Chapter 7, Problem 13QP

Project Analysis You are considering a new product launch. The project will cost $760,000, have a four-year life, and have no salvage value; depreciation is straight-line to zero. Sales are projected at 420 units per year; price per unit will be $17,200; variable cost per unit will be $14,300; and fixed costs will be $640,000 per year. The required return on the project is 15 percent, and the relevant tax rate is 35 percent.

1. a. Based on your experience, you think the unit sales, variable cost, and fixed cost projections given here are probably accurate to within ±10 percent. What are the upper and lower bounds for these projections? What is the base-case NPV? What are the best-case and worst-case scenarios?
2. b. Evaluate the sensitivity of your base-case NPV to changes in fixed costs.
3. c. What is the accounting break-even level of output for this project?

Summary Introduction

To determine: Best-case net present value.

Introduction:

The difference between the present value of cash outflow and the present value of cash inflow is termed as net present value.

Explanation of Solution

Given information:

The project has the cost of $760,000 with no salvage value and 4 years of economic cycle. Depreciation is on straight line method. The sales are at 420 units, the price per unit is $17,200, the variable cost per unit is $14,300, and the fixed cost is$640,000 per year. The required rate of return is 15% and the rate of tax is35%.

The variable cost, unit price, fixed cost projections are accurate within ±10%.

The formula to calculate the net present value:

$\text{Net present value}=-\text{Original cost}+\text{Operating cash flow}\left(\text{PVIFA}\right)$

The formula to calculate the operating cash flow:

$\text{Operating cash flow}=\left(\begin{array}{c}\left[\left(\text{Unit sold}\times \left[\text{Selling price}-\text{Variable price}\right]\right)-\text{Fixed cost}\right]\\ \times \left(1-\text{Tax rate}\right)\\ +\left(\text{Depreciation}\times \text{Tax rate}\right)\end{array}\right)$

Compute the upper and lower bounds of the above projections:

In the best-case, when the cost decreases, the unit sales will increase; whereas in the worst-case, when the cost increases, the unit sales will decrease.

Scenario	Base case	Upper case	Lower case
Unit sales	420	462	378
Variable costs	$14,300	$12,870	$15,730
Fixed costs	$640,000	$576,000	$704,000

Note: Refer excel for the above table.

Compute the base-case operating cash flow:

$\begin{array}{c}\text{Operating cash flow}=\left(\begin{array}{c}\left[\left(\text{Unit sold}\times \left[\text{Selling price}-\text{Variable price}\right]\right)-\text{Fixed cost}\right]\\ \times \left(1-\text{Tax rate}\right)\\ +\left(\text{Depreciation}\times \text{Tax rate}\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\left(420\times \left[\$17,200-\$14,300\right]\right)-\$640,000\right]\\ \times \left(1-0.35\right)\\ +\left(\frac{\$760,000}{4}\times 0.35\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\left(\$1,218,000\right)-\$640,000\right]\times \left(0.65\right)\\ +\left(\$190,000\times 0.35\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\left(\$1,218,000\right)-\$640,000\right]\times \left(0.65\right)\\ +\left(\$190,000\times 0.35\right)\end{array}\right)\end{array}$

$\begin{array}{c}=\left(\$375,700+\$66,500\right)\\ =\$442,200\end{array}$

Hence, the base-case operating cash flow is$442,200.

Compute the base-case net present value:

Note: The increase in the operating cash flow at the present value interest factor of annuity at 15% for 4 years is 2.85498.

Compute the base-case net present value:

$\begin{array}{c}\text{Net present value}=-\text{Original cost}+\text{Operating cash flow}\left(\text{PVIFA}\right)\\ =-\$760,000+\$442,200\times 2.85498\\ =-\$760,000+\$1,262,472.15\\ =\$502,472.15\end{array}$

Hence, the base-case net present value is$502,472.15.

Compute the worst–case operating cash flow:

$\begin{array}{c}\text{Operating cash flow}=\left(\begin{array}{c}\left[\left(\text{Unit sold}\times \left[\text{Selling price}-\text{Variable price}\right]\right)-\text{Fixed cost}\right]\\ \times \left(1-\text{Tax rate}\right)\\ +\left(\text{Depreciation}\times \text{Tax rate}\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\left(378\times \left[\$17,200-\$15,730\right]\right)-\$640,000\right]\\ \times \left(1-0.35\right)\\ +\left(\frac{\$760,000}{4}\times 0.35\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\left(378\times \left[\$1,470\right]\right)-\$704,000\right]\\ \times \left(0.65\right)\\ +\left(\$190,000\times 0.35\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\$555,660-\$704,000\right]\\ \times \left(0.65\right)\\ +\left(\$190,000\times 0.35\right)\end{array}\right)\end{array}$

$\begin{array}{c}=\left(-\$96,421+\$66,500\right)\\ =-\$29,921\end{array}$

Hence, the worst-case operating cash flow is−$29,921.

Compute the worse-case net present value:

Note: The increase in the operating cash flow at the present value interest factor of annuity at 15% for 4 years is 2.85498.

$\begin{array}{c}\text{Net present value}=-\text{Original cost}+\text{Operating cash flow}\left(\text{PVIFA}\right)\\ =-\$760,000-\$29,921\left(2.85498\right)\\ =-\$845,423.85\end{array}$

Hence, the worse-case net present value is −$845,423.85.

