A $10,000 loan is to be amortized for 10 years with quarterly payments of $349.72. If the interest rate is 7%, compounded quarterly, what is the unpaid balance immediately after the sixth payment? (Round your answer to the nearest cent.)
Assuming the payments are made at the end of a period to match the standard formulas
outstanding balance
= 10000(1.0058333...)^6 - 349.72( 1.0085333..^6 - 1)/.00853333...
= .....
To find the unpaid balance immediately after the sixth payment, we need to calculate the remaining principal balance after five payments and then subtract the payment made in the sixth quarter.
Here's how to calculate it step by step:
1. First, we need to find the loan's interest rate per quarter. Given that the annual interest rate is 7%, compounded quarterly, we divide it by 4 to get the quarterly interest rate: 7% / 4 = 0.07 / 4 = 0.0175 (rounded to four decimal places).
2. Next, we calculate the number of payments over the 10-year period. Since payments are made quarterly, we multiply 10 years by 4 quarters per year to get 40 payments.
3. Using the quarterly interest rate, we can calculate the payment amount based on the loan's principal and number of payments. The formula for the payment amount on an amortizing loan can be expressed as:
P = (r * PV) / (1 - (1 + r)^-n)
where:
P = payment amount
r = interest rate per payment period
PV = present value or loan amount
n = total number of payments
Substituting the given values, we have:
$349.72 = (0.0175 * PV) / (1 - (1 + 0.0175)^-40)
4. Solving the equation for the present value (PV), we find that the loan amount is $10,000.
5. Now we can calculate the remaining principal balance after five payments. We need to calculate the remaining balance after each payment and subtract the payment made in the sixth quarter. To do this, we use the formula for the remaining balance on an amortizing loan:
RB = PV * (1 + r)^n - P * ((1 + r)^n - 1) / r
where:
RB = remaining balance
PV = present value or loan amount
r = interest rate per payment period
n = number of payments made
Substituting the given values, we have:
RB = $10,000 * (1 + 0.0175)^5 - $349.72 * ((1 + 0.0175)^5 - 1) / 0.0175
6. Evaluating the expression, we find that the remaining balance after five payments is approximately $9,200.09 (rounded to the nearest cent).
7. Finally, we subtract the payment made in the sixth quarter, $349.72, from the remaining balance to find the unpaid balance immediately after the sixth payment:
Unpaid balance = $9,200.09 - $349.72 = $8,850.37
Therefore, the unpaid balance immediately after the sixth payment is approximately $8,850.37.