Final answer:
The monthly payments on a $300,000 loan at a 6% interest rate, compounded monthly over 30 years, can be calculated using the present value of an annuity formula, and making an extra payment each year will reduce the time and interest required to pay off the loan.
Step-by-step explanation:
To calculate the monthly payments on a $300,000 loan at 6% annual interest rate, compounded monthly, with payments spread over 30 years, we can use the formula for the present value of an annuity. This formula takes into account the compound interest and helps us find the monthly payment necessary to pay off the loan fully over the given period.
The formula for the monthly payment (R) based on the present value (PV) of an annuity is:
PV = R × ¶(1 - (1 + i)^-n) / i
Where:
i = monthly interest rate (annual rate/12)
n = total number of payments (years × 12)
By rearranging the formula to solve for R, we can determine the monthly payment. Once we know the monthly payment for 12 payments per year, we can calculate the effect of making a thirteenth payment annually. This additional payment will reduce the principal more quickly, saving interest over time and thus reducing the overall time to pay off the loan.
To find out how much time and money would be saved with this additional payment, we would compare the amortization schedules: one with 12 payments a year and the other with effectively 13 payments a year (an extra payment every 12 months). By doing so, we can determine the reduced time it takes for the loan balance to reach zero and the total interest saved.