Choose what to calculate. "Compound growth" finds the final amount and interest from the principal, annual rate and time. "Convert a rate" restates the same rate for a different compounding frequency.
Enter numbers only. Enter rates as percentages (for 3%, enter "3"). This is the math of compound interest, and it does not include the fees, taxes or rate changes of real financial products.
Result and graph
Enter the principal, annual rate and time in the fields on the left and press "Calculate". The result and a graph will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and graphs
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the principal, annual rate and time, and you get the final amount with compound interest, \(A = A_0\left(1 + \dfrac{r}{n}\right)^{nt}\), and the total interest right away (compounded annually, semiannually, quarterly, monthly, daily or continuously)
- The result also shows the final amount with simple interest under the same terms and the difference from compound interest. A graph shows how the gap grows over the years
- Switch to "Convert a rate to another compounding frequency" to restate a rate for a different compounding frequency, such as turning an APR compounded monthly into an APY (5 decimal places)
- The Rule of 72, a quick way to estimate how many years it takes for money to double, is explained in plain words in the formula section
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This page does compound interest as math. Real savings accounts, loans and investments involve fees, taxes (interest is generally taxable income) and rates that change, so check the exact amount you will earn or owe with your bank or lender. This page does not recommend any financial product.
What is this calculation used for?
Seeing how your savings will grow, in numbers
For example, $10,000 in a savings account at 4% APY for 10 years grows to \(10000 \times 1.04^{10} \approx 14802\) dollars, about $4,802 of interest. At 0.5% a year, the same 10 years gives only about $10,511. Compounding makes the difference between rates much bigger over time.
In real life, interest is generally taxable and savings rates change over time. Use this calculation as an estimate that assumes "before taxes, with a fixed rate".
Seeing how debt snowballs, in numbers
Compounding works against you when you borrow. If a $5,000 credit card balance at 20% a year is left unpaid for 5 years, compounded once a year for simplicity, it grows to \(5000 \times 1.2^{5} \approx 12442\) dollars, more than double. Credit cards usually compound daily or monthly, which makes it grow even faster (about $13,480 with monthly compounding), and late fees can add more.
Knowing in advance how fast debt grows is one of the best ways to avoid borrowing too much.
Understanding a loan's APR and monthly compounding
On most mortgages, car loans and credit cards, interest is calculated with a monthly rate equal to "the stated annual rate ÷ 12". Adding that interest each month is exactly monthly compounding, so an APR of 12% works out to \((1 + 0.01)^{12} - 1 \approx 12.68\%\) as a rate compounded once a year. The "Convert a rate" mode on this page does exactly this calculation.
Note that the APR a lender discloses under the US Truth in Lending Act can also include certain fees, so it is not only about compounding. The monthly payment on a loan you pay down each month is found with a different formula (amortization) that combines compounding with payments, but compound interest is its foundation.
Estimating how inflation adds up
If prices rise 2% every year (the long-run goal of the US Federal Reserve), the rise follows the same math as compound interest, and prices double in about \(72 \div 2 = 36\) years. Prices doubling means the same money buys half as much, in other words, the real value of cash is cut in half.
Being able to grasp the long-term effect of a small number like "2% a year" with numbers instead of gut feeling is one of the great uses of compound interest math.
Population and bacteria grow by the same math
Anything that grows by the same percentage every year (or every hour) follows the compound interest formula \(A = A_0(1 + r)^t\), even if it is not money. For example, if a population keeps growing 1% a year, it doubles in about 70 years (the exact doubling time is \(\ln 2 \div \ln 1.01 \approx 69.7\) years).
The same idea is widely used to estimate anything where growth builds on growth, such as a bacteria culture, the spread of a virus or social media followers (in reality, the formula stops applying once the growth rate is no longer constant).
Formulas and graphs
The basic compound interest formula (compounded once a year)
Graph
Standard notation
(the usual math form)
\(A\)
\(=\)
\(A_0\)
\(\times\)
\((1 + r)\)
\(t\)
In words
(symbols replaced with words)
④
\(A\): final amount
\(=\)
①
\(A_0\): principal
\(\times\)
②
1 + \(r\): annual rate
③
\(t\): years
The formula in words
①
Take the
\(A_0\): principal
②
multiply it by the growth factor for one year,
"1 + \(r\): annual rate"
③
once for each of the
\(t\): years
④
and you get the
\(A\): final amount
Quick example
The final amount when you deposit $10,000 at 3% a year (compounded once a year) for 10 years is
\(A\): final amount
\(=\)
principal ($10,000)
\(\times\)
1 + rate (1.03)
years (10)
\(10000 \times 1.03^{10} \approx 13439.16\)
Key idea
With simple interest, interest is earned on the principal only, every year, so 10 years gives just \(10000 \times 0.03 \times 10 = 3000\) dollars of interest. With compound interest, the interest you already earned also earns interest the next year, so the same terms give about $3,439, about $439 more. This "interest on interest" effect gets stronger the longer the time.
