How is the effective annual interest rate calculated?

Divide the growth by capital and time. For simple interest, the effective annual rate is the value obtained when the received interests are divided by the product of capital and years. In compound interest, the interest rate is the result of taking the (n times t) root from the ratio between final amount and capital, then multiplying it by n and subtracting 1 afterwards. Where n is the number of interest periods per year. This calculator can perform both calculations simultaneously. Enter the capital, final amount and time, leaving the interest rate blank.

What's the difference between APR and APY?

APR stands for the nominal interest rate without compounding, while lenders usually quote the APR. APY stands for the effective interest rate with compounding, and savings accounts usually quote the APY. When the quoted rates are equal, the APY is either equal to or greater than the APR, and the more frequently interest is compounded, the larger the difference between them.

Should you use simple or compound interest?

For most savings, investment and credit cards you should use compound interest as the interest earned will itself earn interest. For short term fixed rate loans and many bonds simple interest is used as the interest is only calculated on the original capital amount. The comparison table in this calculator shows both results based on the values entered so that you can see how much difference there is.

What is the Rule of 72?

The Rule of 72 is a way to estimate the time it takes for something to double in value. Divide 72 by the annual interest rate as a whole number. At 6 percent, it would take about 12 years; at 9 percent, about 8 years. This is an approximation and works best with rates between about 6 percent and 10 percent. This calculator also shows you the exact doubling time.

Can you determine the interest rate if only the received interest is known?

Yes. Add the interest earned to the principal amount to determine the ending balance. Then enter the principal, ending balance and term and leave the rate field blank. For example: if the principal is 1,000 and the interest earned is 210 then the ending balance would be 1,210.

Annual Interest Rate Calculator

Your numbers

Principal (starting amount)

Final amount

Time (years)

Interest model

How interest builds on the balance.

Compound

Simple

Continuous

Compounding frequency

Dropdown list for Compounding Frequency

Annual interest rate

Adjust for inflation

Show the real annual rate after inflation (the Fisher relation).

Annual interest rate

Interest earned: $
Total return over the term: %
Years to double

At 10% a year, $1,000 grows to $1,210 over 2 years, earning $210 in interest.

Compare and visualize

Show the comparison table

List the final amount and effective rate for simple, every compounding frequency, and continuous.

Show the charts

Plot the principal-versus-interest split and how the balance grows.

Final amount and effective rate by interest model, for your inputs
Interest model	Final amount ($)	Interest ($)	Effective annual rate
Simple interest	1,200	200	10.0000%
Compound annual	1,210	210	10.0000%
Compound semi-annual	1,215.51	215.51	10.2500%
Compound quarterly	1,218.4	218.4	10.3813%
Compound monthly	1,220.39	220.39	10.4713%
Compound daily	1,221.37	221.37	10.5156%
Continuous	1,221.4	221.4	10.5171%

Principal versus interest ($)

How much of the final amount is your original principal versus the interest it earned.

The annual interest rate calculator tool shows the yearly interest rate for loans or investments. You can calculate the annual interest rate by entering principal, ending amount or payment and term. If a field is left blank you can also back-calculate the principal, term or ending amount. You can check the results separately for simple interest, compound interest and continuous compounding as well as showing the corresponding effective interest rate and total interest earned.

What is an annual percentage rate?

The annual interest rate is usually referred to as "per annum" or "p.a." and represents the return or cost of a sum of money over one year expressed in percentage. It is a number that allows savings accounts, bonds and loans to be compared. The stated or quoted interest rate is called the nominal interest rate. When the frequency of compounding is taken into account, it becomes the effective interest rate which is what you actually pay or earn.

Here's how to use this calculator:

Enter three of the four basic values - principal, future value, number of years and annual interest rate. If a field is left blank, the tool will calculate that value. To calculate an interest rate, enter the principal, future value and number of years, leaving the interest rate field blank.

Choose the interest model that best fits your situation. For most savings and investment products it is compound interest, for short term fixed loans and many bonds it is simple interest, and to find a theoretical upper limit choose continuous compounding. With compound interest you can set how often the interest is added. If you turn on "Inflation" you can see what the real rate of return is after taking inflation into account. A comparison table and graph allow you to compare different models side by side.

