Simon has taken a 20-year, $252,000 mortgage on his house at an interest rate of 6% p.a., compounded annually. What is the remaining balance of the mortgage after the payment of the fifth annual instalment?
Question
Simon has taken a 20-year, $252,000 mortgage on his house at an interest rate of 6% p.a., compounded annually. What is the remaining balance of the mortgage after the payment of the fifth annual instalment?
Solution
To solve this problem, we first need to calculate the annual mortgage payment using the formula for an annuity:
PMT = P * [r*(1 + r)^n] / [(1 + r)^n - 1]
Where: P = principal amount = $252,000 r = annual interest rate = 6% or 0.06 n = number of years = 20
PMT = 252,000 * [0.06*(1 + 0.06)^20] / [(1 + 0.06)^20 - 1] PMT = $21,688.96
This is the annual payment Simon needs to make.
After the fifth payment, we need to calculate the remaining balance. We can do this by calculating the future value of an annuity (which is the remaining balance of the loan after 5 payments):
FV = PMT * [(1 + r)^n - 1] / r
Where: PMT = annual payment = $21,688.96 r = annual interest rate = 6% or 0.06 n = number of years remaining = 20 - 5 = 15
FV = 21,688.96 * [(1 + 0.06)^15 - 1] / 0.06 FV = $213,383.05
So, the remaining balance of the mortgage after the payment of the fifth annual instalment is $213,383.05.
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