Question

A firm initiates a 4 year project with an investment of $50,000. Assume that this initial investment is depreciated using the straight line method. There is no salvage value at the end of the project. Under this project a certain product is produced and sold. Key financial information is provided below:

Price/unit	10
Direct Expenses/unit	2
SGA (excl. Depreciation)	7,500
Taxes	30%

1. What is the NI break-even?

2. What is the NPV break-even? Assume that r = 9%.
   
3. Is the NI BE conservative or aggressive compared to the NPV BE? Why? (1-2 sentences)

Answer 1

Solution for a

Let the total number units sold be x

net income = sales - direct expenses - depreciation - SGA - taxes

Sales = number of units sold * price per unit. Hence sales = x*$10 = $10x

direct expenses = number of units sold * direct expenses per unit. Hence direct expenses per unit = x*$2 = $2x

depreciation using straight line = (Investment - salvage value)/ number of years = ($50,000 - $0)/4 = $12,500

taxes = tax rate * (sales - direct expenses - depreciation - SGA) = 0.3 *($10x -$2x - $12,500 - $7,500)

Net income at break even = $0 =sales -direct expenses - depreciation - SGA - taxes

hence, sales = direct expenses + depreciation + SGA + taxes

$10x = $2x + $12,500 + $7500 + [0.3 *($10x -$2x - $12,500 - $7,500)]

$10x = $2x + $20,000 + [0.3 *($8x - $20,000)]

$10x = $2x + $20,000 + [0.3 *$8x - 0.3* $20,000)]

$10x = $2x + $20,000 + [$2.4x - $6,000]

$10x = $2x + $20,000 + $2.4x - $6,000

$10x = $4.4x + $14,000

$10x - $4.4x = $14,000

$5.6x = $14,000

x= $14000/$5.6 = 2,500

The breakeven point for net income is 2,500 units. That is, the company has to sell 2,500 units to cover all its costs.

Solution for b

Similarly, the breakeven point for NPV is the point at which the initial outlay is equal to the future discounted cash flows

Hence initial outlay = cash flow at year n/ (1+ discount rate)ⁿ

Let the undiscounted cash flows for the four years (they are all the same as sales, expenses, depreciation and taxes are constant throughout the 4 years) , be x

Therefore,

$50,000 = x/(1.09)¹ + x/(1.09)² + x/(1.09)³ + x/(1.09)⁴

Using greatest common denominator technique

$50,000 = x*(1.09)³/ [(1.09)¹ *(1.09³)] + x*(1.09)² / [(1.09²) *(1.09²)] + x * (1.09) /(1.09)³* (1.09)   + x/(1.09)⁴

$50,000 = x*(1.09)³/ [(1.09)⁴] + x*(1.09)² / [(1.09)⁴] + x * (1.09) /(1.09)⁴ + x/(1.09)⁴

$50,000 = {x* (1.09)³ + x *(1.09)² + x* (1.09) + x} / (1.09)⁴

$50,000 = (1.295029x + 1.1881x + 1.09x + x) / 1.411582 = 4.573129x/1.411582 = 3.23972x

x= $50,000/ 3.23972 = $15,433.433

Hence, the undiscounted cash flows each year, where NPV = 0 is $15,433.433

Net income= cash flow in each year - depreciation. Since cash flow calculations neglects the noncash expenses like depreciation, we have to subtract depreciation to calculate net income

Hence, net income at which NPV will be $0 is $15,433.433 + $12,500 = $27,933.433

net income = sales - direct expenses - depreciation - SGA - taxes

taxes = tax rate * (sales - direct expenses - depreciation - SGA)

Let the number of units sold be z

Hence, $27,933.433 = $10z - $2z - $12,500 - $7,500 - [ 0.30 ($10z - $2z -$12,500 - $7,500)}

$27,933.433 = $8z - $20,000 - [0.3 ($8z - $20,000)]

$27,933.433 = $8z - $20,000 - (2.4z -$6,000)

$27,933.433 = $8z - $20,000 - 2.4z + $6,000

$27,933.433 = $5.6z - $14,000

5.6z = $27,933.433 + $14,000 = $41,933.433

z = $41,933.433/ 5.6 = 7,488.11 units or 7,489 units since number of goods are in whole numbers

Hence, the company has to sell 7,489 units to breakeven on an NPV basis.

Solution to c

The net income break even point is aggressive compared to its NPV break even point. This is because for the NPV to breakeven, the discounted future cash flows must be equal to the initial outlay. This is not required in net income calculation. Hence, the break even point for NPV is much higher than that of net income.