Question

XYZ, Inc. produces a product that will sell this year for $200 per unit. Fixed costs are expected to total $10,000 this year and remain constant for the following four years. Variable costs are expected to be $75 per unit this year and increase at a rate of 5% in each of the following four years. XYZ, Inc. expects to sell 2,000 units this year and in each of the following four years. To earn an equivalent uniform annual gross profit of $250,000 for the described five year period, the rate at which XYZ, Inc. must increase the selling price each year, for the final four years, is closest to... MARR = 15%.

correct answer: 3.31%

please by hand

Answer 1

revenue in first yr = 200*2000 = 400000

variable cost in first yr = 75 * 2000 = 150000

Present value of geometric series = A *[1 - (1+g)^n /(1+i)^n] /(i-g)

Present value of variable cost = 150000*[1 - (1+0.05)^5 /(1+0.15)^5] /(0.15-0.05)

= 150000*[1 - (1.05)^5 /(1.15)^5] /(0.1)

= 150000*3.65462499

= 548193.75

Present value of total cost = 548193.75 + 10000*(P/A,15%,5)

= 548193.75 + 10000*3.352155

= 581715.30

Present value of total annual profit = 250000*(P/A,15%,5)

= 250000*3.352155

= 838038.75

let increase in price be g, then

Present value of revenue = 400000*[1 - (1+g)^5 /(1+0.15)^5] /(0.15-g)

= 400000*[1 - (1+g)^5 /(1.15)^5] /(0.15-g)

= 400000*[1 - (1+g)^5 / 2.011357] /(0.15-g)

Present value of profit = Present value of revenue - present value of cost

Present value of revenue = 838038.75 + 581715.30 = 1419754.05

400000*[1 - (1+g)^5 / 2.011357] /(0.15-g) = 1419754.05

[1 - (1+g)^5 / 2.011357] /(0.15-g) = 1419754.05 / 400000 = 3.549385

using trail and error method

When i = 3%, value of [1 - (1+g)^5 / 2.011357] /(0.15-g) = [1 - (1+0.03)^5 / 2.011357] /(0.15-0.03) = 3.530299

When i = 4%, value of [1 - (1+g)^5 / 2.011357] /(0.15-g) = [1 - (1+0.04)^5 / 2.011357] /(0.15-0.04) =

3.591895

using interpolation

i = 4% - (3.591895-3.549385) /(3.591895-3.530299)*(4%-3%)

i = 4% - 0.6901422% = 3.30985% ~ 3.31%