Free Compound Interest Calculator

Enter Principle
Enter Time
Enter Rate

The Compound Interest Formula

This calculator uses the compound interest formula to find principal plus interest. It uses this same formula to solve for principal, rate or time given the other known values. You can also use this formula to set up a compound interest calculator in Excel®1.

A = P(1 + r/n)nt

In the formula

Compound Interest Formulas Used in This Calculator

The basic compound interest formula A = P(1 + r/n)nt can be used to find any of the other variables. The tables below show the compound interest formula rewritten so the unknown variable is isolated on the left side of the equation.

Compound Interest Formulas
Calculation
Formula
Calculate accrued amount
Principal + Interest
A = P(1 + r/n)nt
Calculate principal amount
Solve for P in terms of A
P = A / (1 + r/n)nt
Calculate principal amount
Solve for P in terms of I
P = I / ((1 + r/n)nt - 1)
Calculate rate of interest
As a decimal
r = n((A/P)1/nt - 1)
Calculate rate of interest
As a percent
R = r * 100
Calculate time
Solve for t
ln is the natural logarithm
t = ln(A/P) / n(ln(1 + r/n)), then also
t = (ln(A) - ln(P)) / n(ln(1 + r/n))
Formulas where n = 1
(compounded once per period or unit t)
Calculation
Formula
Calculate accrued amount
Principal + Interest
A = P(1 + r)t
Calculate principal amount
Solve for P in terms of A
P = A / (1 + r)t
Calculate principal amount
Solve for P in terms of I
P = I / ((1 + r)t - 1)
Calculate rate of interest
As a decimal
r = (A/P)1/t - 1
Calculate rate of interest
As a percent
R = r * 100
Calculate time
Solve for t
ln is the natural logarithm
t = ln(A/P) / ln(1 + r), then also
t = (ln(A) - ln(P)) / ln(1 + r)
Continuous Compounding Formulas
(n → ∞)
Calculation
Formula
Calculate accrued amount
Principal + Interest
A = Pert
Calculate principal amount
Solve for P in terms of A
P = A / ert
Calculate principal amount
Solve for P in terms of I
P = I / (ert - 1)
Calculate rate of interest
As a decimal
ln is the natural logarithm
r = ln(A/P) / t
Calculate rate of interest
As a percent
R = r * 100
Calculate time
Solve for t
ln is the natural logarithm
t = ln(A/P) / r

How to Use the Compound Interest Calculator: Example

Say you have an investment account that increased from $30,000 to $33,000 over 30 months. If your local bank offers a savings account with daily compounding (365 times per year), what annual interest rate do you need to get to match the rate of return in your investment account?

In the calculator above select "Calculate Rate (R)". The calculator will use the equations: r = n((A/P)1/nt - 1) and R = r*100.

Enter:

Showing the work with the formula r = n((A/P)1/nt - 1):

\[ r = 365 \left(\left(\frac{33,000}{30,000}\right)^\frac{1}{365\times 2.5} - 1 \right) \] \[ r = 365 (1.1^\frac{1}{912.5} - 1) \] \[ r = 365 (1.1^{0.00109589} - 1) \] \[ r = 365 (1.00010445 - 1) \] \[ r = 365 (0.00010445) \] \[ r = 0.03812605 \] \[ R = r \times 100 = 0.03812605 \times 100 = 3.813\% \]

Your Answer: R = 3.813% per year

So you'd need to put $30,000 into a savings account that pays a rate of 3.813% per year and compounds interest daily in order to get the same return as the investment account.

How to Derive A = Pert the Continuous Compound Interest Formula

A common definition of the constant e is that:

\[ e = \lim_{m \to \infty} \left(1 + \frac{1}{m}\right)^m \]

With continuous compounding, the number of times compounding occurs per period approaches infinity or n → ∞. Then using our original equation to solve for A as n → ∞ we want to solve:

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

This equation looks a little like the equation for e. To make it look more similar so we can do a substitution we introduce a variable m such that m = n/r then we also have n = mr. Note that as n approaches infinity so does m.

Replacing n in our equation with mr and cancelling r in the numerator of r/n we get:

\[ A = P \left( \lim_{m\rightarrow\infty} \left(1 + \frac{1}{m}\right)^{mrt} \right) \]

Rearranging the exponents we can write:

\[ A = P \left( \lim_{m\rightarrow\infty} \left(1 + \frac{1}{m}\right)^{m} \right)^{rt} \]

Substituting in e from our definition above:

\[ A = P(e)^{rt} \]

And finally you have your continuous compounding formula.

\[ A = Pe^{rt} \]