Compound interest, worked through slowly
The same deposit at different frequencies and time horizons, with the arithmetic shown.
The short answer
Compound interest is interest paid on interest: each period's earnings join the balance, and the next period's interest is worked out on the larger figure, where simple interest keeps adding the same amount to the original sum. The gap is almost invisible for a year or two, then widens sharply as compounding grows the base it is applied to. The same arithmetic runs in reverse on borrowing, which is why an unpaid balance grows on its own.
Simple interest, first
Simple interest is worked out on the amount originally deposited or borrowed — the principal — and never on the interest that accumulates. Put $1,000 into an account paying 5% simple interest a year and it earns $50 in year one and $50 in every year after that. The base of the calculation never moves.
The arithmetic is a multiplication: principal, times the rate, times the number of periods. After ten years that is $1,000 × 0.05 × 10, or $500 of interest, for a balance of $1,500. The growth is a straight line, with the same slope in year thirty as in year one.
Compound interest changes one thing: the interest is credited to the balance at the end of each period, so the next calculation runs on a larger figure. That turns the straight line into a curve.
The first six years, step by step
Take the same $1,000 at 5%, compounded once a year: each year's interest is 5% of the balance at the start of that year.
Year one earns 5% of $1,000, which is $50, and the balance becomes $1,050. Year two earns 5% of $1,050, which is $52.50 — the extra $2.50 is 5% of the $50 credited in year one. Each year's interest is slightly larger than the last, not because the rate changed but because the balance did.
| Year | Balance at start | Interest at 5% | Balance at end | Simple interest balance |
|---|---|---|---|---|
| 1 | $1,000.00 | $50.00 | $1,050.00 | $1,050.00 |
| 2 | $1,050.00 | $52.50 | $1,102.50 | $1,100.00 |
| 3 | $1,102.50 | $55.13 | $1,157.63 | $1,150.00 |
| 4 | $1,157.63 | $57.88 | $1,215.51 | $1,200.00 |
| 5 | $1,215.51 | $60.78 | $1,276.28 | $1,250.00 |
| 6 | $1,276.28 | $63.81 | $1,340.10 | $1,300.00 |
After six years the compound balance is $1,340.10 against $1,300 simple. The difference of $40.10 is a rounding error in most budgets, and a fair illustration of why compounding looks overstated to anyone checking over a short period.
Extend both calculations to forty years and the compound balance is $7,039.99 against $3,000 simple. The straight line has added $2,000 of interest; the curve has added just over $6,000, at the same rate on the same deposit. The only difference is that one calculation was allowed to work on its own output.
The shorthand for the curve is a power rather than a multiplication: the balance is the principal multiplied by one plus the rate, raised to the number of periods — $1,000 × 1.05 to the power of 40 for forty years. The exponent does the work, and exponents behave in ways intuition underestimates.
What compounding frequency changes
Compounding frequency is how often accrued interest is added to the balance and starts earning in its own right. An account can quote 5% a year and credit it annually, half-yearly, quarterly, monthly or daily. The quoted figure is a nominal annual rate; the frequency decides what a year of it actually produces.
If 5% a year is credited monthly, each month adds 5% divided by twelve, or about 0.4167%, on the balance including all previous months' interest. Over twelve months that compounds to slightly more than 5%.
| Compounded | After one year | Effective annual rate | After 10 years | After 30 years |
|---|---|---|---|---|
| Annually | $1,050.00 | 5.000% | $1,628.89 | $4,321.94 |
| Half-yearly | $1,050.62 | 5.062% | $1,638.62 | $4,399.79 |
| Quarterly | $1,050.95 | 5.095% | $1,643.62 | $4,440.21 |
| Monthly | $1,051.16 | 5.116% | $1,647.01 | $4,467.74 |
| Daily | $1,051.27 | 5.127% | $1,648.66 | $4,481.23 |
Frequency matters, but far less than the phrase "daily compounding" implies: on a nominal 5%, moving from annual to daily adds about 0.13 percentage points to the effective rate. And there is a ceiling — compounding a nominal 5% hourly, or every second, produces almost exactly the daily figure, because the sequence converges.
This is why savings accounts are commonly quoted with an annual equivalent rate beside the nominal one: it shows what a year of that rate at that frequency comes to, putting different frequencies on one basis.
A rough check on compounding: divide 72 by the annual percentage rate and the result approximates the years needed for a balance to double. At 5% that suggests 14.4 years against an exact 14.2; at 3%, 24 years against about 23.5. It drifts at high rates but is close enough to test any claim about doubling.