Compute the best-case operating cash flow:

$\begin{array}{c}\text{Operating cash flow}=\left(\begin{array}{c}\left[\left(\text{Unit sold}\times \left[\text{Selling price}-\text{Variable price}\right]\right)-\text{Fixed cost}\right]\\ \times \left(1-\text{Tax rate}\right)\\ +\left(\text{Depreciation}\times \text{Tax rate}\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\left(462\times \left[17,200-\$12,870\right]\right)-\$576,000\right]\\ \times \left(1-0.35\right)\\ +\left(\frac{\$760,000}{4}\times 0.35\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\left(\$2,000,460\right)-\$576,000\right]\\ \times \left(0.65\right)\\ +\left(\$190,000\times 0.35\right)\end{array}\right)\end{array}$

$\begin{array}{c}=\left(\$925,899+\$66,500\right)\\ =\$992,399\end{array}$

Hence, the best-case operating cash flow is $992,399.

Compute the best-case net present value:

$\begin{array}{c}\text{Net present value}=-\text{Original cost}+\text{Operating cash flow}\left(\text{PVIFA}\right)\\ =-\$760,000+\$992,399\left(2.85498\right)\\ =-\$760,000+\$2,833,279.29\\ =\$2,073,279.29\end{array}$

Hence, the best-case net present value is $2,073,279.29.

Summary Introduction

To determine: The sensitivity of base-case net present value to the changes in fixed costs.

Answer to Problem 13QP

Solution: Therefore, the net present value falls by −$1.856 for an increase in every dollar.

Explanation of Solution

Given information:

The project has the cost of $760,000 with no salvage value and 4 years of economic cycle. Depreciation is on straight line method. The sales are at 420 units, the price per unit is $17,200, the variable cost per unit is $14,300, and the fixed cost is $640,000 per year. The required rate of return is 15% and the rate of tax is 35%.

Compute the base-case operating cash flow to the changes in fixed costs:

Note: To compute the sensitivity of base-case net present value to the changes in fixed costs, the levels of fixed cost need to increase to another level. Assume it to be $650,000.

$\begin{array}{c}\text{Operating cash flow}=\left(\begin{array}{c}\left[\left(\text{Unit sold}\times \left[\text{Selling price}-\text{Variable price}\right]\right)-\text{Fixed cost}\right]\\ \times \left(1-\text{Tax rate}\right)\\ +\left(\text{Depreciation}\times \text{Tax rate}\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\left(420\times \left[\$17,200-\$14,300\right]\right)-\$650,000\right]\\ \times \left(1-0.35\right)\\ +\left(\frac{\$760,000}{4}\times 0.35\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\left(\$1,218,000\right)-\$650,000\right]\times \left(0.65\right)\\ +\left(\$190,000\times 0.35\right)\end{array}\right)\\ =\left(\begin{array}{c}\left[\left(\$1,218,000\right)-\$650,000\right]\times \left(0.65\right)\\ +\left(\$190,000\times 0.35\right)\end{array}\right)\end{array}$

$\begin{array}{c}=\left(\$369,200+\$66,500\right)\\ =\$435,700\end{array}$

Hence, the base-caseoperating cash flow is $435,700.

Compute the base-case net present value:

$\begin{array}{c}\text{Net present value}=-\text{Original cost}+\text{Operating cash flow}\left(\text{PVIFA}\right)\\ =-\$760,000+\$435.700\times 2.85498\\ =-\$760,000+\$1,243,914.78\\ =\$483,914.78\end{array}$

Hence, the base-case net present value is $483,914.78.

Compute the sensitivity of net present value to  the changes in fixed costs:

$\begin{array}{c}\text{Changes in net present value}=\frac{\text{Net present value at base}-\text{Increase in net present value}}{\text{Units sold}-\text{Increase in units sold}}\\ =\frac{\text{\$502,471}\text{.43}-\text{\$483,914}\text{.78}}{\$640,000-\$650,000}\\ =\frac{\$18,556.65}{-\$10,000}\\ =-\$1.856\end{array}$

Hence, the sensitivity of net present value to the changes in fixed cost is −$1.856.

Therefore, the net present value falls by −$1.856for an increase in every dollar.

Summary Introduction

To determine: The accounting break-even level of output.

Introduction:

Accounting break-even point refers to a point where the company faces zero profits.

Answer to Problem 13QP

Solution: The accounting break-even point is 286.21 units.

Explanation of Solution

Given information:

The project has a cost of $760,000 with no salvage value and 4 years of economic cycle. Depreciation is on straight line method. The sales are at 420 units, the price per unit is $17,200, the variable cost per unit is $14,300, and the fixed cost is $640,000 per year. The required rate of return is 15% and the rate of tax is 35%.

The variable cost, unit price, fixed cost projections are accurate within ±10%.

The formula to calculate the accounting break-even level:

$\text{Break-even point}=\frac{\text{Fixed cost}+\text{Depreciation}\left(1-\text{Tax rate}\right)}{\text{Unit price}-\text{Variable price}\left(1-\text{Tax rate}\right)}$

Compute the accounting break-even level:

$\begin{array}{c}\text{Accounting break-even point}=\frac{\text{Fixed cost}+\text{Depreciation}}{\text{Unit price}-\text{Variable price}}\\ =\frac{\$640,000+\left(\frac{\$760,000}{4}\right)}{\$17,200-\$14,300}\\ =\frac{\$640,000+\$190,000}{\$2,900}\end{array}$

$\begin{array}{c}=\frac{\$830,000}{\$2,900}\\ =286.21\text{units}\end{array}$

Hence, the accounting break-even point is286.21 units.