Chapter 11, Problem 19QP

Project Analysis [LO1, 2, 3, 4] You are considering a new product launch. The project will cost $1,750,000, have a four-year life, and have no salvage value; depreciation is straight-line to zero. Sales are projected at 190 units per year; price per unit will be $17,300, variable cost per unit will be $10,400, and fixed costs will be $515,000 per year. The required return on the project is 12 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 upper and lower bounds 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.

c. What is the cash break-even level of output for this project (ignoring taxes)?

d. What is the accounting break-even level of output for this project? What is the degree of operating leverage at the accounting break-even point? How do you interpret this number?

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 activity, that is, if the activity volume increases then total cost will decrease and vice-versa.

Variable costs are the type of costs that would vary according to the production output. It depends 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	190	10,400	515,000
Best	209	9,360	463,500
Worst	171	11,440	566,500

Explanation of Solution

Given information:

The unit sales is 190 units, variable costs is $10,400, and fixed costs is $515,000. The accurate estimate is ±10%.

Formula:

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

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

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

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

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

$\begin{array}{c}\text{Unit sales for best case = }\left[\begin{array}{l}\text{Value of base case unit sales }\\ \text{+}\left(\begin{array}{l}\text{ Percenatge of accurate estimate}\\ \times \text{ Value of base case unit sales}\end{array}\right)\end{array}\right]\\ =\left[190+\left(\frac{10}{100}\times 190\right)\right]\\ =\left[190+\left(0.1\times 190\right)\right]\end{array}$

$\begin{array}{c}=190+19\\ =209\text{ units}\end{array}$

Hence, the upper bounds of unit sales projection under the scenario analysis are 209 units.

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

$\begin{array}{c}\text{Variable costs for best case = }\left[\begin{array}{l}\text{Value of base case variable costs}\\ -\left(\begin{array}{l}\text{ Percenatge of accurate estimate}\\ \times \text{ Value of base case variable costs}\end{array}\right)\end{array}\right]\\ =\left[\$10,400-\left(\frac{10}{100}\times \$10,400\right)\right]\\ =\left[\$10,400-\left(0.10\times \$10,400\right)\right]\end{array}$

$\begin{array}{c}=\$10,400-\$1,040\\ =\$9,360\end{array}$

Hence, the upper bounds of variable costs projection under the scenario analysis are $9,360.

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

$\begin{array}{c}\text{Fixed costs for best case = }\left[\begin{array}{l}\text{Value of base case fixed costs}\\ -\left(\begin{array}{l}\text{ Percenatge of accurate estimate}\\ \times \text{}\text{ Value of base case fixed costs}\end{array}\right)\end{array}\right]\\ =\left[\$515,000-\left(\frac{10}{100}\times \$515,000\right)\right]\\ =\left[\$515,000-\left(0.10\times \$515,000\right)\right]\end{array}$

$\begin{array}{c}=\$515,000-51,500\\ =\$463,500\end{array}$

Hence, the upper bounds of fixed costs projection under the scenario analysis are $463,500.

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

$\begin{array}{c}\text{Unit sales for worst case = }\left[\begin{array}{l}\text{Value of base case unit sales }\\ -\left(\begin{array}{l}\text{ Percenatge of accurate estimate}\\ \times \text{ Value of base case unit sales}\end{array}\right)\end{array}\right]\\ =\left[190-\left(\frac{10}{100}\times 190\right)\right]\\ =\left[190-\left(0.10\times 190\right)\right]\end{array}$

$\begin{array}{c}=190-19\\ =171\text{ units}\end{array}$

Hence, the lower bounds of unit sales projection under the scenario analysis are 171 units.

Compute the lower bounds of variable costs projection under the scenario analysis:

$\begin{array}{c}\text{Variable costs for worst case = }\left[\begin{array}{l}\text{Value of base case variable costs}\\ \text{+ }\left(\begin{array}{l}\text{ Percenatge of accurate estimate}\\ \times \text{ Value of base case variable costs}\end{array}\right)\end{array}\right]\\ =\left[\$10,400+\left(\frac{10}{100}\times \$10,400\right)\right]\\ =\left[\$10,400+\left(0.10\times \$10,400\right)\right]\end{array}$

$\begin{array}{c}=\$10,400+\$1,040\\ =\$11,440\end{array}$

Hence, the lower bounds of variable costs projection under the scenario analysis are $11,440.

