Question 7.7: A borrower is paying €1200 monthly on a 20-year mortgage whe......
A borrower is paying €1200 monthly on a 20-year mortgage where the interest rate is 4% compounded monthly. Determine the following:
1. The principal borrowed
2. The principal amount outstanding at the start of year 20.
The number of payment periods is 20 ∗ 12 = 240 and the interest rate per period is 4%/12 = 0.33% = 0.0033.
1. The principal borrowed may be determined from the formula for calculating the regular payment to amortize the loan, and so we solve for the unknown value P in the formula.
We manipulate the formula to get that
A = \frac{P i}{\left[1-\frac{1}{(1+i)^{n}}\right]}
P = A\left[1{-}(1+r)^{-n}\right]/i
= 1200\,[1-(1.0033)^{-240}]/0.0033
= €198,716.72
2. The principal outstanding at the start of year 20 is the present value of the 12 payments made in year 20:
P = A\left[1{-}(1+r)^{-n}\right]/i
= 1200\,[1-(1.0033)^{-12}]/0.0033
= 1200 * 0.03876/0.0033
= €14,095.82