Chapter 11, Problem 19QP

a)

Summary Introduction

To determine: The upper and lower bounds of unit sales, variable costs, and fixed costs projections under the scenario analysis.

Introduction:

Fixed costs remain the same as the total costs despite of the changes in the level of activity. However, the fixed cost per unit has a negative relationship with the activity, that is, if the activity volume increases then the total cost will decrease and vice-versa.

Variable costs are the type of costs that would vary according to the production output. They depend on the production volume.

Answer to Problem 19QP

Under the scenario analysis, the lower (worst scenario) and upper (best scenario) bounds of unit sales, variable costs, and fixed costs projections are as follows:

Scenarios	Unit sales (in units)	Variable cost (in $)	Fixed cost (in $)
Base	180	$9,800	$430,000
Best	198	$8,820	$387,000
Worst	162	$10,780	$473,000

Explanation of Solution

Given information:

The unit sales is 180 units, variable costs is $9,800, price per unit is $16,000, fixed costs is $430,000 and accurate estimate is ±10%. The cost of project is $1,400,000 and life of the project is 4 years. The required rate on the project is 12% and tax rate is 35%.

Formulae:

The formula to calculate the upper bounds of unit sales projection under the scenario analysis:

$\text{Unit sales for best case}=\left[\begin{array}{l}\text{Value of base case unit sales }\\ +\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{Value of base case unit sales}\end{array}\right)\end{array}\right]$

The formula to calculate the upper bounds of variable costs projection under the scenario analysis:

$\text{Variable costs for best case}=\left[\begin{array}{l}\text{Value of base case variable costs}\\ -\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{Value of base case variable costs}\end{array}\right)\end{array}\right]$

The formula to calculate the upper bounds of fixed costs projection under the scenario analysis:

$\text{Fixed costs for best case}=\left[\begin{array}{l}\text{Value of base case fixed costs}\\ -\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{}\text{Value of base case fixed costs}\end{array}\right)\end{array}\right]$

The formula to calculate the lower bounds of unit sales projection under the scenario analysis:

$\text{Unit sales for worst case}=\left[\begin{array}{l}\text{Value of base case unit sales }\\ -\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{Value of base case unit sales}\end{array}\right)\end{array}\right]$

The formula to calculate the lower bounds of variable costs projection under the scenario analysis:

$\text{Variable costs for worst case}=\left[\begin{array}{l}\text{Value of base case variable costs}\\ +\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{Value of base case variable costs}\end{array}\right)\end{array}\right]$

The formula to calculate the lower bounds of fixed costs projection under the scenario analysis:

$\text{Fixed costs for worst case}=\left[\begin{array}{l}\text{Value of base case fixed costs}\\ +\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{}\text{Value of base case fixed costs}\end{array}\right)\end{array}\right]$

Compute theupper bounds of unit sales projection under the scenario analysis:

$\begin{array}{c}\text{Unit sales for best case}=\left[\begin{array}{l}\text{Value of base case unit sales }\\ +\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{Value of base case unit sales}\end{array}\right)\end{array}\right]\\ =\left[180+\left(\frac{10}{100}\times 180\right)\right]\\ =\left[180+\left(0.1\times 180\right)\right]\\ =180+18\\ =198\text{ units}\end{array}$

Hence, the upper bounds of unit sales projection under the scenario analysis are 198 units.

Computethe upper bounds of variable costs projection under the scenario analysis:

$\begin{array}{c}\text{Variable costs for best case}=\left[\begin{array}{l}\text{Value of base case variable costs}\\ -\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{Value of base case variable costs}\end{array}\right)\end{array}\right]\\ =\left[\$9,800-\left(\frac{10}{100}\times \$9,800\right)\right]\\ =\left[\$9,800-\left(0.10\times \$9,800\right)\right]\\ =\$9,800-\$980\\ =\$8,820\end{array}$

Hence, the upper bounds of variable costs projection under the scenario analysis are $8,820.

Computethe upper bounds of fixed costs projection under the scenario analysis:

$\begin{array}{c}\text{Fixed costs for best case}=\left[\begin{array}{l}\text{Value of base case fixed costs}\\ -\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{}\text{Value of base case fixed costs}\end{array}\right)\end{array}\right]\\ =\left[\$430,000-\left(\frac{10}{100}\times \$430,000\right)\right]\\ =\left[\$430,000-\left(0.10\times \$430,000\right)\right]\\ =\$430,000-\$43,000\\ =\$387,000\end{array}$

Hence, the upper bounds of fixed costs projection under the scenario analysis are $387,000.

