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Compound Interest Calculator

Q: What is the compound interest formula?

The formula is A = P(1 + r/n)^(nt), where A is future value, P is principal, r is annual rate, n is compounding periods per year, and t is years.

Q: How do you calculate interest earned?

First calculate the future value A, then subtract the principal: Interest Earned = A - P.

Use this Compound Interest Calculator to calculate future value, interest earned, principal, interest rate, or time. Enter your starting amount, annual interest rate, compounding frequency, and investment term to see how money grows when interest earns interest.

Future valueCalculate $A=P\left(1+\frac{r}{n}\right)^{nt}$.

Interest earnedFind $\text{Interest}=A-P$.

Solve missing valuesFind principal, rate, or time from known values.

Enter compound interest details

Calculate future value from principal, rate, time, and compounding Calculate principal from future value, rate, time, and compounding Calculate annual interest rate from principal, future value, time, and compounding Calculate time from principal, future value, rate, and compounding

Future value mode selected.
Enter $P$, $r$, $n$, and $t$. The calculator uses $A=P\left(1+\frac{r}{n}\right)^{nt}$.

Principal $P$

Future value $A$

Annual interest rate $r$ (%)

Time $t$, in years

Compounding frequency

Rounding

Currency symbol

This calculator assumes no additional deposits or withdrawals during the term. For recurring deposits, use a future value of annuity method or a savings calculator.

Results

Enter values and calculate.

Main result

$16,470.09

Interest earned

$6,470.09

Total growth

$64.70\%$

Effective APY

$5.12\%$

Compound interest formula

Compound interest is interest calculated on both the original principal and the interest that has already been added. The standard compound interest formula is:

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

Where:

$A$ = future value or final balance
$P$ = principal or starting amount
$r$ = annual interest rate written as a decimal
$n$ = number of compounding periods per year
$t$ = time in years

The interest earned is the final balance minus the starting principal:

\[ \text{Interest Earned}=A-P \]

If the annual rate is given as a percentage, convert it to a decimal before using the formula:

\[ r=\frac{\text{annual rate percentage}}{100} \]

For example, $5\%$ becomes $0.05$. If interest compounds monthly, then $n=12$. If it compounds quarterly, then $n=4$. If it compounds daily, a common approximation is $n=365$.

How to use the Compound Interest Calculator

Choose what you want to calculate. Select future value, principal, annual rate, or time.
Enter the known values. For the most common calculation, enter principal, annual interest rate, time, and compounding frequency.
Select compounding frequency. Choose annual, semiannual, quarterly, monthly, weekly, daily, or continuous compounding.
Choose rounding and currency. Select the number of decimal places and the currency symbol that fits your example.
Click Calculate Compound Interest. The calculator displays the result, interest earned, total growth, effective APY, and step-by-step formulas.
Interpret the output carefully. The result assumes the rate stays constant and no additional deposits or withdrawals are made during the term.

This tool is useful for savings accounts, certificates of deposit, investment growth, fixed deposits, classroom finance problems, APY comparisons, and long-term growth examples.

What is compound interest?

Compound interest means interest earns interest. In a simple interest calculation, interest is calculated only on the original principal. In compound interest, interest is added to the balance, and future interest is calculated on the larger balance. This creates growth that can accelerate over time.

Suppose $1000$ earns $10\%$ interest per year. After the first year, the balance becomes:

\[ 1000(1.10)=1100 \]

In the second year, the $10\%$ interest is calculated on $1100$, not only on the original $1000$:

\[ 1100(1.10)=1210 \]

After the third year, the interest is calculated on $1210$:

\[ 1210(1.10)=1331 \]

This is compounding. The balance grows because the base keeps increasing. Over long periods, this effect can become powerful, especially when money remains invested or saved without being withdrawn.

