CI Formula
A savings account advertises 5% annual interest, compounded monthly. You deposit $1,000. How much will you have in 10 years? The answer comes from the CI (compound interest) formula–a cornerstone of financial math. It explains why money grows faster when interest earns its own interest.
What is the Compound Interest Formula?
Compound interest calculates interest on both the original principal and all accumulated interest from previous periods. Unlike simple interest, which only looks at the principal, compound interest creates exponential growth. The standard CI formula is:
A = P (1 + r/n)^(n t)
Where:
- A = the future value (final amount)
- P = the principal (initial amount)
- r = annual interest rate, expressed as a decimal (e.g., 5% = 0.05)
- n = number of compounding periods per year
- t = time the money is invested or borrowed, in years
This formula works for savings accounts, investment returns, inflation-adjusted values, and even certain debt scenarios.
Variable Breakdown
- Principal (P): The starting sum. A $10,000 deposit is P = 10,000.
- Rate (r): Always use the decimal form. A 7.2% rate becomes r = 0.072.
- Compounding frequency (n): Annual = 1, semi-annual = 2, quarterly = 4, monthly = 12, daily = 365. Some accounts use 360 for simplicity, but 365 is standard.
- Time (t): Duration in years. 18 months = 1.5 years.
For educational purposes only; actual returns depend on taxes, fees, and market conditions.
How to Calculate Step by Step
Take a $5,000 investment at 6% compounded quarterly for 5 years:
- Convert r: 6% → 0.06.
- Divide r by n: 0.06 / 4 = 0.015.
- Multiply n by t: 4 × 5 = 20.
- Add 1 to the rate portion: 1 + 0.015 = 1.015.
- Raise to the power: 1.015^20 ≈ 1.346855.
- Multiply by P: 5,000 × 1.346855 = $6,734.28.
So the investment grows by $1,734.28 purely through compounding.
The calculator below automates this process. Enter any values to see the impact of different rates, frequencies, or timeframes.
The calculator above applies the standard CI formula. Provide the principal, annual rate, compounding frequency, and time. It instantly computes the future value and shows the total interest earned. You can adjust the compounding dropdown to compare monthly versus annual compounding effects.
Continuous Compounding: The Mathematical Limit
When compounding happens infinitely often, we use the continuous compound interest formula:
A = P e^(r t)
Here e (Euler’s number) ≈ 2.71828, and r and t are the same as before. This formula is common in theoretical finance, derivatives pricing, and certain bank products.
Example: A $2,000 deposit at 4% for 3 years with continuous compounding gives:
A = 2,000 × e^(0.04 × 3) = 2,000 × e^0.12 ≈ 2,000 × 1.1275 = $2,255.
The difference from daily compounding is small but measurable over longer horizons. For the same scenario with daily compounding (n=365), the future value would be $2,254.88–only $0.12 less.
Simple vs. Compound Interest
Simple interest formula: I = P r t. With simple interest, you earn exactly the same dollar amount each period.
Compare a $10,000 investment at 5% over 10 years:
| Type | Formula | Interest Earned | Final Balance |
|---|---|---|---|
| Simple | I = 10,000 × 0.05 × 10 | $5,000 | $15,000 |
| Compound (annually) | A = 10,000 (1+0.05)^10 | $6,288.95 | $16,288.95 |
The compound version yields an extra $1,288.95 because each year’s interest is calculated on a growing balance. With monthly compounding, the final amount rises further to $16,470.09–the power of frequency.
When to Use the CI Formula
- Savings accounts and CDs: To project how deposits grow given the APY.
- Investment returns: Estimate retirement fund growth assuming a fixed annual return.
- Inflation adjustment: Model how purchasing power erodes if money sits idle.
- Debt accumulation: Calculate unpaid credit card balances that charge compound interest on the outstanding amount (though most loans use amortization schedules, not pure compound growth).
For loan payments, the CI formula provides the balance if no payments are made. To find monthly installments, you need the annuity or amortization formula, which is a different tool.
Common Mistakes to Avoid
- Using the rate as a percentage: Always divide the advertised rate by 100 first. 3.75% → 0.0375.
- Mixing time units: If n is monthly (12), t must be in years, not months. For 18 months, t = 1.5.
- Forgetting the power order: Apply the exponent to the entire (1+r/n), not just (r/n).
- Ignoring compound frequency: An 8% annual rate compounded quarterly yields a higher return than 8% compounded annually. The effective annual rate (EAR) differs: EAR = (1 + r/n)^n – 1.
The CI formula remains a fundamental tool in personal finance and investment planning. Understanding its components gives you control over how you project growth, compare financial products, and make informed decisions.