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What is compound interest?

Compound interest is interest calculated on the initial principal and also on the accumulated interest from previous periods. Unlike simple interest, which is calculated only on the principal, compound interest allows your investment to grow exponentially over time.

How is compound interest calculated?

The formula is A = P(1 + r/n)^(nt), where P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years. When regular contributions are included, each contribution compounds separately from the date it's added.

The Math, Worked Out

The classic formula:

A = P × (1 + r/n)^(nt)

• A = final amount
• P = principal (initial deposit)
• r = annual interest rate as a decimal (7% = 0.07)
• n = compounding periods per year (12 for monthly, 365 for daily)
• t = number of years

When you add regular contributions, each one starts compounding from the moment it's deposited. The future value of those contributions is calculated separately and added to the principal's growth: PMT × [((1+r/n)^(nt) − 1) / (r/n)]. The calculator does both pieces and shows the combined future value, total contributions paid in, and total interest earned.

Simple vs Compound: A Concrete Example

Take $10,000 at 5% for 30 years:

• Simple interest (P × r × t): $10,000 × 0.05 × 30 = $15,000 in interest. Final: $25,000.
• Compound interest (annual): $10,000 × (1.05)³⁰ = $43,219. The interest portion alone is $33,219.

Same starting capital, same rate, same time, but compound earns more than double the interest. This gap widens dramatically as the time horizon grows. At 50 years it's $35k vs $115k. At 70 years it's $45k vs $295k. Time and compounding multiply each other.

Why Compounding Frequency Matters Less Than You'd Think

The intuition that "daily compounding is much better than annual" is mostly wrong. With a 7% annual rate over 30 years on a $10,000 principal:

The jump from annual to monthly is meaningful (~6%); after that the gains diminish quickly. Daily and continuous are essentially identical for practical purposes. Continuous compounding uses the elegant formula A = P × e^rt. Jacob Bernoulli discovered the constant e (~2.71828) in 1683 while studying exactly this problem.

The Rule of 72 (And When to Trust It)

A famous mental shortcut: years to double your money ≈ 72 ÷ annual rate %.

• At 6%: 72 / 6 = 12 years to double
• At 8%: 72 / 8 = 9 years
• At 10%: 72 / 10 ≈ 7.2 years
• At 4%: 72 / 4 = 18 years

It's an approximation, accurate to within ~1 year for rates between 4% and 12%. At very high rates (say 25%) the rule starts to under-estimate the actual doubling time; at very low rates (1-2%) the error is small but noticeable. Useful for back-of-the-envelope sanity checks; the calculator above gives you the exact answer.

Where People Use This in Real Life

• Savings accounts, typically the lowest rates (~4 to 5% APY in 2026 high-yield accounts; far less in standard accounts). Daily compounding, but rate matters more than frequency.
• CDs and bonds, fixed-rate, compounded periodically (typically semi-annually for US Treasury bonds).
• Retirement accounts, 401(k), traditional IRA, Roth IRA in the US; ISAs in the UK; RRSPs in Canada. Decades of compounding turn modest contributions into substantial wealth.
• Index funds and stock investing, compounding through reinvested dividends and capital gains. The S&P 500's long-term annualised total return is roughly 10% nominal, ~7% after inflation, depending on the start year you measure from.
• Education planning, 529 plans for college savings, with 18 years of compounding from birth.
• Long-horizon goals, paying off a mortgage early via extra principal payments, building a down payment, planning a sabbatical.

The Three Things This Calculator Doesn't Show You

The headline future value is honest math, but the real number you'll spend in retirement (or wherever) is lower because of three factors a basic calculator can't model:

• Inflation. A 7% nominal return with 3% inflation is closer to 4% in real (inflation-adjusted) terms. The Fisher equation: real rate ≈ nominal rate − inflation. Over 30 years, that's a huge difference. To see real returns, mentally subtract your inflation expectation (3% is a reasonable long-term US assumption) from the nominal rate before running the calculation.
• Taxes. US 401(k) and traditional IRA: tax-deferred until withdrawal. Roth IRA: post-tax contributions, tax-free withdrawals. Taxable brokerage: dividends and realised gains taxed each year, which drags on the effective compounding rate (often 0.5 to 1.5% per year of drag depending on tax rate and turnover). UK ISAs are tax-free; outside an ISA, income tax and capital gains tax apply.
• Fees. A 1% expense ratio over 30 years can cost 25%+ of total returns. Index funds at 0.03 to 0.10% expense ratios are the right choice for most long-term investors; actively-managed funds at 1%+ are usually a worse deal even when they outperform briefly.

