Chapter 10, Problem 32QP

Summary Introduction

To find: The net present value and the internal rate of return.

Introduction:

The variation between the present value of the cash outflows and the present value of the cash inflows are known as the net present value. In capital budgeting, the net present value is utilized to analyze the profitability of a project or investment. The rate of return (which compares the initial investment and the present value of net cash inflows)isreferred to as internal rate of return. This is also called actual rate of return.

Answer to Problem 32QP

The net present value is $2,098,569.18 and the internal rate of return is 21.54%.

Explanation of Solution

Given information:

Company A projects the unit sale for the new 7 octave voice emulation implant as follows:

• Year 1’s unit sales is 81,000
• Year 2’s unit sales is 94,000
• Year 3’s unit sales is 108,000
• Year 4’s unit sales is 103,000
• Year 5’s unit sales is 84,000

The production implant needs $1,600,000 in the net working capital to begin their production activities. The extra net working capital investment for every year is equivalent to the 15% of the sales that is projected, which has to rise for the following year. The total fixed cost is $1,500,000 for a year, the unit price is $380, and the variable production cost is $265. The installation cost of the equipment is $21,000,000.

The equipment is qualified in the 7 Year MACRS depreciation under the property class. In 5 years, the equipment sale can be sold for 20% of its acquisition cost. The marginal tax bracket is 35% and has the required rate of return,which is 18%.

MACRS depreciation table for 7 year:

MACRS Depreciation table for seven year
Year	Seven year
1	14.29%
2	24.49%
3	17.49%
4	12.49%
5	8.93%
6	8.92%
7	8.93%
8	4.46%

Computation of the net present value:

Computation of the cash outflow:

$\begin{array}{c}\text{Cash outflow = }\left(\begin{array}{l}-\text{Capital spending}\left(\text{Installed cost}\right)+\\ -\text{Initial net working capital}\end{array}\right)\\ =\left(-\$21,000,000\right)+\left(-\$1,600,000\right)\\ =-\$22,600,000\end{array}$

Table showing the cash inflows:

Year	1	2	3	4	5
Ending book value	$17,999,100	$12,856,200	$9,183,300	$6,560,400	$4,685,100
Sales	$30,780,000	$35,720,000	$41,040,000	$39,140,000	$31,920,000
Less: Variable costs	$21,465,000	$24,910,000	$28,620,000	$27,295,000	$22,260,000
Fixed costs	$1,500,000	$1,500,000	$1,500,000	$1,500,000	$1,500,000
Depreciation	$3,000,900	$5,142,900	$3,672,900	$2,622,900	$1,875,300
EBIT	$4,814,100	$4,167,100	$7,247,100	$7,722,100	$6,284,700
Less: Taxes	$1,684,935	$1,458,485	$2,536,485	$2,702,735	$2,199,645
Net income	$3,129,165	$2,708,615	$4,710,615	$5,019,365	$4,085,055
Add: Depreciation	$3,000,900	$5,142,900	$3,672,900	$2,622,900	$1,875,300
Operating cash flow	$6,130,065	$7,851,515	$8,383,515	$7,642,265	$5,960,355
Net cash inflows:
Operating cash flow	$6,130,065	$7,851,515	$8,383,515	$7,642,265	$5,960,355
Change in net working capital	-$741,000	-$798,000	$285,000	$1,083,000	$1,771,000
Capital spending	$0	$0	$0	$0	$4,369,785
Total cash inflows	$5,389,065	$7,053,515	$8,668,515	$8,725,265	$12,101,140

Computations for the above table:

Formula to calculate the ending book value:

$\text{Ending book value = }\left[\begin{array}{l}\text{ Installed cost of an equipment}-\\ \left(\text{Installed cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]$

Computation of the ending book value for year 1:

$\begin{array}{c}\text{Ending book value for year 1 = }\left[\begin{array}{l}\text{ Installed cost of an equipment}-\\ \left(\text{Installed cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]\\ \text{=\$21,000,000}-\left(\text{\$21,000,000}\times \text{14}\text{.29\%}\right)\\ =\$17,999,100\end{array}$

$\begin{array}{c}\text{Ending book value for year 2 = }\left[\begin{array}{l}\text{ Ending book value of an equipment for year 1}-\\ \left(\text{Initial cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]\\ \text{=\$17,999,100}-\left(\text{\$21,000,000}\times 2\text{4}\text{.49\%}\right)\\ =\$12,856,200\end{array}$

$\begin{array}{c}\text{Ending book value for year 3 = }\left[\begin{array}{l}\text{ Ending book value of an equipment for year 2}-\\ \left(\text{Initial cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]\\ \text{=\$12,856,200}-\left(\text{\$21,000,000}\times 17.\text{49\%}\right)\\ =\$9,183,300\end{array}$

$\begin{array}{c}\text{Ending book value for year 4 = }\left[\begin{array}{l}\text{ Ending book value of an equipment for year 3}-\\ \left(\text{Initial cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]\\ \text{=\$9,183,300}-\left(\text{\$21,000,000}\times 12.\text{49\%}\right)\\ =\$6,560,400\end{array}$

$\begin{array}{c}\text{Ending book value for year 5 = }\left[\begin{array}{l}\text{ Ending book value of an equipment for year 4}-\\ \left(\text{Initial cost of an equipment}\times \text{MACRS rate }\right)\end{array}\right]\\ \text{=\$6,560,400}-\left(\text{\$21,000,000}\times 8.93\text{\%}\right)\\ =\$4,685,100\end{array}$

Note:

• Sales are calculated by multiplying the price per unit with the unit sales of each year.
• Depreciation is calculated by multiplying the equipment’s installation cost with the MACRS depreciation rate for the particular year.
• The taxes are calculated by multiplying the earnings before tax for the specific year with the marginal tax rate.
• The change in the net working capital is calculated by subtracting the current year’s sales with the next year’s sales and multiplying it with the 15% (increased percentage).

