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).

\[ X + X \cdot \frac{Y}{100} \qquad{(1)}\]

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

\[ X + X \cdot Y \qquad{(2)}\]

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

Simplifying the algebra:

\[ X \cdot \left( 1 + Y \right) \qquad{(3)}\]

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

You would use eq. 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:

\[ \begin{aligned} Y &= X \cdot \left( 1 + i_1 \right) \\ Z &= Y \cdot \left( 1 + i_2 \right) \end{aligned} \qquad{(4)}\]

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

\[ Z = X \cdot \left( 1 + i_1 + i_2 \right) \qquad{(5)}\]

The solution is relatively easy, solve for \(Z\) in terms of \(X\):

\[ Z = X \cdot \left( 1 + i_1 \right) \cdot \left( 1 + i_2 \right) \qquad{(6)}\]

Simplify the interest rate terms:

\[ \begin{aligned} \left( 1 + i_1 \right) \cdot \left( 1 + i_2 \right) \\ \left( 1 + i_1 \right) \cdot \left( 1 + i_2 \right) \\ 1 + i_2 + i_1 + i_1 \cdot i_2 \end{aligned} \qquad{(7)}\]

Therefore:

\[ Z = X \cdot \left( 1 + i_2 + i_1 + i_1 \cdot i_2 \right) \neq X \cdot \left( 1 + i_1 + i_2 \right) \qquad{(8)}\]

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