Most people are quite familiar with the concept of percentage. If we were not, shops would not bother putting up mathematical symbols tempting us with 50% discount on a nice sweater. Not the least on today’s Black Friday.
Repeated percentage change is less familiar territory for most. Compound interest for instance, is the result of positive percentage change only, and is a form of exponential growth. But in this post we will look at alternating positive and negative percentage change.
We are used to the operation +1 followed by -1 resulting in no change, as the sum is zero. But what about the operation +1% followed by -1%? Well, there is a significant difference between the two types of operation. The addition and subtraction of +1 and -1 are independent of the value they operate on, the difference is always 1. Percentage change on the other hand is not independent of the value it operates on. 1% of something large is larger than 1% of something small.
An Asymmetry Shows Up
An immediate effect of percentage change dependency on the value it operates on is that the “chronology” matters. The second change can only be applied after the first change has been calculated etc. With this in mind, we take a closer look at the result of alternating positive/negative percentage change.
Consider equal positive and negative 20% changes. This is equivalent to multiplying by 1.2 for increase and by 0.8 for decrease. In other words, regardless of the order (increase or decrease first) we end up with a change of 1.2*0.8 = 0.8*1.2 = 0.96, which is a decrease.
In order not to change the original value from one cycle to the next, the percentage increase must be slightly larger than the decrease. Assume that the increase is x and the decrease is y. An unchanged value requires that x*y = 1. For example, with y = 0.8 it follows that x = 1.25. This means that a 20% decrease must be followed by a 25% increase.
The two cases discussed above are illustrated in the figure below, where alternating percentage changes are applied to an original value of 100. Note that when alternating with -20% and +20% the results (blue line) get smaller and smaller. In contrast, when alternating with -20% and +25% the result oscillates but the trend never drops.
Even though decreasing and increasing by the same percentage lowers the result, it will never reach zero. This is another consequence of percentage change scaling with the value it acts on. As the value gets smaller, so does the change. To become zero, the change must be -100%, from which there is also no recovery, as a percentage change of zero is zero.