This will be elementary to those who are sound in high school maths, but might easily be overlooked by others. Since it is counter-intuitive to the untrained mind, as well as relevant in many day-to-day situations, hope this discussion will be valued at least by some. Yes, value is the topic of this discussion – specifically, the curious outcome resulting from a series of percentage changes applied on a value.

First, take a look at absolute changes applied to a value. For our example, let’s assume I hold 10 grams of a not-so-precious metal, each gram costing Rs 10. Clearly, my holding is worth Rs 100. If the price of this metal were to fall by a rupee, and then appreciate by a rupee, the value of my holding remains unchanged at Rs 100. Now, let us assume that the price falls by 10% instead. On the next day, the metal appreciates by 10%. What do you think has happened to the value of the metal in my possession?

Bewildering as it might appear to some, the metal in my possession is worth less than what I started off with. 10% of Rs 100 was Rs 10, and so when the price fell, I ended up owning Rs 90 worth of metal. A 10% increase in this will take the price of my metal to Rs 99 – which is 1% less than its original value. In fact, if you hold stock worth a million rupees, and if you sell it after the value first drops by 10% and then gains by 11%, you still end up losing thousand rupees!

All right, so depreciation followed by appreciation in same percentage figures will lead to us losing value. Then, perhaps, we may be in for some profit in case of an appreciation followed by a depreciation? Well, turns out that the this reversal of order does **not** change the way it affects our fortune. A 10% increase followed by a 10% decrease still brings us to 1% below the figure that we start with. So, if a shopkeeper marks up the price by 11% and then gives a discount of 10% on it, he stands to lose 0.1%.

The reason for this (possibly surprising) result is that the actual value of a percentage change depends on the base value – and no matter what order you use, if you increase and then decrease, the base used for applying the reduction is always higher!

There are at least two interesting points to note from the above graph which compares the impact of applying depreciation and appreciation cycles of different percentages, and each of these leads us to a corresponding logical limit that will help us appreciate the observation even more.

Firstly, it will be noted that the reduction in value increases with the percentage value of change. The logical limits for this are easily seen by looking at cases in which we apply 0% and 100% changes. (most everyday scenarios fall in this range, so let’s ignore any value outside of it). If we apply 0% change – be it increase or decrease, in any order, as many times as we wish, there will be no change in the value. On the other hand, if we apply a 100% decrease once, the value will be down to zero irrespective of how many times we increase the value by 100% before or after this. Both of these make intuitive sense and are thus easy to understand.

The next observation is that the absolute change in value keeps decreasing with each cycle applied. This does not come as a surprise because we have already noted that this change depends on the base, and the base keeps getting smaller with each cycle. This makes us wonder if it would be possible to bring down the value to make it smaller than an arbitrary (non-negative) value arbitrarily increasing the number of cycles.

Before you read on, take a deep breath, and make up your mind on what you want to bet on!

We have already seen that with 0% change cycles this is not possible, and with 100% change cycles, this is achieved in a single cycle. So let us now focus on the values in this range. To understand this, let us recall that the result of a 10% change cycle is a 1% reduction. So the question of whether a large number of 10% change cycles can bring down the value to an arbitrarily small positive number is equivalent to whether a large number of 1% reductions can bring down the value arbitrarily. Each 1% reduction has the effect of multiplying the value by 0.99. So the value at the end of *n* such cycles is going to be the *n*-th power of 0.99. And basic Calculus tells us that this value can be made as close to zero as you wish by making *n* sufficiently large.

Before ending this discussion, let’s look at a different, perhaps more common, scenario. I’m a shopkeeper and intend to give some discount on a specific product. At first, I think of giving a 30% discount, which would still allow me to make a reasonable profit. If I want greater profit, but give my customers an impression of getting the same discount, I can give it as three separate discounts – say, 10% on the brand, 10% for the season and another 10% loyalty discount. If the cost is Rs 100, a straight 30% discount reduces the cost to Rs 70. The cost at the end of applying three 10% discounts, on the other hand, will be Rs 72.9, and I increase my profit by 2.9% of the original price!

Thus, if one clearly understands how percentages work, it can be used to their advantage; if not, it is easy to be misled to making incorrect assessments which lead to wrong decisions.

*If you like this article, you might want to check out our post on Misleading Uses of Measures of Central Tendency*

very interesting!