You must’ve noticed that percentages are ubiquitous in finance. For example, credit cards advertise an Annual Percentage Rate (“APR”), whereas bank accounts do the same for an Annual Percentage Yield (“APY”) or “interest rate”¹. Have you wondered why financial institutions are so interested in advertising these annual percentage quantities?

The fundamental reason for the importance of annual percentages is that these quantities make it possible to directly compare competing offers from different financial institutions in a rigorous quantitative way. To understand how, let’s break down this quantity into its individual components, and understand each one carefully.

Percentage

First let’s start with percentage. Basically in this context, percentage indicates the amount of interest on a balance of 100 LCU. In the case of credit card APR, it indicates the interest that will be charged on a balance of 100 LCU². In the case of a bank account APY, it indicates the interest you’ll earn on a balance of 100 LCU in that account. But, you may wonder, in real life, we will almost never have a balance of exactly 100 LCU in our credit card or bank accounts. So why do we care so much about the interest on 100 LCU and not, say, 200 LCU?

According to the law of interest proportionality, we saw that the interest you earn on a sum of money is proportional to the amount of money. This is also true for the interest you’ll be charged on a credit card balance.

In fact, all interest computations in finance³ follow this *law of interest proportionality.

Therefore, by specifying the interest on a balance of 100 LCU, you’re indirectly specifying the interest on any other amount. So, for example, if you had a balance of 200 LCU, to compute the corresponding interest charged (in the case of a credit card balance)/earned (in the case of a bank account balance), you’d simply double the quoted APR/APY, since 200 LCU is double of 100 LCU.

Analogy with decimal number system

To gain further intuition about percentages, we can make an analogy with using the decimal system, a.k.a. “base-10 system”, for writing down numbers. Mathematically, a percentage is a fraction⁴ with a fixed denominator, viz., 100. To see the analogy of percentage with the decimal number system, consider the number 1,234. In the decimal system, 1,234 is really a shorthand:

$$ \begin{equation} 1234 = 1 \times 10^3 + 2 \times 10^2 + 3 \times 10^1 + 4 \times 10^0 \end{equation} $$

Notice that in this number system, it is implicitly understood that each of the digits 1, 2, 3, and 4 in the number 1,234 is multiplied by a power of 10 – the power of 10 that multiplies each of the digits 1, 2, 3, and 4 is determined by its position in the number 1,234 – 10 being the “base” of the decimal system.

Similarly in the case of a percentage: for example, 25% is the same as $\frac{25}{100}$ (which is another way of writing $25 \div 100$). So, 25% denotes a fraction whose numerator is 25, and the denominator is implicitly understood as 100. Thus, percentage is simply a way of specifying a fraction with a “base”, i.e., denominator of 100. This analogy extends further – just as the same number 1,234 has a different representation in a number system using a different base, so does a fraction when using a different denominator. For example, 1,234 in a base-8, a.k.a. “octal”, number system is written as 2,322, and similarly, 25% can also be expressed as the fraction $\frac{1}{4}$, where the denominator is now 4 instead of 100.

Annualization

Now let’s understand the second part, viz., “annual”. This one refers to the period, viz., one year, over which the interest on the balance is calculated. Just like in the case of percentage, it’s a way to standardize the time period over which the interest is applied (in the case of APR) or earned (in the case of APY).

Many quantities in finance, e.g., the interest earned/charged on a balance in a bank account or credit card respectively, depend on the time frame over which these quantities are computed.

So in order to facilitate comparison, it’s standard practice to express these quantities as if they are computed for a full year, whether or not the time period of interest is a full year⁵. This practice is known as “annualization”.

Thus, annual percentage provides a means of succinctly comparing different offers for credit cards, bank accounts, etc. For two credit cards with different APRs, the one with the lower one will charge you a lower interest than the other one for the same balance over the same time period. Similarly, for two bank accounts with different APYs, the one with the higher APY will earn you higher interest than the other one for the same balance over the same time period.

ROI

Annual percentages are used widely as standard metrics in many other financial matters, not just bank accounts and credit cards. For example, if you want to see which investments did better than others, you’d simply see which one has a higher “return on investment” (ROI).

ROI is mathematically analogous to APR and APY, i.e., it denotes the return, i.e., profit or loss, on an investment of 100 LCU, and it serves a purpose similar too, viz., allowing you to quantitatively compare different investments on the same basis.

Now you can see why annual percentages are so important in finance – they allow you to make an apples-to-apples comparison among comparable financial offerings.