Worked example: monthly compound interest

Suppose you deposit $10{,}000$ at an annual interest rate of $5\%$, compounded monthly, for $10$ years. Identify the values:

\[ P=10{,}000,\qquad r=0.05,\qquad n=12,\qquad t=10 \]

Use the compound interest formula:

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

Substitute the values:

\[ A=10{,}000\left(1+\frac{0.05}{12}\right)^{12\cdot10} \]

The monthly rate is:

\[ \frac{0.05}{12}=0.0041667 \]

The number of compounding periods is:

\[ 12\cdot10=120 \]

So:

\[ A=10{,}000(1.0041667)^{120} \]

The future value is approximately:

\[ A\approx16{,}470.09 \]

The interest earned is:

\[ 16{,}470.09-10{,}000=6{,}470.09 \]

Compound interest versus simple interest

Simple interest is calculated only on the original principal. The simple interest formula is:

\[ A=P(1+rt) \]

Compound interest is calculated on the growing balance. Its formula is:

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

Using $P=10{,}000$, $r=5\%$, and $t=10$, simple interest gives:

\[ A=10{,}000(1+0.05\cdot10)=15{,}000 \]

Monthly compound interest gives about $16{,}470.09$. The compound result is higher because interest is repeatedly added to the balance and begins earning more interest. The longer the time period, the larger the difference between simple and compound interest can become.

Compounding frequency

Compounding frequency describes how often interest is added to the balance. Common options include annual, semiannual, quarterly, monthly, weekly, daily, and continuous compounding. The more frequently interest compounds, the higher the final balance becomes for a positive interest rate, although the increase becomes smaller as frequency rises.

Compounding frequency	Value of $n$	Formula example
Annually	$1$	$A=P(1+r)^t$
Semiannually	$2$	$A=P(1+\frac{r}{2})^{2t}$
Quarterly	$4$	$A=P(1+\frac{r}{4})^{4t}$
Monthly	$12$	$A=P(1+\frac{r}{12})^{12t}$
Weekly	$52$	$A=P(1+\frac{r}{52})^{52t}$
Daily	$365$	$A=P(1+\frac{r}{365})^{365t}$

For most everyday savings calculations, the difference between monthly and daily compounding is not huge at normal rates, but it still matters when comparing accounts. APY is often used because it includes the compounding effect and makes comparison easier.

Continuous compounding

Continuous compounding is the theoretical case where interest compounds infinitely often. Instead of using the standard $n$-period formula, continuous compounding uses Euler’s number $e$:

\[ A=Pe^{rt} \]

If $P=10{,}000$, $r=0.05$, and $t=10$, then:

\[ A=10{,}000e^{0.05\cdot10} \]

So:

\[ A=10{,}000e^{0.5}\approx16{,}487.21 \]

Continuous compounding is common in advanced finance and calculus because it creates elegant exponential formulas. Most real savings accounts and certificates of deposit compound periodically, such as monthly or daily, but continuous compounding is useful for understanding the mathematical limit of increasingly frequent compounding.

How to calculate principal

If you know the future value, rate, compounding frequency, and time, you can solve for the principal. Start with:

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

Divide by the compound factor:

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

For continuous compounding, the principal formula is:

\[ P=\frac{A}{e^{rt}} \]

This is useful when you have a future savings goal. For example, if you want $50{,}000$ after $8$ years at $6\%$ compounded monthly, the calculator can estimate how much principal would be needed today, assuming no additional deposits are made.

How to calculate interest rate

If you know the principal, future value, time, and compounding frequency, you can solve for the annual nominal interest rate. Start with:

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

Divide by $P$:

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

Raise both sides to the power $1/(nt)$:

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

Solve for $r$:

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

For continuous compounding, the rate formula is:

\[ r=\frac{\ln(A/P)}{t} \]

The calculator converts the decimal rate into a percentage so the result is easier to read.

