Compound interest is interest earned on interest. Instead of paying a return only on the original deposit, each period's interest is added to the balance and earns a return in the next period. Over short horizons the difference is small. Over decades it dominates everything else.
This guide sets out the formula, works through it by hand, shows what compounding frequency actually changes, and covers regular contributions and inflation adjustment.
Reproduce every figure with the Compound Interest Calculator.
The Compound Interest Formula
Where:
- A — final amount, principal plus interest
- P — initial principal
- r — annual interest rate as a decimal (7 percent = 0.07)
- n — compounding periods per year
- t — time in years
Interest earned is simply A − P.
The formula assumes a constant rate, no withdrawals, no taxes and no fees.
Worked Example 1: Annual Compounding
Interest earned = ₹96,715.
Simple interest on the same deposit would be 1,00,000 × 0.07 × 10 = ₹70,000. Compounding adds ₹26,715 without any extra deposit.
Worked Example 2: The First Few Years, Step by Step
Watching the balance build makes the mechanism obvious.
| Year | Opening balance | Interest at 7 % | Closing balance |
|---|---|---|---|
| 1 | 1,00,000 | 7,000 | 1,07,000 |
| 2 | 1,07,000 | 7,490 | 1,14,490 |
| 3 | 1,14,490 | 8,014 | 1,22,504 |
| 4 | 1,22,504 | 8,575 | 1,31,080 |
| 5 | 1,31,080 | 9,176 | 1,40,255 |
The rate never changes. The interest amount rises every year because the base it applies to keeps growing. By year ten the annual interest is over ₹12,800 — almost double the first year's.
Worked Example 3: Compounding Frequency
Same ₹1,00,000 at 7 percent for 10 years, varying n:
| Compounding | n | Final amount | Effective annual rate |
|---|---|---|---|
| Annually | 1 | ₹1,96,715 | 7.000 % |
| Half-yearly | 2 | ₹1,98,979 | 7.123 % |
| Quarterly | 4 | ₹2,00,160 | 7.186 % |
| Monthly | 12 | ₹2,00,966 | 7.229 % |
| Daily | 365 | ₹2,01,360 | 7.250 % |
| Continuous | ∞ | ₹2,01,375 | 7.251 % |
Two things are worth noticing. Frequency does help — but the gain from annual to monthly is about 2 percent of the final balance, while the gain from monthly to daily is negligible. Beyond monthly, frequency is a marketing detail rather than a financial one.
Effective annual rate
To compare products quoted at different frequencies, convert to the effective annual rate:
Always compare EARs, never nominal rates.
Adding Regular Contributions
Most real saving is a monthly deposit rather than a single lump sum. The future value of a series of equal end-of-period payments is:
Where PMT is the payment per period, i the rate per period, and m the number of payments.
Worked example
Total contributed = 10,000 × 180 = ₹18,00,000. Growth accounts for ₹23,44,704 — more than the contributions themselves.
If both a lump sum and monthly contributions exist, calculate each separately and add the results.
The Rule of 72
A quick mental estimate of doubling time:
Years to double ≈ 72 ÷ annual interest rate in percent
At 8 percent: 72 ÷ 8 = 9 years. The exact figure from the formula is 9.006 years.
The approximation is accurate to within a few months for rates between roughly 4 and 12 percent, and drifts at higher rates.
Time Matters More Than Rate
Two savers, both investing ₹1,00,000 once at 9 percent:
| Saver | Years invested | Final amount |
|---|---|---|
| Starts at 25, stops at 65 | 40 | ₹31,40,942 |
| Starts at 35, stops at 65 | 30 | ₹13,26,768 |
A ten-year head start — a quarter of the time — produces more than double the outcome. Because growth is exponential, the last years contribute the largest absolute gains, which is exactly why an early start cannot be recovered later by contributing more.
Inflation: The Real Return
Nominal growth overstates purchasing power. The real rate is:
Subtracting the two rates gives 3 percent, close enough for a rough estimate but consistently slightly optimistic. Use the Inflation Calculator to express a future balance in today's money.
Assumptions and Limitations
- A constant rate for the whole period. Market returns are not constant, and an average return does not produce the same result as a steady one.
- No taxes. Interest and capital gains are taxable in most jurisdictions, and tax drag compounds too.
- No fees. An annual expense ratio of 1 percent reduces a 9 percent return to 8 percent, which over 30 years costs roughly a quarter of the final balance.
- No withdrawals, and contributions made exactly on schedule.
These calculations are educational, not investment advice or a projection of any specific product's returns.
Common Mistakes
- Entering 7 instead of 0.07 for the rate.
- Using years for the exponent when compounding monthly — the exponent is n × t, not t.
- Comparing a monthly-compounded product with an annually-compounded one using nominal rates.
- Confusing the annual rate with the periodic rate in the contributions formula.
- Forgetting that the same compounding works against you on outstanding debt.
Related Calculations
- Compound Interest Calculator — any principal, rate, frequency and term
- Simple Interest Calculator — the flat-interest comparison
- SIP Calculator — monthly contribution growth
- CAGR Calculator — the compound rate implied by a past result
- Inflation Calculator — future value in today's money
Frequently Asked Questions
What is the compound interest formula?
A = P × (1 + r/n)^(n × t), where r is the annual rate as a decimal, n the compounding periods per year and t the number of years.
Does daily compounding make a meaningful difference?
Rarely. At 7 percent over 10 years, daily compounding beats monthly by about 0.2 percent of the final balance. The rate and the time horizon matter far more.
How do I compare two rates with different compounding?
Convert both to the effective annual rate using EAR = (1 + r/n)^n − 1, then compare.
Does compound interest apply to loans?
Yes. Credit-card balances and unpaid interest compound against you, which is why a revolving balance grows so quickly.
What is a realistic long-term return assumption?
There is no single correct figure, and no honest one can be guaranteed. Model a range — a conservative case and an optimistic case — rather than a single number.
Summary
Compound interest rewards time far more than it rewards frequency. Get the periodic rate and the exponent right, compare products on their effective annual rate, adjust for inflation and fees, and start as early as the money allows. Run your own scenarios on the Compound Interest Calculator.