Enter the annual rate \(r\) as a decimal, the percentage divided by 100 (for 3%, \(r = 0.03\)).
One exception: when interest compounds once a year and the time is less than 1 year (such as 0.5 years), the formula calculates "part of one year's growth" as \((1+r)^{0.5}\), and simple interest comes out slightly larger (this is known as Bernoulli's inequality). That is why the calculator may show a negative difference in this case.
The formula with a compounding frequency (semiannual, monthly and so on)
Graph
Standard notation
(the usual math form)
\(A\)
\(=\)
\(A_0\)
\(\times\)
\(\left(1 + \dfrac{r}{n}\right)\)
\(nt\)
In words
(symbols replaced with words)
④
\(A\): final amount
\(=\)
①
\(A_0\): principal
\(\times\)
②
1 + rate per period
③
\(nt\): number of periods
The formula in words
①
Take the
\(A_0\): principal
②
multiply it by
\(1 + r/n\): 1 plus the rate per period
③
once for each of the
\(nt\): number of periods (\(n\) a year × \(t\) years)
④
and you get the
\(A\): final amount
Quick example
The final amount when you deposit $10,000 at 3% a year, compounded monthly (12 times a year), for 10 years is
\(A\): final amount
\(=\)
principal ($10,000)
\(\times\)
1 + 0.03 ÷ 12
periods (120)
\(10000 \times \left(1 + \dfrac{0.03}{12}\right)^{120} \approx 13493.54\)
Key idea
The annual rate \(r\) divided by the number of times per year \(n\), \(r/n\), is the rate per period, and \(n \times t\) is the total number of times interest is added. At the same 3% a year, compounding once a year gives about $13,439, while compounding monthly gives about $13,494. The more often interest compounds, the slightly larger the final amount.
If you compound more and more often, all the way to "continuous compounding", the final amount approaches \(A = A_0 e^{rt}\), using Euler's number \(e\) (about 2.718). Daily and continuous compounding are almost the same: in the example above, daily compounding gives about $13,498.42 and continuous compounding about $13,498.59, only 17 cents apart.
Converting a rate to another compounding frequency (APR ⇔ APY)
Graph
Standard notation
(the usual math form)
\(F\)
\(=\)
\(\left(1 + \dfrac{r_{\mathrm{in}}}{m}\right)\)
\(m\)
\(r_{\mathrm{out}}\)
\(=\)
\(n\)
\(\times\)
\((\)
\(F\)
\(1/n\)
\(-\)
\(1\)
\()\)
In words
(symbols replaced with words)
③
\(F\): annual growth factor
\(=\)
①
1 + rate per period
②
\(m\): times per year
⑦
\(r_{\mathrm{out}}\): converted rate
\(=\)
⑥
\(n\): target times per year
\(\times\)
\((\)
\(F\): annual growth factor
④
to the power \(1/n\)
\(-\)
⑤
\(1\): the principal part
\()\)
The formula in words
①
Take
\(1 + r_{\mathrm{in}}/m\): 1 plus the rate per period
②
and multiply it by itself
\(m\): times per year
times
③
to find the
\(F\): annual growth factor
(how many times larger the money gets in 1 year)
④
Raise \(F\)
to the power \(1/n\)
to turn it into the growth factor for one target period
⑤
subtract
\(1\): the principal part
to keep only the rate for one period
⑥
multiply by the
\(n\): target times per year
⑦
and you get the
\(r_{\mathrm{out}}\): converted rate
Quick example
Converting a 6% APR compounded monthly into an annual rate compounded once a year (APY)
\(F\): annual growth factor
\(=\)
1 + 0.06 ÷ 12
times (12)
\(r_{\mathrm{out}}\): converted rate
\(=\)
target times (1)
\(\times\)
\((\)
\(F\) to the power 1
\(-\)
principal part 1
\()\)
\(F = \left(1 + \dfrac{0.06}{12}\right)^{12} \approx 1.0616778\)
\(r_{\mathrm{out}} = 1 \times (1.0616778 - 1) = 0.0616778\ \ (6.16778\%)\)
Key idea
"6% a year, compounded monthly" actually grows by more than 6% in a year, because the interest added each month also earns interest. This formula restates the rate so that the growth over a year (the annual growth factor \(F\)) stays the same. In the US, the nominal annual rate before compounding is called the APR, and the effective annual rate that includes compounding is called the APY. Converting between the two is exactly this calculation.