Comparison of simple and compound interest

With simple interest, the interest is calculated only on the original principal amount. As the annual interest yield remains constant, the balance grows in a straight line.

A = P\left(1 + r\,t\right) \qquad r = \frac{A/P - 1}{t}

With compound interest, the interest earned in each period is added to the principal amount so that in the next period it also earns interest. This is why with compound interest the growth curve deviates upwards over time.

A = P\left(1 + \frac{r}{n}\right)^{nt} \qquad r = n\left[\left(\frac{A}{P}\right)^{\frac{1}{nt}} - 1\right]

In this formula P is the principal amount, A is the final amount, r is the annual rate of interest expressed as a decimal number, n is the number of compounding periods per year, and t is the number of years. Assuming that an amount of 1,000 grows to 1,210 in two years, then at simple interest the interest rate would be 10.5 percent. This is necessary to spread a total increase of 21 percent evenly over two years. At annual compounding, the interest rate would be 10 percent. Calculating a 10 percent increase for two consecutive years on an amount of 1,000 results in a final amount of 1,210. If some of this increase is due to compound interest, then the required interest rate will be slightly lower.

Symbol	Meaning	Example
P	Principal (starting amount)	1,000
A	Final amount	1,210
r	Annual interest rate	10 percent
n	Compounding periods per year	12
t	Time in years	2

Nominal and effective interest rates (APR and APY)

The nominal rate gives the interest rate itself but not how often it is capitalized. The effective annual rate contains the capitalization effect and thus represents a true yearly value. For a nominal rate that is capitalized n times per year, we have:

\text{Effective rate} = \left(1 + \frac{r}{n}\right)^{n} - 1

Lenders usually state the nominal interest rate as an annual percentage (APR), since this appears lower. Savings accounts are usually stated with the effective interest rate as an annual return (APY), since this appears higher. Because the two statements use different compounding methods, comparing only the nominal rates can be misleading. So you should convert both to effective interest rates first.

The Effects of Compounding Frequency on Interest Rate

The more frequent the compounding of interest, the greater the difference between the effective rate and the nominal rate, but each time this increase is smaller than the previous one. The following table shows the effective rates for various nominal rates.

Nominal rate	Annual	Monthly	Daily	Continuous
2%	2.000%	2.018%	2.020%	2.020%
5%	5.000%	5.116%	5.127%	5.127%
10%	10.000%	10.471%	10.516%	10.517%
20%	20.000%	21.939%	22.134%	22.140%

Continuous interest accrual

As the frequency of compounding increases, the effective interest rate approaches an upper limit that is determined by the constant e. This limit is continuous compounding and represents the maximum possible effective interest rate that can be achieved with compound interest for a given nominal interest rate.

A = P\,e^{rt} \qquad r = \frac{\ln(A/P)}{t}

If the nominal interest rate is 5 percent, continuous compounding gives a value of e to the power of 0.05 minus 1, which is approximately 5.127 percent. Daily compounding already comes very close to this value. This explains why changes in results are gradually diminishing as we go from monthly compounding to daily compounding and finally continuous compounding.

The Rule of 72.

The Rule of 72 is a simple way to estimate the time it takes for an amount to double. Divide 72 by the annual interest rate as a whole number. If the interest rate is 8 percent, then it will take approximately nine years for the amount to double when you divide 72 by 8. This calculation works best for interest rates between about 6 percent and 10 percent. In addition to providing an estimate, this calculator also shows the exact doubling time based on the entered interest rate and selected model.

Nominal and real interest rates.

As inflation reduces the purchasing power of money, a nominal interest rate overestimates the true return on an investment. The real interest rate accounts for the effects of inflation according to the Fisher equation.

r_{\text{real}} = \frac{1 + r}{1 + i} - 1

Here r is the nominal annual interest rate and i is the inflation rate. An interest rate of 10 percent with an inflation rate of 3 percent results in a real interest rate of about 6.8 percent, rather than 7 percent. Over long periods this difference compounds through the effect of compound interest so that the real interest rate represents the true measure of growth.