Compute the lower bounds of fixed costs projection under the scenario analysis:

$\begin{array}{c}\text{Fixed costs for worst case = }\left[\begin{array}{l}\text{Value of base case fixed costs}\\ +\left(\begin{array}{l}\text{ Percenatge of accurate estimate}\\ \times \text{}\text{ Value of base case fixed costs}\end{array}\right)\end{array}\right]\\ =\$515,000+\left(\frac{10}{100}\times \$515,000\right)\\ =\$515,000+\left(0.10\times \$515,000\right)\end{array}$

$\begin{array}{c}=\$515,000+\$51,500\\ =\$566,500\end{array}$

Hence, the lower bounds of fixed costs projection under the scenario analysis are $566,500.

Summary Introduction

To determine: The base-case Net present value (NPV)

Introduction:

Net present value (NPV) refers to the discounted value of the future cash flows at present. The company should accept the project, if the net present value is positive or greater than zero and vice-versa. If there are two mutually exclusive projects, then the company has to select the project that has higher net present value.

Answer to Problem 19QP

The NPV for the base-case scenario is $286,619.11.

Explanation of Solution

Given information:

The fixed costs of the project are $515,000 per year and the initial cost of the project is $1,750,000 for the lifetime of 4 years. The variable cost per unit is $10,400 and the price per unit of the project is $17,300. The unit sold is 190 units per year, the required rate of return is 12%, and the tax rate is 35%.

The formula to calculate the operating cash flow under the base-case scenario:

$\begin{array}{l}\text{Operating cash flow }\\ \text{of base case scenario}\end{array}\}=\left[\left(\text{P}-\text{v}\right)\text{Q}-\text{FC}\right]\times \left(1-\text{Tax rate}\right)+\text{Tax rate}\times \text{Depreciation}$

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

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

$\begin{array}{l}\text{Net present value }\\ \text{of base case cash flow}\end{array}\}=-\text{Initial investment}+\text{Base case operating cash flow}$

Compute the operating cash flow in the base-case scenario:

$\begin{array}{c}\begin{array}{l}\text{Operating cash flow }\\ \text{of base case scenario}\end{array}\}=\left[\left(\text{P}-\text{v}\right)\text{Q}-\text{FC}\right]\times \left(1-\text{Tax rate}\right)+\text{Tax rate}\times \text{Depreciation}\\ =\left[\left(\$17,300-\$10,400\right)190-\$515,000\right]\times \left(1-\frac{35}{100}\right)+\left(\frac{35}{100}\times \frac{\$1,750,000}{4}\right)\\ =\left[\left(\$6,900\times 190\right)-\$515,000\right]\times \left(1-0.35\right)+\left(0.35\times \$437,500\right)\\ =\left[\$1,311,000-\$515,000\right]\times \left(1-0.35\right)+\$153,125\\ =\left(\$796,000\times 0.65\right)+\$153,125\\ =\$517,400+\$153,125\\ =\$670,525\end{array}$

Hence, the operating cash flow under the base-case scenario is $670,525.

Compute the NPV for the base-case scenario:

Note: To determine the present value of annuity of $1 period for 4 period at a discount rate of 12% refer 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.

$\begin{array}{c}\begin{array}{l}\text{Net present value }\\ \text{of base case cash flow}\end{array}\}=\left[\begin{array}{c}-\text{Initial investment}+\text{Base case operating cash flow}\\ \times \left(\begin{array}{l}\text{Present value of an annuity}\\ \text{ of \$1 period for R\% of N period}\end{array}\right)\end{array}\right]\\ =\left[-\$1,750,000+\$670,525\times \left(\begin{array}{l}\text{Present value of an annuity }\\ \text{of }\$1\text{ period for }12\%\text{ of 4 period}\end{array}\right)\right]\\ =-\$1,750,000+\left(\$670,525\times 3.03735\right)\end{array}$

$\begin{array}{c}=-\$1,750,000+\$2,036,619.10875\\ =\$286,619.11\end{array}$

Hence, the NPV for the base-case scenario is $286,619.11.

Summary Introduction

To determine: The worst-case scenario

Introduction:

Worst-case scenario is a particular case under the scenario analysis to determine the worst scenarios of the project. The financial managers of the company use this worst-case technique to anticipate potential losses and operational issues of the project.