How to calculate time

If you know the principal, future value, interest rate, and compounding frequency, you can solve for time. Start with:

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

Divide by $P$:

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

Use logarithms:

\[ \ln(A/P)=nt\ln\left(1+\frac{r}{n}\right) \]

Solve for $t$:

\[ t=\frac{\ln(A/P)}{n\ln\left(1+\frac{r}{n}\right)} \]

For continuous compounding:

\[ t=\frac{\ln(A/P)}{r} \]

This is useful for questions such as, “How long will it take to double my money?” At an annual rate of $8\%$, compounded annually, the exact doubling time is:

\[ t=\frac{\ln(2)}{\ln(1.08)}\approx9.01 \]

Effective APY and compound interest

APY stands for annual percentage yield. It measures the effective annual rate after compounding. For a nominal annual interest rate $r$ compounded $n$ times per year, APY is:

\[ APY=\left(1+\frac{r}{n}\right)^n-1 \]

If $r=5\%=0.05$ and $n=12$, then:

\[ APY=\left(1+\frac{0.05}{12}\right)^{12}-1\approx5.12\% \]

APY is helpful because it lets you compare accounts that compound at different frequencies. If two accounts show the same nominal rate but one compounds more often, the APY may be slightly higher for the more frequent compounding account. If a bank already quotes APY, that number already includes compounding for one year.

Compound interest in savings

In savings accounts, compound interest helps deposits grow without additional work from the saver. The account earns interest, the interest is added to the balance, and the next interest calculation uses the larger balance. Over short periods, growth may look small. Over long periods, the accumulated interest can become significant.

Suppose $5000$ is saved at $4\%$ compounded monthly for $20$ years:

\[ A=5000\left(1+\frac{0.04}{12}\right)^{12\cdot20} \]

The final balance is approximately:

\[ A\approx11{,}113 \]

The original $5000$ more than doubles because interest remains in the account and continues earning interest. This is why time is one of the most important inputs in compound interest calculations.

Compound interest in investing

Investments can also compound when gains are reinvested. If an investment earns returns and those returns remain invested, future returns can be earned on both the original principal and earlier gains. However, investment returns are usually not fixed like a savings account rate. They may rise, fall, or fluctuate from year to year.

The compound interest formula is still useful as a model:

\[ A=P(1+r)^t \]

When $r$ represents an annualized return, the formula estimates the future value. But a modeled return is not a guarantee. Real investments involve market risk, fees, taxes, inflation, liquidity risk, and timing risk. Use compound interest results as projections, not promises.

The Rule of 72

The Rule of 72 is a quick mental shortcut for estimating how long it takes money to double at a given annual rate. The rule is:

\[ \text{Doubling Time}\approx\frac{72}{\text{annual rate percentage}} \]

At $8\%$, the Rule of 72 gives:

\[ \frac{72}{8}=9 \]

The exact annual-compounding result is about $9.01$ years, so the shortcut is very close in this case. The Rule of 72 is not exact, but it is useful for quick estimates. For exact calculations, use the logarithmic time formula or the calculator above.

Compound interest and inflation

Compound interest shows nominal growth, but inflation affects purchasing power. If your savings grow at $5\%$ per year and inflation is $3\%$, the real growth in purchasing power is much smaller than $5\%$. A simple approximation is:

\[ \text{Real Return}\approx \text{Nominal Return}-\text{Inflation Rate} \]

A more precise relationship is:

\[ r_{\text{real}}=\frac{1+r_{\text{nominal}}}{1+i}-1 \]

Where $i$ is the inflation rate. For example, with $r_{\text{nominal}}=0.05$ and $i=0.03$:

\[ r_{\text{real}}=\frac{1.05}{1.03}-1\approx1.94\% \]

This means the money grows by $5\%$ in nominal terms but only about $1.94\%$ in real purchasing-power terms.

Taxes, fees, and compound interest

The calculator shows mathematical growth based on the inputs. Real-world results may be lower if taxes, account fees, trading costs, withdrawal penalties, or inflation apply. If interest is taxable, an after-tax interest estimate can be written as:

\[ \text{After-Tax Interest}=\text{Interest Earned}\times(1-\text{Tax Rate}) \]

If fees are charged annually, they can reduce the effective growth rate. For example, an investment earning $7\%$ before fees with a $1\%$ annual fee may behave closer to a $6\%$ net growth model before tax effects. For accurate financial planning, always use net assumptions that reflect the real costs of the product.

