FinToolBox - Compounding Interest Rate

Compounding Interest Rate

Now let's switch the gear and talk about how to compare different interest rates. What do we mean by "different" interest rates? It refers to interest rates compounded at different time intervals.

It sounds terrifying indeed. But, don't start to panic now — you are just getting cold feet because you are not familiar with the term "compound" yet. I promise you, after this chapter, you will be a "compounding guru".

As usual, let's start by an example.

Learning by Example: Compound Interest Rate

You go to a bank to save a check. The clerk offers you a interest rate of 10% compounded semiannually. You ask yourself, what does it mean?

Well, the word "compound" in our financial context basically means "to put interest into your principal" (where the prefix "com" means "together", and the suffix "pound" means "to put"; so, "com-pound" means "to put together", just FYI).

So, translation for "10% compounded semiannually" is "you earn 5% every 6 months". What does it really means is, you can get paid with your 5% interest by the end of the first 6 months, and put the interest rate into your principle, so that your interest rate will generate more interest rate by the end of the second 6 months.

Doesn't this sounds too nice for a bank to give you this offer? Well, this term is most likely invented when the bank offer the loans and charge interests hundreds of years ago, so you may not need to worry too much about the well-being of the banks. Besides, what matters is the level of real interest rate, rather than how it is compounded, as you will see later how we convert different interest rates and compare them.

Back to your offer, given this "5% every 6 months", let's calculate of the worth of $100 in a year. Remember, the unit of interest and time period should always be consistent and here it is 6 months. For convenience, we denote the interest rate for 6 months as r6.

Knowing the Future Value is $110.25, the Present Value is $100, and the time period is 1 year, by applying the formula of interest rate, we can find the interest rate for 1 year, which we denote by r12 for now:

Notice that the interest rate for 6 months, r6, is 5%; the interest rate for 1 year, r12, is 10.25%. As a matter of fact that these two interest rates are in essence the same interest rate just written in two different formats. It is because, with the same amount of money (let say $100), it doesn't matter if you compound every 6 months with 5% per 6 months, or you compound every 1 year with 10.25% per year, you will always end up with the same future value.

Let's try 6 half-years, or 3 years, this time. Therefore, for r6, T will be 6; for r12, T will be 3.

So, either with 5% per 6 months, or with 10.25% per year, the future value is the same, $134.01. That's why we say r6 and r12 are equivalent — they are the same interest rate given different time periods. We will introduce how to convert the same interest rate given different time periods later this chapter.

Last but not the least, it is obvious that "10% compounded twice a year" and 10.25% per year are the same thing. However, "10% compounded twice a year" is different from 10% per year, that is for sure, because the former is 10.25% per year and the latter is 10% per year.

I guess the takeaway is, when you see "compounded twice a year" behind 10%, you should realize immediately it is not 10% per year in terms of the real interest rate. We will run into the similar jargons, when we talk about the APR and EAR next session.

Anecdote: Finance & Jargons

Why is it called "10% compounded semiannually" instead of "5% compounded semiannually"? No particular reason, and you can consider it just as a jargon (I guess it is for historical reason though).

For many cases, you will probably realize finance is more of a language than a science, and you are probably right on that. Some ancient business rituals and habits are preserved in the financial world pretty well. For example, it was not until the year 2001 did US stock market start to use cent as the minimal tick size. Prior to April 2001, the minimum tick size was 1/16th of a dollar, which meant that a stock could only move in increments of $0.0625. Weird, isn't it? It is actually because when the New York Stock Exchange began over 200 years ago, it was based on the Spanish trading system.

In Spain in the 1600s, Spanish investors traded with gold doubloons, which were split in half, quarter or even one-eighth pieces so traders could count them on their fingers. And they obviously skipped their thumbs!

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