Final answer:
The rate of the annuity for the given loan can be calculated using the formula PV = R(1 - (1 + i)-n)/i, where PV is the present value of the loan, R is the monthly payment, i is the monthly interest rate, and n is the number of months.
Step-by-step explanation:
To calculate the monthly payment of an annuity, you can use the formula PV = R(1 - (1 + i)-n)/i, where PV is the present value of the loan, R is the monthly payment, i is the monthly interest rate, and n is the number of months. In this case, the loan amount is $300,000 with a 6% interest rate, convertible monthly, and the loan term is 30 years. Plugging these values into the formula, we get:
300,000 = R(1 - (1 + 0.06/12)⁽⁻³⁰ˣ¹²⁾)/(0.06/12)
Simplifying the equation gives us:
R ≈ $1,798.65
So the monthly payments for the loan would be approximately $1,798.65.
If the payments were larger by a fraction of 12, it means making 13 payments a year instead of just 12. To calculate the monthly payments for the larger payments, we can use the same formula, but with a modified loan term of 30 × 12/13 months:
300,000 = R(1 - (1 + 0.06/12)⁽⁻³⁰ˣ¹²/¹³⁾)/(0.06/12)
Simplifying the equation gives us:
R ≈ $1,887.29
So with the larger payments, the monthly payments for the loan would be approximately $1,887.29.
By comparing the two monthly payments, we can see that the larger payments save an additional $88.64 per month. Over the course of the loan, which is 30 years or 360 months, the total amount saved would be approximately $31,910.40.