Vanna has just financed the purchase of a home for $200 000. She agreed to repay the loan by making equal monthly blended payments of $3000 each at 4%/a, compounded monthly.
a. How long will it take to repay the loan?
b. How much will be the final payment?
c. Determine how much interest she will pay for her loan.
To answer these questions, we can use the formula for the present value of an ordinary annuity:
\[PV = \frac{P(1 - (1 + r)^{-n})}{r}\]
Where:
PV = Present value (loan amount)
P = Monthly payment
r = Monthly interest rate
n = Number of periods
a. How long will it take to repay the loan?
We need to solve for n in the equation above. Let's substitute the given values into the equation:
\[200000 = \frac{3000(1 - (1 + 0.04/12)^{-n})}{0.04/12}\]
Simplifying the equation, we get:
\[1 - (1 + 0.04/12)^{-n} = \frac{200000 \cdot (0.04/12)}{3000}\]
\[1 + 0.04/12)^{-n} = 1 - \frac{200000 \cdot (0.04/12)}{3000}\]
Taking the natural logarithm of both sides to solve for n:
\[-n \cdot ln(1 + 0.04/12) = ln \left(1 - \frac{200000 \cdot (0.04/12)}{3000}\right)\]
Once we solve for n, we'll have the number of periods needed to repay the loan.
b. How much will be the final payment?
The final payment will be the same as the monthly payment amount. In this case, $3000.
c. Determine how much interest she will pay for her loan.
The total amount of interest paid can be calculated by subtracting the loan amount from the sum of all the payments made. We can find the sum of payments with the formula:
\[Sum\ of\ Payments = Monthly\ Payment \times Number\ of\ Periods\]
Then, the interest paid can be found by subtracting the loan amount from the sum of payments:
\[Interest\ Paid = Sum\ of\ Payments - Loan\ Amount\]
Let's solve these calculations step by step.