Simple vs. Compound Interest: The Real Difference (With the Math)
Simple vs compound interest: simple interest grows in a straight line, compound interest curves upward. Full derivations, worked tables across decades and compounding frequencies, the Rule of 72, and how tax quietly erodes compounding.
Simple vs compound interest comes down to one question: does interest already earned earn interest of its own? Simple interest and compound interest start from the same basic idea, interest paid on money over time, and diverge completely in how that interest behaves afterward. Simple interest only ever applies to the original amount, so it grows the same fixed amount every period, forever. Compound interest lets each period's interest join the balance and start earning interest of its own, so it grows by more each period than the one before. Over a short term the two are nearly indistinguishable. Over decades, the gap between them becomes the entire story.
This guide works through both formulas from scratch, with worked numbers across five different timeframes and five different compounding frequencies, then covers the Rule of 72's real accuracy, where simple interest still shows up in practice, and exactly how much a tax rate costs beyond what a flat-percentage guess would suggest, the practical heart of any simple vs compound interest comparison. Every figure below is computed directly, not estimated, so the tables can be checked or rerun with the interest calculator this site builds.
Simple interest, from the formula up
Simple interest is defined by one short formula: interest equals principal times rate times time, or I = P × r × t, where r is the annual rate as a decimal and t is the number of years. Nothing about the formula changes based on how much interest has already accrued, which is exactly the point: the original principal is the only thing the calculation ever touches.
Take $1,000 at an 8% annual rate. After one year, the interest is $1,000 × 0.08 × 1 = $80, for a total of $1,080. After a second year, the interest is again $1,000 × 0.08 × 1 = $80, not 8% of $1,080, because simple interest never looks at the current balance, only the original $1,000. Extend that out and the pattern is exactly linear: $1,000 plus $80 for every year that passes, with no acceleration at any point.
Compound interest, from the formula up
Compound interest changes one part of that calculation: each period's interest is added to the balance before the next period's interest is calculated. The formula reflects that directly: A = P(1 + r/n)^(nt), where n is how many times per year interest compounds and the balance A already includes every period's interest folded back in.
Run the same $1,000 at 8%, compounded annually this time. Year one still adds $80, reaching $1,080, exactly like simple interest so far. Year two calculates 8% of $1,080 rather than 8% of $1,000, which is $86.40, for a total of $1,166.40. That extra $6.40 over what simple interest would have produced looks small on its own, but it is the entire mechanism: from this point on, every year's interest calculates against a slightly larger number than the year before, and that effect keeps stacking.
Simple vs compound interest: watching the two curves separate over decades
The clearest way to see the simple vs compound interest gap is to run them side by side over a long timeframe. Starting from the same $1,000 at 8%:
| Years | Simple interest total | Compound interest total | Compound's advantage |
|---|---|---|---|
| 5 | $1,400.00 | $1,469.33 | $69.33 |
| 10 | $1,800.00 | $2,158.92 | $358.92 |
| 20 | $2,600.00 | $4,660.96 | $2,060.96 |
| 30 | $3,400.00 | $10,062.66 | $6,662.66 |
| 40 | $4,200.00 | $21,724.52 | $17,524.52 |
Simple interest's column climbs in perfectly even $400 steps every 5 years, exactly what a straight line does. Compound interest's column does not just climb, it accelerates: the gain from year 30 to year 40 alone ($11,661.86) is larger than compound interest's entire total at year 20. This is the whole case for compound interest in one table: given enough time, the curve does not just beat the straight line, it leaves it behind by an order of magnitude.
How much compounding frequency actually matters
A separate question from how LONG interest compounds is how OFTEN it compounds within each year. Running $1,000 at 6% for 10 years at five different compounding frequencies shows the effect clearly:
| Compounding frequency | Ending balance |
|---|---|
| Annually | $1,790.85 |
| Semi-annually | $1,806.11 |
| Quarterly | $1,814.02 |
| Monthly | $1,819.40 |
| Daily | $1,822.03 |
Moving from annual to monthly compounding added $28.55 to the total. Moving from monthly all the way to daily, a much bigger jump in how often interest is calculated, added only $2.63 more. Each step toward more frequent compounding helps, but by a rapidly shrinking amount: the difference between compounding once a year and compounding twelve times a year is far larger than the difference between compounding twelve times a year and compounding 365 times a year, because there is a mathematical ceiling (continuous compounding) that more frequent compounding only ever approaches and never actually exceeds.