Common mistakes when calculating compound interest

Forgetting to convert percentages to decimals. Use $5\%=0.05$, not $5$, inside the formula.
Using the wrong compounding frequency. Monthly uses $n=12$, quarterly uses $n=4$, and daily often uses $n=365$.
Mixing months and years. The standard formula uses $t$ in years when $r$ is an annual rate.
Confusing simple interest with compound interest. Simple interest does not add interest back into the base.
Ignoring APY. APY is often better for comparing accounts because it includes compounding.
Assuming rates never change. Some savings and investment rates are variable.
Ignoring fees, taxes, and inflation. The calculator gives a gross mathematical result unless inputs are adjusted.

Compound interest formula summary table

Calculation	Formula	Use it when
Future value	$A=P\left(1+\frac{r}{n}\right)^{nt}$	You know principal, rate, compounding frequency, and time.
Interest earned	$\text{Interest}=A-P$	You want only the growth portion.
Principal	$P=\frac{A}{\left(1+\frac{r}{n}\right)^{nt}}$	You know target balance, rate, compounding, and time.
Annual rate	$r=n\left[\left(\frac{A}{P}\right)^{1/(nt)}-1\right]$	You know starting amount, ending amount, time, and compounding.
Time	$t=\frac{\ln(A/P)}{n\ln(1+r/n)}$	You know starting amount, ending amount, rate, and compounding.
Continuous compounding	$A=Pe^{rt}$	You want the theoretical continuous compounding result.
APY	$APY=\left(1+\frac{r}{n}\right)^n-1$	You want the effective annual yield after compounding.

Related calculators and study tools

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Compound Interest Calculator FAQs

What is compound interest?

Compound interest is interest calculated on both the original principal and previously earned interest. It is often described as interest on interest.

What is the compound interest formula?

The formula is $A=P\left(1+\frac{r}{n}\right)^{nt}$, where $A$ is the future value, $P$ is principal, $r$ is the annual rate, $n$ is compounding periods per year, and $t$ is years.

How do you calculate interest earned?

First calculate the future value $A$, then subtract the principal: $\text{Interest Earned}=A-P$.

Does compounding frequency matter?

Yes. More frequent compounding usually increases the future value for a positive rate, although the difference becomes smaller as frequency increases.

What is continuous compounding?

Continuous compounding is the theoretical case where interest compounds infinitely often. The formula is $A=Pe^{rt}$.

Is APY the same as the interest rate?

Not always. APY includes the effect of compounding, while a nominal interest rate may not. APY is often better for comparing savings products.

Compounding frequency	Value of \(n\)	Formula example
Annually	\(1\)	\(A=P(1+r)^t\)
Semiannually	\(2\)	\(A=P(1+\frac{r}{2})^{2t}\)
Quarterly	\(4\)	\(A=P(1+\frac{r}{4})^{4t}\)
Monthly	\(12\)	\(A=P(1+\frac{r}{12})^{12t}\)
Weekly	\(52\)	\(A=P(1+\frac{r}{52})^{52t}\)
Daily	\(365\)	\(A=P(1+\frac{r}{365})^{365t}\)

Calculation	Formula	Use it when
Future value	\(A=P\left(1+\frac{r}{n}\right)^{nt}\)	You know principal, rate, compounding frequency, and time.
Interest earned	\(\text{Interest}=A-P\)	You want only the growth portion.
Principal	\(P=\frac{A}{\left(1+\frac{r}{n}\right)^{nt}}\)	You know target balance, rate, compounding, and time.
Annual rate	\(r=n\left[\left(\frac{A}{P}\right)^{1/(nt)}-1\right]\)	You know starting amount, ending amount, time, and compounding.
Time	\(t=\frac{\ln(A/P)}{n\ln(1+r/n)}\)	You know starting amount, ending amount, rate, and compounding.
Continuous compounding	\(A=Pe^{rt}\)	You want the theoretical continuous compounding result.
APY	\(APY=\left(1+\frac{r}{n}\right)^n-1\)	You want the effective annual yield after compounding.