Dime a Dozen Diamonds makes synthetic diamonds by treating carbon.
Each diamond can be sold for $100.
The materials cost for a standard diamond is $30. The fixed costs
incurred each year for factory upkeep
and administrative expenses are $200,000. The machinery costs $1
million a year and is depreciated
straight-line over 10 years to a salvage value of zero.							
							
a. What is the accounting break-even level of sales in terms of number
of diamonds sold?
							
b. What is the NPV break-even sales assuming a tax rate of 35 percent,
a 10-year project life and a discount
rate of 12 percent?							
							
							
Solution							
							
Problem 8-9 							
Instructions							
							
In part a, enter the formula to calculate the break-even point. In
part b, enter the formulas to calculate
all the unknown items (you will know that your formulas are correct if
the NPV is approximately equal to 0.
							
a. What is the accounting break-even level of sales in terms of number
of diamonds sold?
							
Accounting break-even point			diamonds				
							
b. What is the NPV break-even sales assuming a tax rate of 35 percent,
a 10-year project life and a discount
rate of 12 percent?							
							
Number of diamonds	5,978						
Annuity factor	5.650 						
Revenue							
Variable Expenses							
Depreciation							
Fixed expenses	$0 						
Cash Flow							
Present value of cash flow	$0.00 						
Net present value	($1,000,000.00)

Hi mbastu!!


a. What is the accounting break-even level of sales in terms of number
of diamonds sold?

The machinery cost of $1 million is depreciated straight-line over 10
years to a salvage value of zero. That means the Depreciation per year
is:
D = $1,000,000 / 10 = $100,000 

The break-even level of sales is the sales point at which EBIT = 0 ;
in other words:
sales break-even point = level of sales necessary to cover operating costs.

At the break-even level of sales:
EBIT = Revenues - Variable costs - Fixed costs - Depreciation = 0 
then:
Revenues - Variable costs = Fixed costs + Depreciation

If we call:
q = quantity sold; 
p = price per unit; 
v = variable cost per unit, 
then
Revenues = q.p   and   Variable costs = q.v   

We will have:
Revenues - Variable costs = q.p - q.v = q.(p - v) 
Revenues - Variable costs = q.(p - v) = Fixed costs + Depreciation

Then:
q = (Fixed costs + Depreciation) / (p - v)
q = ($200,000 + $100,000) / ($100 - $30) = $300,000 / $70 = 4285.71

The break-even level of sales in terms of number of diamonds sold is
4,286 diamonds.

                      -----------------------

b. What is the NPV break-even sales assuming a tax rate of 35 percent,
a 10-year project life and a discount rate of 12 percent?

At the break even point:

Revenues = q.p = 5,978 x $100 = $597,800

Variable costs = q.v = 5,978 x $30 = $179,340

Fixed costs = $200,000

Total Costs = $179,340 + $200,000 = $379,340

Depreciation = $1,000,000 / 10 years = $100,000 per year.

Tax rate (T) = 0.35

Cash Flow = (1-T)*(Revenues - Total Expenses) + T * Depreciation = 
          = 0.65*($597,800 - $379,340) + 0.35*$100,000 =
          = $141,999 + $35,000 =
          = $176,999

Present value of cash flow (PV)= Cash Flow * Annuity factor 
 
The 12%, 10-year annuity factor is 5.650 , then:
PV = $176,999 x 5.650 = $1,000,044.35


NPV = PV - Initial Investment =
    = $1,000,044.35 - $1,000,000 =
    = $44.35


NOTE:
The above value is a good aproximation to zero, and the error is a
result of some hidden roundings, you can see that taking the reverse
way:

At the breakeven point the NPV is zero, then:
NPV = 5.650 x (0.65 .(q.p - q.v - FC) + 0.35 . D ) - $1,000,000 = 0

then:
$1,000,000/5.650 - 0.35*$100,000 = 0.65*q*(p-v) - 0.65*$200,000
then:
$141,991.15 = 0.65*q*($100 - $30) - $130,000  
then:
$271,991.15 = 0.65*q*$70
==> q = $271,991.15 / (0.65*$70) = 5977.83 diamonds; this result must
be rounded to 5978 diamonds (this is the origin of the non zero result
of the NPV).



I hope this helps. If you find something unclear or missed please do
not hesitate to request for a clarification, I will be glad to give
you further assistance on this before you rate this answer.

Best regards.
livioflores-ga

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Clarification of Answer by livioflores-ga on 09 Apr 2005 21:33 PDT

Hi!!

Just one note about NPV break-even: Do not confuse it with the NPV of
the break-even level of sales point.
NPV break-even analysis calculates the level of sales that generates
an NPV of zero, known also as the financial break-even. Sales higher
than the zero NPV sales will produce positive NPV. This is equivalent
to finding the sales level for which the cost of the project equals
the present value of the cash flows. This is why we use:
NPV break-even = PV - I = 0  or  PV = I 

Hope this helps you to understand the second part of the problem.

Regards.
livioflores-ga

I had all the correct formulas but when the NPV did not equal 0, I
thought it was wrong. Thanks for your hard work and effort.