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 a certificate of depositA savings certificate, usually issued by a commercial bank, that has a stated maturity and interest rate.  Certificates of deposit, often called CDs, are insured by the Federal Deposit Insurance Corp... More (CD) or you put money in an account that pays interest, the counterpartyThe other person involved in a financial transaction. Examples of counterparties include the bank issuing your mortgage, the bank at which you have your checking and savings accounts or the company is... More (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 principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More is the amount you deposited into the account. For other investments, the definition of principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More 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 rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More but are making monthly payments, you need to divide the interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More 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 InterestThe amount of interest paid is calculated as the interest rate times the principal. – the amount of interest paid is calculated each period (often each month) as the interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More for that period times the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
  • Credit cards: The amount of purchases you have made but not paid on your credit card ... More.
• Compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. –the amount of interest paid is calculated as the interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More for that period (e.g., each month) times the sum of the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
  • Credit cards: The amount of purchases you have made but not paid on your credit card ... More 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 interestThe amount of interest paid is calculated as the interest rate times the principal. is the same as compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. because there is only one interest payment so there is no interest that can be added to the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More. 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 interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed..

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 interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed.. If a contract says that interest will be calculated based solely on the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More, it uses simple interestThe amount of interest paid is calculated as the interest rate times the principal.. If interest is calculated based on the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More plus any accumulated interest, it uses compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed..

Simple InterestThe amount of interest paid is calculated as the interest rate times the principal.

A certificate of depositA savings certificate, usually issued by a commercial bank, that has a stated maturity and interest rate.  Certificates of deposit, often called CDs, are insured by the Federal Deposit Insurance Corp... More (CD) is a type of investment that pays simple interestThe amount of interest paid is calculated as the interest rate times the principal.. You can buy a CD from a financial institution. The amount you pay for the CD is the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More. The financial institution promises to pay you interest over the termThe time period over which you re-pay the loan of the CD at a stated rate and will return your principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More at the end of the termThe time period over which you re-pay the loan. Let’s assume you buy a $1,000 one-year CD that has a 6% annual interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More 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 interestThe amount of interest paid is calculated as the interest rate times the principal. = interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More/12 months x principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More = 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 interestThe amount of interest paid is calculated as the interest rate times the principal. = 6%/12 months x principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More x 12 months

The two “12 months” values cancel out and the interest is then 6% times the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More. 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 InterestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed.

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 principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More. 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 interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. = 6%/12 months x principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More

This formula is the same as for simple interestThe amount of interest paid is calculated as the interest rate times the principal.. But, in the second month, we replace the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More with the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More plus the first month’s interest. The math for the second month becomes:

second month’s compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. = 6%/12 months x [principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More + 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 interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. through n^th month = principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More 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 interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. = (1 + 6%/12 months)¹² x principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More

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 = 2⁴ = 16.

Comparison

If we started the year with $1,000, the balance at the end of the year would be $1,061.68 using compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. or $1.68 more than in the simple interestThe amount of interest paid is calculated as the interest rate times the principal. example. This difference doesn’t seem like very much, but it adds up as the interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More goes up, the beginning balance increases or the time frame increases.

The graph below shows how compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. accumulates slightly faster on a month-by-month basis.

There is a bit more terminology to know about compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed.. The “interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More” 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 rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More 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 yieldThe annual percentage rate after consideration of the impact of compounding. Annual percentage yield is the same thing as the effective annual rate.” or “effective annual rateThe annual percentage rate after consideration of the impact of compounding. Effective annual rate is the same thing as the annual percentage yield.” is the rate you actually earn or pay or (1 + 6%/12 months)¹² – 1 = 6.17% in our example.

A Longer TermThe time period over which you re-pay the loan Example of Compounding

We can look at the difference between simple and compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. 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 bondA form of debt issued by government entities and corporations. that pays 8%. The bondA form of debt issued by government entities and corporations. itself pays simple interestThe amount of interest paid is calculated as the interest rate times the principal., 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 principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More 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 interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. accumulates faster on a year-by-year basis.

All of a sudden, the difference between simple and compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. becomes meaningful.

