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Online Calculators

Compound Interest Calculator Resources

This page contains a comprehensive list of formulas for solving compound interest calculations. You can use these formulas to solve for principal, interest, rate, time, and total principal plus interest when doing calculations by hand, or when creating formulas to use in Excel or other spreadsheets. We include formulas for when compounding happens once per year, and also when compounding is continuous.

Compound Interest Reference Formulas

The 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 with the unknown variable on the left side of the equation.

Compound Interest Formulas
Calculation
Formula
Calculate total 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 interest amount
Solve for I in terms of P
I = P((1 + r/n)nt - 1)
Calculate compound interest rate
As a decimal
r = n((A/P)1/nt - 1)
Calculate compound interest rate
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 total 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 interest amount
Solve for I in terms of P
I = P((1 + r)t - 1)
Calculate compound interest rate
As a decimal
r = (A/P)1/t - 1
Calculate compound interest rate
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 total 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 interest amount
Solve for I in terms of P
I = P(ert - 1)
Calculate compound interest rate
As a decimal
ln is the natural logarithm
r = ln(A/P) / t
Calculate compound interest rate
As a percent
R = r * 100
Calculate time
Solve for t
ln is the natural logarithm
t = ln(A/P) / r

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.

\[ A = Pe^{rt} \]