Computation of the net working capital for year 5:

$\begin{array}{c}\text{NWC for year 5 = Recovery of all NWC}\\ \text{= Initial NWC – }\left[\begin{array}{l}\text{Change in NWC for year 1 }\\ \text{+ Change in NWC for Year 2 }\\ \text{+}\text{Change in NWC for year 3 }\\ \text{+ Change in NWC for Year 4}\end{array}\right]\end{array}$

$\begin{array}{c}=\$1,600,000-\left[\begin{array}{l}-\$741,000-\$798,000\\ +285,000+\$1,083,000\end{array}\right]\\ =\$1,771,000\end{array}$

Computation of the ending book value:

$\begin{array}{c}\text{Ending book value = }\left[\begin{array}{l}\text{Installation cost of an equipment}-\\ \left(\begin{array}{l}\text{Depreciation for year 1}+\text{Depreciation for year 2+}\\ \text{Depreciation for year}3+\text{Depreciation for year}4+\\ \text{Depreciation for year}5\end{array}\right)\end{array}\right]\\ =\$21,000,000-\left(\begin{array}{l}\$3,000,900+\$5,142,900+\\ \$3,672,900+\$2,622,900+\$1,875,300\end{array}\right)\\ =\$21,000,000-\$16,314,900\\ =\$4,685,100\end{array}$

Formula to calculate the after-tax salvage value:

$\text{After-taxsalvage value = }\left[\begin{array}{l}\text{Total fixed cost +}\left(\begin{array}{l}\text{Ending book value}-\\ \text{Total fixed cost}\end{array}\right)\\ \times \text{Marginal tax rate}\end{array}\right]$

Computation of the after-tax salvage value:

$\begin{array}{c}\text{After-taxsalvage value = }\left[\begin{array}{l}\text{Total fixed cost +}\left(\begin{array}{l}\text{Ending book value}-\\ \text{Total fixed cost}\end{array}\right)\\ \times \text{Marginal tax rate}\end{array}\right]\\ =\$1,500,000+\left(\$4,685,100-\$1,500,000\right)\times .35\\ =\$1,500,000+\$1,114,785\\ =\$2,614,785\end{array}$

Formula to calculate the net present value:

$\begin{array}{c}\text{NPV = Cash outflow +<x-custom-btb-me data-me-id="2578" class="microExplainerHighlight">Cash inflow</x-custom-btb-me>}\\ \text{=}-\text{Total cash outflow +}\left(\begin{array}{l}\frac{\text{Cash inflow for year 1}}{{\left(1+\text{r}\right)}^1}\text{+}\\ \frac{\text{Cash inflow for year 2}}{{\left(1+\text{r}\right)}^2}\text{+}\\ \frac{\text{Cash inflow for year 3}}{{\left(1+\text{r}\right)}^3}\text{+}\\ \frac{\text{Cash inflow for year 4}}{{\left(1+\text{r}\right)}^4}\text{+}\\ \frac{\text{Cash inflow for year 5}}{{\left(1+\text{r}\right)}^5}\end{array}\right)\end{array}$

Computation of the net present value:

$\begin{array}{c}\text{NPV = Cash outflow +Cash inflow}\\ \text{=}-\text{Total cash outflow +}\left(\begin{array}{l}\frac{\text{Cash inflow for year 1}}{{\left(1+\text{r}\right)}^1}\text{+}\\ \frac{\text{Cash inflow for year 2}}{{\left(1+\text{r}\right)}^2}\text{+}\\ \frac{\text{Cash inflow for year 3}}{{\left(1+\text{r}\right)}^3}\text{+}\\ \frac{\text{Cash inflow for year 4}}{{\left(1+\text{r}\right)}^4}\text{+}\\ \frac{\text{Cash inflow for year 5}}{{\left(1+\text{r}\right)}^5}\end{array}\right)\\ =-\$22,600,000+\left(\begin{array}{l}\frac{\$5,389,065}{{\left(1.18\right)}^1}+\frac{\$7,053,515}{{\left(1.18\right)}^2}+\\ \frac{\$8,668,515}{{\left(1.18\right)}^3}+\frac{\$8,725,265}{{\left(1.18\right)}^4}+\frac{\$12,101,140}{{\left(1.18\right)}^5}\end{array}\right)\\ =\$2,098,569.18\end{array}$

Hence, the net present value is $2,098,569.18.

Computation of the internal rate of return:

The internal rate of return is calculated by the spreadsheet method.

Step 1:

• Type the formula of the internal rate of return in H6 in the spreadsheet and consider the IRRvalue as H8.

Step 2:

• Assume the IRRvalue as 0.10%.

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 H8 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.

Step 5:

• The IRRvalue appears to be 21.5372255343952%.

Hence, the internal rate of return is 21.54%.