Compound interest is the reason a savings balance grows faster in its twentieth year than in its first, even when the rate never changes. The worked tables below use the same month-by-month method as our calculator, so you can check every figure and then swap in your own. They cover one year done by hand, daily against monthly and annual compounding, regular deposits, a lump sum left for 30 years, and what taxes and inflation take. The safe investments hub lists the accounts and securities these numbers apply to.
How does compound interest work?
Compound interest pays interest on your earlier interest as well as on your deposits, so each period's earnings are a little larger than the last.
Investor.gov, the SEC's site for individual investors, defines it as "interest paid on principal and on accumulated interest." The alternative is simple interest, which is paid only on the original deposit. At 4.5% a year, $10,000 earning simple interest gains $450 every year and reaches $19,000 after 20 years. Compounded annually, the same deposit earns $450 in year one, $470.25 in year two and $1,038.54 in year 20, ending at $24,117.
Four inputs decide the result:
- The rate, which sets the percentage added each period
- The compounding frequency, which sets how often interest is added to the balance
- Time, which sets how many periods the growth runs
- Deposits and withdrawals, which change the base that earns interest
Of the four, frequency matters least and time matters most, as the tables below show.
What is the compound interest formula?
For a single deposit, the balance after t years is A = P × (1 + r ÷ n)^(n × t), where n is the number of compounding periods a year.
A = P × (1 + r ÷ n)^(n × t)
P is the starting deposit, r the annual rate as a decimal and n the periods a year: 1 for annual, 12 for monthly, 365 for daily. Regular deposits add a second piece. For a fixed deposit D at the end of every month, with a monthly rate i over m months:
Value of the deposits = D × ((1 + i)^m − 1) ÷ i
Our compound interest calculator combines the two. It converts any compounding frequency into an equivalent monthly rate, (1 + r ÷ n)^(n ÷ 12) − 1, applies it to the balance every month and adds each deposit at the end of the month. At 4.5% compounded daily, that equivalent monthly rate is about 0.3757%, compared with exactly 0.375% when interest is compounded monthly.
How do you calculate one year of compound interest by hand?
Work month by month: multiply the balance by the monthly rate, add that interest to the balance, and repeat with the new, larger balance.
Worked example: $10,000 at 4.5% compounded monthly for one year
The monthly rate is 4.5% ÷ 12 = 0.375%. Each month's interest is rounded to the cent and added to the balance, as a bank would credit it.
| Month | Interest | Balance |
|---|---|---|
| 1 | $37.50 | $10,037.50 |
| 2 | $37.64 | $10,075.14 |
| 3 | $37.78 | $10,112.92 |
| 4 | $37.92 | $10,150.84 |
| 5 | $38.07 | $10,188.91 |
| 6 | $38.21 | $10,227.12 |
| 7 | $38.35 | $10,265.47 |
| 8 | $38.50 | $10,303.97 |
| 9 | $38.64 | $10,342.61 |
| 10 | $38.78 | $10,381.39 |
| 11 | $38.93 | $10,420.32 |
| 12 | $39.08 | $10,459.40 |
- Total interest: $459.40, against $450.00 of simple interest, so compounding added $9.40
- Formula check: $10,000 × 1.00375^12 = $10,459.40
- Effective yearly rate: $459.40 ÷ $10,000 = 4.59%, which is this account's APY
Add $200 at the end of each month and the same steps give $37.50 of interest in month 1, $38.39 in month 2 and $47.48 in month 12, for $509.51 in the year and a balance of $12,909.51. The formulas give $10,459.40 for the deposit plus $2,450.12 for the twelve deposits, or $12,909.52; the extra cent comes from rounding each month. Converting the 4.59% yearly rate back into the rate that produced it is covered in APY vs APR.
How much does daily, monthly or annual compounding change the result?