Fields of application for the annual interest rate.

When comparing growing amounts of money under different conditions the effective annual rate should be used. For savings it shows the true return on a savings account or fixed deposit. For loans it shows the true cost of a loan or credit card including the effect of compound interest. If the starting and ending amount of an investment are already known then an annual interest rate can be calculated that links those two amounts together. As everything is converted into one effective annual rate different options can be fairly compared.

Tips for accurate interest rate calculation:

Match the interest model to the product. Most savings and investment products use compound interest, short term personal loans and many bonds use simple interest while credit cards usually charge daily interest. Make sure that the time unit is years and divide the number of months by twelve to convert it. If only the interest earned but not the ending amount is known, add the interest to the principal first. The ending amount is the sum of the principal and the interest. For longer periods enable inflation to get more realistic results.

This calculator is for general educational and planning purposes only and does not constitute financial advice. As interest rates, fees and terms can vary between products, please check with lenders or banks before making a decision.

Frequently asked questions

How is the effective annual interest rate calculated?: Divide the growth by capital and time. For simple interest, the effective annual rate is the value obtained when the received interests are divided by the product of capital and years. In compound interest, the interest rate is the result of taking the (n times t) root from the ratio between final amount and capital, then multiplying it by n and subtracting 1 afterwards. Where n is the number of interest periods per year. This calculator can perform both calculations simultaneously. Enter the capital, final amount and time, leaving the interest rate blank.
What's the difference between APR and APY?: APR stands for the nominal interest rate without compounding, while lenders usually quote the APR. APY stands for the effective interest rate with compounding, and savings accounts usually quote the APY. When the quoted rates are equal, the APY is either equal to or greater than the APR, and the more frequently interest is compounded, the larger the difference between them.
Should you use simple or compound interest?: For most savings, investment and credit cards you should use compound interest as the interest earned will itself earn interest. For short term fixed rate loans and many bonds simple interest is used as the interest is only calculated on the original capital amount. The comparison table in this calculator shows both results based on the values entered so that you can see how much difference there is.
What is the Rule of 72?: The Rule of 72 is a way to estimate the time it takes for something to double in value. Divide 72 by the annual interest rate as a whole number. At 6 percent, it would take about 12 years; at 9 percent, about 8 years. This is an approximation and works best with rates between about 6 percent and 10 percent. This calculator also shows you the exact doubling time.
Can you determine the interest rate if only the received interest is known?: Yes. Add the interest earned to the principal amount to determine the ending balance. Then enter the principal, ending balance and term and leave the rate field blank. For example: if the principal is 1,000 and the interest earned is 210 then the ending balance would be 1,210.

Related calculators

Compound Interest CalculatorCalculate compound interest and the future value of a deposit, with optional regular contributions, any compounding frequency, the effective annual rate (APY), a simple-versus-compound comparison, a growth chart, and a year-by-year table. Works in reverse to solve for the rate, time, or starting amount.

Simple Interest CalculatorCalculate simple interest and the total repayment from a principal, rate, and term, solve for any missing value, and compare it against compound interest.

Deposit Interest CalculatorWork out the interest your deposit earns with compound interest, regular top-ups, inflation, and tax. Compare simple and compound interest and solve for the deposit, rate, or term.

Real Interest Rate CalculatorFind the real interest rate from the nominal rate and inflation with the Fisher equation. Solve for any of the three rates, compare the exact and approximate formulas, and add purchasing-power and after-tax layers.

APY CalculatorCalculate annual percentage yield from a nominal rate and compounding frequency, solve backward for the stated rate, and project savings growth with optional monthly deposits.

Interest Coverage Ratio CalculatorCalculate the interest coverage ratio from EBIT and interest expense, or solve for either figure. Adds the interest and cushion share of EBIT, EBITDA and EBITDA-less-Capex coverage, a target planner, and industry benchmarks.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Investopedia: Interest Rate
Definition of the interest rate, how it is set, and nominal versus real rates.
U.S. Securities and Exchange Commission: Compound Interest
Regulator explainer on how compounding builds a return over time.