# Interest Rates - Interesting Properties#

I was working on some wage modelling in Excel and was looking at multiple increases within a given period. I wondered if the result of applying individual increases throughout the year is the same as applying the sum of the increases to the initial value (Spoiler: it is not). I work the algebra to satisfy my curiosity. It isn’t difficult, but it can be a mistake that is difficult to detect.

## Background - Preamble#

Every year, if you work for a good company, you should get a rate increase on your base salary. It usually is in the form of a percentage, e.g. $$5\%$$. Let’s say your base salary is $$X$$, and the percentage increase is $$Y$$ (in percentage).

(1)#$X + X \cdot \frac{Y}{100}$

To simplify things, let’s transform the percentage to a real number ratio. That will simplify (1) to:

(2)#$X + X \cdot Y$

Note

We are reusing $$Y$$ and letting it be the ratio instead of the percentage. That is $$5\%$$ becomes $$0.05$$.

Simplifying the algebra:

(3)#$X \cdot \left( 1 + Y \right)$

Note

We are not going to use numbers for examples, just algebra.

You would use (3) to solve for the new wage after the increase. Pretty straightforward.

## Multiple Small Increases in a Given Time Period#

However, what happens if you do two or more rate increases in a year? Will that equal the same as summing the interest rates and using that against the initial balance?

Let’s see what happens with the two increases. Here are the actors:

• $$X$$ - The salary at the start of the year

• $$i_1$$ - Rate increase 1; $$\left( i_1 \in \mathbb{R}, 0 \leq i_1 \leq 1 \right)$$

• $$i_2$$ - Rate increase 2; $$\left( i_2 \in \mathbb{R}, 0 \leq i_2 \leq 1 \right)$$

• $$Y$$ - The new salary after the first increase

• $$Z$$ - The new salary after the second increase

Given:

(4)#\begin{align}\begin{aligned}\begin{split}Y = X \cdot \left( 1 + i_1 \right) \\\end{split}\\Z = Y \cdot \left( 1 + i_2 \right)\end{aligned}\end{align}

Can we apply the sum of the interest rates to the actual wage and obtain $$Z$$? In other words, does:

(5)#$Z = X \cdot \left( 1 + i_1 + i_2 \right)$

The solution is relatively easy, solve for $$Z$$ in terms of $$X$$:

(6)#$Z = X \cdot \left( 1 + i_1 \right) \cdot \left( 1 + i_2 \right)$

Simplify the interest rate terms:

(7)#\begin{align}\begin{aligned}\left( 1 + i_1 \right) \cdot \left( 1 + i_2 \right)\\\left( 1 + i_1 \right) \cdot \left( 1 + i_2 \right)\\1 + i_2 + i_1 + i_1 \cdot i_2\end{aligned}\end{align}

Therefore:

(8)#$Z = X \cdot \left( 1 + i_2 + i_1 + i_1 \cdot i_2 \right) \neq X \cdot \left( 1 + i_1 + i_2 \right)$

From the algebra, we can see that they are not equal. We have to be careful when working with Excel and rates.