Maths SolverThe textbook explanation of simple vs compound interest goes something like: simple interest is calculated on the original amount; compound interest is calculated on the original amount plus any interest already accrued. End of explanation. Move on to the formulas.
This is correct, but it understates how much the two methods diverge over time. The difference between the two formulas looks small algebraically and is dramatic numerically. The whole point of this post is to show what that difference actually looks like, year by year, with real numbers.
The formulas, briefly
Simple interest:
$$A = P(1 + rt).$$
Compound interest, compounded once a year:
$$A = P(1 + r)^t.$$
$P$ is the principal (the starting amount), $r$ is the annual interest rate as a decimal, and $t$ is the time in years. Same inputs, very different formulas: simple is linear in time, compound is exponential in time.
$\pounds 1000$ at $5\%$, year by year
Let’s plug in $P = \pounds 1000$, $r = 0.05$, and watch what happens over thirty years.
| Year | Simple | Compound | Gap |
|---|---|---|---|
| 0 | 1000.00 | 1000.00 | 0.00 |
| 1 | 1050.00 | 1050.00 | 0.00 |
| 2 | 1100.00 | 1102.50 | 2.50 |
| 5 | 1250.00 | 1276.28 | 26.28 |
| 10 | 1500.00 | 1628.89 | 128.89 |
| 15 | 1750.00 | 2078.93 | 328.93 |
| 20 | 2000.00 | 2653.30 | 653.30 |
| 25 | 2250.00 | 3386.35 | 1136.35 |
| 30 | 2500.00 | 4321.94 | 1821.94 |
After one year, the two are identical. After two years, compound has pulled ahead by $\pounds 2.50$ — nothing dramatic. After ten years, the compound balance is $\pounds 130$ ahead. After thirty years, the gap is over $\pounds 1{,}800$, which is more than the original principal. Compound interest has produced a balance more than $70\%$ larger than simple interest, on the same starting amount at the same rate.
This is the entire reason financial advisors talk about “the power of compounding” and the entire reason credit-card debt is so dangerous. The gap is small in the early years and overwhelming in the late ones. If you ignore it for the first decade because the numbers look modest, you have already missed the moment where it mattered.
Why the gap grows the way it does
The simple-interest formula adds $rP$ to the balance every year. The balance grows by the same amount each year — a straight line. After $t$ years it has gained $rPt$, full stop.
The compound formula multiplies the balance by $(1 + r)$ every year. After year one the balance is $P(1 + r)$. After year two it is $P(1 + r)^2$. After thirty years it is $P(1 + r)^{30}$. Each year the “next $5\%$” is computed on a larger starting balance than the previous year, so the absolute amount of interest accrued each year keeps going up.
The technical name for this distinction is “linear growth” vs “exponential growth.” Linear growth gains the same amount per year. Exponential growth gains the same fraction per year, which means the absolute amount keeps getting bigger.
Where each one applies in real life
Simple interest is rarer than you might think. The places I’ve seen it used:
- Some short-term consumer loans (especially in the US, governed by Truth in Lending Act disclosures).
- Some bonds, where the “coupon” payment is a fixed amount per period.
- Most car loans in the US (technically the loan amortises, but the interest charge per period is calculated as simple interest on the remaining balance).
Compound interest is the default for almost everything else:
- Savings accounts and ISAs (typically compounding monthly or daily).
- Mortgages (compounded monthly).
- Credit-card balances (compounded daily, which is why they are so painful).
- Investment returns over multi-year horizons (the standard way of comparing performance is the “compound annual growth rate”).
A practical rule: if you are borrowing, you want simple interest if you can get it (you pay less). If you are lending or saving, you want compound interest (you earn more). Most products are not designed in your favour.
The doubling time and the rule of 72
A surprisingly useful mental shortcut for compound interest: “72 divided by the percentage rate” gives you, approximately, how many years it takes for an investment to double.
At $5\%$, doubling takes about $72/5 = 14.4$ years. (Exact value: $\ln 2 / \ln 1.05 \approx 14.21$ years. The rule is close.)
At $8\%$, doubling takes about $72/8 = 9$ years. (Exact: $9.01$.)
At $12\%$, doubling takes about $6$ years. And so on. The rule comes from the fact that $\ln 2 \approx 0.693 \approx 70/100$, but $72$ divides nicely against more interest rates so it ends up being the useful version.
Once you have the doubling time, you can do quick mental projections. $\pounds 10{,}000$ at $5\%$ for thirty years: that is roughly two doublings ($14.4$ years each) plus a bit, so somewhere around $\pounds 40{,}000$ to $\pounds 50{,}000$. Exact value is about $\pounds 43{,}219$. The mental ballpark is good enough to sanity-check anything you read in a savings advert.
The compounding-frequency rabbit hole
The numbers above assumed compounding once a year. If you compound more often — monthly, daily, every second — the gap between simple and compound widens further, but only slightly. The difference between annual and monthly compounding at $5\%$ over thirty years is about $\pounds 250$ on a $\pounds 1{,}000$ starting balance. The difference between monthly and continuous compounding is about another $\pounds 5$. Past that, you are in diminishing returns.
The interesting limit is “continuous compounding,” in which interest is added at every instant. The formula is $A = Pe^{rt}$, where $e$ is the constant from the $e$ post. For our $\pounds 1000$ at $5\%$ for $30$ years, continuous compounding gives $1000 \cdot e^{1.5} \approx \pounds 4{,}481.69$. Compare with $\pounds 4{,}321.94$ for annual compounding — about $\pounds 160$ more, after thirty years. Compounding more often than annually is a nice-to-have, not a transformative force.
What this means for personal finance
Two practical implications.
First: starting early matters more than the rate. Someone who invests $\pounds 100$/month from age $25$ to $65$ at a modest $5\%$ ends up with substantially more money than someone who invests $\pounds 200$/month from age $40$ to $65$ at a generous $7\%$. Time in the market is the dominant variable. The maths is just compound interest applied to a series of contributions, and the answer is “the early person wins by a margin you would not believe without the spreadsheet.”
Second: credit-card debt is much worse than the headline rate suggests. A $20\%$ APR on a credit card is compounded daily, which gives an effective annual rate of about $22\%$. On a balance you never pay down, the debt doubles in about $3.5$ years. This is the mathematical reason why the standard advice to “pay off credit-card debt before investing in anything” is correct.
The compound-interest calculator on this site lets you plug in your own numbers and see the side-by-side comparison for any combination of starting amount, rate, time, and compounding frequency.