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Compound Interest Formula With Examples

Thumbnail image of Alastair Hazell By Alastair Hazell. Reviewed by Chris Hindle.

Compound interest, or 'interest on interest', is calculated using the compound interest formula. In this article we'll look at how the formula works and how to use it for manual calculations or within a spreadsheet.

The formula for compound interest is A = P(1 + r/n)^nt where P is the principal balance, r is the interest rate, n is the number of times interest is compounded per year and t is the number of years.

The compound interest formula: A = P(1 + r/n)^nt

The concept of compound interest is that interest is added back to the principal sum so that further interest is gained on that already-accumulated interest during the next compounding period. How important is compound interest? Just ask Warren Buffett, one of the world's most successful investors:

“My wealth has come from a combination of living in America, some lucky genes, and compound interest.”

Warren Buffett, 2010 1

Variations of the compound interest formula

Here are some useful variations of the compound interest formula. We'll discuss each variation individually later in the article.

Calculation Formula
Calculate future value of principal+interest (A) A = P(1 + r/n)^nt
Annual compound interest formula (1x compound per year) A = P(1 + r)^t
Quarterly compound interest formula A = P(1 + r/4)^4t
Monthly compound interest formula A = P(1 + r/12)^12t
Daily compound interest formula A = P(1 + r/365)^365t
Calculate principal (P) based upon future value P = A / (1 + r/n)^nt
Calculate interest rate as a percentage (R) R = n[(A/P)^(1/nt)-1] × 100
Calculate time factor (how long it takes to reach a target figure) (t) t = ln(A/P) / n[ln(1 + r/n)]

Where:

  • A = future value of the investment/loan
  • P = principal amount
  • r = annual interest rate (decimal)
  • R = annual interest rate (percentage)
  • n = number of times interest is compounded per year
  • t = time in years
  • ^ = ... to the power of ...
  • ln = the natural logarithm

In the remainder of this article, we'll take a look at how to calculate compound interest on your savings or loan using the formula. We'll include instructions for how to do this both with and without an Excel spreadsheet. We'll also look at other variations of the formula that can help you to calculate the interest rate and time factor, or to incorporate monthly deposits or withdrawals.

How to use the compound interest formula

To use the compound interest formula you will need the figures for your initial balance, annual interest rate (as a decimal) and the number of time periods (e.g. the number of years). Let's take a look at the calculation process...

How to calculate compound interest

Multiply your initial balance by one plus the annual interest rate (as a decimal) raised to the power of the number of time periods (years). Subtract the initial balance from the result if you want to see only the interest earned.

The above set out as a formula is:

A = P(1+r)^t

This simplified formula assumes that interest is compounded once per time period, rather than multiple times per time period (e.g. once per year).

If you want to compound more than once per time period (e.g. monthly compounding for a number of years), you'll need to use the advanced formula which incorporates the number of compounds per time period:

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

Where:

  • A = future value of the investment/loan
  • P = principal investment or loan amount
  • r = annual interest rate (decimal)
  • n = number of times interest is compounded per year
  • t = time in years
  • ^ = ... to the power of ...

It's worth noting that this formula gives you the future value of an investment or loan, which is compound interest plus the principal. Should you wish to calculate the compound interest only, you need to deduct the principal from the result. So, your formula looks like this:

Earned interest only (without principal)

Interest = P(1 + r/n)^nt - P

Let's look at how we can use this formula for monthly compounding, and we can then go through an example calculation...

Monthly compound interest formula

The formula for calculating compound interest with monthly compounding is:

A = P(1 + r/12)^12t

Where:

  • A = future value of the investment
  • P = principal investment amount
  • r = annual interest rate (decimal)
  • t = time in years
  • ^ = ... to the power of ...

How to use the formula in Excel or Google Sheets

If you're using Excel, Google Sheets or Numbers, you can copy and paste the following into your spreadsheet and adjust your figures for the first four rows as you see fit. This example shows monthly compounding (12 compounds per year) with a 5% interest rate.

Principal 10,000
Interest rate (%) 5
Compounds per year 12
Years 10
Future value = ROUND(INDIRECT(ADDRESS(ROW()-4,COLUMN())) * (1+(INDIRECT(ADDRESS(ROW()-3,COLUMN())) / 100)/INDIRECT(ADDRESS(ROW()-2,COLUMN()))) ^ (INDIRECT(ADDRESS(ROW()-2,COLUMN())) * INDIRECT(ADDRESS(ROW()-1,COLUMN()))),2)

Here's how it will look in Excel or Google Sheets...

