# Compound Interest Calculator

The Compound Interest Calculator below can be used to compare or convert the interest rates of different compound periods. Please use our Interest Calculator to do actual calculations on compound interest.

 Input Interest Compound Output Interest Compound Annually (APY)SemiannuallyQuarterlyMonthly (APR)SemimonthlyBiweeklyWeeklyDailyContinuously = 6.16778% Annually (APY)SemiannuallyQuarterlyMonthly (APR)SemimonthlyBiweeklyWeeklyDailyContinuously

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Albert Einstein once described compound interest as "the greatest mathematical discovery of all time."

Unlike physics however, compound interest isn't rocket science. It's simply interest earned on both the original amount and on existing interest accumulated, unlike regular interest which is only on the original amount. Because interest is also being earned on interest, earnings compound over time like an exponentially-growing, avalanching snowball.

However, things can get complicated when different frequencies of compounding are involved. There's even such a thing as compounding interest for an infinite number of periods! But don't worry, that is what our Compound Interest Calculator is here for. It allows comparisons between many popularly used frequencies used to compound interest to help users visualize the differences between them all.

### Compound Interest Formulas

Basic Compound Interest

Basic formula for compound interest:

At = A0(1+r)n

where:
A0 : principal amount, or initial investment
At : amount after time t
r : interest rate
n : number of compounding periods, usually expressed in years

The following is an example of \$1000 in a savings account for two years advertised at 6% APY compounded once a year. Use the equation above to find the total due at maturity:

At = \$1000 × (1 + 6%)2 = \$1,123.60

For other compounding frequencies (such as monthly, weekly, or daily), the situation calls for the formula below.

At = A0 × (1 + r/n)nt

where:
A0 : principal amount, or initial investment
At : amount after time t
n : number of compounding periods in a year
r : interest rate
t : number of years

Assume that the \$1000 in the savings account in the previous example comes with a 6% rate with interest accumulated daily. The actual interest earned every day is:

6% ÷ 365 = 0.0164384%

Using the formula above, it is possible to find the value at the end.

At = \$1000 × (1 + 0.0164384%)(365 × 2)
At = \$1000 × 1.12749
At = \$1,127.49

\$1,127.49 will be the end value of a 2-year savings account containing \$1000 that has a 6% interest rate compounded daily.

Continuous Compound Interest

Continuously compounding interest is interest that is perpetually accumulated. It represents the mathematical limit that compound interest can reach within a specified time period. The continuous compound equation is as follows:

At = A0ert

where:
A0 : principal amount, or initial investment
At : amount after time t
r : interest rate
t : number of years
e : mathematical constant e, ~2.718

The equation utilizes something called Euler's Constant, also known as e. Although e is widely used in many areas such as probability theory and Newton's law of cooling/heating, it was discovered when Jacob Bernoulli was studying compound interest in 1683. The mathematician understood that, within a specified finite time period, the more compounding periods involved, the faster the compounding principal was able to grow. It didn't matter whether it was in intervals of years, months, days, hours, minutes, seconds, nanoseconds, picoseconds, or femtoseconds, each additional period generated higher returns (for the lender). Because it is still based on the same interest rate that gets divided up further and further, the surplus derived from each compounded period becomes exponentially smaller. Bernoulli discerned that this sequence eventually snowballed towards a limit, thus defined as e which describes the relationship between the plateau and the interest rate when compounding.

Say for instance, we wanted to find the maximum interest that could possibly be earned on the \$1000 savings account in two years.

Using the equation above:

At = \$1000e(6% × 2)
At = \$1000e0.12
At = \$1,127.50

It is very clear from all 3 examples given on compound interest that, the more compounding frequencies involved, ceteris paribus, the more interest will be earned.

For a savings account with \$1,000 at a 6% interest rate, here are the results of end value at the different compound frequencies:

 Compound Times End Value Compounded annually for two years 2 times \$1,123.60 Compounded daily (365) for two years 730 times \$1,127.49 Compounded continuously for two years Infinity \$1,127.50

Notice that the interest earned as compounding frequencies reach higher stages are marginally less and less. Going from 730 periods to infinite number of periods is only a \$0.01 increase in interest, hardly noticeable.