Chapter 7, Problem 1QP

Sensitivity Analysis and Break-Even Point We are evaluating a project that costs $588,000, has an eight-year life, and has no salvage value. Assume that depreciation is straight-line to zero over the life of the project. Sales are projected at 70,000 units per year. Price per unit is $36, variable cost per unit is $20, and fixed costs are $695,000 per year. The tax rate is 35 percent, and we require a return of 15 percent on this project.

1. a. Calculate the accounting break-even point.
2. b. Calculate the base-case cash flow and NPV. What is the sensitivity of NPV to changes in the sales figure? Explain what your answer tells you about a 500-unit decrease in projected sales.
3. c. What is the sensitivity of OCF to changes in the variable cost figure? Explain what your answer tells you about a $1 decrease in estimated variable costs.

Summary Introduction

To determine: Accounting break-even point

Introduction:

Accounting break-even point refers where the company faces zero profits. This technique determines the variable’s value (independent) affecting another variable (dependent) under some set of assumptions is termed as sensitivity analysis.

Answer to Problem 1QP

Solution: The accounting break-even point is $48,031.25.

Explanation of Solution

Given information:

The cost of the project is $588,000 and the life is 8 years with no salvage value and is depreciated under straight line method. A sale of the project is 70,000 units, price per unit is $36, variable cost is $20 per unit and fixed cost is $695,000. The tax is at 35% and the rate of return is 15%.

The formula to calculate accounting break-even point:

$\text{Break-even point}=\frac{\text{Fixed cost}+\text{Depreciation}}{\text{Unit price}-\text{Variable price}}$

The formula to calculate the depreciation:

$\text{Depreciation}=\frac{\text{Original cost}}{\text{Number of years}}$

Compute the value of depreciation:

$\begin{array}{c}\text{Depreciation}=\frac{\text{Original cost}}{\text{Number of years}}\\ =\frac{\$588,000}{8}\\ =\$73,500\end{array}$

Hence, the depreciation is $73,500.

Compute the accounting break-even point:

$\begin{array}{c}\text{Break-even point}=\frac{\text{Fixed cost}+\text{Depreciation}}{\text{Unit price}-\text{Variable price}}\\ =\frac{\$695,000+\$73,500}{\$36-\$20}\\ =\frac{\$768,500}{\$16}\\ =\$48,031.25\end{array}$

Hence, the accounting break-even point is $48,031.25.

Summary Introduction

To determine: The base-case flow and net present value

Introduction:

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

Answer to Problem 1QP

Solution:

Base-case operating cash flow is $301,975 and the net present value is $767,058.45.

The sensitivity of net present value is $813,726.58 and the net present value drops $23,330.

Explanation of Solution

Given information:

The cost of the project is $588,000 and the life is 8 years with no salvage value and is depreciated under straight line method. A sale of the project is $70,000 units, price per unit is $36, variable cost is $20 per unit and fixed cost is $695,000. The tax is at 35% and the rate of return is 15%.

The formula to calculate the base-case cash flow:

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

The formula to calculate the net present value:

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

Compute the base-case cash flow:

$\begin{array}{c}\text{Net present value}=\left(\begin{array}{l}\left[\text{Unit sold}\times \left(\text{Selling price}-\text{Variable price}\right)\right]-\text{Fixed cost}\\ \times \left(1-\text{Tax rate}\right)+\text{Depreciation}\times \text{Tax rate}\end{array}\right)\\ =\left(\left[\$70,000\times \left(\$36-\$20\right)-\$695,000\right]\left(1-0.35\right)+\left(\$73,500\times 0.35\right)\right)\\ =\$425,000\times 0.65+\$25,725\\ =\$276,250+\$25,725\\ =\$301,975\end{array}$

Hence, base-case operating cash flow is $301,975.

Compute net present value:

Note: The increase in operating cash flow at present value of interest factor annuity at 15% for 8 years is 4.48732.

$\begin{array}{c}\text{Net present value}=-\text{ Original cost}+\text{Operating cash flow(PVIFA)}\\ =-\$588,000+\$301,975\left(4.48732\right)\\ =\$767,058.45\end{array}$

Hence, the net present value is $767,058.45.

Note: The following calculations are required to compute the net present value when sales decrease by 500 units.

