Chapter 9, Problem 23QP

a)

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 $1,100,782.054 and the worst case of the net present value is -$509,642.76.

Explanation of Solution

Given information:

A project with 4-year life is being evaluated, the cost of the project is $875,000 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 190 units for one year.

The price for a unit is $19,200, the fixed cost for a year is $345,000, and the variable cost for a unit is $15,100. 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	190	$15,100	$345,000
Best	209	$13,590	$310,500
Worst	171	$16,610	$379,500

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:

$\begin{array}{c}{\text{OCF}}_{\text{base}}\text{ = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)\\ =\left[\left[\left[\left(\$19,200-\$15,100\right)\left(190\right)-\left(345,000\right)\right]\left(0.65\right)+\frac{\$875,000}{4}\left(0.35\right)\right]\right]\\ =\$358,662.5\end{array}$

Hence, the base case of the operating cash flow is $358,662.5.

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:

$\begin{array}{c}{\text{NPV}}_{\text{base}}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}\\ =-\text{\$875,000+\$358,662}\text{.5}\left({\text{PVIFA}}_{11\%\text{ 4years}}\right)\\ =-\text{\$875,000+\$358,662}\text{.5}\left(3.1024\right)\\ =\$237,714.54\end{array}$

Hence, the base case of the net present value is $237,714.54.

Computation of the worst case of the operating cash flow:

$\begin{array}{c}{\text{OCF}}_{\text{worst}}\text{ = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)\\ =\left[\left[\left[\left(\$19,200-\$16,610\right)\left(171\right)-\left(379,500\right)\right]\left(0.65\right)+\frac{\$875,000}{4}\left(0.35\right)\right]\right]\\ =\$117,766\end{array}$

Hence, the worst case of the operating cash flow is $117,766.

Computation of the worst case of the net present value:

$\begin{array}{c}{\text{NPV}}_{\text{worst}}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}\\ =-\text{\$875,000+\$117,766}\left({\text{PVIFA}}_{11\%\text{ 4years}}\right)\\ =-\text{\$875,000+\$117,766}\left(3.1024\right)\\ =-\$509,642.76\end{array}$

Hence, the worst case of the net present value is -$509,642.76.

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:

$\begin{array}{c}{\text{OCF}}_{\text{best}}\text{ = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)\\ =\left[\left[\left[\left(\$19,200-\$13,590\right)\left(209\right)-\left(310,500\right)\right]\left(0.65\right)+\frac{\$875,000}{4}\left(0.35\right)\right]\right]\\ =\$636,856\end{array}$

Hence, the best case of the operating cash flow is $636,856.

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:

$\begin{array}{c}{\text{NPV}}_{\text{best}}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}\\ =-\text{\$875,000+\$636,856}\left({\text{PVIFA}}_{11\%\text{ 4years}}\right)\\ =-\text{\$875,000+\$636,856}\left(3.1024\right)\\ =\$1,100,782.054\end{array}$

Hence, the best case of the net present value is $1,100,782.054.

b)

Summary Introduction

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

Note: To find the sensitivity of the net present value to the variations in the fixed costs, other level of fixed costs is chosen. The chosen fixed cost is $355,000.

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 $875,000 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 190 units for one year.

The price for a unit is $19,200, the fixed cost for a year is $345,000, and the variable cost for a unit is $15,100. 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	190	$15,100	$345,000
Best	209	$13,590	$310,500
Worst	171	$16,610	$379,500

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, Q refers to quantity, and FC is the fixed cost.

Computation of the operating cash flow:

$\begin{array}{c}\text{OCF = }\left[\text{P}-\text{FC}\right]\left(1-{\text{T}}_c\right)+\text{Depreciation}\left({\text{T}}_c\right)\\ =\left[\left[\left[\left(\$19,200-\$15,100\right)\left(190\right)-\left(355,000\right)\right]\left(0.65\right)+\frac{\$875,000}{4}\left(0.35\right)\right]\right]\\ =\$352,162.5\end{array}$

Hence, the operating cash flow is $352,162.5.

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:

$\begin{array}{c}\text{NPV}=-\text{Present value of cash outflow}+\text{Present value of cash inflow}\\ =-\text{\$875,000+\$352,162}\text{.5}\left({\text{PVIFA}}_{11\%\text{ 4years}}\right)\\ =-\text{\$875,000+\$352,162}\text{.5}\left(3.1024\right)\\ =\$217,548.94\end{array}$

Hence, the net present value is $217,548.94.

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:

$\begin{array}{c}\frac{\Delta \text{NPV}}{\Delta \text{FC}}=\frac{\left({\text{NPV}}_{\text{Base}}-\text{NPV}\right)}{\left({\text{FC}}_{\text{Base}}-\text{FC}\right)}\\ =\frac{\left(\$237,714.54-\$217,548.94\right)}{\left(\$345,000-\$355,000\right)}\\ =\frac{\$20,165.6}{-\$10,000}\\ =-\$2.02\end{array}$

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