Ms. Jones borrows 3.120M for 5.5% interest compounded monthly. What is the periodic payment needed after 15yrs?
To find the periodic payment needed after 15 years for a loan with an interest rate of 5.5% compounded monthly, you can use the formula for the future value of an annuity.
The formula is:
A = P * (1 - (1 + r)^(-n)) / r
Where:
A is the future value of the annuity (the total amount repaid),
P is the periodic payment,
r is the interest rate per compounding period,
n is the number of compounding periods.
Let's break down the problem and solve it step by step:
1. Convert the interest rate to its decimal form:
5.5% = 0.055.
2. Convert the loan amount to its decimal form:
3.120M = 3,120,000.
3. Determine the number of compounding periods:
Since the loan term is 15 years and it is compounded monthly, multiply the number of years by 12:
15 years * 12 months/year = 180 months.
4. Substitute the values into the formula:
A = P * (1 - (1 + r)^(-n)) / r
3,120,000 = P * (1 - (1 + 0.055) ^ (-180)) / 0.055
5. Solve for P:
P = 3,120,000 * (0.055) / (1 - (1 + 0.055) ^ (-180))
P = 3,120,000 * (0.055) / (1 - 0.0966)
P = 171,600 / 0.9034
P ≈ 189,910.942
So, the periodic payment needed after 15 years would be approximately $189,910.942.