Computethelower bounds of unit sales projection under the scenario analysis:

$\begin{array}{c}\text{Unit sales for worst case}=\left[\begin{array}{l}\text{Value of base case unit sales }\\ -\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{Value of base case unit sales}\end{array}\right)\end{array}\right]\\ =\left[180-\left(\frac{10}{100}\times 180\right)\right]\\ =\left[180-\left(0.10\times 180\right)\right]\end{array}$

$\begin{array}{c}=180-18\\ =162\text{ units}\end{array}$

Hence, the lower bounds of unit sales projection under the scenario analysis are 162 units.

Computethelower bounds of variable costs projection under the scenario analysis:

$\begin{array}{c}\text{Variable costs for worst case}=\left[\begin{array}{l}\text{Value of base case variable costs}\\ +\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{Value of base case variable costs}\end{array}\right)\end{array}\right]\\ =\left[\$9,800+\left(\frac{10}{100}\times \$9,800\right)\right]\\ =\left[\$9,800+\left(0.10\times \$9,800\right)\right]\\ =\$9,800+\$980\\ =\$10,780\end{array}$

Hence, the lower bounds of variable costs projection under the scenario analysis are $10,780.

Computethelower bounds of fixed costs projection under the scenario analysis:

$\begin{array}{c}\text{Fixed costs for worst case}=\left[\begin{array}{l}\text{Value of base case fixed costs}\\ +\left(\begin{array}{l}\text{Percenatge of accurate estimate}\\ \times \text{}\text{Value of base case fixed costs}\end{array}\right)\end{array}\right]\\ =\$430,000+\left(\frac{10}{100}\times \$430,000\right)\\ =\$430,000+\left(0.10\times \$430,000\right)\\ =\$430,000+\$43,000\\ =\$473,000\end{array}$

Hence, the lower bounds of fixed costs projection under the scenario analysis are $473,000.

b)

Summary Introduction

To determine: The sensitivity of base-case NPV to change in fixed costs

Introduction:

Sensitivity analysis analyzes the impact of changing only one variable of the net present value. It helps to identify the areas in which the forecasting errors would be severe. The basic concept of sensitivity analysis is to freeze all the variables except one variable.

Answer to Problem 19QP

The sensitivity of NPV to change in the sales value is −$1.97.

Explanation of Solution

Given information:

The fixed costs of the project are $430,000 per year and the initial cost of the project is $1,400,000 for the lifetime of 4 years. The variable cost per unit is $9,800 and the price per unit of the project is $16,000. The unit sold is 180 units per year, the required rate of return is 12%, and the tax rate is 35%.

Formulae:

The formula to calculate the sensitivity of NPV to make changes in the sales value:

$\begin{array}{l}\text{Sensitivity of net present value to }\\ \text{change in the sales value}\end{array}\}=\frac{\text{Change in net present value}}{\text{Change in fixed cost}}$

The formula to calculate the NPV of the new case operating cash flow:

$\text{NPV of the new case cash flow}=\left[\begin{array}{c}-\text{Initial investment}\\ +\text{ New case cash flow}\times \left(\begin{array}{l}\text{Present value of an <x-custom-btb-me data-me-id="2167" class="microExplainerHighlight">annuity</x-custom-btb-me>}\\ \text{ of \$1 period for R\% of N period}\end{array}\right)\end{array}\right]$

The formula to calculate the new case operating cash flow:

$\text{New case operating cash flow}=\left[\left(\text{P}-\text{v}\right)\text{Q}-\text{FC}\right]\times \left(1-\text{Tax rate}\right)+\text{Tax rate}\times \text{<x-custom-btb-me data-me-id="391" class="microExplainerHighlight">Depreciation</x-custom-btb-me>}$

Where,

P refers to the price per unit of the project,

v refers to the variable cost per unit,

Q refers to the number of unit sold,

FC refers to the fixed costs.

Note: To compute the NPV to change in fixed costs, assume the next level of fixed cost as $431,000.

Compute thenew case operating cash flow:

$\begin{array}{c}\text{New case operating cash flow}=\left[\left(\text{P}-\text{v}\right)\text{Q}-\text{FC}\right]\times \left(1-\text{Tax rate}\right)+\text{Tax rate}\times \text{Depreciation}\\ =\left[\begin{array}{c}\left[\left(\$16,000-\$9,800\right)\times 180-\$431,000\right]\\ \times \left(1-\frac{35}{100}\right)+\frac{35}{100}\times \left(\frac{\$1,400,000}{4}\right)\end{array}\right]\\ =\left[\left(\$6,200\times 180\right)-\$431,000\right]\times \left(1-0.35\right)+\left(0.35\times \$350,000\right)\\ =\left[\$1,116,000-\$431,000\right]\times 0.65+\$122,500\\ =\left(\$685,000\times 0.65\right)+\$122,500\\ =\$445,250+\$122,500\\ =\$567,750\end{array}$

Hence, the new case operating cash flow is $567,750.

