Big Company purchased a machine on February 1, 2013, and will make seven semiannual payments of $26,000 beginning five years from the date of purchase. The interest rate will be 12%, compounded semiannually. Determine the purchase price of the machine.
To determine the purchase price of the machine, we need to calculate the present value of the seven semiannual payments.
First, let's determine the semiannual interest rate. Since the interest rate is given as 12% compounded semiannually, the semiannual interest rate is 12% divided by 2, which is 6%.
Now, let's calculate the present value of the seven semiannual payments using the formula for calculating the present value of an annuity:
PV = PMT × (1 - (1 + r)^-n) / r
Where:
PV = Present value
PMT = Payment amount
r = Interest rate per period
n = Number of periods
In this case, PMT is $26,000, r is 6% (0.06 as a decimal), and n is 7.
Let's plug the values into the formula:
PV = $26,000 × (1 - (1 + 0.06)^-7) / 0.06
Calculating this expression will give us the present value of the seven semiannual payments, which represents the purchase price of the machine.