Example of compound interest formula calculation on Excel and Google Sheets

Now that we've looked at how to use the formula for calculations in Excel, let's go through a step-by-step example to demonstrate how to make a manual calculation using the formula...

Example calculation

If an amount of $10,000 is deposited into a savings account at an annual interest rate of 3%, compounded monthly, the value of the investment after 10 years can be calculated as follows...

  • P = 10000
  • r = 3/100 = 0.03 (decimal)
  • n = 12
  • t = 10

If we plug those figures into the formula, we get the following:

A = 10000 × (1 + 0.03 / 12)^(12 * 10) = 13493.54.

So, the investment balance after 10 years is $13,493.54.

Formula methodology

Let's go through, step-by-step, how we get the 13493.54 result. Our methodology revolves around the PEMDAS order of operations. That is to say that we have to perform operations such as multiplication and addition in the right order.

PEMDAS - the order of operations

Let's start off with our equation again:

A = 10000 (1 + 0.03 / 12) ^ (12(10))

(note that ^ means 'to the power of')

Using the order of operations we work out the totals in the brackets first.

Within the first set of brackets, you need to do the division first and then the addition (division and multiplication should be carried out before addition and subtraction). We can also work out the 12(10), which is 12 × 10. This gives us...

A = 10000 (1 + 0.0025)^120

Then:

A = 10000 (1.0025)^120

The exponent goes next. So, we calculate 1.0025^120.

This means we end up with:

10000 × 1.34935355

= 13,493.54.

The benefits of compound interest

I think it's worth taking a moment to mention the monetary gain that interest compounding can offer.

Looking back at our example, with simple interest (no compounding), your investment balance at the end of the term would be $13,000, with $3,000 interest. With regular interest compounding, however, you would stand to gain an additional $493.54 on top.

Interest for $10,000 at 3% for 10 years:

  • With simple interest: $3,000
  • With compound interest: $3,493.54

Interest for $10,000 at 5% for 10 years:

  • With simple interest: $5,000
  • With compound interest: $6,470.09

Interactive compound interest formula

I created the calculator below to show you the formula and resulting accrued investment/loan value (A) for the figures that you enter.




P ( 1 + r n ) ( n t ) = A

It may be that you want to manipulate the compound interest formula to work out the interest rate for IRR or CAGR, or a principal investment/loan figure. Here are the formulae you need.

Formula for calculating interest rate (r)

Compound interest formula - how to calculate interest rate

This formula can help you work out the yearly interest rate you're getting on your savings, investment or loan. Note that you should multiply your result by 100 to get a percentage figure (%).

r = n[(A/P)^(1/nt)-1]

Where:

  • r = interest rate (decimal)
  • A = future value of the investment
  • P = principal investment amount
  • n = number of times interest is compounded per year
  • t = time in years
  • ^ = ... to the power of ...

Formula for calculating principal (P)

Compound interest formula - how to calculate principal (starting amount)

This formula is useful if you want to work backwards and calculate how much your starting balance would need to be in order to achieve a future monetary value.

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

Where:

  • P = principal investment amount
  • A = future value of the investment
  • r = interest rate (decimal)
  • n = number of times interest is compounded per year
  • t = time in years
  • ^ = ... to the power of ...

Example: Let's say your goal is to end up with $10,000 in 5 years, and you can get an 8% interest rate on your savings, compounded monthly. Your calculation would be: P = 10000 / (1 + 0.08/12)^(12×5) = $6712.10. So, you would need to start off with $6712.10 to achieve your goal.

Formula for calculating time factor (t)

This variation of the formula works for calculating time (t), by using natural logarithms. You can use it to calculate how long it might take you to reach your savings target, based upon an initial balance and interest rate. You can see how this formula was worked out by reading this explanation on algebra.com.

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

Where:

  • A = value of the accrued investment/loan
  • P = principal amount
  • r = annual interest rate (decimal)
  • n = number of times interest is compounded per year
  • t = time in years
  • ln = the natural logarithm

Monthly contributions formula

I've received a lot of requests over the years to provide a formula for compound interest with monthly contributions. So, let's go over how we do this.

In order to work out calculations involving regular contributions, you will need to combine two formulae: our original compound interest formula — P(1+r/n)^(nt) — plus the future value of a series formula for the monthly deposits.

These formulae assume that your frequency of compounding is the same as the periodic payment interval (monthly compounding, monthly contributions, etc).

If you would like to try a version of the formula that allows you to have a different periodic payment interval to the compounding frequency, please see the 'periodic payments' section below.