Compute net present value when sales is 71,000:

Note: Net present value at different quantities is calculated to determine the sensitivity of the net present value. Therefore, consider 71,000 as units sold.

$\begin{array}{c}\text{Net present value}=\left(\begin{array}{l}\left[\text{Unit sold}\times \left(\text{Selling price}-\text{Variable price}\right)\right]-\text{Fixed cost}\\ \times \left(1-\text{Tax rate}\right)+\text{Depreciation}\times \text{Tax rate}\end{array}\right)\\ =\left(\left[\$71,000\times \left(\$36-\$20\right)-\$695,000\right]\left(1-0.35\right)+\left(\$73,500\times 0.35\right)\right)\\ =\$312,375\end{array}$

Hence, the operating cash flow when sales are 71,000 is $312,375.

Compute net present value when sales is 71,000:

Note: The increase in operating cash flow at present value of interest factor annuity at 15% for 8 years is 4.48732.

$\begin{array}{c}\text{Net present value}=-\text{ Original cost}+\text{Operating cash flow(PVIFA)}\\ =-\$588,000+\$312,375\left(4.48732\right)\\ =\$813,726.58\end{array}$

Hence, the sensitivity of net present value is $813,726.58.

Compute changes in net present value:

$\begin{array}{c}\text{Change in net present value}=\frac{\text{Net present value at base}-\text{Net present value}}{\text{Unit sold}-\text{Increase in units sold}}\\ =\frac{\$767,058.45-\$813,726.58}{\$70,000-\$71,000}\\ =\frac{-\$46,668.13}{-\$1,000}\\ =\$46.66\end{array}$

Hence, the changes in net present value are $46.66.

Note: When sales drop by 500 units, then eventually the net present value will drop.

Compute the NPV drop value:

$\begin{array}{c}\text{Amount of NVP drop}=\text{Change in net present value}\times \text{Sales drop value}\\ =\$46.66\times 500\\ =\$23,330\end{array}$

Hence, the net present value drops by $23,330.

Summary Introduction

To determine: The sensitivity of operating cash flow to changes in the variable cost and a decrease of $1 in estimated variable costs.

Answer to Problem 1QP

Solution: The change in net present value is −$45,500 when variable cost decreases by $1.

Explanation of Solution

Given information:

The cost of the project is $588,000 and the life is 8 years with no salvage value and is depreciated under straight line method. A sale of the project is 70,000 units, price per unit is $36, variable cost is $20 per unit and fixed cost is $695,000. The tax is at 35% and the rate of return is 15%.

The formula to calculate the operating cash flow:

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

The formula to calculate the net present value:

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

The formula to calculate changes in operating cash flow for a $1 change in variable cost:

$\text{Changes in net present value}=\frac{\text{Base-case operating cash flow}-\text{Operating csah flow}}{\text{Unit of variable cost}-\text{Increase in variable cost}}$

Compute sensitivity of operating cash flow change in variable cost:

Note: Assume variable cost as $21 since choosing another number is irrelevant because the same ratio of operating cash flow will be determined when the dollar decreases by $1. Using ‘tax shield approach’, determine the operating cash flow when the variable cost is $21.

$\begin{array}{c}\text{Operating cash flows}=\left(\begin{array}{l}\left[\text{Unit sold}\times \left(\text{Selling price}-\text{Variable price}\right)\right]-\text{Fixed cost}\\ \times \left(1-\text{Tax rate}\right)+\text{Depreciation}\times \text{Tax rate}\end{array}\right)\\ =\left(\left[\$70,000\times \left(\$36-\$21\right)-\$695,000\right]\left(1-0.35\right)+\left(\$73,500\times 0.35\right)\right)\\ =\$230,750+\$25,725\\ =\$256,475\end{array}$

Hence, the operating cash flow is $256,475.

Compute change in operating cash flow for a $1 change in variable cost:

$\begin{array}{c}\text{Changes in net present value}=\frac{\text{Base-case operating cash flow}-\text{Operating csah flow}}{\text{Unit of variable cost}-\text{Increase in variable cost}}\\ =\frac{\$301,975-\$256,475}{\$20-\$21}\\ =-\$45,500\end{array}$

Hence, the change in the net present value is −$45,500