Why compounding frequency has a ceiling
The frequency table above shows the gains shrinking fast: annual to monthly gained $28.55, but monthly to daily only gained $2.63 more. That is not a coincidence of these particular numbers; it reflects a real mathematical ceiling called continuous compounding, the limit a nominal rate approaches as the compounding frequency increases without bound, calculated as A = Pe^(rt) rather than P(1 + r/n)^(nt). At 6% for one year, that ceiling caps the effective annual growth at about 6.18%, and daily compounding at 6.18% is already close enough to that ceiling that compounding even more often, hourly or by the minute, would add only a fraction of a cent more on a typical balance. This is also exactly why a bank's advertised Annual Percentage Yield, or APY, is usually a little higher than its stated nominal rate: APY already bakes the compounding frequency into a single effective number, so a 6% nominal rate compounded monthly is advertised as roughly a 6.17% APY, and the same rate compounded daily as roughly 6.18%, letting two accounts with different compounding schedules be compared on equal footing without needing to run the formula by hand.
The Rule of 72, and where its accuracy actually holds up
The Rule of 72 estimates doubling time at a given annual rate without running the full compound interest formula: divide 72 by the rate. Checking that estimate against the real, exact doubling time across a range of rates shows where the shortcut is trustworthy and where it starts to drift:
| Rate | Rule of 72 estimate | Actual doubling time | Error |
|---|---|---|---|
| 2% | 36.0 years | 35.0 years | 1.0 year |
| 4% | 18.0 years | 17.67 years | 0.33 years |
| 6% | 12.0 years | 11.9 years | 0.1 years |
| 8% | 9.0 years | 9.01 years | -0.01 years |
| 10% | 7.2 years | 7.27 years | -0.07 years |
| 12% | 6.0 years | 6.12 years | -0.12 years |
| 18% | 4.0 years | 4.19 years | -0.19 years |
| 24% | 3.0 years | 3.22 years | -0.22 years |
The shortcut is remarkably close right around 8%, off by only about a hundredth of a year, which is not a coincidence: 72 was chosen specifically because it divides evenly by many small numbers (2, 3, 4, 6, 8, 9, 12) while still landing close to the mathematically exact constant (which is closer to 69.3 for continuous compounding) across the range of rates most savings and investment planning actually uses. It drifts to about a year off at the low end, 2%, and grows steadily less accurate, though still usably close, as the rate climbs into the high teens and twenties.
Where simple interest still shows up in practice
Compound interest dominates savings accounts, certificates of deposit, and investment growth, anywhere money sits and grows for an extended period. Simple interest has not disappeared, though. Many auto loans, personal installment loans, and short-term or payday-style loans calculate interest on a simple basis across the loan's stated term. Some short-term bonds and promissory notes also quote a simple annual rate, since a term measured in months rather than years leaves little room for compounding to meaningfully diverge from it anyway.
A borrower's version of the earlier worked tables makes the practical upside of simple interest concrete. A $5,000 loan at 8% simple annual interest over a 3-year term accrues $1,200 in interest across the full term. Paying that same loan off after just one year instead accrues only $400, a saving of $800, because simple interest only ever counts the time the loan was actually outstanding, with no compounding penalty and no early-payoff complexity to work through. This is the one place the straight-line behavior that makes simple interest grow slower as a saver also makes it easier to reason about as a borrower: the interest owed scales in direct, predictable proportion to how long the money was borrowed, nothing more.