On $10,000 at 4.5%, daily compounding beats annual compounding by about $153 over 10 years and $1,118 over 30 years, and beats monthly by far less.
| After | Annual | Monthly | Daily | Daily minus annual |
|---|---|---|---|---|
| 1 year | $10,450.00 | $10,459.40 | $10,460.25 | $10.25 |
| 5 years | $12,461.82 | $12,517.96 | $12,523.05 | $61.23 |
| 10 years | $15,529.69 | $15,669.93 | $15,682.69 | $152.99 |
| 20 years | $24,117.14 | $24,554.66 | $24,594.67 | $477.53 |
| 30 years | $37,453.18 | $38,476.98 | $38,571.05 | $1,117.86 |
| APY | 4.50% | 4.59% | 4.60% | 0.10 point |
The gap grows over time because the extra interest compounds too, but it stays small next to the effect of the rate. A 5% rate compounded only once a year turns the same $10,000 into $43,219 after 30 years, $4,648 more than 4.5% compounded daily. That is why comparing accounts by APY, which already folds in the compounding schedule, is enough for most decisions.
Daily compounding does not mean interest is paid daily. Many banks compound daily but credit interest monthly, and Regulation DD's APY formula assumes the interest stays in the account. If you withdraw the interest as it is paid, the balance stops compounding and you earn closer to the simple rate.
How do regular deposits and time change the outcome?
Regular deposits add to the base that earns interest, and time lets that interest build on itself, so the interest share of the balance keeps rising.
This table is computed from the inputs in its caption, with deposits at the end of each month:
| After | Total deposited | Interest earned | Balance |
|---|---|---|---|
| 1 year | $12,400 | $510 | $12,910 |
| 3 years | $17,200 | $1,936 | $19,136 |
| 5 years | $22,000 | $3,947 | $25,947 |
| 10 years | $34,000 | $11,910 | $45,910 |
| 15 years | $46,000 | $24,898 | $70,898 |
| 20 years | $58,000 | $44,180 | $102,180 |
Read down the interest column rather than the balance. Interest is about 15% of the balance after 5 years, 26% after 10 years and 43% after 20 years. The last five years alone add about $19,281 of interest, well above the $11,910 earned in the first ten.
Deposits on their own tell the same story. Here is $200 a month at 4.5% compounded monthly with no starting balance:
| Years of deposits | Total deposited | Interest earned | Balance |
|---|---|---|---|
| 10 | $24,000 | $6,240 | $30,240 |
| 20 | $48,000 | $29,625 | $77,625 |
| 30 | $72,000 | $79,877 | $151,877 |
| 40 | $96,000 | $172,230 | $268,230 |
Doubling the time from 20 to 40 years doubles the deposits but multiplies the balance by about 3.5. Starting earlier can also rival saving more: $200 a month for 30 years ends near $151,877 on $72,000 of deposits, close to the $155,250 that $400 a month reaches in 20 years on $96,000 of deposits. Timing within the month matters far less. Making each $200 deposit at the start of the month instead of the end adds only about $291 over 20 years.
How does a single deposit grow when you add nothing?
A lump sum left alone grows by the same percentage each year, so the dollar gains get bigger every decade even though the rate never changes.
This second table starts with $25,000, adds nothing and compounds daily at 4%, roughly where a CD or a high-yield savings account might sit:
| After | Total deposited | Interest earned | Balance |
|---|---|---|---|
| 1 year | $25,000 | $1,020 | $26,020 |
| 3 years | $25,000 | $3,187 | $28,187 |
| 5 years | $25,000 | $5,535 | $30,535 |
| 10 years | $25,000 | $12,295 | $37,295 |
| 15 years | $25,000 | $20,551 | $45,551 |
| 20 years | $25,000 | $30,636 | $55,636 |
| 25 years | $25,000 | $42,953 | $67,953 |
| 30 years | $25,000 | $57,997 | $82,997 |
The first decade adds about $12,295, the second $18,341 and the third $27,361, more than double the first. The balance passes $50,000, twice the deposit, after about 17.3 years, because the 4% rate compounded daily is an APY of 4.08%. The rule of 72 estimates that doubling time without a calculator: 72 ÷ 4.08 = 17.6 years.
The 4% is held constant only to show the mechanics. A real 30-year path would involve many rates: a CD fixes the rate only until maturity, a T-bill ladder resets as each bill comes due, and a savings account can change its rate at any time. A Treasury bill pays its interest once, at maturity, so it compounds only when you reinvest the proceeds.
