How to find the rate of interest

Interest on interest—also referred to as compound interest—is the interest earned when interest payments are reinvested. Compound interest is used in the context of bonds. Coupon payments from bonds are assumed to be reinvested at some interest rate and held until the bond is sold or matures.

Compound interest refers to the interest owed or received on an investment, and it grows at a faster rate than simple interest.

Key Takeaways:

• Interest on interest is the interest earned when interest payments are reinvested, particularly in the context of bonds.
• This is also known as compound interest, or compounding.
  
• Compound interest grows at a faster rate than basic interest, and it will be fastest when compounding periods are most frequent.
• Simple interest, in contrast, only credits the original amount of principal.
• Coupon payments from bonds can be reinvested at some compound interest rate and held until the bond is sold or matures. Dividends can also be reinvested to compound stock returns.

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Compounding: My Favorite Term

How Interest on Interest Works

Interest on interest works, as the term implies, by paying interest on past interest payments received as well as on the initial amount of principal invested or saved.

For example, U.S. Savings bonds are financial securities that pay interest on interest to investors with interest that compounds semi-annually and accrues monthly every year for 30 years. Most savings accounts at banks also pay interest on interest, with payments compounded on a monthly basis.

Interest on interest differs from simple interest. Simple interest is only charged on the original principal amount while interest on interest applies to the principal amount of the bond or loan and to any other interest that has previously accrued.

How to Calculate Interest on Interest

When calculating interest-on-interest, the compound interest formula determines the amount of accumulated interest on the principal amount invested or borrowed. The principal amount, the annual interest rate, and the number of compounding periods are used to calculate the compound interest on a loan or deposit.

The formula to calculate compound interest is to add 1 to the interest rate in decimal form, raise this sum to the total number of compound periods, and multiply this solution by the principal amount. The original principal amount is subtracted from the resulting value.

Compound interest:

The "rule of 72" estimates the number of years it will take for the value of an investment or savings to double when there is interest on interest. Divide the number 72 by the interest rate to get the approximate number of years.

Example

For example, assume you want to calculate the compound interest on a $1 million deposit. The principal is compounded annually at a rate of 5%. The total number of compounding periods is five, representing five one-year periods.

The resulting compounded interest on the deposit is as follows:

What Is Interest on Interest?

Interest on interest refers to an investment or deposit whereby interest that has been credited in the past is also used for calculating future interest payments. Because interest on interest compounds over time, it can grow exponentially as time passes.

What Is Another Name for Interest on Interest?

Interest on interest is also known as compound interest, or simply as compounding.

What Is Interest?

Interest refers to payments made on investments, loans, or deposits. In particular, interest is payment received due to the opportunity cost of lending, depositing, or investing money to somebody else rather than utilizing yourself over some period of time. The greater the time period or the greater the perceived risk involved with handing over your money, the higher the rate of interest required. Interest may, therefore, be thought of as the "cost of money" or the cost of the "time value of money."

What Is an Effective Annual Interest Rate?

An effective annual interest rate is the real return on a savings account or any interest-paying investment when the effects of compounding over time are taken into account. It also reflects the real percentage rate owed in interest on a loan, a credit card, or any other debt.

It is also called the effective interest rate, the effective rate, or the annual equivalent rate (AER).

Key Takeaways

• The effective annual interest rate is the true interest rate on an investment or loan because it takes into account the effects of compounding.
• The more frequent the compounding periods, the higher the rate.
• A savings account or a loan may be advertised with both a nominal interest rate and an effective annual interest rate.
• The effective annual interest rate is the rate that should be compared between loans and investment rates of return.

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The Effective Annual Interest Rate

Understanding the Effective Annual Interest Rate 

The effective annual interest rate describes the true interest rate associated with an investment or loan. The most important feature of the effective annual interest rate is that it takes into account the fact that more frequent compounding periods will lead to a higher effective interest rate.

Suppose, for instance, you have two loans, and each has a stated interest rate of 10%, in which one compounds annually and the other compounds twice per year. Even though they both have a stated interest rate of 10%, the effective annual interest rate of the loan that compounds twice per year will be higher. 

The effective annual interest rate is important because, without it, borrowers might underestimate the true cost of a loan. And investors need it to project the actual expected return on an investment, such as a corporate bond.

