In this article, we'll look at the compound interest formula, give you some worked examples and provide you with the tools to calculate it yourself.
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What is the compound interest formula?
Compound interest is the concept of adding accumulated interest back to the principal sum, so that interest is earned on top of interest from that moment on. The compound interest formula determines the amount of interest earned based on the initial principal, the interest rate, the compounding frequency and the total time period.
This formula gives you the future value of an investment or loan, which is the initial principal plus all of the compound interest earned over the period.
Understanding the variables
Let's break down each variable in the formula:
Example 1: Basic compound interest
Let's say you invest $5,000 at an annual interest rate of 5%, compounded annually, for 10 years. How much will you have at the end?
Worked Example
Using the formula A = P(1 + r/n)nt with our values:
A = 5000 × (1.05)10
A = 5000 × 1.62889...
A = $8,144.47
So, after 10 years your investment would be worth $8,144.47. The total compound interest earned is $8,144.47 − $5,000 = $3,144.47.
Example 2: Different compounding periods
What if the same investment ($5,000, 5%, 10 years) is compounded monthly instead of annually?
Monthly Compounding
A = 5000 × (1 + 0.004167)120
A = 5000 × (1.004167)120
A = 5000 × 1.64701...
A = $8,235.05
With monthly compounding you earn an extra $90.58 compared to annual compounding. This is because interest is being calculated and added more frequently.
The more frequently interest is compounded, the more you earn. Here's a comparison:
| Compounding | n value | Final Amount | Interest Earned |
|---|---|---|---|
| Annually | 1 | $8,144.47 | $3,144.47 |
| Quarterly | 4 | $8,218.10 | $3,218.10 |
| Monthly | 12 | $8,235.05 | $3,235.05 |
| Daily | 365 | $8,243.04 | $3,243.04 |
Compound interest formula with regular contributions
If you're making regular deposits (or withdrawals) alongside your investment, the formula becomes more complex. The formula for compound interest with regular contributions is:
Where PMT is the regular contribution (deposit) amount made at the end of each compounding period.
Example 3: With monthly contributions
You invest $5,000 initially and contribute $100 per month at 5% annual interest, compounded monthly, for 10 years.
Worked Example with Contributions
A₁ = 5000 × (1 + 0.05/12)(12×10)
A₁ = 5000 × 1.64701 = $8,235.05
A₂ = 100 × [((1 + 0.05/12)120 − 1) / (0.05/12)]
A₂ = 100 × [(1.64701 − 1) / 0.004167]
A₂ = 100 × [0.64701 / 0.004167]
A₂ = 100 × 155.282 = $15,528.23
A = $8,235.05 + $15,528.23
A = $23,763.28
Your total deposits were $5,000 + ($100 × 120) = $17,000. So the total interest earned is $23,763.28 − $17,000 = $6,763.28.
Continuous compounding formula
As the compounding frequency approaches infinity, we arrive at continuous compounding. The formula for continuous compounding uses Euler's number (e ≈ 2.71828):
Continuous Compounding Example
Using the same $5,000 at 5% for 10 years:
A = 5000 × e0.5
A = 5000 × 1.64872...
A = $8,243.61
Note that continuous compounding ($8,243.61) gives only a slight increase over daily compounding ($8,243.04).
Finding other variables
You can rearrange the compound interest formula to solve for other variables:
Where ln is the natural logarithm.
Compound interest growth chart
The following table shows how $5,000 grows at 5% interest with different compounding frequencies over various time periods:
| Year | Annual | Monthly | Daily |
|---|---|---|---|
| 1 | $5,250.00 | $5,255.81 | $5,256.31 |
| 2 | $5,512.50 | $5,524.71 | $5,525.70 |
| 5 | $6,381.41 | $6,416.79 | $6,419.74 |
| 10 | $8,144.47 | $8,235.05 | $8,243.04 |
| 15 | $10,394.64 | $10,568.39 | $10,584.98 |
| 20 | $13,266.49 | $13,563.20 | $13,590.55 |
| 25 | $16,931.77 | $17,404.55 | $17,448.64 |
| 30 | $21,609.71 | $22,332.96 | $22,399.47 |
The Rule of 72
A handy shortcut for estimating how long it takes to double your money is the Rule of 72. Simply divide 72 by the annual interest rate:
For example, at a 6% annual interest rate, it would take approximately 72 ÷ 6 = 12 years to double your money. This is an approximation but works remarkably well for interest rates between 4% and 12%.
Simple interest vs compound interest
With simple interest, interest is calculated only on the original principal. With compound interest, interest is calculated on both the principal and all previously accumulated interest. Over time, compound interest results in significantly higher returns.
Comparison: $10,000 at 8% for 20 years
Compound interest (annual): $10,000 × (1.08)20 = $46,609.57
Compound interest earns you an extra $20,609.57 over the same period — nearly double what simple interest provides.
If you wish to calculate your figures with the formulas from this article, you can do so with our compound interest calculator. You may also want to check our separate articles on compound interest tables and how to calculate compound interest.