Final answer:
Calculating the monthly payment for a $14,794 loan with a 4% nominal interest rate over 10 years, plus a final balloon payment, involves finding the present value of both the regular payments and the balloon payment, and then solving for the regular monthly payment.
Step-by-step explanation:
To calculate the amount of each monthly payment for a loan of $14,794 with a 4% nominal annual interest rate, to be repaid over 10 years with a final balloon payment of $4,994, we will apply the concept of present value for both the annuity (regular payments) and the single sum (balloon payment). Since the interest rate is nominal and compounded annually, we need to adjust it to a monthly rate by dividing by 12 (the number of months in a year).
First, we find the present value of the balloon payment. Using the formula PV = FV / (1 + i)^n, where PV is the present value, FV is the future value (the balloon payment), i is the monthly interest rate, and n is the number of periods. The monthly interest rate (i) is 4% divided by 12, which is 0.0033, and n is 120 (10 years times 12 months). The present value of the balloon payment can be calculated accordingly.
Next, we calculate the present value of the regular monthly payments using the annuity present value formula PV = R * [(1 - (1 + i)^-n) / i], where R is the regular monthly payment. We then subtract the present value of the balloon payment from the initial loan amount to find the present value of the regular payments.
Finally, we solve for R (the regular monthly payment), by rearranging the annuity present value formula. With the present value known and all other variables (i and n) being fixed, the calculation will yield the required monthly repayment amount excluding the balloon payment. Remember to add the balloon payment back to the final month's payment.
It is essential to understand the total amount of interest paid over the life of the loan, including both regular payments and the balloon payment, to fully grasp the cost of borrowing. As demonstrated in various scenarios, the interest can sometimes exceed the principal amount borrowed, emphasizing the importance of considering the total cost of a loan over time.