Question

Dime a Dozen Diamonds makes synthetic diamonds by treating carbon. Each diamond can be sold for $130. The materials cost for a standard diamond is $80. The fixed costs incurred each year for factory upkeep and administrative expenses are $206,000. The machinery costs $1.2 million 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? Break-even sales b. What is the NPV break-even level of sales assuming a tax rate of 40%, a 10-year project life, and a discount rate of 12%? (Do not round intermediate calculations. Round your answer to the nearest whole number.) Break-even sales

Answer 1

Solution:

a.Calculation of Accounting Break even level sales :

The formula for calculating the Accounting Break – even level is

= ( Fixed costs per year + Depreciation per year ) / Contribution per unit

Where Contribution per unit = Sales price per unit - Variable costs per unit

As per the information given in the question we have

Fixed costs per year = $ 206,000 ;

Sales price per unit = $ 130 ;   Variable costs per unit = Materials cost per unit = $ 80

Thus Contribution per unit = $ 130 - $ 80 = $ 50

Cost of the machinery = $ 1,200,000   ; No. of years of project life = 10 years ;

Salvage value = Nil

We know that Annual Depreciation as per straight line method

= ( Cost of the machinery – Salvage value ) / No. of years of project life

Applying the available information in the formula for straight line depreciation we have

= ( $ 1,200,000 – $ 0 ) / 10

= $ 120,000

Depreciation per year = $ 120,000

Applying the above information in the formula for accounting break even level we have

Accounting Break even level sales = ($ 206,000   + $ 120,000 ) / $ 50

= $ 326,000 / $ 50

= 6,520

Thus the accounting break even level of sales in terms of the number of diamonds sold = 6,520

b. Calculation of NPV Break even level sales :

Calculation of Annual After cash Inflows :

The formula for calculating the annual after tax cash Inflow is

= [ (Sales - Materials cost - Fixed Cost - Depreciation ) * ( 1 - Tax rate ) ] + Depreciation

As per the information given in the question we have

Sales price per unit = $ 130   ; Let the units of sales be “x” units

Thus sales value = $ 130 * x = 130x

Materials cost per unit = $ 80   ; Let the units of sales be “x” units

Thus total Materials costs = $ 80 * x = 80x

Fixed Cost = $ 206,000 ; Tax rate = 40 % = 0.40

Cost of the machinery = $ 1,200,000   ; No. of years of project life = 10 years ;

Salvage value = Nil

We know that Annual Depreciation as per straight line method

= ( Cost of the machinery – Salvage value ) / No. of years of project life

Applying the available information in the formula for straight line depreciation we have

= ( $ 1,200,000 – $ 0 ) /10

= $ 120,000

Thus annual straight line depreciation = $ 120,000

Applying the above information we have the annual after tax cash inflows

= [ ( 130x – 80x - $ 206,000 - $ 120,000 ) * ( 1 - 0.40 ) ] + $ 120,000

= [ ( 50x - $ 326,000 ) * 0.60 ) ] + $ 120,000                   

= [ 30x - $ 195,600 ] + $ 120,000                                  

= 30x - $ 75,600

Thus the annual after tax cash inflows = 30x - $ 75,600

Calculation of present value of after cash Inflows :

As per the information given in the question

Discount rate for the project = 12 %    ; No. of years of project life = 10 Years

The present value factor at 12 % for 10 years is = PVIFA(12 %, 10) = 5.650223

Thus the present value of after tax cash inflows of the project = Annual after tax cash inflows * PVIFA(12 %, 10)

= ( 30x - $ 75,600 ) * 5.650223

= 169.506691x – $ 427,156.860948

The present value of after tax cash inflows of the project = 169.506691x – $ 427,156.860948

We know that at the NPV break even Level of sales is the present value of after tax cash inflow of the project = Initial Investment

Thus we have

169.506691x – $ 427,156.860948 = $ 1,200,000

169.506691x = $ 427,156.860948 + $ 1,200,000

169.506691x = $ 1,627,156.860948

x = 1,627,156.860948 / 169.506691

x = 9,599.366566

x = 9,599 units ( when rounded off to the nearest whole number )

Thus the NPV break even level of sales in terms of the number of diamonds sold = 9,599