<p>If you assume that interest is not reinvested until the end of the quarter, then, given a principal of <span class="math-container">$w$</span> at the beginning of August, at the end of September (after two months or <span class="math-container">$1/6$</span> of a year) you would have earned interest of</p>
<p><span class="math-container">$$x=\frac{3.5\%}{6}w$$</span></p>
<p>This plus the principal (i.e. <span class="math-container">$w+x$</span>) is reinvested and earns <span class="math-container">$3.5\%$</span> for three months giving interest of</p>
<p><span class="math-container">$$y=\frac{3.5\%}{4}(w+x)$$</span></p>
<p>Then <span class="math-container">$y$</span> is reinvested along with the <span class="math-container">$w+x$</span> and earns <span class="math-container">$3.5\%$</span> for one month giving interest of:</p>
<p><span class="math-container">$$z_1=\frac{3.5\%}{12}(w+x+y)$$</span></p>
<p>and <span class="math-container">$1\%$</span> for two months, giving interest of</p>
<p><span class="math-container">$$z_2=\frac{1\%}{6}(w+x+y)$$</span></p>
<p>So at the end you have</p>
<p><span class="math-container">$$\begin{align*}z_1+z_2&amp;=\frac{3.5\%}{12}(w+x+y)+\frac{1\%}{6}(w+x+y)\\
&amp;=\left(\frac{3.5\%}{12}+\frac{1\%}{6}\right)(w+x+y)\\
&amp;=\left(\frac{3.5\%+2(1\%)}{12}\right)(w+x+y)\\
&amp;=\left(\frac{\frac{1}{3}(3.5\%)+\frac{2}{3}(1\%)}{4}\right)(w+x+y)\\
\end{align*}$$</span></p>
<p>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.</p>
<p>More generally, if you had a length of time <span class="math-container">$T=t_1+t_2$</span> with <span class="math-container">$t_1$</span> periods of per-<span class="math-container">$T$</span> interest rate <span class="math-container">$r_1$</span> and <span class="math-container">$t_2$</span> periods of per-<span class="math-container">$T$</span> interest rate <span class="math-container">$r_2$</span>, then the effective interest rate over <span class="math-container">$T$</span> is</p>
<p><span class="math-container">$$\frac{t_1}{T}r_1+\frac{t_2}{T}r_2$$</span></p>
<p>In the example above, <span class="math-container">$T=3$</span>, <span class="math-container">$t_1=1$</span>, <span class="math-container">$r_1=3.5\%/4$</span>, <span class="math-container">$t_2=2$</span> and <span class="math-container">$r_2=1\%/4$</span>.</p>

CC BY-SA 4.0
2023-07-31T16:34:23.200Z
2023-07-31T18:26:49.673Z
2023-07-31T18:26:49.673Z
smcc
<p>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).</p>
<p>The next table would summarize the issue:</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th>Aug</th>
<th>Sep</th>
<th>Oct</th>
<th>Nov</th>
<th>Dec</th>
<th>Jan</th>
<th>Feb</th>
<th>Mar</th>
</tr>
</thead>
<tbody>
<tr>
<td>3.5% p.a.</td>
<td>3.5% p.a.</td>
<td>3.5% p.a.</td>
<td>3.5% p.a.</td>
<td>3.5% p.a.</td>
<td>3.5% p.a.</td>
<td>1% p.a.</td>
<td>1% p.a.</td>
</tr>
</tbody>
</table>
</div>
<p>I've tried to calculate the rates by transforming the yearly interest rate into a monthly interest rate and then adding them:</p>
<blockquote>
<ol>
<li>Transform the 3.5% p.a. to a monthly rate, which equals ~ 0.28709%</li>
<li>Applied this rate from Aug to Jan</li>
<li>Applied monthly rate equivalent to 1% p.a. (~ 0.08295) to Feb and Mar</li>
</ol>
</blockquote>
<p>Nonetheless, I'm quite sure this is not a correct procedure, because the interests are compounded quarterly and not monthly.</p>
<p>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.</p>
<p>How can I correctly calculate the generated interest?</p>
<p>Thanks a lot in advance for your help!</p>

<interest-rate><investment>
How to calculate generated interests while having two different nominal rates compunded into a single quarter
Amateur_Investor
# How to calculate generated interests while having two different nominal rates compunded into a single quarter

(Asked by: username_0 on 2023-07-31)

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!

**Tags:** interest-rate, investment


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## 1 answers

### Answer by username_1 on 2023-07-31. Score: 0 | Accepted answer

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 $r\_2=1\%/4$.


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