# Compound Interest Calculator

## Calculator Use

The compound interest calculator lets you see how your money can grow using interest compounding.

Calculate compound interest on an investment, 401K or savings account with annual, quarterly, daily or continuous compounding.

We provide answers to your compound interest calculations and show you the steps to find the answer. You can also experiment with the calculator to see how different interest rates or loan lengths can affect how much you'll pay in compounded interest on a loan.

Read further below for additional compound interest formulas to find principal, interest rates or final investment value. We also show you how to calculate continuous compounding with the formula A = Pe^rt.

## 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

- A = Accrued amount (principal + interest)
- P = Principal amount
- r = Annual nominal interest rate as a decimal
- R = Annual nominal interest rate as a percent
- r = R/100
- n = number of compounding periods per unit of time
- t = time in decimal years; e.g., 6 months is calculated as 0.5 years. Divide your partial year number of months by 12 to get the decimal years.
- I = Interest amount
- ln = natural logarithm, used in formulas below

### 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.

Principal + Interest

^{nt}

Solve for P in terms of A

^{nt}

Solve for P in terms of I

^{nt}- 1)

As a decimal

^{1/nt}- 1)

As a percent

Solve for t

ln is the natural logarithm

t = (ln(A) - ln(P)) / n(ln(1 + r/n))

Principal + Interest

^{t}

Solve for P in terms of A

^{t}

Solve for P in terms of I

^{t}- 1)

As a decimal

^{1/t}- 1

As a percent

Solve for t

ln is the natural logarithm

t = (ln(A) - ln(P)) / ln(1 + r)

Principal + Interest

^{rt}

Solve for P in terms of A

^{rt}

Solve for P in terms of I

^{rt}- 1)

As a decimal

ln is the natural logarithm

As a percent

Solve for t

ln is the natural logarithm

### 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:

- Total P+I (A): $33,000
- Principal (P): $30,000
- Compound (n): Daily (365)
- Time (t in years): 2.5 years (30 months equals 2.5 years)

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

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 = Pe^{rt} the Continuous Compound Interest Formula

A common definition of the constant *e* is that:

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:

And finally you have your continuous compounding formula.

\[ A = Pe^{rt} \]Further Reading

Tree of Math: Continuous Compounding

^{1}Excel^{®} is a registered trademark of Microsoft Corporation

**Cite this content, page or calculator as:**

Furey, Edward "Compound Interest Calculator"; CalculatorSoup,
*https://www.calculatorsoup.com* - Online Calculators