Annuities over Non-Integer Periods/ Loan Repayment Question

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

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

Remark:

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

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

• 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. Aug 1 '19 at 11:01
• I have made an edit. Aug 1 '19 at 14:37
• <<Is it that simple?>> Yes. See here for more information. Aug 1 '19 at 14:43