The power of compounding

Not all interest rates tell you how fast your capital will grow. Let's deconstruct a common investment instrument offered by a local bank to see how capital does/doesn't grow over time.

If you’re looking to invest, you’ll need to know the right interest rate to use. Interest rates are reported in a variety of ways, but only one tells you how much return on your investment you’ll actually receive. That particular interest rate is often hidden in a mix of other rates that can cause confusion.

Key Ingredients

There are a few ingredients that you need to know in order to calculate how much of a return you will earn:

1. your initial investment - determined primarily by you

2. the number of compounding periods in a year

   1. why “a year”? It is easier to compare anticipated returns between various investment instruments if you use the same time period. The standard time period used in the investment world is one year.

3. the effective compounding interest rate

   1. this is, likely, something you will have to calculate (which we will do in this exposition)

In the above example, we have two CD options for anyone wanting to invest < $100K: a 7- and 12-month commitment. What these commitments mean is that you will earn interest on your investment for 7 or 12 compounding periods in one year.

✓ we know the number of compounding periods. This information satisfies one of the three ingredients that we need to calculate our return on investment.

We will assume that we invest $10K.

✓ we know the starting investment value at t = 0. This information satisfies another of the three ingredients that we need to calculate our return on investment.

Now that we have 2 of the 3 ingredients, our focus can turn to the most complicated part: finding the right interest rate.

Types of interest rates: compounding and non-compounding

Interest rates come in two large varieties: non-compounding and compounding. Each type of interest rate has its own synonyms.

• Non-compounding interest rates: coupon, yield to maturity, annual percentage rate (APR), interest rate (not otherwise specified).

• Compounding interest rates: annual percentage yield (APY), effective rate (almost always, this interest rate is one you must calculate)

In our example, we see two columns: interest rate (not otherwise specified) and APY. How are these numbers related to each other and, ultimately, which one tells us how much we will earn as our return.

Interest rate (not otherwise specified)

This interest rate is a non-compounding rate. It is considered an “artificial” rate because it assumes that you will remove any interest you collect in each and every compounding period. In reality, very few people actually take this action - if you did, you’d lose all the benefit of compounding interest. So why do financial institutions give you the non-compounding interest rate?

Honestly, I don’t know. What we do know, however, is that we can easily calculate the effective compounding interest rate using a non-compounding interest rate. To do that, divide the non-compounding interest rate by the number of compounding periods.

Effective interest rate

The effective interest rate is, indeed, the interest rate that we earn on our initial investment ($10K) plus the interest that we earn during each compounding period.

For example, let’s say you invest $10K into the 7-month instrument whose interest rate (not otherwise specified) is 4.88%.

The table above shows you how much you earn from the:

1. interest on your principal of $10K, plus

2. interest you earn on the accumulated interest from the previous compounding periods - a.k.a. interest-on-interest

If you invest $10K at t = 0, by the time you complete 7 compounding periods (remember, this is a 7-month term instrument), you’ll have $10498.33. The actual return that you earned is 4.98% → 5% (the APY).

The APY (a form of a compounding interest rate) tells you how much you will earn at the end of all the compounding periods in a year.

Compounding periods: 7 versus 12

Now, let’s compare the other instrument available to us: the 12-month term instrument.

Calculating the indifference time point

Now that you know what an APY is, you can compare various financial instruments against one another. In our example, the 7-month term instrument provides you a greater APY (5%) in a shorter period of time (after 7 compounding periods) than the 12-month instrument (APY 4.5% after 12 compounding periods). It appears that the 7-month instrument is more valuable: greater returns in a shorter period of time.

So how can the 12-month instrument compete? One way is to change the term of the instrument and extend the number of compounding periods. How many additional compounding periods, at an effective rate of 0.37% (unchanged), do we need to have an APY of 5% (like in the 7-month instrument)?

There you have it. A real life example of interest rates and the power of compounding. Leave us questions or comments below.