Compute the NPV of new case operating cash flow:

Note: To determine the present value of annuity of $1 period for 4 period at a discount rate of 12% refer to the PV of an annuity of $1 table. Then find out 12% discount rate and period of 8 years value from the table. Here, the value for the rate 12% and 4 years period is 3.03735.

$\begin{array}{c}\begin{array}{c}\text{NPV of the new case }\\ \text{operating cash flow}\end{array}\}=\left[\begin{array}{c}-\text{Initial investment}\\ +\text{ New case cash flow}\times \left(\begin{array}{l}\text{Present value of an annuity}\\ \text{ of \$1 period for R\% of N period}\end{array}\right)\end{array}\right]\\ =-\$1,400,000+\$567,750\times \left(\begin{array}{l}\text{Present value of an annuity }\\ \text{of }\$1\text{ period for }12\%\text{ of 4 period}\end{array}\right)\\ =-\$1,400,000+\left(\$567,750\times 3.03735\right)\\ =-\$1,400,000+\$1,724,455.46\\ =\$324,455.46\end{array}$

Hence, the NPV of the new case operating cash flow is $324,455.46.

Compute the sensitivity of NPV to change in the sales value:

$\begin{array}{c}\begin{array}{l}\text{Sensitivity of NPV to }\\ \text{change in the sales value}\end{array}\}=\frac{\text{Change in NPV}}{\text{Change in fixed cost}}\\ =\frac{\left(\$326,429.74-\$324,455.46\right)}{\left(\$430,000-\$431,000\right)}\\ =\frac{\$1,974.28}{-\$1,000}\\ =-\$1.97\end{array}$

Hence, the sensitivity of NPV to change in the sales value is -$1.97. As a result, for every dollar of fixed cost, the NPV decrease by $1.97.

c)

Summary Introduction

To determine: The cash break-even level of output of the project.

Introduction:

Cash break-even point specifies a sales level which can result in zero operating cash flow.

Answer to Problem 19QP

The cash break-even level of output is 69.35 units.

Explanation of Solution

Given information:

The unit price is $16,000, unit variable costs are $9,800, and fixed costs are $430,000.

The formula to calculate the cash break-even quantity:

$\text{Cash break-even quantity}=\frac{\text{Fixed cost}}{\left(\text{Selling price per unit}-\text{Variable cost per unit}\right)}$

Compute thecash break-even quantity:

$\begin{array}{c}\text{Cash break-even quantity}=\frac{\text{Fixed cost}}{\left(\text{Selling price per unit}-\text{Variable cost per unit}\right)}\\ =\frac{\$430,000}{\left(\$16,000-\$9,800\right)}\\ =\frac{\$430,000}{\$6,200}\\ =69.35\text{ units}\end{array}$

Hence, the cash break-even quantity is 69.35 units.

d)

Summary Introduction

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

Introduction:

Accounting break-even is a sales point at which there is no profit or no loss. It is most widely used to measure the break-even point.

Answer to Problem 19QP

The accounting break-even level of output is 125.80 units.

Explanation of Solution

Given information:

The fixed costs of the project are $430,000 per year. The initial cost of the project is $1,400,000 for the life time of 4 years. The variable cost per unit is $9,800 and price per unit of the project is $16,000.

The formula to calculate the accounting break-even quantity:

$\text{Accounting break-even quantity}=\frac{\left(\text{Fixed cost}+\text{Depreciation}\right)}{\left(\text{Unit price}-\text{Unit variable cost}\right)}$

Compute the accounting break-even quantity:

$\begin{array}{c}\text{Accounting break-even quantity}=\frac{\left(\text{Fixed cost}+\text{Depreciation}\right)}{\left(\text{Unit price}-\text{Unit variable cost}\right)}\\ =\frac{\$430,000+\left(\frac{\$1,400,000}{4}\right)}{\left(\$16,000-\$9,800\right)}\\ =\frac{\left(\$430,000+\$350,000\right)}{\$6,200}\\ =\frac{\$780,000}{\$6,200}\\ =125.80\text{ units}\end{array}$

Hence, the accounting break-even quantity is 125.80 units.