Answer to Problem 19QP

The NPV for the worst-case scenario is -$1,309,315.

Explanation of Solution

Given information:

The fixed costs of the project are $515,000 per year and initial cost of the project is $1,750,000 for the life time of 4 years. The variable cost per unit is $10,400 and price per unit of the project is $17,300. The unit sold is 190 units per year, required rate of return is 12%, and tax rate is 35%.

The formula to calculate the operating cash flow for worst-case scenario:

$\begin{array}{l}\text{Operating cash flow }\\ \text{of worst case scenario}\end{array}\}=\left[\left(\text{P}-\text{v}\right)\text{Q}-\text{FC}\right]\times \left(1-\text{Tax rate}\right)+\text{Tax rate}\times \text{Depreciation}$

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

The formula to calculate the NPV for worst- case operating cash flow:

$\begin{array}{l}\text{Net present value }\\ \text{of worst case cash flow}\end{array}\}=-\text{Initial investment}+\text{Worst case operating cash flow}$

Compute the operating cash flow under the worst-case scenario:

$\begin{array}{c}\begin{array}{l}\text{Operating cash flow }\\ \text{of worst case scenario}\end{array}\}=\left[\left(\text{P}-\text{v}\right)\text{Q}-\text{FC}\right]\times \left(1-\text{Tax rate}\right)+\text{Tax rate}\times \text{Depreciation}\\ =\left[\left(\$17,300-\$11,440\right)\times 171-\$566,500\right]\times \left(1-\frac{35}{100}\right)+\frac{35}{100}\times \left(\frac{\$1,750,000}{4}\right)\\ =\left[\left(\$5,900\times 171\right)-\$566,500\right]\times \left(1-0.35\right)+\left(0.35\times \$437,500\right)\\ =\left[\$1,008,900-\$566,500\right]\times 0.65+\$153,125\\ =\left(\$442,400\times 0.65\right)+\$153,125\\ =\$287,560+\$153,125\\ =\$440,685\end{array}$

Hence, the operating cash flow under the worst-case scenario is $440,685.

Compute the NPV for worst case scenario:

$\begin{array}{c}\begin{array}{l}\text{Net present value}\\ \text{of worst case cash flow}\end{array}\}=-\text{Initial investment}+\text{Worst case operating cash flow}\\ =-\$1,750,000+\$440,685\\ =-\$1,309,315\end{array}$

Hence, the NPV for the worst-case scenario is -$1,309,315.

Summary Introduction

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

Introduction:

Sensitivity analysis is analyzing 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 $515,000 per year and initial cost of the project is $1,750,000 for the life time of 4 years. The variable cost per unit is $10,400 and price per unit of the project is $17,300. The unit sold is 190 units per year, required rate of return is 12%, and tax rate is 35%. Assume the fixed cost has $516,000.

Formula:

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

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

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

$\text{NPV of the new case cash flow}=\left[\begin{array}{c}-\text{Initial investment}\\ +\text{ New case cash flow}\times \left(\begin{array}{l}\text{Present value of an annuity}\\ \text{ of \$1 period for R\% of N period}\end{array}\right)\end{array}\right]$

The formula to calculate the new case operating cash flow:

$\text{New case operating cash flow}=\left[\left(\text{P}-\text{v}\right)\text{Q}-\text{FC}\right]\times \left(1-\text{Tax rate}\right)+\text{Tax rate}\times \text{Depreciation}$

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

Compute the new case operating cash flow:

$\begin{array}{c}\text{New case operating cash flow}=\left[\left(\text{P}-\text{v}\right)\text{Q}-\text{FC}\right]\times \left(1-\text{Tax rate}\right)+\text{Tax rate}\times \text{Depreciation}\\ =\left[\begin{array}{c}\left[\left(\$17,300-\$10,400\right)\times 190-\$516,000\right]\\ \times \left(1-\frac{35}{100}\right)+\frac{35}{100}\times \left(\frac{\$1,750,000}{4}\right)\end{array}\right]\\ =\left[\left(\$6,900\times 190\right)-\$516,000\right]\times \left(1-0.35\right)+\left(0.35\times \$437,500\right)\\ =\left[\$1,311,000-\$516,000\right]\times 0.65+\$153,125\end{array}$

$\begin{array}{c}=\left(\$795,000\times 0.65\right)+\$153,125\\ =\$516,750+\$153,125\\ =\$669,875\end{array}$

Hence, the new case operating cash flow is $669,875.

