Question

The payment necessary to amortize a 6.7% loan of $84,000 compounded annually, with 10 annual payments is $11,794.40. The total of the payments is $117,944.00 with a total interest payment of $33,944,00. The borrower made larger payments of $12,000.00. Calculate

(a) the time needed to pay off the loan

Answer 1

(a) The time needed to pay off the loan, given larger payments of $12,000.00, can be determined using the formula for the number of periods in a loan. Utilizing the formula
`\(n = (\log(PV \cdot r / PMT + 1))/(\log(1 + r))\)`, where
`\(n\)` is the number of periods,
`\(PV\)` is the present value (loan amount),
`\(r\)` is the interest rate per period, and
`\(PMT\)` is the periodic payment, the time is approximately 7.5 years.

Step-by-step explanation:

To calculate the time needed to pay off the loan, we use the formula for the number of periods
`(\(n\))` in a loan. The formula is
`\(n = (\log(PV \cdot r / PMT + 1))/(\log(1 + r))\),` where
`\(PV\)` is the present value (loan amount),
`\(r\)` is the interest rate per period, and
`\(PMT\)` is the periodic payment. In this case,
`\(PV = $84,000\), \(r = 6.7\%\)` (converted to decimal form as
`\(0.067\))`, and
`\(PMT = $12,000.00\).` Substituting these values into the formula, we find
`\(n \approx 7.5\)`years.

The result indicates that, with larger payments of $12,000.00, the borrower would need approximately 7.5 years to fully pay off the loan. It's important to note that this calculation assumes regular, consistent payments over time. The formula accounts for compounding interest and allows for the determination of the time needed to repay the loan based on the given parameters.

In summary, by applying the formula for the number of periods, we can calculate the time required to pay off the loan with larger payments, providing a clear understanding of the repayment timeline.