Andy needs to pay off a loan of $18,000 in 5 years. Find the amortization payment he would need to make each semi-monthly pay period (twice a month)at 6% compounded bi-monthly, in order to pay off the loan.
I understand bi-monthly to mean every other month, or 6 times a year.
The main problem here is that the payment period does not coincide with the interest period, so my normal formula does not work
so we will convert the 6% compounded bi-monthly to one of i compounded semi-monthly since the payment period is semi-monthly
so (1+i)24 = (1.-1)^6
take the 24th root of both sides
1+i = 1.01^(6/24) = 1.01^(1/4)
1+i = 1.00249
(check:
1.00249^24 = 1.06152..
1.01^6 = 1.06152.. OK )
let the payment be P
he will make 30 of these
P( 1 - 1.00249^-30)/.00249 = 18000
P = 623.44
To calculate the amortization payment, we need to use the formula for the monthly payment on a loan. In this case, since the payments are semi-monthly, we will need to adjust the formula accordingly.
The formula to calculate the monthly payment on a loan, also known as the amortization payment, is:
P = (r * PV) / (1 - (1 + r) ^ -n)
Where:
P = Monthly Payment
PV = Loan Amount (Present Value)
r = Monthly Interest Rate
n = Total Number of Payments
First, we need to determine the values for the formula. Let's calculate them step by step:
1. Loan Amount (Present Value):
The loan amount is given as $18,000.
2. Monthly Interest Rate:
The interest rate is given as 6%, compounded bi-monthly. Since there are 12 months in a year, we need to adjust the interest rate accordingly. We divide the annual interest rate by 12 to get the monthly interest rate.
Monthly Interest Rate = 6% / 12 = 0.06 / 12 = 0.005
3. Total Number of Payments:
The loan is to be paid off in 5 years. Since there are two payments per month, the total number of payments would be 5 years * 12 months/year * 2 payments/month = 120 payments.
Now that we have all the values, let's calculate the amortization payment (P) using the formula mentioned above:
P = (r * PV) / (1 - (1 + r) ^ -n)
P = (0.005 * $18,000) / (1 - (1 + 0.005) ^ -120)
Calculating this using a calculator, the amortization payment comes out to be approximately $287.77 per semi-monthly pay period.
Therefore, Andy would need to make an amortization payment of approximately $287.77 every semi-monthly pay period (twice a month) to pay off the loan of $18,000 in 5 years at 6% compounded bi-monthly.