If the additional deposits are made at the END of the period (end of month, year, etc), here are the two formulae you need:

Compound interest for principal:

P(1+r/n)^(nt)

Future value of a series:

PMT × {[(1 + r/n)^(nt) - 1] / (r/n)}

If the additional deposits are made at the BEGINNING of the period (beginning of year, etc), here are the two formulae you need:

Compound interest for principal:

P(1+r/n)^(nt)

Future value of a series:

PMT × {[(1 + r/n)^(nt) - 1] / (r/n)} × (1+r/n)

Where:

  • A = future value of the investment/loan
  • P = principal investment amount
  • PMT = monthly payment amount
  • r = annual interest rate (decimal)
  • n = number of times interest is compounded per year
  • t = time in years
  • ^ = ... to the power of ...

Example calculation

If an amount of $5,000 is deposited into a savings account at an annual interest rate of 5%, compounded monthly, with additional deposits of $100 per month (made at the end of each month). The value of the investment after 10 years can be calculated as follows...

P = 5000. PMT = 100. r = 3/100 = 0.03 (decimal). n = 12. t = 10.

Let's plug those figures into our formulae and use our PEMDAS order of operations to create our calculation...

  • Total = [ Compound interest for principal ] + [ Future value of a series ]
  • Total = [ P(1+r/n)^(nt) ] + [ PMT × (((1 + r/n)^(nt) - 1) / (r/n)) ]
  • Total = [ 5000 × (1 + 0.03 / 12)^(12 × 10) ] + [ 100 × (((1 + 0.03 / 12)^(12 × 10) - 1) / (0.03 / 12)) ]
  • Total = [ 5000 × (1 + 0.0025)^(12 × 10) ] + [ 100 × (((1 + 0.0025)^(12 × 10) - 1) / (0.0025)) ]
  • Total = [ 5000 × (1.0025)^120 ] + [ 100 × (((1.0025^120) - 1) / 0.0025) ]
  • Total = [ 6746.77 ] + [ 100 × (0.34935354719 / 0.0025) ]
  • Total = [ 6746.77 ] + [ 13974.14 ]
  • Total = [ $20,720.91 ]

Our investment balance after 10 years therefore works out at $20,720.91.

Calculating different periodic payments

A few people have requested a version of the above compounding formula that takes into account the number of periodic contributions (both formulae above assume your periodic payments match the frequency of compounding). For example, your money may be compounded monthly but you're making contributions quarterly.

In this case, you may wish to try this version of the formula, originally suggested by Darinth Douglas, and then expanded upon by Jean-Baptiste Delaroche. I'm most grateful for their input. This formula assumes that regular deposits are paid at the beginning rather than at the end of the period.

Compound interest for principal:

P(1+r/n)^(nt)

Future value of a series:

PMT × p {[(1 + r/n)^(nt) - 1] / (r/n)}

(With 'p' being the number of periodic payments in the time period, divided by n)

For more information about what to do with the formula when calendar intervals are irregular, see this useful section from Jon Wittwer at Vertex42.

Example calculation

An amount of $100 is deposited quarterly into a savings account at an annual interest rate of 10%, compounded monthly. The value of the investment after 12 months can be calculated as follows...

PMT = 100. r = 0.1 (decimal). n = 12. p = 4/n = 4/12 = 0.3333333.

Let's plug those figures into our formula...

  • Total = PMT × p {[(1 + r/n)^(nt) - 1] / (r/n)}
  • Total = 100 × 0.3333333 × {[(1 + 0.1 / 12) ^ (12 × 1) - 1] / (0.1 / 12)}
  • Total = 100 × 0.3333333 × {[1.008333 ^ (12) - 1] / 0.008333}
  • Total = 100 × 0.3333333 × {0.104709 / 0.008333}
  • Total = 100 × 0.3333333 × 12.565583
  • Total = 418.85

So, the investment balance after 12 months is $418.85 (or $418.84 if you round the numbers during the calculation).

To conclude

This article about the compound interest formula has expanded and evolved based upon your requests for adapted formulae and examples. So, I appreciate it's now quite long and detailed. That said, I hope you've found it helpful. If you have, I would be very grateful if you would consider sharing a link to it on social media or on your website/blog. Thank you.

If you need any help with checking your calculations, please make use of our popular compound interest calculator and daily compounding calculator.

By Alastair Hazell Updated: December 1, 2022

Disclaimer: Whilst every effort has been made in building our calculator tools, we are not to be held liable for any damages or monetary losses arising out of or in connection with their use. Full disclaimer.

References

  1. My philanthropic pledge. CNN Money.
  2. Discounting and compounding. John A. Dutton e-Education Institute.

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