# BM4.4 – Basics of Compound Interest

Chapter 4, Lesson 4

In this lesson students will:

• understand the difference between simple interest and compound interest
• calculate a periodic interest rate given a nominal interest rate
• interpret the terminology associated with compound interest rates
• determine a future value after one payment period, given a compounding interest rate

## Introduction to Compound Interest

#### Simple Versus Compound Interest

Recall in Lesson 3.2 we briefly discussed the difference between simple and compound interest. In general, simple interest is a form of interest in which only the principle amount can earn interest; whereas with compound interest, the interest your earn during one payment period is permitted to earn interest during future payment periods. The focus of the rest of this chapter will be on compound interest.

Let’s begin again by looking at the simple case of borrowing $\1500$, but let’s follow the amount owing over the span of two years. With a simple interest rate of $5\%$, we know that after one year we would owe $FV = P(1+rt) = \1500(1+0.05) = \1575$ (we would have accrued $\75$ in interest). After two years, we would owe $FV = P(1+rt) = \1500(1+0.05 \times 2) = \1650$ (we have accrued $\150$ in interest). In other words, we expect to accrue $\75$ in interest at the end of each year.

Next, let’s contrast this with compound interest. If we assume assume a compound interest rate of $5\%$ at the end of each year, then at the end of the first year we would have to add $\1500 \times 5\% = \75$ in interest to our amount owing. Our amount owing at the end of the first year is similar to simple interest: $\1575$. It is in this next year that the compounding will occur – it is the new total of $\1575$ that will accrue interest this second year! The amount we have to add at the end of the second year is $\1575 \times 5\% = \78.75$. This gives a final total owing of $\1653.75$.

Notice that the final amount is slightly higher for compound interest compared to simple interest. This is true in general: since interest is permitted to earn interest in future periods, compound interest will always either make your investment more money, or cause your amount owing on debt to be larger, in the long-term when compared to simple interest.

#### Nominal Versus Periodic Interest Rates

In our example above, we assumed that compound interest was calculated at the end of each year. This does not have to be the case – in fact, most major banks and credit card companies offer interest rates that are compounded every month (compounded monthly). What exactly does this mean? This means that interest is earned at the end of every month rather than at the end of the year. This sounds like a great deal, right?! The ability to get $5\%$ at the end of each month would definitely be an awesome investment! In practice, what institutions actually do is take the posted rate, or the nominal rate, and divide it into equal parts to form a periodic interest rate much smaller than the posted rate. It is the periodic interest rate that consumers actually receive each interest period.

Let’s analyze a couple examples to help clarify. Suppose the bank posts a nominal rate of $5\%$ compounded monthly. The periodic rate that we would earn at the end of each month can be found by dividing $5\%$ into twelve equal parts: $5\% / 12 = 0.4167\%$. This means the periodic interest rate we actually get at the end of each month is $0.4167\%$; much lower than we initially thought!

The periods in which we earn compound interest are sometimes referred to as compounding periods. The compounding periods do not have to be monthly. In fact, there are many instances in which we might be interested in using a different structure. For example, suppose we get paid bi-weekly (every other week), and take some of this money to invest into a pension fund that earns a nominal rate of $3.5\%$ compounded bi-weekly. What is the periodic interest rate we earn at the end of each bi-weekly period? If we know that there are $52$ weeks in one year, and we get paid every other week, this means there are $26$ times per year in which we earn interest. Dividing into $26$ equal parts, we obtain a periodic interest rate of $3.5\% / 26 = 0.1346\%$.

In general, the nominal interest rate $r$ and the periodic interest rate $i$ are connected through the following formula

\begin{aligned} i &= \frac{r}{f} \end{aligned}

where $f$ is known as the frequency number. The frequency number is the number of compounding periods per year. Some common frequency numbers are listed below.

$\begin{tabular}{l|l|l} Name & Compounds per Year & Frequency Number\\ \hline Annually & one & 1\\ \hline Semi-annually & two & 2\\ \hline Quarterly & four & 4\\ \hline Monthly & twelve & 12\\ \hline Bi-weekly & twenty-six & 26\\ \hline Weekly & fifty-two & 52 \end{tabular}$

The words semi and bi can be applied to the other names as well. In general, adding the suffix semi means double the frequency number; whereas adding the suffix bi means half the frequency number. For instance, semi-monthly would have $f=24$ (twice per month, so twenty-four times per year), and bi-monthly would have $f=6$ (once every other month, so six times per year).

#### Calculating Interest Earned Over One Compound

Now that we have an idea of how to calculate compound interest over a compounding period, let’s work on a few examples to help solidify the concepts. In general, we can find the amount of interest earned $I$ in a particular compounding period by multiplying the principle amount by the periodic interest rate:

\begin{aligned} I &= P \times i \\ &= P \times \frac{r}{f}. \end{aligned}

If the nominal rate is given, it is often easier to use the second formula. The following video will guide you through a couple examples of calculating interest.

#### Calculating the Future Value After One Compound

Using the interest formula above, we can think of the future value after one compound period as the principle amount plus the accrued interest: $FV = P + I$. But since $\displaystyle I = P \times \frac{r}{f}$, we can simplify the formula to

\begin{aligned} FV &= P + P \times \frac{r}{f}\\ FV &= P \left( 1+ \frac{r}{f} \right). \end{aligned}

Most of the time in practice, the nominal rate is given, so the above formula is favored. If the periodic interest rate is given instead, the formula $FV = P(1+i)$ can be applied. The video below will work through a couple examples of calculating a future value after one period, given a compounding interest rate.