Time set against rate
Because the number of periods sits in the exponent, time and rate are not interchangeable levers: both raise the outcome, but not in the same proportion.
| Rate | 10 years | 20 years | 30 years | 40 years |
|---|---|---|---|---|
| 3% | $1,343.92 | $1,806.11 | $2,427.26 | $3,262.04 |
| 5% | $1,628.89 | $2,653.30 | $4,321.94 | $7,039.99 |
| 7% | $1,967.15 | $3,869.68 | $7,612.26 | $14,974.46 |
Read the table diagonally and the trade-off is concrete. Twenty years at 7% ($3,869.68) beats forty years at 3% ($3,262.04): the higher rate wins despite half the time. But thirty years at 5% ($4,321.94) beats twenty years at 7%: the extra decade wins despite the lower rate. Neither lever dominates in the abstract.
What is reliable is where the growth arrives. Of the roughly $6,040 of interest earned over forty years at 5%, about $629 accrues in the first ten years and about $2,718 in the final ten — on the same money at the same rate. The curve is flat where people tend to watch it and steep where they have stopped looking.
Adding to the balance each month
Most saving is not one deposit left alone but a smaller sum added regularly, each deposit compounding for however long it has left: the first for the whole term, the last for a month.
Take $100 a month into an account paying a nominal 5% credited monthly. After ten years, $12,000 paid in and a balance of about $15,528. After twenty, $24,000 and about $41,103. After thirty, $36,000 and about $83,226 — at which point the interest of roughly $47,226 exceeds every deposit made.
The same figures show what a delay costs. Starting a decade later, so twenty years of identical deposits rather than thirty, leaves roughly half the balance having paid in two-thirds of the money. The missing contributions are not the main loss; the lost compounding on the earliest deposits is.
The same mechanism, on debt
Interest charged on a balance that includes unpaid interest compounds identically, with the direction reversed. Revolving credit shows this most plainly, because interest is quoted per month and added monthly.
Take $2,000 on a card charging 1.5% a month. Twelve times 1.5% is a nominal 18% a year, but compounding makes the effective annual rate about 19.56%: $1,000 left untouched for a year at 1.5% a month grows to $1,195.62, not $1,180. Month one adds $30 of interest, taking the balance to $2,030. If the minimum payment is 2% of the balance subject to a $25 floor — a common shape, though formulas vary by provider — that month's minimum is $40.60, of which $30 covers the interest just added and $10.60 reduces what is owed.
Because the minimum is a percentage of a shrinking balance, it falls as the balance does, and the share going to interest stays high for years.
| Monthly payment | Time to clear | Total paid | Interest paid |
|---|---|---|---|
| Minimum only (2%, $25 floor) | about 15 years | about $5,198 | about $3,198 |
| $50 fixed | about 5 years 2 months | about $3,077 | about $1,077 |
| $75 fixed | about 2 years 11 months | about $2,573 | about $573 |
| $100 fixed | about 2 years | about $2,396 | about $396 |
Doubling the payment from $50 to $100 does not halve the interest; it cuts it by roughly two-thirds, because it removes both the balance and the time it compounds over. Rates on borrowing are typically a multiple of the rates available on cash savings, so a debt usually compounds faster than a deposit of the same size. Where both exist at once, which one to feed depends on the two rates, any tax or employer treatment attached to the savings, and how much accessible cash a household needs to keep.
Compounding also works on prices. If costs rise a few per cent a year, the buying power of a fixed sum shrinks by the same exponential arithmetic in reverse. A balance growing at a nominal rate close to the rate at which prices rise may be roughly flat in real terms, even though the statement shows a larger number each year.
What this means in practice
The arithmetic above applies to any balance where interest is added to interest, at any frequency, in either direction. A few things follow.
- Short checks understate it. Over a year or two, compound and simple interest are nearly identical, so judging compounding by one statement makes it look trivial.
- Periods and rate are not equivalent. Time enters as an exponent, so a longer horizon can outweigh a higher rate, or fail to. Only the actual numbers settle it.
- Frequency is a small effect with a ceiling. An effective or annual equivalent rate is the figure that makes different frequencies comparable.
- Early deposits do most of the work. So a delay costs more than the deposits missed during it.
- On debt the curve runs against the balance. A percentage-based minimum can stretch a modest balance over many years, with most of each early payment covering interest.
- Nominal is not real. A balance can grow in cash terms while standing still in buying power.
Every figure here is arithmetic on chosen round rates, shown to illustrate the mechanism rather than describe any account or agreement. Real products differ in how interest is calculated and credited, whether it is taxed, what fees apply and how minimums are set, and published rate ranges move over time. A specific decision — particularly one setting savings against debt repayment — turns on details this arithmetic leaves out, and is properly discussed with a qualified professional who knows the full circumstances.
