## 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.

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 = Pe^{rt}

Calculate principal amount

Solve for P in terms of A

P = A / e^{rt}

Calculate principal amount

Solve for P in terms of I

P = I / (e^{rt} - 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:

- 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):

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

\begin{align} R&= r \times 100 \\[0.5em] &= 0.03812605 \times 100 \\[0.5em] &= 3.813\% \end{align}

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:

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

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.

\[ A = Pe^{rt} \]

## Excel: Calculate Compound Interest in Spreadsheets

Use the tables below to copy and paste compound interest formulas you need to make these calculations in a spreadsheet such as Microsoft Excel, Google Sheets and Apple Numbers.

To copy correctly, start your mouse outside the table upper left corner. Drag your mouse to the outside of the lower right corner. Be sure all text inside the table is selected. **Using Control + C and Control + V** ; Paste the copied information into cell **A1** of your spreadsheet. Formulas will only work starting in A1. You can modify the formulas and formatting as you wish.

### Calculate Accrued Amount (Future Value FV) using A = P(1 + r/n)^nt

In this example we start with a principal investment of 10,000 at a rate of 3% compounded quarterly (4 times a year) for 5 years. If you paste this correctly you should see the answer Accrued Amount (FV) = 11,611.84 in cell B1. Change the values in B2, B3, B4 and B5 to your specific problem.

Copy and paste this table into spreadsheets as explained in the above section.

Accrued Amount (FV) $ | = ROUND(B3 * POWER(( 1 + ((B2/100)/B4)),(B4*B5)),2) |

Rate % | 3 |

Principal $ | 10000 |

Compounding per year | 4 |

Years | 5 |

### Calculate Rate using Rate Percent = n[ ( (A/P)^(1/nt) ) - 1] * 100

In this example we start with a principal of 10,000 with interest of 500 giving us an accrued amount of 10,500 over 2 years compounded monthly (12 times per year). If you paste this correctly you should see the answer for Rate % = 2.44 in cell B1. Change the values in B2, B3, B4 and B5 to your specific problem.

Copy and paste this table into spreadsheets as explained in the above section.

Rate % | = ROUND(B4*((POWER((B2/B3),(1/(B4*B5))))-1)*100,2) |

Accrued Amount $ | 10500 |

Principal $ | 10000 |

Compounding per year | 12 |

Years | 2 |

### Further Reading

Tree of Math: Continuous Compounding

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