How do you work out the time or savings a goal needs?
Run the formula backwards: logarithms give the years, a root gives the rate, and dividing by the deposit factor gives the monthly saving.
Years = ln(target ÷ deposit) ÷ (n × ln(1 + r ÷ n))
Rate = n × ((target ÷ deposit)^(1 ÷ (n × years)) − 1)
Monthly deposit = (target − starting balance × (1 + i)^m) × i ÷ ((1 + i)^m − 1)
Worked example: Three goal questions, compounded monthly
- How long does $10,000 take to reach $15,000 at 4.5%? ln(1.5) ÷ (12 × ln(1.00375)) = 0.4055 ÷ 0.0449 = 9.03 years
- What rate turns $10,000 into $15,000 in 8 years? 12 × (1.5^(1 ÷ 96) − 1) = 5.08%, an APY of 5.20%
- What monthly deposit builds $50,000 in 10 years at 4.5%, starting from zero? 1.00375^120 = 1.5670, so $50,000 × 0.00375 ÷ 0.5670 = $330.69
- The same goal with $10,000 already saved: the $10,000 grows to $15,669.93 by itself, leaving $34,330.07 to build, which takes $227.05 a month
Plug any of these answers into the calculator to confirm the final balance. The time formula with a target of twice the deposit gives the exact doubling time, which the rule of 72 approximates.
How do taxes and inflation reduce compounding?
Taxes skim each year's interest before it can compound, and inflation shrinks what the final balance buys, so the after-tax real growth rate is much lower.
Worked example: $10,000 at 4.5% for 20 years, before and after tax and inflation
Assume annual compounding, a 22% federal tax rate (it applies to 2026 taxable income over $50,400 for single filers and $100,800 for married couples filing jointly) and 3% inflation a year.
- No tax: $10,000 × 1.045^20 = $24,117
- Taxed every year: the rate drops to 4.5% × 0.78 = 3.51%, and $10,000 × 1.0351^20 = $19,936
- Tax deferred, then taxed once at 22% at the end: $10,000 + 0.78 × $14,117 = $21,011
- Prices after 20 years of 3% inflation: 1.03^20 = 1.8061 times today's level
- In today's dollars: $24,117 ÷ 1.8061 = $13,353 with no tax, and $19,936 ÷ 1.8061 = $11,038 taxed yearly
- After-tax real growth rate: 1.0351 ÷ 1.03 − 1 = 0.50% a year
Over 30 years the yearly-taxed balance reaches $28,149, which buys only about $11,597 of today's goods and services.
Deferral helps because the money that would have gone to tax keeps compounding, which is the main appeal of tax-advantaged retirement accounts. Tax rules and brackets change, and a large withdrawal can fall in a higher bracket than 22%, so check the details with a tax professional. The real returns guide shows how to adjust any rate for inflation with the exact formula, using the latest CPI data.
What are the risks of counting on compound growth?
The biggest risk is assuming the rate stays fixed, because compounding magnifies every change in the rate over long periods, whether up or down.
- Rates change. Run the first table at 3.5% instead of 4.5% and the 20-year balance falls from $102,180 to about $89,491, a gap of $12,689. Savings rates are variable, and bills and CDs expose you to reinvestment risk when they mature.
- Fees compound too. A 0.5% yearly fee on a 4.5% return leaves 4.0%, and over 30 years that turns $10,000 into $32,434 instead of $37,453.
- Debt compounds against you. A 24% credit card APR charged daily doubles an unpaid balance in about 2.9 years. APY vs APR shows how lenders quote that cost.
- Market returns are not smooth. A steady rate is a feature of deposits and bills. Stocks and bond funds can lose value in some years, and a loss shrinks the base that later gains compound on.
- Insurance has limits. Compounding only helps if the money is safe: FDIC insurance covers $250,000 per depositor, per insured bank, per ownership category, explained in the FDIC insurance guide, and Treasury securities are backed by the full faith and credit of the US government.
- Withdrawals stop the engine. Taking out interest or principal early removes the base that later growth depends on, and a CD may charge a penalty for it.
Projections like the tables above are arithmetic, not forecasts. They show what a given rate would do if it held.