Question

Big Company purchased a machine on February 1, 2013, and will make seven semiannual payments of $23,500 beginning five years from the date of purchase. The interest rate will be 12%, compounded semiannually. Determine the purchase price of the machine.

Answer 1

the purchase price of machine will be the present value of the stream of payments:

first we need to know the present value of stream of payments starting 5 years from date of purchase:

the are a form of annuity immediate at the end of 5th year:

the present value of annuity immediate = P + P[1 -(1+r)^-(n-1)]/r

here,

P=23,500

r = 12% per annum =>12%*6/12=>6% since semi annual compounding.

n = 7 payments.

so

value of annuity at the end of 5 th year = 23,500 + 23,500[1 - (1.06)^-(7-1)]/0.06

=>23,500 + 23500[0.2950395/0.06]

=>23,500 + 23,500*[4.917325]

=> 23,500+115,557.14

=>139,057.14

so,

the purchase price of the machine will be the present value of this single amount of $139,057.14 at the end of fifth year.

present value = future amount / (1+r)^n

here,

r=6%=>0.06

n = 5 years * 2 semi annual periods

=>10 periods

=> 139,057.14 / (1.06)^10

=>$77,648.78.

purchase price of machine will be =$77,648.78