A brief, honest history
Interest on lent money is ancient, with lending and debt records going back thousands of years, long before anyone wrote down a formula distinguishing simple from compound. For much of history, many legal and religious traditions restricted or banned charging interest at all, which complicates any simple timeline of when compound interest specifically was first used versus simply tolerated. The now-famous line calling compound interest the eighth wonder of the world is commonly attributed to Albert Einstein, but no verified original source for Einstein actually saying or writing it has ever surfaced; it is one of the better-known examples of a finance quote whose popularity outran its documentation. The underlying point still holds regardless of who first said it, and the worked tables above make the case with real numbers rather than a borrowed quote.
The same exponential shape shows up well outside finance, which is part of why the formula itself is so well studied. Population growth under ideal conditions, radioactive decay running in reverse, and the spread of a contagious illness in its early stage all follow the same A = P(1 + r/n)^(nt)-style relationship compound interest does, just with a different meaning attached to each symbol. Money is simply the context most people encounter the formula in first, since a bank statement makes the effect visible and personal in a way an abstract growth curve in a textbook usually does not.
Tax: the part of the compounding story that gets underestimated
Every table so far assumed no tax on the interest earned, which is rarely true for a real, decades-long account. Applying a tax rate to $10,000 at 7% for 30 years, taxed on each year's interest as it is earned rather than on the final total, shows how much that assumption actually costs:
| Tax rate | Ending balance | Total tax paid |
|---|---|---|
| 0% | $76,122.55 | $0.00 |
| 15% | $56,627.69 | $8,228.42 |
| 25% | $46,415.51 | $12,138.50 |
| 35% | $37,994.53 | $15,073.98 |
At a 25% tax rate, the ending balance falls by $29,707.04 compared to the tax-free version, but the actual tax collected along the way only totals $12,138.50. The remaining $17,568.54 of the drop is not tax at all: it is the compounding that never got to happen on money that was taxed away in early years and therefore never had the chance to keep growing for the decades still remaining. This is the same mechanism that makes compound interest so powerful when it is working for you; taxed early enough and often enough, it works just as forcefully against you instead. A tax-advantaged account, one where interest is not taxed annually as it accrues, avoids this drag entirely, which is a meaningfully different outcome than simply paying a lower tax rate on the same taxable account.
APR vs APY: the same rate under two labels
APR is the stated annual rate before compounding, and APY includes the effect of compounding within the year, so APY is what a saver actually earns. The link between them is APY = (1 + APR/n)^n - 1. A 5% APR compounded daily works out to a 5.13% APY, compounded monthly it is 5.12%, and compounded once a year it is exactly 5%. When two accounts quote the same APR, the one that compounds more often pays slightly more, which is why banks must advertise the APY on savings so the comparison is fair. Loans work the other way around: lenders quote an APR that includes some fees, and the interest on a car loan or mortgage is charged on the declining balance rather than compounding on unpaid interest.
A worked comparison with monthly deposits
Most savers add money regularly instead of depositing one lump sum, so here is the same contrast with deposits. Take $200 at the end of every month for 20 years at 5%. The deposits total $48,000. If each deposit earned simple interest only for the time it sat in the account, the interest would total $23,900 and the pot would reach $71,900. With interest compounding monthly, the same deposits grow to $82,207, so compounding adds another $10,307 on top of simple interest. The deposits are identical in both cases; the whole gap comes from interest earning interest.
Checking the numbers in a spreadsheet
A spreadsheet reproduces every figure in this article. For a lump sum in cell A1, simple interest at rate r over t years is =A1*(1+rt) and annual compounding is =A1(1+r)^t. Monthly compounding is =A1*(1+r/12)^(12*t). For regular deposits, the FV function does the compounding for you: =FV(0.05/12, 240, -200) returns the $82,207 from the example above. To estimate doubling time, divide 72 by the percentage rate, which is the Rule of 72 covered earlier.
Running your own numbers
Every figure in this guide came from the same underlying compounding math the interest calculator uses, extended with a tax rate and an inflation-adjusted buying power figure the plain formulas above do not include on their own. Plug in a real starting amount, a real rate, and a real tax rate to see how the two forces, compounding and taxation, play out together for your own numbers rather than the ones worked through here.