Effective Annual Interest Rate Formula 

The following formula is used to calculate the effective annual interest rate:

Effective Annual Interest Rate=(1+in)n−1where:i=Nominal interest raten=Number of periods\begin{aligned} &Effective\ Annual\ Interest\ Rate=\left ( 1+\frac{i}{n} \right )^n-1\\ &\textbf{where:}\\ &i=\text{Nominal interest rate}\\ &n=\text{Number of periods}\\ \end{aligned}​Effective Annual Interest Rate=(1+ni​)n−1where:i=Nominal interest raten=Number of periods​

What the Effective Annual Interest Rate Tells You

A certificate of deposit (CD), a savings account, or a loan offer may be advertised with its nominal interest rate as well as its effective annual interest rate. The nominal interest rate does not reflect the effects of compounding interest or even the fees that come with these financial products. The effective annual interest rate is the real return.

That’s why the effective annual interest rate is an important financial concept to understand. You can compare various offers accurately only if you know the effective annual interest rate of each one.

Example of Effective Annual Interest Rate

Consider these two offers: Investment A pays 10% interest, compounded monthly. Investment B pays 10.1%, compounded semiannually. Which is the better offer?

In both cases, the advertised interest rate is the nominal interest rate. The effective annual interest rate is calculated by adjusting the nominal interest rate for the number of compounding periods that the financial product will undergo in a period of time. In this case, that period is one year. The formula and calculations are as follows:

• Effective annual interest rate = (1 + (nominal rate ÷ number of compounding periods)) ^ (number of compounding periods) - 1
• For investment A, this would be: 10.47% = (1 + (10% ÷ 12)) ^ 12 - 1
• And for investment B, it would be: 10.36% = (1 + (10.1% ÷ 2)) ^ 2 - 1

Investment B has a higher stated nominal interest rate, but the effective annual interest rate is lower than the effective rate for investment A. This is because Investment B compounds fewer times over the course of the year. If an investor were to put, say, $5 million into one of these investments, the wrong decision would cost more than $5,800 per year.

Effect of the Number of Compounding Periods

As the number of compounding periods increases, so does the effective annual interest rate. Quarterly compounding produces higher returns than semiannual compounding, monthly compounding produces higher returns than quarterly, and daily compounding produces higher returns than monthly. Below is a breakdown of the results of these different compound periods with a 10% nominal interest rate:

• Semiannual = 10.250%
• Quarterly = 10.381%
• Monthly = 10.471%
• Daily = 10.516%

Limits to Compounding

There is a ceiling to the compounding phenomenon. Even if compounding occurs an infinite number of times—not just every second or microsecond, but continuously—the limit of compounding is reached.

With 10%, the continuously compounded effective annual interest rate is 10.517%. The continuous rate is calculated by raising the number “e” (approximately equal to 2.71828) to the power of the interest rate and subtracting one. In this example, it would be 2.171828 ^ (0.1) - 1.

How do you calculate the effective annual interest rate?

The effective annual interest rate is calculated using the following formula:

Effective Annual Interest Rate=(1+in)n−1where:i=Nominal interest raten=Number of periods\begin{aligned} &Effective\ Annual\ Interest\ Rate=\left ( 1+\frac{i}{n} \right )^n-1\\ &\textbf{where:}\\ &i=\text{Nominal interest rate}\\ &n=\text{Number of periods}\\ \end{aligned}​Effective Annual Interest Rate=(1+ni​)n−1where:i=Nominal interest raten=Number of periods​

Although it can be done by hand, most investors will use a financial calculator, spreadsheet, or online program. Moreover, investment websites and other financial resources regularly publish the effective annual interest rate of a loan or investment. This figure is also often included in the prospectus and marketing documents prepared by the security issuers.

What is a nominal interest rate?

A nominal interest rate does not take into account any fees or compounding of interest. It is often the rate that is stated by financial institutions.

What is compound interest?

Compound interest is calculated on the initial principal and also includes all of the accumulated interest from previous periods on a loan or deposit. The number of compounding periods makes a significant difference when calculating compound interest.

The Bottom Line

Banks and other financial institutions typically advertise their money market rates using the nominal interest rate, which does not take fees or compounding into account. The effective annual interest rate does take compounding into account and results in a higher rate than the nominal. The more the periods of compounding involved, the higher the ultimate effective interest rate will be.