When You Borrow Money

When you borrow money, you pay simple interestThe amount of interest paid is calculated as the interest rate times the principal. unless you miss a payment. If you miss a payment, the entity that loaned you the money will always charge you using compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed.. 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 rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More. 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 rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More. 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 interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed., it will state that the interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More 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 rateThe interest rate adjusted to reflect any associated expenses, such as closing costs, mortgage insurance or loan origination fees.   Annual percentage rates are always stated before consideration of... More is the interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More adjusted to reflect any expenses associated with the loan, such as closing costs, mortgage insurance or loan origination fees.

The annual percentage yieldThe annual percentage rate after consideration of the impact of compounding. Annual percentage yield is the same thing as the effective annual rate. or effective annual rateThe annual percentage rate after consideration of the impact of compounding. Effective annual rate is the same thing as the annual percentage yield. is the actual yield you will earn after compounding or, in the case of a loan, the annual percentage rateThe interest rate adjusted to reflect any associated expenses, such as closing costs, mortgage insurance or loan origination fees.   Annual percentage rates are always stated before consideration of... More 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 InterestThe amount of interest paid is calculated as the interest rate times the principal. – One Year tab allows you to calculate the amount of interest you would earn or pay in one year for a financial instrumentAny investment that you purchase.  Examples include an exchange-traded fund, a mutual fund, stock in an individual company, a bond and a money market fund.  There are also many more complex financia... More that uses simple interestThe amount of interest paid is calculated as the interest rate times the principal..

Here are the inputs:

• Cell B1 – Enter the annual interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More. This value was 6% in the illustration above.
• Cell B2 – Enter the face amount or principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
  • Credit cards: The amount of purchases you have made but not paid on your credit card ... More for the financial instrumentAny investment that you purchase.  Examples include an exchange-traded fund, a mutual fund, stock in an individual company, a bond and a money market fund.  There are also many more complex financia... More. 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 principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
  • Credit cards: The amount of purchases you have made but not paid on your credit card ... More 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 principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
  • Credit cards: The amount of purchases you have made but not paid on your credit card ... More until the end of the year).

Compound Interest (The Compound Interest – One Year Tab)

The Compound InterestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed. – One Year tab allows you to calculate the amount of interest you would earn or pay in one year for a financial instrumentAny investment that you purchase.  Examples include an exchange-traded fund, a mutual fund, stock in an individual company, a bond and a money market fund.  There are also many more complex financia... More that uses compound interestThe amount of interest paid is calculated as the interest rate times the sum of the principal and any interest earned or owed..

Here are the inputs:

• Cell B1 – Enter the annual interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More. This value was 6% in the illustration above.
• Cell B2 – Enter the face amount or principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
  • Credit cards: The amount of purchases you have made but not paid on your credit card ... More for the financial instrumentAny investment that you purchase.  Examples include an exchange-traded fund, a mutual fund, stock in an individual company, a bond and a money market fund.  There are also many more complex financia... More. 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 principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
  • Credit cards: The amount of purchases you have made but not paid on your credit card ... More 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 principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
  • Credit cards: The amount of purchases you have made but not paid on your credit card ... More 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 rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More. This value was 8% in the illustration above.
• Cell B2 – Enter the face amount or principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
  • Credit cards: The amount of purchases you have made but not paid on your credit card ... More for the financial instrumentAny investment that you purchase.  Examples include an exchange-traded fund, a mutual fund, stock in an individual company, a bond and a money market fund.  There are also many more complex financia... More. 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 interestThe amount of interest paid is calculated as the interest rate times the principal. 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 instrumentAny investment that you purchase.  Examples include an exchange-traded fund, a mutual fund, stock in an individual company, a bond and a money market fund.  There are also many more complex financia... More that has the same interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More. This amount reflects the benefit of the compounding of interest over time.

Download Interest Practice Spreadsheet

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[1] There is a 1 inside the inner parentheses (the termThe time period over which you re-pay the loan that has the exponent) to allow the interest to compound on both the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More and interest. If we excluded the -1 in the outer parentheses, the result would include the principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More as well as the interest. Send me an e-mail if you’d like to see the details of the math.