Chapter 9, Problem 20QP

Summary Introduction

To calculate: The MIRR (Modified internal rate of return) for the project utilizing all three methods at a rate of discount and rate of reinvestment of 11% and 8% respectively

Introduction:

MIRR is the Modified internal rate of return, which is a financial measure of attracting the investments. It is utilized in capital budgeting to rank the alternative investments of same size.

Answer to Problem 20QP

The MIRR for the project using the discounted approach is 22.97%, reinvestment approach is 15.50%, and combination approach is 15%.

Explanation of Solution

Given information:

Company R is assessing a project, where the cash flows are$15,700, $19,400, $24,300, $18,100, and -$9,400 for year 1, 2, 3, 4, and 5 respectively. The initial cost is $41,000. The rate of discount and the rate of reinvestment are 11% and 8% respectively.

Discounted approach:

In this approach, compute the negative cash outflows value at the year 0. On the other hand, the positive cash flows remain at its time of occurrence. The rate of discount is 11%. Hence, discount the cash outflows to year 0.

$\begin{array}{c}\text{Time 0 cash flow}=\text{Initial cost}+\frac{\text{Cash outflows}}{{\left(1+r\right)}^t}\\ =-\$41,000+\frac{-\$9,400}{{\left(1+0.11\right)}^5}\\ =-\$46,578.44\end{array}$

Hence, the discounted cash flow at time 0 is -$46,578.44.

Equation of MIRR in discounted approach:

$0=-\$46,578.44+\frac{\$15,700}{\left(1+\text{MIRR}\right)}+\frac{\$19,400}{{\left(1+\text{MIRR}\right)}^2}+\frac{\$24,300}{{\left(1+\text{MIRR}\right)}^3}+\frac{\$18,100}{{\left(1+\text{MIRR}\right)}^4}$

Compute MIRR 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 the what-if analysis.
• In what-if analysis, select goal seek.
• In “Set cell”, select H6 (the formulae).
• 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 IRRvalue.

Step 5:

• Thevalue appears to be 22.9719885171979%.

Hence, the MIRRvalue is 22.97%.

Reinvestment approach:

In this approach, compute the future value of all the cash flows excluding the initial cost at the closure of the project. The rate of reinvestment is 8%. Hence, compute the reinvesting cash flows to year 5 as:

$\begin{array}{c}\text{Time }5\text{ cash flow}=\text{Cash flows }\left(\begin{array}{l}\text{year}1{\left(1+r\right)}^4+\text{year}2{\left(1+r\right)}^3+\text{year}3{\left(1+r\right)}^2+\\ \text{year}4\left(1+r\right)+\text{year}5\end{array}\right)\\ =\left(\begin{array}{l}\$15,700{\left(1+0.08\right)}^4+\$19,400{\left(1+0.08\right)}^3+\$24,300{\left(1+0.08\right)}^2+\\ \$18,100\left(1+0.08\right)-\$9,400\end{array}\right)\\ =\left(\begin{array}{l}\$15,700\left(1.36048896\right)+\$19,400\left(1.259712\right)+\$24,300\left(1.1664\right)+\\ \$18,100\left(1.08\right)-\$9,400\end{array}\right)\\ =\$84,289.61\end{array}$

Hence, the reinvesting cash flow at time 5 is $84,289.61.

Equation of MIRR in reinvestment approach:

$0=-\$41,000+\frac{\$84,289.61}{{\left(1+\text{MIRR}\right)}^5}$

Compute the MIRR:

$\begin{array}{c}0=-\$41,000+\frac{\$84,289.61}{{\left(1+\text{MIRR}\right)}^5}\\ \frac{\$84,289.61}{\$41,000}={\left(1+\text{MIRR}\right)}^5\\ \text{MIRR}={\left(\frac{\$84,289.61}{\$41,000}\right)}^{1/5}-1\\ \text{MIRR}=0.1550\text{ or }15.50\%\end{array}$

Hence, the MIRR is 15.50%.

Combination approach:

In this approach, compute all the cash outflows at year 0 utilizing the rate of discount, and all the cash inflows at the closure of the project utilizing the rate of investment. Hence, the value of the cash flows is as follows:

$\begin{array}{c}\text{Time 0 cash flow}=\text{Initial cost}+\frac{-\text{Cash outflows}}{{\left(1+r\right)}^t}\\ =-\$41,000+\frac{-\$9,400}{{\left(1+0.11\right)}^5}\\ =-\$46,578.44\end{array}$

Hence, the total cash outflow at year 0 is -$46.578.44.

$\begin{array}{c}\text{Time }5\text{ cash flow}=\text{Cash flows }\left(\begin{array}{l}\text{year}1{\left(1+r\right)}^4+\text{year}2{\left(1+r\right)}^3+\text{year}3{\left(1+r\right)}^2+\\ \text{year}4\left(1+r\right)+\text{year}5\end{array}\right)\\ =\left(\begin{array}{l}\$15,700{\left(1+0.08\right)}^4+\$19,400{\left(1+0.08\right)}^3+\$24,300{\left(1+0.08\right)}^2+\\ \$18,100\left(1+0.08\right)\end{array}\right)\\ =\left(\begin{array}{l}\$15,700\left(1.36048896\right)+\$19,400\left(1.259712\right)+\\ \$24,300\left(1.1664\right)+\$18,100\left(1.08\right)\end{array}\right)\\ =\$93,689.61\end{array}$

Hence, the value of total cash inflows is $93,689.61.

Equation of MIRR in combination approach:

$0=-\$46,836.66+\frac{\$98,120.77}{{\left(1+\text{MIRR}\right)}^5}$

Compute the MIRR:

$\begin{array}{c}0=-\$46,578.44+\frac{\$93,689.61}{{\left(1+\text{MIRR}\right)}^5}\\ \frac{\$93,689.61}{\$46,578.44}={\left(1+\text{MIRR}\right)}^5\\ \text{MIRR}={\left(\frac{\$93,689.61}{\$46,578.44}\right)}^{1/5}-1\\ \text{MIRR}=0.1500\text{ or }15\%\end{array}$

Hence, the MIRR is 15%.