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.

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