Chapter 9, Problem 23QP

Project Analysis. You are considering a new product launch. The project will cost $780,000, have a four-year life, and have no salvage value; depreciation is straight-line to zero. Sales are projected at 180 units per year; price per unit will be $16,300, variable cost per unit will be $11,100, and fixed costs will be $535,000 per year. The required return on the project is 11 percent, and the relevant tax rate is 35 percent.

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 best and worst cases for these projections? What is the base-case NPV? What are the best-case and worst-case scenarios?

b. Evaluate the sensitivity of your base-case NPV to changes in fixed costs.

Summary Introduction

To calculate: The best and the worst case of the net present value.

Introduction:

The process of analyzing the future proceedings compared to the figures of the net present value is scenario analysis. A project often experiences the best and the worst case of scenarios.

Answer to Problem 23QP

The best case of the net present value is $980,214.89 and the worst case of the net present value is -$418,874.44.

Explanation of Solution

Given information:

A project with 4-year life is being evaluated, the cost of the project is $780,000 and it has no salvage value. The depreciation is assumed to be a straight-line to zero over the project’s life. The projected sales is 180 units for one year.

The price for a unit is $16,300, the fixed cost for a year is $535,000, and the variable cost for a unit is $11,100. The rate of tax is 35% and the required return is 11%. The projections for the quantity, price, variable cost, and fixed costs are within 11%. The rate of tax is 35%.

Scenario	Unit sales	Variable cost	Fixed costs
Base	180	$11,100	$535,000
Best	198	$9,990	$481,500
Worst	162	$12,120	$588,500

Formula to calculate the operating cash flow using the tax shield approach:

$\text{OCF = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)$

Here, ${\text{T}}_c$ is the tax rate, P is the price per unit, v is the variable cost, Q refers to quantity, and FC is the fixed cost.

Computation of the base case of the operating cash flow:

$\begin{array}{c}{\text{OCF}}_{\text{base}}\text{ = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)\\ =\left[\left[\left[\left(\$16,300-\$11,100\right)\left(180\right)-\left(535,000\right)\right]\left(0.65\right)+\frac{\$780,000}{4}\left(0.35\right)\right]\right]\\ =\$328,900\end{array}$

Hence, the base case of the operating cash flow is $328,900.

Formula to calculate the net present value:

$\text{NPV}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}$

Note: As there are many years, PVIFA is used. PVIFA is the present value interest factor of annuity.

Computation of the base case of the net present value:

$\begin{array}{c}{\text{NPV}}_{\text{base}}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}\\ =-\text{\$780,000+\$328,900}\left({\text{PVIFA}}_{11\%\text{ 4years}}\right)\\ =-\text{\$780,000+\$328,900}\left(3.1024\right)\\ =\$240,379.36\end{array}$

Hence, the base case of the net present value is $240,379.36.

Computation of the worst case of the operating cash flow:

$\begin{array}{c}{\text{OCF}}_{\text{worst}}\text{ = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)\\ =\left[\left[\left[\left(\$16,300-\$12,210\right)\left(162\right)-\left(588,500\right)\right]\left(0.65\right)+\frac{\$780,000}{4}\left(0.35\right)\right]\right]\\ =\$116,402\end{array}$

Hence, the worst case of the operating cash flow is $116,402.

Computation of the worst case of the net present value:

$\begin{array}{c}{\text{NPV}}_{\text{worst}}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}\\ =-\text{\$780,000+\$116,402}\left({\text{PVIFA}}_{11\%\text{ 4years}}\right)\\ =-\text{\$780,000+\$116,402}\left(3.1024\right)\\ =-\$418,874.44\end{array}$

Hence, the worst case of the net present value is -$418,874.44.

Formula to calculate the operating cash flow using the tax shield approach:

$\text{OCF = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)$

Here, ${\text{T}}_c$ is the tax rate, P is the price per unit, Q refers to quantity, and FC is the fixed cost.

Computation of the best case of the operating cash flow:

$\begin{array}{c}{\text{OCF}}_{\text{best}}\text{ = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)\\ =\left[\left[\left[\left(\$16,300-\$9,990\right)\left(198\right)-\left(481,500\right)\right]\left(0.65\right)+\frac{\$780,000}{4}\left(0.35\right)\right]\right]\\ =\$567,372\end{array}$

Hence, the best case of the operating cash flow is $567,372.

Formula to calculate the net present value:

$\text{NPV}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}$

Computation of the best case of the net present value:

$\begin{array}{c}{\text{NPV}}_{\text{best}}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}\\ =-\text{\$780,000+\$567,372}\left({\text{PVIFA}}_{11\%\text{ 4years}}\right)\\ =-\text{\$780,000+\$567,372}\left(3.1024\right)\\ =\$980,214.89\end{array}$

Hence, the best case of the net present value is $980,214.89.

Summary Introduction

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

Introduction:

The process of analyzing the future proceedings compared to the figures of the net present value is scenario analysis. A project often experiences the best and the worst case of scenarios.

Answer to Problem 23QP

For each dollar, the fixed costs will increase and the NPV declines by $2.02.

Explanation of Solution

Given information:

A project with 4-year life is being evaluated, the cost of the project is $780,000 and it has no salvage value. The depreciation is assumed to be a straight-line to zero over the project’s life. The projected sales is 180 units for one year.

The price for a unit is $16,300, the fixed cost for a year is $535,000, and the variable cost for a unit is $11,100. The rate of tax is 35% and the required return is 11%. The projections for the quantity, price, variable cost, and fixed costs are within 11%. The rate of tax is 35%.

Note: To find the sensitivity of the net present value to the changes in fixed costs, other level of fixed costs is chosen. The chosen fixed cost is $545,000.

Formula to calculate the operating cash flow using the tax shield approach:

$\text{OCF = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)$

Here, ${\text{T}}_c$ is the tax rate, P is the price per unit, Q refers to quantity, and FC is the fixed cost.

Computation of the operating cash flow:

$\begin{array}{c}\text{OCF = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)\\ =\left[\left[\left[\left(\$16,300-\$11,100\right)\left(180\right)-\left(545,000\right)\right]\left(0.65\right)+\frac{\$780,000}{4}\left(0.35\right)\right]\right]\\ =\$322,400\end{array}$

Hence, the operating cash flow is $322,400.

Formula to calculate the net present value:

$\text{NPV}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}$

Computation the net present value:

$\begin{array}{c}\text{NPV}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}\\ =-\text{\$780,000+\$322,400}\left({\text{PVIFA}}_{11\%\text{ 4years}}\right)\\ =-\text{\$780,000+\$322,400}\left(3.1024\right)\\ =\$220,213.76\end{array}$

Hence, the net present value is $220,213.76.

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

$\frac{\Delta \text{NPV}}{\Delta \text{FC}}=\frac{\left({\text{NPV}}_{\text{Base}}-\text{NPV}\right)}{\left({\text{FC}}_{\text{Base}}-\text{FC}\right)}$

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

$\begin{array}{c}\frac{\Delta \text{NPV}}{\Delta \text{FC}}=\frac{\left({\text{NPV}}_{\text{Base}}-\text{NPV}\right)}{\left({\text{FC}}_{\text{Base}}-\text{FC}\right)}\\ =\frac{\left(\$240,379.36-\$220,213.76\right)}{\left(\$535,000-\$545,000\right)}\\ =\frac{\$20,165.6}{-\$10,000}\\ =-\$2.02\end{array}$

Hence, the sensitivity of the NPV to the changes in fixed costs is -$2.02.