Suppose I have a loan of M dollars. At the end of each year, I am charged interest at rate R and make a repayment of P. The loan is repaid after n years.
- How long (n) does it take to repay the loan if I am given the other variables?
- How much are the repayments of P if I am given the other variable?
- Suppose that the payments were at the start of the year. How would this change the problem?
Basic Theory
The way to solve this problem is to calculate how much each payment reduces your debt after you have been repaying your loan for $n$ years. Let $r=1+R/100$, ie. this converts the interest rate from a percentage to a value you can multiply your debt by to calculate how much you owe after adding one time period's interest.
If I make a payment of $P$ at the end of the $k$th year, then we avoid paying interest on this money $n-k$ times and so we reduce our debt by $Pr^{n-k}$. We sum up the future values of all our payments:
$\sum\limits_{k=1}^n Pr^{n-k}$
If we reverse this, it is equivalent to:
$\sum\limits_{k=0}^{n-1} Pr^k$
This is a geometric series, which can be solved using the formula $\frac{ar^{n-1}}{r-1}$ where $a$ is the first term, $r$ is the factor and $n$ is the number of terms being summed. We then attempt to equate this with the debt owed after $n$ years, which is $Mr^n$.
We now compare the two equations:
$\frac{Pr^{n-1}}{r-1} = Mr^n$
Calculating $n$
We group the $r^n$ terms:
$\frac{P}{r-1} = r^n\frac{M-P}{r-1}$
$r^n = \frac{P}{M(r-1)-P}$
So we just take the $n$th log of the right hand side.
Calculating repayments
Given the principal ($M$) and the interest rate ($r$), what will my payment-per-term ($P$) be over $n$ accruation terms?
$P=\frac{Mr^n(r-1)}{r^{n-1}}$
Payments made at the start of the year
In this case, the future values of our interest payment simply become:
$\sum\limits_{k=1}^n Pr^k$
We proceed as we did before.
Notes
We could also solve this problem using present value instead of future value.