Let's say a loan of £1000 was to be repaid by payments of £100 at the end of each quarter, and a smaller final payment made one quarter after the last regular payment. If the interest rate is 10% p.a, but it drops to 8% p.a once the outstanding balance is less than £500, then how would I calculate which payment causes the outstanding balance to drop to less than £500?

I know that in theory this could be done by drawing up a loan schedule, but how could I approach this question algebraically?


Hint: The outstanding balance after $n$ quarterly made payments is

$$B_n=1000\cdot \left( 1+\frac{0.1}4 \right)^n-100\cdot \frac{\left( 1+\frac{0.1}4 \right)^n-1}{\frac{0.1}4 }$$

Therefore the outstanding balance is less than 500, if $B_n<500$. You have to solve it for $n$. If we multiply the inequality by $\frac{0.1}4 =\frac1{40}$ and substitute $1+\frac{0.1}4$ by $q$ we obtain

$$25\cdot q^n-100\cdot \left( q^n-1\right)<12.5$$


In this context I can only refer to problems where a loan ($C_0$) is taken out and the repayments are constant. If the repayments are made $m$ times in a year and the (constant) repayments (r) are made at the end of each period we have the following term:

$$B_n=C_0\cdot \left( 1+\frac{i}m \right)^{n}-r\cdot \frac{\left( 1+\frac{i}m \right)^{n}-1}{\frac{i}m }$$

$B_n$: Outstanding balance after $n$ periods.

$i$: nominal interest rate

Usually we have a constant interest rate i and a constant repayment (annuity). In this case we can set $B_n$ equal to zero and we are able to calculate the missing value of $i, n,r$ or $C_0$

If the repayments are made at the $\color{blue}{\textrm{beginning}}$ of each period the formula changes to

$$B_n=C_0\cdot \left( 1+\frac{i}m \right)^{n}-r\cdot \color{blue}{ \left( 1+\frac{i}m \right)}\cdot \frac{\left( 1+\frac{i}m \right)^{n}-1}{\frac{i}m }$$

  • $\begingroup$ Great work. This brings the correct answer which is the 7th payment. One question is: when you have an 8% interest rate p.a, why does it work that you can just divide it by 4 to obtain the quarterly interest rate? Is it that simple? My main problem in financial mathematics is understanding when and when not to use nominal rates/ effective rates/ rates per conversion period. I know I may be asking a lot for one comment/ post, but can you offer any clarity to me? Thank you. $\endgroup$
    – jpatrickd
    Aug 1 '19 at 11:01
  • $\begingroup$ I have made an edit. $\endgroup$ Aug 1 '19 at 14:37
  • $\begingroup$ <<Is it that simple?>> Yes. See here for more information. $\endgroup$ Aug 1 '19 at 14:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.