By Alastair Hazell.Reviewed by Chris Hindle.
Compound interest, or 'interest on interest', is calculated using the compound interest formula A = P*(1+r/n)^(nt),where P is the principal balance, r is the interest rate (as a decimal), n represents the number of times interest is compounded per year and t is the number of years.
How to use the formula
To better understand the formula, let's create a simple example (we'll look into more complex variations, involving different compound intervals and extra contributions, later on in our article). Let's say that you invest $10,000 into a savingsaccount for 20 years at an annual interest rate of 6%, compounded monthly. Here's how our calculation looks:
- P = 10000
- r = 6/100 = 0.06 (decimal)
- n = 12
- t = 20
Plugging those figures into the formula, we can calculate as follows:
- A = 10000 × (1 + 0.06 / 12)^(12 × 20)
- A = 10000 × (1 + 0.005)^240
- A = 10000 × 3.310204
- A = 33,102.04
The investment balance after 20 years is therefore $33,102.04. Checking this figure againstThe Calculator Site's compound interest calculator, we can see that we have calculated correctly.
Keep scrolling to see variations of the formula for annual, quarterly, monthly and daily compounding, a step-by-step explanation of how to use the formula,and some information on how to integrate it within a spreadsheet...
- Table of contents:
- Formula variations
- How to use the formula
- Monthly compound interest formula
- How to use the formula in Excel or Google Sheets
- Example calculation
- Interactive compound interest formula
- Formula for calculating interest rate (%)
- Formula for calculating principal
- Formula for calculating time factor
- Monthly contributions formula
Formula variations
To assist those looking for a convenient formula reference, I've included a concise list of compound interest formula variations applicable to common compounding intervals. Later in the article, we will delve into each variation separately for a comprehensive understanding.
| Calculation | Formula |
|---|---|
| Annual compounding (1x compound per year) | A = P(1 + r)^t |
| Quarterly compounding | A = P(1 + r/4)^4t |
| Monthly compounding | A = P(1 + r/12)^12t |
| Daily compounding | A = P(1 + r/365)^365t |
| Calculate principal (P) based upon future value | P = A / (1 + r/n)^nt |
| Calculate interest rate % (R) | R = n[(A/P)^(1/nt)-1] × 100 |
| Calculate time factor (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
How to use the compound interest formula
In order to use the compound interest formula you will require specific values for your initial balance (principal), annual interest rate (expressed as a decimal), the number of compounds per year and the number ofyears you wish to calculate for. Let's take a look at how we put these into our formula...
Start by multiply your initial balance by one plus the annual interest rate (expressed as a decimal) divided by the number of compounds per year. Next, raise the result to the power of the number of compounds per year multiplied by the number of years. Subtract the initial balancefrom the result if you want to see only the interest earned.How to calculate compound interest using the formula
The above set out as a formula is:
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 tocalculate 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 fourrows 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...
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 manualcalculation 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.
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. 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 × 1.34935355Formula methodology
= 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 balanceat 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 +rn)(nt)=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)
This formula can help you work out the yearly interest rate you're getting on your savings, investment or loan. Note that youshould 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)
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 calculatehow long it might take you to reach your savings target, based upon an initial balance and interest rate. Youcan 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 originalcompound 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 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 3%, 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.
To conclude
This article about the compound interest formula has expanded and evolved based upon your requests for adapted formulae andexamples. So, I appreciate it's now quite long and detailed. That said, I hope you've found it useful. Please feel free to share any thoughts in the comments section below.
Should you need any help with checking your calculations, please make use of our regular interest compoundingcalculator and daily compounding calculator.
By Alastair HazellUpdated: July 29, 2024
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