The question reads:
A savings plan requires you to make payments of £250 each at the end of every month for a year. The bank will then make six equal monthly payments to you, with its first payment due one month after the last payment you make to the bank. Compute the size of each monthly payment made by the bank, assuming a nominal interest rate of 4% p.a. payable monthly.
Answer: £515.23 (=£3055.62/5.930618).
So I've been pondering on this question for a while now, and I'm not making any progress. I've tried various methods without any confidence in what I'm doing. Every method I try brings me a figure not mentioned in the answer.
The only parts I'm confident in is the AER = 4.074%. What is the line of logic I should take with this question? Help is much appreciated as always.
Best Answer
Firstly we need a reference date for your payments and the payments of the bank. I´ve made a time line. The converted montly interest rate is $i_{12}=\frac{0.04}{12}$
Your payments start at the last day of January. We calculate the future value of your 12 payments at 31.12. We pretend that you start at 01.01, but make the payments at the end of each month. Then the future value is at $\color{red}{t=0} \ (31.12)$ This is the left hand side of the equation. Then we calculate the present value of the payments made by the bank. Since the payments ($+x$) start one month after $t=0$ we just discount the future value 6 times. The equation is
$$250\cdot\frac{(1+\frac{0.04}{12})^{12}-1}{\frac{0.04}{12}}=x\cdot\frac{(1+\frac{0.04}{12})^{6}-1}{\frac{0.04}{12}}\cdot\frac1{\left(1+\frac{0.04}{12}\right)^6}$$
$$3055.6157=x\cdot 5.930618 \Rightarrow x=\large{\color{grey}{£}} \ \normalsize 515.23 \ \ $$