I am having terrible trouble with the following question:
Suppose that $x_n$ is the amount owed on a mortgage after n years, £$m$ is the monthly repayment and $r$ is the annual percentage interest rate charged on the amount of the mortgage outstanding.
i) Derive a difference equation satisfied by $x_n$.
ii) Solve the difference equation derived in i) for a loan of $£m$ to be repaid over $N$ years and hence determine what the monthly payment should be.
iii) If the interest rate is 5% show that the monthly repayment on a loan of £50,000 to be repaid over $25$ years is £295.64.
iv) What is the total amount paid back on the loan?
Any help would be great thanks 🙂
Best Answer
As stated in the problem, $£m$, or for simplicity's sake, $x_0$, is the loan amount. We want to derive an equation for $x_n$, which is the amount owed after $n$ years. That's why we make the loan amount $x_0$, because it is the amount owed after 0 years. We do this by writing it in terms of $x_{n-1}$, the amount owed from the previous year. Doing this is the same as writing a difference equation; they are the same thing.
We start with $x_n$: $$x_n = ?$$
So we know that $x_{n-1}$ is the amount owed from the previous year. So we will say that $x_n$, the amount owed for the $n$th year, is related to $x_{n-1}$, the amount owed for the previous year: $$x_n = f(x_{n-1}) = x_{n-1} + \text{adjustments}$$
We make 12 monthly payments of $m$, so we will subtract $12m$ from this: $$x_n = x_{n-1} - 12m + \text{adjustments}$$
Then we are charged $r$ times the amount owed the previous year, which is $x_{n-1}$: $$x_n = x_{n-1} - 12m + r\cdot x_{n-1}$$
That gives (i).
An alternate form of (i) can be obtained by just using arithmetic: $$x_n = (1 + r)x_{n-1} - 12m$$