When the input rate is compounded continuously, \(F = e^{r_{\mathrm{in}}}\). When the target is continuous compounding, \(r_{\mathrm{out}} = \ln F\) (the natural logarithm).
The Rule of 72 (a quick estimate of the years to double)
Graph
Standard notation
(the usual math form)
\(Y\)
\(\approx\)
\(72\)
\(\div\)
\(r\)
In words
(symbols replaced with words)
③
\(Y\): doubling time
\(\approx\)
①
the number 72
\(\div\)
②
\(r\): annual rate (%)
The formula in words
①
Divide
the number 72
②
by the
\(r\): annual rate (the percentage as it is)
③
and you get the
\(Y\): doubling time (about how many years it takes the principal to double)
Quick example
At 6% a year compounded, the approximate number of years for the principal to double is
\(Y\): doubling time
\(\approx\)
the number 72
\(\div\)
annual rate (6%)
\(72 \div 6 = 12\)
Key idea
The exact doubling time is \(Y = \ln 2 \div \ln(1 + r)\), which is about 11.9 years at 6%. The Rule of 72 is an approximation of it and works well for rates of about 2% to 10%. In theory, 69.3 (\(100 \ln 2\)) would be more accurate, but 72 divides evenly by 2, 3, 4, 6, 8, 9 and 12, which makes it easy to use in your head.
The rule works for anything that grows. If prices keep rising 2% a year, they double in about \(72 \div 2 = 36\) years, which means the real value of your money is cut in half.
With compound interest, the interest you earn also earns interest. The final amount is found by multiplying by "1 + the rate per period" once for every period. The key points: the longer the time, the wider the gap with simple interest, and the years it takes to double can be estimated as "72 ÷ annual rate".
Symbols and terms
Symbols
| \(A_0\) | A naught (A sub zero) | The principal, the amount you deposit (or borrow) at the start. (Example - $10,000) |
| \(A\) | A | The final amount, the principal plus the interest. |
| \(r\) | r | The annual interest rate, written as a decimal. (Example - for 3%, \(r = 0.03\)) |
| \(t\) | t | The time in years, how long the money is saved or borrowed. |
| \(n\) | n | The compounding frequency, the number of times per year interest is added. (Example - for monthly compounding, \(n = 12\)) |
| \(m\) | m | In the rate conversion formula, the compounding frequency of the rate you start with, in times per year. (Example - for monthly compounding, \(m = 12\)) |
| \(r_{\mathrm{in}}\) | r in | In the rate conversion formula, the rate you start with (before converting). |
| \(r_{\mathrm{out}}\) | r out | In the rate conversion formula, the rate you find (after converting). |
| \(Y\) | Y | The doubling time, about how many years it takes the principal to double. It is what the Rule of 72 estimates. |
| \(F\) | F | The annual growth factor, how many times larger the money gets in 1 year. (Example - at 6% compounded monthly, \(F \approx 1.0617\)) |
| \(e\) | e (Euler's number) | Euler's number, the base of the natural logarithm, about 2.71828. It appears as the limit when interest is compounded more and more often (continuous compounding), and it was first found by Jacob Bernoulli while he studied compound interest. |
| \(\ln\) | natural log (L N) | The natural logarithm, the logarithm with base \(e\). It appears in the exact doubling time \(\ln 2 \div \ln(1+r)\). |
Terms
| compound interest | Interest that is added to the principal, so the next interest is paid on the total. Because interest earns interest, it grows much more than simple interest over a long time. |
| simple interest | Interest paid only on the original principal. The interest is the same every year, so the growth is a straight line. |
| principal | The money you deposit or borrow at the start. Interest is calculated from it. |
| final amount | The principal plus the interest, also called the future value or ending balance. It is what the compound growth mode on this page finds. |
| interest | Money paid in return for depositing or lending money. It is the final amount minus the principal. |
| compounding frequency | How often interest is added to the principal (annually, semiannually, monthly and so on). At the same annual rate, more frequent compounding grows the money slightly more. |
