Understanding the Mathematics of Personal Finance

Compound Interest

The most common, if not universal, way to express the amount of interest to be paid on a loan is the annual percentage rate (APR). The interest is expressed as a per­centage or a fraction of the amount of money loaned if the money were to be loaned, with no intermediate payments or corrections, for a year.

Calculating interest is very simple. An important point to remember is that while interest is usually expressed as a percentage, for example, “6% per year,” calcula­tions must always use the decimal or fractional equivalent of this percentage:

6% = 6 = 0.06.

100

The interest due after a year on a $1,200 loan, for example, is then Interest = ($1,200.00)(0.06) = $72.00.

A type of interest calculation that is rarely used is called simple interest. In a simple interest calculation, the interest is equal to the amount borrowed, or the principal, times the rate of interest, times the amount of time the money is borrowed. The formula is

Interest = (Principal)(Rate)(Time).

Time must be expressed in the same units as rate. In other words, if the rate is expressed in percentage per year, then the time must also be expressed in years. If the rate is in percentage per month, then the time must be in months. The interest calculated, of course, is for the time period that the rate is expressed in.

The same example as above would therefore be written as

Interest = ($1,200.00) (0.06)(1.0) = $72.00.

The following are a few more examples:

The simple interest on an $800 loan with an interest rate of 8% per year after 2 years is

Interest = ($800.00)(0.08)(2) = $128.00.

Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by

Lawrence N. Dworsky

Copyright © 2009 John Wiley & Sons, Inc.

The simple interest on a $2,500 loan with an interest rate of 5% per year after 18 months is (remember that 18 months is 1.5 years)

Interest = ($2,500.00)(0.05)(1.5) = $187.50.

As I have stated above, simple interest is rarely used in real transactions. For that matter, the term interest when used alone does not mean simple interest. It means compound interest.

In a compound interest loan, there is a period of time called the compounding interval or the compounding period. Suppose you are told that your loan will be compounded monthly, starting 1 month after you originate the loan. Starting 1 month after you originate the loan, and monthly thereafter, interest is calculated on the total amount of money that you owe. This is very different from simple interest in that the interest from each compounding calculation is added to the amount of money you owe, now called your balance. and subsequent interest calculations are calcu­lated based on this balance.

I need to introduce one other convention before presenting some interest calcu­lation examples. It is conventional to present the terms of a loan as the annual interest rate and the compounding intervals. The actual interest rate used to calculate the interest is just the APR divided by the compounding interval (remember to put both of these numbers into the same units).

Examples:

1. If interest is compounded monthly, then it is compounded 12 times a year. If a 6% APR loan is compounded monthly, then each month the interest is 6%/12 = 0.5% of the balance of the loan.

2. Interest compounded quarterly is compounded 4 times a year. A 10% loan compounded quarterly earns 10%/4 = 2.5% interest every quarter (every 3 months).[4]

3. Interest compounded annually is compounded once a year. In this case, the stated APR is identically the interest rate used for calculation.

Now let’s look at some real compound interest loan calculations. I’ll start by explaining the different columns in the tables that detail these examples.

Consider a $5,000.00 loan is taken at 5% interest, compounded annually.

In Table 2.1, the first column is a list of compounding periods. The loan is taken at a starting compounding period of 0, with compounding periods occurring annually afterwards. The second column in Table 2.1 shows actual dates. In this example, I assumed that the loan was taken on July 14, 2007, and that it will be paid back after 15 years, on July 14, 2022.

Also shown in Table 2.1 is a third column labeled interest and a fourth column labeled balance. Since the loan was originated on July 14, 2007, at that date there have been no interest compoundings yet. The interest column therefore shows $0, and the balance (how much you owe) shows the principal, $5,000.00.

The first interest calculation is performed on July 14, 2008. Since this is the first such calculation, column 1 shows this as compounding interval #1. Five percent APR compounded annually is just 5% interest per calculation. This interest is therefore

Interest = ($5,000.00) (0.05) = $250.00.

The balance, that is, how much you owe, is now the previous balance ($5,000.00) plus this interest ($250.00), which is $5,250.00.

July 14, 2009, is the date of the second compound interest calculation. Here’s where compound interest is very different from simple interest. The interest calcula­tion is based not on the principal ($5,000.00) but on the balance at the time of the calculation, in this case $5,250.00:

Interest = ($5,250.00) (0.05) = $262.50.

Notice that this interest value is higher than the first interest value ($262.50 vs. $250.00), and that the balance grows to $5,612.50.

I’ll repeat this calculation in detail one more time. On July 14, 2010, the third compound interest calculation is

Interest = ($5,512.50)(0.05) = $275.63, and the new balance is $5,788.13.

Each time the calculation is performed, the interest is larger. This is sometimes called the “miracle of compound interest.” If these calculations are for the $5,000

deposit you put into a savings bank, then you can see how the interest takes on a life of its own and, over the years, earns you more money than the principal you started with. After 15 years, the balance is slightly more than twice the original deposit (principal).

What has been presented so far is really the total subject of compound interest, and the basis for almost everything else in this book. What has to be added to this chapter are several practical cases that come up frequently, some formulas for working with these cases efficiently, and techniques for “running the numbers” yourself.

Table 2.2 shows the calculated interest and balance over time for a $1,000.00 loan with a 10% annual interest rate, compounded monthly for 2 years (24 months). Since there are 12 months in a year, each month’s interest is calculated based on a monthly rate of

8.333%.[5]

Since only the elapsed time since the origination of the loan matters for interest calculations, not the actual date of the loan, I omitted the actual dates in this table.

One last reminder: When performing the division shown above, make sure that units agree. If the interest rate is percentage per year, then the compounding interval must be the number of compounding intervals per year, and so on.

The calculations required to produce Table 2.2 are the same as those used to produce Table 2.1, so I won’t repeat them here.

Table 2.2 shows that after 1 year (12 months), the balance is $11,047.13. Isn’t this a bit odd? If I borrow $10,000 at an APR of 10%, shouldn’t I just owe $10,000 + (0.1)($10,000) = $11,000 at the end of a year? This discrepancy is a result of the way things are quoted. Because the loan is compounded many times over the course of the year, the “miracle of compound interest” causes the balance to grow to a larger number than we’d see if we just compounded once at the end of the year. In order to owe $11,047.13 if the balance were compounded only once a year, the interest rate would have had to be 10.4713% (work it out; it’s pretty straightforward). This latter number is referred to as the EAPR, or effective annual percentage rate.[6]