Compound Interest Formula
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Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| $A$ | Final amount | currency | Total value after interest is applied. |
| $P$ | Principal | currency | Initial investment or loan amount. |
| $r$ | Annual interest rate | decimal | Annual rate as a decimal (e.g., 5% = 0.05). |
| $n$ | Compounding frequency | times/year | Number of times interest is compounded per year. |
| $t$ | Time | years | Duration of the investment in years. |
What Is Compound Interest?
Compound interest is interest calculated not just on the original principal, but also on the interest that has already accumulated. This creates an exponential growth curve rather than the linear growth of simple interest:
$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
The Power of Compounding
The dramatic effect of compounding becomes apparent over long time periods. Consider $10,000 invested at 7% annual return:
| Years | Simple Interest | Compound (Annual) |
|---|---|---|
| 10 | $17,000 | $19,672 |
| 20 | $24,000 | $38,697 |
| 30 | $31,000 | $76,123 |
After 30 years, compound interest yields 2.45× the simple-interest result.
Compounding Frequency
The more frequently interest compounds, the faster the growth:
| Frequency | n | $10,000 at 6%, 10 years |
|---|---|---|
| Annual | 1 | $17,908 |
| Semi-annual | 2 | $18,061 |
| Monthly | 12 | $18,194 |
| Daily | 365 | $18,220 |
The difference between monthly and daily compounding is small; the biggest jump is from annual to more frequent intervals.
Einstein's "Eighth Wonder"
The compound interest formula is often attributed a famous quote: "Compound interest is the eighth wonder of the world. He who understands it, earns it; he who doesn't, pays it." While attribution to Einstein is disputed, the sentiment captures the asymmetric power this formula gives patient investors over time.
How to Use This Formula
- Convert the annual rate to decimal: r = rate% ÷ 100
- Divide r by compounding frequency n
- Multiply n × t for the exponent
- Raise the bracket to that power
- Multiply by principal P
The result A includes both principal and interest. Subtract P to isolate the interest earned: Interest = A − P.
Derivation & History
Compound interest was documented by Babylonian mathematicians as early as 2000 BCE in cuneiform tablets describing grain loans that "grew." The mathematical formulation evolved through the Italian merchant banking tradition of the 13th–16th centuries, formalized by Luca Pacioli in Summa de arithmetica (1494).
The modern formula A = P(1 + r/n)^(nt) emerged from the development of calculus in the 17th century. Jacob Bernoulli investigated the limiting case as n → ∞ in 1683, discovering the mathematical constant e ≈ 2.71828, which underpins continuous compounding. The formula as used today was standardised in actuarial mathematics during the 18th century.
Worked Examples
Annual compounding, 5 years
- A = 5000 × (1 + 0.06/1)^(1×5)
- A = 5000 × (1.06)^5
- A = 5000 × 1.33823
- A = $6,691.13
Result: Final amount: $6,691.13 (interest earned: $1,691.13)
Monthly compounding, 10 years
- A = 10000 × (1 + 0.05/12)^(12×10)
- A = 10000 × (1.004167)^120
- A = 10000 × 1.64701
- A = $16,470.09
Result: Final amount: $16,470.09 (interest earned: $6,470.09)
Edge Cases & Limitations
Negative interest rates: Substitute r < 0 to model depreciation or deflationary environments; A will be less than P.
Very high n (daily vs continuous): The difference between daily (n=365) and continuous compounding is negligible (<0.01%) for most practical rates.
Inflation adjustment: Nominal compound interest does not account for purchasing-power erosion; use the real return formula to adjust.
Loan context: When this formula applies to a loan, A represents the total repayable amount — use EMI or loan balance formulas for instalment planning.
Real-World Applications
Bank savings accounts, fixed deposits, government bonds, and equity investments all grow via compound interest. Retirement fund projections use this formula to show the long-term value of consistent contributions. Central banks use compound growth models in monetary policy decisions. Credit card balances compound at high monthly rates — the same formula explains why carrying a balance is expensive. Warren Buffett's wealth accumulation is often cited as the most famous real-world demonstration of long-term compounding.