| continuous compounding | The theoretical limit of compounding infinitely often. The final amount becomes \(A = A_0 e^{rt}\). It is often used in theoretical finance. |
| APR | Annual Percentage Rate. A nominal yearly rate that does not include the effect of compounding. For loans, the APR that lenders must disclose under the US Truth in Lending Act also includes certain fees, so it can be higher than the interest rate alone. |
| APY | Annual Percentage Yield. The effective yearly rate that includes the effect of compounding. At the same nominal rate (APR), the more often interest compounds, the higher the APY. US banks show the APY on savings accounts and CDs so that accounts can be compared. |
| effective annual rate | A rate compounded semiannually, monthly and so on, restated as the rate compounded once a year. It lets you compare products with different compounding frequencies fairly. For savings, it is the same idea as the APY. |
| Rule of 72 | A mental math shortcut that estimates the years it takes money to double as "72 ÷ annual rate (%)". At 6% a year, it is about 12 years. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Percents (Grade 6) |
|
| Exponents (Grades 6–8) |
|
| Geometric sequences (Algebra 1 and 2, going further) |
|
| Exponential and logarithmic functions and e (Algebra 2 and precalculus, going further) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table for the final amount (compounded once a year)
| Principal | 10000 |
| Annual rate (decimal; 3% is 0.03) | 0.03 |
| Time (years) | 10 |
| Final amount (compound) | =B1*(1+B2)^B3 |
Table for the final amount (with a compounding frequency)
| Principal | 10000 |
| Annual rate (decimal; 3% is 0.03) | 0.03 |
| Times per year (12 for monthly) | 12 |
| Time (years) | 10 |
| Final amount (compound) | =B1*(1+B2/B3)^(B3*B4) |
Table to convert a rate to another compounding frequency
| Rate to convert (decimal; 6% is 0.06) | 0.06 |
| Times per year, input rate (12 for monthly) | 12 |
| Times per year, target (1 for annual) | 1 |
| Annual growth factor F | =(1+B1/B2)^B2 |
| Converted rate (decimal) | =B3*(B4^(1/B3)-1) |
Table for the Rule of 72 (estimated doubling time)
| Annual rate (the percentage as it is) | 6 |
| Approximate years to double | =72/B1 |
After pasting, the upper rows of column B are your inputs and the last row is calculated automatically.
"^" means "to the power of" (multiply this many times), "*" is multiplication and "/" is division.
The first table shows about 13,439.16 (dollars), the second about 13,493.54 (dollars), the third about 0.0617 (= 6.16778%) and the fourth 12 (years). Note that rates are entered as decimals, the percentage divided by 100 (3% is 0.03), not as the percentage itself.
"^" means "to the power of" (multiply this many times), "*" is multiplication and "/" is division.
The first table shows about 13,439.16 (dollars), the second about 13,493.54 (dollars), the third about 0.0617 (= 6.16778%) and the fourth 12 (years). Note that rates are entered as decimals, the percentage divided by 100 (3% is 0.03), not as the percentage itself.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table for the final amount (compounded once a year)
| Principal | 10000 |
| Annual rate (decimal; 3% is 0.03) | 0.03 |
| Time (years) | 10 |
| Final amount (compound) | =B1*(1+B2)^B3 |
Table for the final amount (with a compounding frequency)
| Principal | 10000 |
| Annual rate (decimal; 3% is 0.03) | 0.03 |
| Times per year (12 for monthly) | 12 |
| Time (years) | 10 |
| Final amount (compound) | =B1*(1+B2/B3)^(B3*B4) |
Table to convert a rate to another compounding frequency
| Rate to convert (decimal; 6% is 0.06) | 0.06 |
| Times per year, input rate (12 for monthly) | 12 |
| Times per year, target (1 for annual) | 1 |
| Annual growth factor F | =(1+B1/B2)^B2 |
| Converted rate (decimal) | =B3*(B4^(1/B3)-1) |
Table for the Rule of 72 (estimated doubling time)
| Annual rate (the percentage as it is) | 6 |
| Approximate years to double | =72/B1 |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the input values in column B with your own.
