ =\left[\$869,567.01\times 1.08\right]+\$400,000\\ =\$1,339,132.37\end{array}$

Conclusion:

The IRR for the gold mine is 19%.

Summary Introduction

To explain: Project should be opened or not if company requires 11% return.

Internal Rate of Return (IRR):

The internal rate of return is that discounting rate of capital budgeting in which the NPV of cash flows of the project comes equal to zero. IRR is the discounting rate at which present value of cash inflow is equal to present value of cash outflow.

Answer to Problem 24QP

Solution:

Yes, the project should be opened.

Explanation of Solution

The IRR form the gold mine is 19% and the return that company requires is 11%. Since the IRR of the project is higher than the return requires by the company therefore the company should opened the project.

Conclusion

Project should be opened