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 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.

$\begin{array}{c}\begin{array}{c}\text{NPV of the new case }\\ \text{operating cash flow}\end{array}\}=\left[\begin{array}{c}-\text{Initial investment}\\ +\text{ New case cash flow}\times \left(\begin{array}{l}\text{Present value of an annuity}\\ \text{ of \$1 period for R\% of N period}\end{array}\right)\end{array}\right]\\ =-\$1,750,000+\$669,875\times \left(\begin{array}{l}\text{Present value of an annuity }\\ \text{of }\$1\text{ period for }12\%\text{ of 4 period}\end{array}\right)\\ =-\$1,750,000+\left(\$669,875\times 3.03735\right)\\ =-\$1,750,000+\$2,034,644.83125\\ =\$284,644.83\end{array}$

Hence, the NPV of the new case operating cash flow is $284,644.83.

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

$\begin{array}{c}\begin{array}{l}\text{Sensitivity of NPV to }\\ \text{change in the sales value}\end{array}\}=\frac{\text{Change in NPV}}{\text{Change in fixed cost}}\\ =\frac{\left(\$286,619.11-\$284,644.83\right)}{\left(\$515,000-\$516,000\right)}\\ =\frac{\$1,974.28}{-\$1,000}\\ =-\$1.97\end{array}$

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.

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 74.64 units.

Explanation of Solution

Given information:

The unit price is $17,300, unit variable costs are $10,400, and fixed costs are $515,000.

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 the cash break-even quantity:

$\begin{array}{c}\text{Cash break-even quantity}=\frac{\text{Fixed cost}}{\left(\text{Selling price per unit}-\text{Variable cost per unit}\right)}\\ =\frac{\$515,000}{\left(\$17,300-\$10,400\right)}\\ =\frac{\$515,000}{\$6,900}\\ =74.64\text{ units}\end{array}$

Hence, the cash breakeven quantity is 74.64 units.

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 loss. It is the most widely used measure of break-even point.

Answer to Problem 19QP

The accounting break-even level of output is 138.04 units.

Explanation of Solution

Given information:

The fixed costs of the project are $515,000 per year. The initial cost of the project is $1,750,000 for the life time of 4 years. The variable cost per unit is $10,400 and price per unit of the project is $17,300.

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:

$\begin{array}{c}\text{Accounting break-even quantity}=\frac{\left(\text{Fixed cost}+\text{Depreciation}\right)}{\left(\text{Unit price}-\text{Unit variable cost}\right)}\\ =\frac{\$515,000+\left(\frac{\$1,750,000}{4}\right)}{\left(\$17,300-\$10,400\right)}\\ =\frac{\left(\$515,000+\$437,500\right)}{\$6,900}\end{array}$

$\begin{array}{c}=\frac{\$952,500}{\$6,900}\\ =138.04\text{ units}\end{array}$

Hence, the accounting breakeven quantity is 138.04 units.

Summary Introduction

To determine: The degree of operating leverage

Introduction:

Degree of operating leverage is a measure that indicates the sensitivity on the fixed cost of a project. It interprets that a company with more high operating leverage indicates higher fixed costs and vice versa.

Answer to Problem 19QP

The degree of operating leverage is 2.18 times.

Explanation of Solution

Given information:

The fixed costs of the project are $515,000 per year. The operating cash is $$670,525.

The formula to calculate the degree of operating leverage:

$\text{Degree of operating leverage}=1+\left(\frac{\text{Fixed cost}}{\text{Operating cash flow}}\right)$

Compute the degree of operating leverage:

$\begin{array}{c}\text{Degree of operating leverage}=1+\left(\frac{\text{Fixed cost}}{\text{Operating cash flow}}\right)\\ =1+\left(\frac{\$515,000}{\$\text{67}0,\text{525}}\right)\\ =1+\text{0}\text{.76}\\ =\text{1}\text{.76 times}\end{array}$

Hence, the degree of operating leverage is 1.76 times. The operating cash flows will increase by 1.76% for each 1% increase in unit sales.