# Fundamentals of Interest

Interest is a basic financial concept. It applies anytime you borrow money or someone borrows money from you. You pay interest on any loan, including any credit card balances you don’t pay off in full each month, student loans, car loans, mortgages and the like. Whenever someone borrows money from you, such as when you buy a bond (click here to learn more about bonds or here about credit cards) or a certificate of deposit (CD) or you put money in an account that pays interest, the counterparty (in this case the person to whom you loaned the money) pays you interest. In this post, I’ll provide you with the fundamentals of interest, focusing on different types of interest and how they affect your finances.

Interest is commonly calculated as a percentage or rate multiplied by the principal. In the case of a loan, the principal is the amount that you borrowed from the bank, reduced by the portion of payments you have made that cover the principal (i.e., excluding any interest you owe). For more information on loans, see this post. In the case of a savings account, the principal is the amount you deposited into the account. For other investments, the definition of principal varies and will be discussed in the future posts on those topics.

**Different “kinds” of interest: Simple and Compound**

Interest rates are frequently quoted as annual rates, but it is important to make sure you know the period over which the rates apply before you use them in any calculations. For example, if you have an annual interest rate but are making monthly payments, you need to divide the interest rate by 12 in calculating the monthly interest. Even then, annual interest rates can have different impacts on the amount of interest paid depending on how the interest is applied.

- Simple Interest – the amount of interest paid is calculated each period (often each month) as the interest rate for that period times the principal
*.* - Compound interest –the amount of interest paid is calculated as the interest rate for that period (e.g., each month) times the sum of the principal and any interest earned or owed.

If there is only one period, such as looking at your savings account interest for a single month, simple interest is the same as compound interest because there is only one interest payment so there is no interest that can be added to the principal. The two concepts differ when there is more than one interest payment, such as when you are looking at the interest deposited in your savings account over a full year. In that case, you earn interest on the interest deposited in previous months. In other words, you earn compound interest.

When reading a contract or a description of how interest will be calculated, you’ll want to focus on whether you will pay or receive simple or compound interest. If a contract says that interest will be calculated based solely on the principal, it uses simple interest. If interest is calculated based on the principal plus any accumulated interest, it uses compound interest.

### Simple Interest

A certificate of deposit (CD) is a type of investment that pays simple interest. You can buy a CD from a financial institution. The amount you pay for the CD is the principal. The financial institution promises to pay you interest over the term of the CD at a stated rate and will return your principal at the end of the term. Let’s assume you buy a $1,000 one-year CD that has a 6% annual interest rate that is paid monthly. (Please note that I chose 6% to keep the math simple. There are very few CDs that pay as much as 6% in the current economic environment.)

The interest paid *each month* is the same every month and is calculated as:

one-month simple interest = interest rate/12 months x principal = 6%/12 x $1,000

If you want to calculate the interest you would receive in a full year, you multiply the above formula by 12 months, leading to:

Twelve-month simple interest = 6%/12 months x principal x 12 months

The two “12 months” values cancel out and the interest is then 6% times the principal. Since you paid $1,000 for the CD at the beginning of the year, you will have $1,060 at the end of the year including interest.

### Compound Interest

In a savings account, interest compounds, as long as there are no withdrawals. For simplicity, I’ll assume that you deposit $1,000 in a savings account and don’t make any other deposits or withdrawals during the year. In this case, your $1,000 deposit is the principal. As with the CD, assume that the savings accounts pays 6%.

Because the interest is calculated on a compound basis, the math is a little more complicated. The interest earned in the first month is calculated as:

first month’s compound interest = 6%/12 months x principal

This formula is the same as for simple interest. But, in the second month, we replace the principal with the principal plus the first month’s interest. The math for the second month becomes:

second month’s compound interest = 6%/12 months x [principal + the first month’s interest]

With a little algebra, the general formula for the interest you will earn through the n^{th} month[1] is:

compound interest through n^{th} month = principal x ((1 + 6%/12 months)^{n }– 1)

By the end of the year, the total amount in your account is:

account balance at end of year with compound interest = (1 + 6%/12 months)^{12} x principal

For those of you who aren’t as comfortable with math as I am, don’t panic when you see the exponent in this formula. Just remember, when you add something to itself over and over again, it is the same as using multiplication. For example, 2+2+2+2 = 4 x 2 = 8. When you multiply something by itself over and over again, it is the same as using an exponent. For example, 2 x 2 x 2 x 2 = 2^{4} = 16. To see this formula in action, check out my compound interest calculator.

### Comparison

If we started the year with $1,000, the balance at the end of the year would be $1,061.68 using compound interest or $1.68 more than in the simple interest example. This difference doesn’t seem like very much, but it adds up as the interest rate goes up, the beginning balance increases or the time frame increases.

The graph below shows how compound interest accumulates slightly faster on a month-by-month basis.

There is a bit more terminology to know about compound interest. The “interest rate” corresponds to the 6% in the illustration above. It is often the rate that banks and lenders mention in their advertising. It is the interest rate you would pay or earn if there is no compounding (e.g., you make your full loan payment every month or take all of the interest you earn out of your savings account every month). The “annual percentage yield” or “effective annual rate” is the rate you actually earn or pay or (1 + 6%/12 months)^{12} – 1 = 6.17% in our example.

