Question

The business plan for KnowIt, LLC, a start-up company that manufactures portable multigas detectors, showed equivalent annual cash flows of $400,000 for the first 5 years. If the cash flow in year 1 was $307,000 and the constant increase thereafter was $50,000 per year, what interest rate was used in the calculation? The interest rate used in the calculation was

Answer 1

Present Value of Amount payed after n years is given by:

P = A/(1 + r)ⁿ

where r = interest rate

We have to find r such that Present Value of both the cash flows are equal

There is annual cash flow of 400,000 for 5 years

Hence In this case we have net Present Value is given by:

NPV = 400,000/(1 + r) + 400,000/(1 + r)² + 400,000/(1 + r)³ + 400,000/(1 + r)⁴ + 400,000/(1 + r)⁵

In second case we have following cash flow 307,000 for 1st year and There is constant increase thereafter was $50,000 per year.

Now NPV' = 307,000/(1 + r) + 357,000/(1 + r)² + 407,000/(1 + r)³ + 457,000/(1 + r)⁴ + 507,000/(1 + r)⁵

Hence equating these we get

NPV = NPV'

=> 400,000/(1 + r) + 400,000/(1 + r)² + 400,000/(1 + r)³ + 400,000/(1 + r)⁴ + 400,000/(1 + r)⁵

= 307,000/(1 + r) + 357,000/(1 + r)² + 407,000/(1 + r)³ + 457,000/(1 + r)⁴ + 507,000/(1 + r)⁵

=> 93,000/(1 + r) + 43,000/(1 + r)² - 7,000/(1 + r)³ - 57,000/(1 + r)⁴ - 107,000/(1 + r)⁵ = 0

Let (1 + r) = t

=> 93,000/t + 43,000/t² - 7,000/t³ - 57,000/t⁴ - 107,000/t⁵ = 0

=> 93,000t⁴ + 43,000t³ - 7,000t² - 57,000t - 107,000 = 0

Solving this we get:

t = 1.07267

=> (1 + r) = 1.07267

=> t = 0.07267 ~ 7.267%

Hence,

The interest rate used in the calculation was 7.267%