Why Starting Early Matters More Than You'd Guess

The classic illustration: two investors, both contribute $200/month to an account earning 7% annual, both stop contributing at age 65.

• Early Edith contributes from age 25 to 65 (40 years, $96,000 total contributed).
• Late Larry contributes from age 35 to 65 (30 years, $72,000 total contributed).

At age 65, Edith ends with roughly $525,000; Larry ends with roughly $245,000. Larry only put in $24,000 less than Edith but finishes with less than half her balance. The difference is that Edith's earliest contributions had four full decades to compound, while Larry's earliest contributions had only three. Compounding rewards time more than it rewards amount, especially over long horizons.

Common Mistakes

1. Confusing the nominal rate with APY. APY (Annual Percentage Yield) accounts for compounding frequency; the nominal rate doesn't. A 6% nominal rate compounded monthly produces an APY closer to 6.17%. Banks usually advertise APY for deposits and APR for loans, not always consistently.
2. Ignoring inflation. A "7% return" is misleading without context. After typical 3% inflation, the real purchasing-power return is closer to 4%. Plan in real terms for retirement.
3. Modelling unrealistic returns. The S&P 500's long-term real return is around 7%, not 12%, not 15%. Plans that assume 15% returns are setting up disappointment.
4. Forgetting fees. A 1% fund expense ratio compounds against you the same way returns compound for you. Over 30 years, 1% can cost a quarter of your portfolio.
5. Underestimating tax drag. Outside tax-advantaged accounts, dividend and gain taxation each year reduces effective compounding by half a percent or more depending on your bracket.
6. Confusing dollar weight with time weight. "Saving $1,000 a month for the last decade is better than saving $200 for the last three" is true on contributions, but the compounding bonus on early money is dramatic.
7. Treating predictable savings interest the same as variable stock returns. A CD's 5% is guaranteed; the S&P 500's average 10% is the long-run average through brutal swings. Over short horizons, stocks can return -40% in a year.
8. Quoting Einstein on compound interest. The famous "eighth wonder of the world" quote is widely attributed to him but has no source. Earliest known print citation is 1983, well after his death. The math is real even when the attribution isn't.

More Frequently Asked Questions

What's a realistic interest rate to use?

Depends on the account. In 2026: top high-yield US savings accounts pay around 4 to 5% APY; CDs in similar range; US 10-year Treasuries 3 to 5%; the S&P 500's long-run nominal annualised return is around 10% (about 7% real after inflation), but with significant year-to-year variance. Plug in conservative numbers if the calculator output drives a real-life decision; the math doesn't lie about the future, but the input rate is an assumption you're making.

Does the calculator include taxes or inflation?

No. The result is the nominal future value before taxes and inflation. To see the real (inflation-adjusted) value, run the calculation with the rate reduced by your expected inflation rate (e.g. 7% nominal − 3% inflation = 4% real). For tax-advantaged accounts (401k, IRA, ISA), the nominal calculation is closer to your actual outcome; for taxable accounts, expect 10 to 25% drag depending on your tax bracket and asset mix.

When are contributions assumed to be made?

The calculator uses end-of-month contributions, the prevailing convention for retirement-account projections. The difference between "start" and "end" of period is small over short periods but compounds noticeably over decades; beginning-of-period would credit one extra month's interest per year on each contribution.

Are my financial figures sent anywhere?

No. The calculation runs entirely in your browser. Principal, monthly contribution, rate, time, and the resulting projection are computed locally. Nothing is uploaded; no analytics endpoint sees the values; no marketing list captures your inputs. Many bank-branded calculators monetise by capturing exactly this kind of demographic data.

What's the difference between continuous and daily compounding?

Continuous compounding is the mathematical limit as the compounding interval shrinks toward zero: A = P × e^rt, where e ≈ 2.71828. Daily compounding (n = 365) is so close to continuous that the difference rarely matters: about a hundredth of a percent extra over 30 years on a $10k principal at 7%. Both are essentially the "maximum" compounding rates for practical purposes.

Can I model a one-time deposit and ongoing contributions together?

Yes, that's the default mode. The "Initial Investment" field is your starting balance; the "Monthly Contribution" field is what you add each month after that. Set Monthly Contribution to 0 if you only want to model the lump sum; set Initial Investment to 0 if you're starting from nothing and only modelling regular contributions.

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