#### A Longer Term Example of Compounding

We can look at the difference between simple and compound interest in a different context. Let’s say you have $10,000 you want to invest for 10 years and that you are able to buy a 10-year bond that pays 8%. The bond itself pays simple interest, so you will receive $8,000 (8% times $10,000 times 10 years) over the 10-year period. If you take the interest out of the account or leave it in cash, you’ll have $18,000 at the end of the ten years when the principal is re-paid. If, on the other hand, you are able to re-invest your interest at the same 8% rate, you will have earned $11,589 in interest, so you would have $21,589 at the end of 10 years! The graph below shows how compound interest accumulates faster on a year-by-year basis.

All of a sudden, the difference between simple and compound interest becomes meaningful.

#### When You Borrow Money

When you borrow money, you pay simple interest *unless you miss a payment*. If you miss a payment, the entity that loaned you the money will always charge you using compound interest. That is, if you miss a payment or two, the entity making the loan will charge you interest on the interest you didn’t pay (and may also charge you a penalty for late payment). As such, missing payments can be very expensive.

For example, let’s say you owe $1,000 on your credit card and it charges a 15% annual interest rate. The credit card issuer will charge you 1.25% (15% / 12 months) of any amount you don’t pay in the month you made the charges. So, if you pay off the full amount of your credit card balance, it will cost you $1,000. If you pay only half (50%) of the balance this month and half next month, you will pay $1,006.25. The extra $6.25 is the interest and is calculated as $6.25 = 1,000 x 50% x 1.25%. That’s worse than $1,000, but not too bad. If, however, you don’t make any payments until the third month, you will owe the credit card issuer $1,038 before consideration of any finance charges or fees for not having made the minimum payment or any additional charges you make. You can see how that amount could increase very quickly.

#### A Little More Vocabulary

You will frequently see both loans and savings accounts refer to the interest rate. That amount corresponds to the 6% in the examples above, so is the rate *before* compounding. That is, even if the savings account pays compound interest, it will state that the interest rate is 6%, even though it pays 6.17% if you keep all of your money and interest in the account for the full year.

When looking at loans, the annual percentage rate is the interest rate adjusted to reflect any expenses associated with the loan, such as closing costs, mortgage insurance or loan origination fees.

The annual percentage yield or effective annual rate is the actual yield you will earn *after* compounding or, in the case of a loan, the annual percentage rate *after* considering the impact of compounding (equivalent to the 6.17% in the example above).

## Still A Bit Confused?

I’ve created a spreadsheet in which you can enter some values and see how they impact the amount of interest you will get or pay. Here’s a brief guide through the spreadsheet.

When you first open the spreadsheet, it will be populated with the values in the illustrations above. All input cells are highlighted in light green. To allow you to more easily look at the formulas used to calculate each of the values, I have not protected the spreadsheet. If you think you might have changed a formula, you can test the formulas by entering the values discussed above and see if you still get the right answer. If you do not, you’ll want to download a fresh copy of the spreadsheet.

### Simple Interest (The Simple Interest – One Year Tab)

The Simple Interest – One Year tab allows you to calculate the amount of interest you would earn or pay in one year for a financial instrument that uses simple interest.

#### Here are the inputs:

- Cell B1 – Enter the annual interest rate. This value was 6% in the illustration above.
- Cell B2 – Enter the face amount or principal for the financial instrument. This value was $1,000 in the illustration above.
- Cell B3 – Enter the number of interest payments during the year. This value was 12 in the illustration above.

#### Here are the outputs:

- Cell B5 – The amount of interest you will earn or pay each period (assuming you do not withdraw or pay any principal until the end of the year).
- Cell B6 – The amount of interest you will earn or pay over the full year (assuming you do not withdraw or pay any principal until the end of the year).

### Compound Interest (The Compound Interest – One Year Tab)

The Compound Interest – One Year tab allows you to calculate the amount of interest you would earn or pay in one year for a financial instrument that uses compound interest.

#### Here are the inputs:

- Cell B1 – Enter the annual interest rate. This value was 6% in the illustration above.
- Cell B2 – Enter the face amount or principal for the financial instrument. This value was $1,000 in the illustration above.
- Cell B3 – Enter the number of interest payments during the year. This value was 12 in the illustration above.

#### Here are the outputs:

- Cells B6 through B17 – The amount of interest you will earn or pay in every period (assuming you do not withdraw or pay any principal until the end of the year) for the first 12 periods.
- Cell B19 – The amount of interest you will earn or pay over the full year (assuming you do not withdraw or pay any principal until the end of the year). This amount includes all of the interest payments, even any payments made in the 13
^{th}and subsequent periods, if there are any, that are not shown individually. - Cell B20 – The annual percentage or effective annual yield, assuming that there are no additional costs.

### Benefit of Compounding of Returns (The Multi-Year Compounding Tab)

Benefit of Compounding of Returns – The Multi-Year Compounding tab allows you to calculate the amount of interest you would earn over several years.

#### Here are the inputs:

- Cell B1 – Enter the effective annual interest rate. This value was 8% in the illustration above.
- Cell B2 – Enter the face amount or principal for the financial instrument. This value was $10,000 in the illustration above.
- Cell B3 – Enter the number of years you will hold the investment. This value was 10 in the illustration above.

#### Here are the outputs:

- Cell B6 – The amount of interest you will earn if you leave the interest payments in cash or withdraw them from the account. This amount correspond to simple interest being earned over the life of the investment.
- Cell B9 – The amount of interest you will earn if you reinvest the interest payments in the same or another financial instrument that has the same interest rate. This amount reflects the benefit of the compounding of interest over time.

Download Interest Practice Spreadsheet

[1] There is a 1 inside the inner parentheses (the term that has the exponent) to allow the interest to compound on both the principal and interest. If we excluded the -1 in the outer parentheses, the result would include the principal as well as the interest. Send me an e-mail if you’d like to see the details of the math.