# Compound Interest Calculator

## Calculator Use

The compound interest calculator shows you how your money can grow with interest compounding.

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

The calculator computes compound interest calculations and shows you the steps including the math. You can also use the calculator to try different interest rates or investment terms and see how those factors affect compound interest earnings over time.

Input any 4 values of principal, interest, rate, time or compounding frequency and calculate the missing value. Remember to input time in terms of years. So if your time value is in months just divide by 12 to get the equivalent value in years.

## What is Compound Interest?

Compound interest is when interest you earn on a savings account or investment is rolled back into your balance to earn additional interest. The compound interest calculation accounts for interest you earn over time and adds it back into the amount being invested or saved. So while you are earning interest on your original principal you are also earning interest on accumulated interest.

Interest compounding can also happen with loans or credit card debt. In this case you are charged interest on interest that accumulates and is not paid off. Compounding means that interest charged is added to the balance owed so that subsequent interest is calculated from your unpaid balance **plus** accrued interest.

Compound interest is different from simple interest where the interest amount is calculated at the beginning of the investment or loan. Simple interest means that interest earned is not rolled back into the balance for future interest calculations.

### Additional Resources

Visit InterestMagician.com for interactive compound interest calculations. There, you can experiment with investment values, interest rates and lengths of deposit and visualize your earnings over time.

Visit the CalculatorSoup® Simple Interest Calculator to read more about simple interest. Learn that without exponents, interest calculations using the simple interest formula A = P(1 + rt) are very different from compound interest calculations.

## The Compound Interest Formula

This calculator uses the compound interest formula to find the total principal plus accrued interest. It uses this same formula to solve for principal, rate or time given the other known values. You can also use the compound interest equation to set up a compound interest calculator in an Excel^{1} spreadsheet.

**A = P(1 + r/n) ^{nt}**

In the formula

- A = Final amount (principal + accrued interest)
- P = Principal starting 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. For example 6 months is equal to 0.5 years. Divide your 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 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.

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 match the rate of return in your investment account?

In the compound interest 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 divided by 12 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.

\[ A = Pe^{rt} \]## Excel: Calculate Compound Interest in Spreadsheets

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