How to calculate generated interests while having two different nominal rates compunded into a single quarter

Question

I would like to calculate the interest generated from an investment within a time period of 8 months. The bank manages an interest rate of 3.5% p.a. for six months, and the last two months is 1%. The interests are compounded quarterly. Furthermore, the special rate (3.5%) starts in the middle of the third quarter (August).

The next table would summarize the issue:

Aug	Sep	Oct	Nov	Dec	Jan	Feb	Mar
3.5% p.a.	3.5% p.a.	3.5% p.a.	3.5% p.a.	3.5% p.a.	3.5% p.a.	1% p.a.	1% p.a.

I've tried to calculate the rates by transforming the yearly interest rate into a monthly interest rate and then adding them:

1. Transform the 3.5% p.a. to a monthly rate, which equals ~ 0.28709%
2. Applied this rate from Aug to Jan
3. Applied monthly rate equivalent to 1% p.a. (~ 0.08295) to Feb and Mar

Nonetheless, I'm quite sure this is not a correct procedure, because the interests are compounded quarterly and not monthly.

While looking through the internet, I searched for a formula that takes into account two different rates within the same compounding period, but I couldn't find anything.

How can I correctly calculate the generated interest?

Thanks a lot in advance for your help!

Answer 1

If you assume that interest is not reinvested until the end of the quarter, then, given a principal of $w$ at the beginning of August, at the end of September (after two months or $1/6$ of a year) you would have earned interest of

$$x=\frac{3.5\%}{6}w$$

This plus the principal (i.e. $w+x$) is reinvested and earns $3.5\%$ for three months giving interest of

$$y=\frac{3.5\%}{4}(w+x)$$

Then $y$ is reinvested along with the $w+x$ and earns $3.5\%$ for one month giving interest of:

$$z_1=\frac{3.5\%}{12}(w+x+y)$$

and $1\%$ for two months, giving interest of

$$z_2=\frac{1\%}{6}(w+x+y)$$

So at the end you have

$$\begin{align*}z_1+z_2&=\frac{3.5\%}{12}(w+x+y)+\frac{1\%}{6}(w+x+y)\\ &=\left(\frac{3.5\%}{12}+\frac{1\%}{6}\right)(w+x+y)\\ &=\left(\frac{3.5\%+2(1\%)}{12}\right)(w+x+y)\\ &=\left(\frac{\frac{1}{3}(3.5\%)+\frac{2}{3}(1\%)}{4}\right)(w+x+y)\\ \end{align*}$$

So for the January to March quarter, you just take a weighted average of the interest rates to get the correct quarterly interest rate to apply.

More generally, if you had a length of time $T=t_1+t_2$ with $t_1$ periods of per-$T$ interest rate $r_1$ and $t_2$ periods of per-$T$ interest rate $r_2$, then the effective interest rate over $T$ is

$$\frac{t_1}{T}r_1+\frac{t_2}{T}r_2$$

In the example above, $T=3$, $t_1=1$, $r_1=3.5\%/4$, $t_2=2$ and $r_2=1\%/4$.