How to calculate it in Python
principal = 10000 # principal (dollars)
annual_rate = 0.03 # annual rate (0.03 for 3%)
years = 10 # time (years)
times_per_year = 1 # compounding frequency (annually=1, semiannually=2, monthly=12, daily=365)
compound_amount = principal * (1 + annual_rate / times_per_year) ** (times_per_year * years)
simple_amount = principal * (1 + annual_rate * years)
print(f"Final amount (compound): ${compound_amount:,.2f}")
print(f"Final amount (simple): ${simple_amount:,.2f}")
print(f"Compound minus simple: ${compound_amount - simple_amount:,.2f}")
Runs with the standard library only. "**" means "to the power of" (multiply this many times). Change the principal, rate, time and frequency at the top, then run the code. For continuous compounding, add "import math" and use "principal * math.exp(annual_rate * years)".
How to write it in LaTeX and other math languages (copy and paste)
The basic compound interest formula (compounded once a year)
A = A₀(1 + r)ᵗ
A = A_0 (1 + r)^{t}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>A</mi>
<mo>=</mo>
<msub><mi>A</mi><mn>0</mn></msub>
<msup>
<mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>r</mi><mo>)</mo></mrow>
<mi>t</mi>
</msup>
</mrow>
</math>
A = A_0 (1 + r)^t
a0 (1 + r)^t
A := A0*(1 + r)^t;
A = A0*(1 + r)^t;
A = A_0 (1 + r)^t
The formula with a compounding frequency (semiannual, monthly and so on)
A = A₀(1 + r/n)ⁿᵗ
A = A_0 \left(1 + \dfrac{r}{n}\right)^{nt}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>A</mi>
<mo>=</mo>
<msub><mi>A</mi><mn>0</mn></msub>
<msup>
<mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mi>r</mi><mi>n</mi></mfrac><mo>)</mo></mrow>
<mrow><mi>n</mi><mi>t</mi></mrow>
</msup>
</mrow>
</math>
A = A_0 (1 + r/n)^(n t)
a0 (1 + r/n)^(n t)
A := A0*(1 + r/n)^(n*t);
A = A0*(1 + r/n)^(n*t);
A = A_0 (1 + r/n)^(nt)
Converting a rate to another compounding frequency (APR ⇔ APY)
F = \left(1 + \dfrac{r_{\mathrm{in}}}{m}\right)^{m}, \quad r_{\mathrm{out}} = n\left(F^{1/n} - 1\right)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>F</mi>
<mo>=</mo>
<msup>
<mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><msub><mi>r</mi><mtext>in</mtext></msub><mi>m</mi></mfrac><mo>)</mo></mrow>
<mi>m</mi>
</msup>
<mo>,</mo>
<msub><mi>r</mi><mtext>out</mtext></msub>
<mo>=</mo>
<mi>n</mi>
<mrow>
<mo>(</mo>
<msup><mi>F</mi><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></msup>
<mo>−</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</math>
F = (1 + r_(in)/m)^m, r_(out) = n(F^(1/n) - 1)
f = (1 + rin/m)^m; rout = n (f^(1/n) - 1)
F := (1 + r_in/m)^m; r_out := n*(F^(1/n) - 1);
F = (1 + r_in/m)^m; r_out = n*(F^(1/n) - 1);
F = (1 + r_in/m)^m, r_out = n(F^(1/n) - 1)
The Rule of 72 (a quick estimate of the years to double)
Y ≈ 72 ÷ r
Y \approx \dfrac{72}{r}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>Y</mi>
<mo>≈</mo>
<mfrac><mn>72</mn><mi>r</mi></mfrac>
</mrow>
</math>
Y ~~ 72 / r
72 / r
Y := 72 / r;
Y = 72 / r;
Y ≈ 72/r
How to have ChatGPT do the calculation
You are an interest calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). 1. Find the final amount and the total interest when $10,000 is deposited at 3% a year, compounded once a year, for 10 years. 2. Also find the final amount with simple interest under the same terms, and the difference from compound interest. 3. What is 6% a year compounded monthly as an annual rate compounded once a year (APY)? Give it to 5 decimal places. Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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Related Features
Percentage Change Calculator (Increase and Decrease)
"Up 3%" means "× 1.03". This percentage change is one step of compound interest
Geometric Sequence Calculator (nth Term and Sum)
Money growing at compound interest forms a geometric sequence with common ratio 1 + r
Exponent Calculator (Powers, a to the nth Power)
Does the core step of compound interest on its own - multiplying the same number many times
Simple Interest Calculator (Interest, Principal, Rate and Time)
Simple interest, where interest is earned only on the original principal. It can also solve for the principal, rate or time
Average Return Calculator (Annualized Geometric Average and Cumulative Return)
When returns differ from year to year, finds the single yearly compound rate that gives the same result
