1. Simple interest
Interest computed only on the original principal, never on accumulated interest. The interest earned per period is the same forever.
If you put principal $P$ in an account paying rate $r$ per year (as a decimal, so 6% is $r = 0.06$) for $t$ years, the interest is
$$ I = P \, r \, t $$and the final amount is
$$ A = P + I = P(1 + rt). $$That's the entire model. The interest doesn't care that the account is larger this year than it was last year — it always computes on the starting principal. The amount $A$ grows as a straight line in $t$ with slope $Pr$.
Put $\$2{,}000$ in a 4%-simple account for 3 years and you earn $I = 2000 \times 0.04 \times 3 = \$240$. Final balance $\$2{,}240$. Wait 30 years and you've earned $\$2{,}400$ — exactly ten times what you earned in three. Linear in time, full stop.
Simple interest is rare in modern banking — almost every savings account, loan, and credit card compounds. You'll see it on some short-term commercial loans, in bond coupon arithmetic before reinvestment, and in textbook examples. Default-assume "compound" unless told otherwise.
2. Compound interest
Compound interest puts last period's interest into the principal so it earns interest too. Each period, the balance is multiplied by the same growth factor. If interest is added once a year at rate $r$, after $t$ years
$$ A = P(1 + r)^{t}. $$Put $\$5{,}000$ in a 6%-annual-compounding fund for 5 years and you get $A = 5000 \times 1.06^{5} \approx \$6{,}691$. Simple interest at the same rate over the same span would have given $5000 + 5000 \cdot 0.06 \cdot 5 = \$6{,}500$. The compounding advantage is small at first — \$191 over 5 years — but it grows exponentially. Run the same comparison out to 30 years and compound delivers $\$28{,}717$ versus simple's $\$14{,}000$.
With compounding $n$ times per year at nominal annual rate $r$ for $t$ years,
$$ A = P\!\left(1 + \frac{r}{n}\right)^{nt}. $$Each period, the per-period rate is $r/n$ and there are $nt$ periods.
The exponent $nt$ counts compounding events; the base $1 + r/n$ is the per-period growth factor. The annual case is $n = 1$; everything else is just a finer subdivision of the same idea.
In year 2 of a compound account, last year's interest is itself earning interest. Simple interest only pays on the original principal forever. The gap is interest on interest — and that gap, year over year, also earns interest. The difference doesn't grow; it compounds.
3. The compounding-frequency lever
Banks advertise rates per year, but they don't all credit interest at the same cadence. The same nominal rate $r$ gives different amounts depending on $n$:
| Frequency | $n$ | Growth factor over 1 year at $r = 12\%$ |
|---|---|---|
| Annually | 1 | $1.12^{1} = 1.1200$ |
| Semi-annually | 2 | $1.06^{2} = 1.1236$ |
| Quarterly | 4 | $1.03^{4} \approx 1.1255$ |
| Monthly | 12 | $1.01^{12} \approx 1.1268$ |
| Daily | 365 | $(1 + 0.12/365)^{365} \approx 1.1275$ |
| Continuously | $\infty$ | $e^{0.12} \approx 1.1275$ |
Two things to notice. Going from annual to monthly added about 68 basis points (0.68%). Going from monthly to continuous added… seven. The function $n \mapsto (1 + r/n)^{n}$ rises quickly at first and then flattens out into an asymptote — there is a hard ceiling on what frequency can buy you.
That ceiling is the rate $r$ itself. More frequent compounding helps a little; a higher rate helps a lot. If you have a choice between "5% compounded monthly" and "5.1% compounded annually", the second wins.
4. Continuous compounding (the limit)
If you keep slicing the year finer and finer, the growth factor approaches a limit:
$$ \lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^{nt} = e^{rt}. $$So continuous compounding gives
$$ A = P\, e^{rt}. $$This is the cleanest of all the formulas because the awkward "per-period" rate disappears — there are no periods. It's also the upper bound: no compounding schedule can do better than $e^{rt}$ at a given nominal rate.
$\$1{,}000$ at 5% continuously for 10 years grows to $1000 \times e^{0.5} \approx \$1{,}648.72$. Annual compounding at the same rate gives $1000 \times 1.05^{10} \approx \$1{,}628.89$ — a \$20 difference over a decade.
The number $e \approx 2.71828$ was discovered by Bernoulli in 1683 precisely while studying compound interest. The derivation of why that specific limit gives $e^{rt}$ — and why $e$ deserves to be called "the natural base" — is the subject of the algebra topic on exponential and logarithmic functions. Here we just use the result.
Where you'll meet continuous compounding outside textbooks: bond pricing, derivatives (Black–Scholes uses $e^{rt}$ as the risk-free growth factor), and any physical model where the rate of change is proportional to current size — radioactive decay, cooling bodies, unconstrained population growth. The arithmetic is identical; only the story changes.
5. Simple vs compound, on one chart
The two formulas live on the same axes. Simple interest is the straight line $A = P(1 + rt)$. Compound interest is the curve $A = P(1 + r)^{t}$. They start at the same point and have the same slope at $t = 0$ — for a moment they're indistinguishable. Then the curve takes off.
Read off the chart: at 10 years compound has pulled barely \$200 ahead of simple. By 20 years the gap is around \$1{,}000. By 30 years compound has roughly doubled simple's gain. The same rate, the same principal — the curve wins, and it wins by more every passing year.
6. The rule of 72
How long does money take to double at compound interest? Exactly:
$$ t_{\text{double}} = \frac{\ln 2}{\ln(1 + r)} \approx \frac{0.6931}{\ln(1 + r)}. $$For small $r$, $\ln(1 + r) \approx r$, so $t_{\text{double}} \approx 0.693 / r \approx 69.3 / r\%$. People use 72 instead of 69.3 because it's close enough and divides cleanly by 2, 3, 4, 6, 8, 9, and 12 — perfect for mental math.
Doubling time at compound annual rate $r\%$ is approximately
$$ t_{\text{double}} \approx \frac{72}{r\%}. $$Best between roughly 5% and 12%; at higher rates use 73 or 75 for a closer fit.
| Rate | Rule of 72 | Exact (CI) |
|---|---|---|
| 3% | 24 years | 23.45 |
| 6% | 12 years | 11.90 |
| 8% | 9 years | 9.01 |
| 12% | 6 years | 6.12 |
| 20% | 3.6 years | 3.80 |
There's a cousin for tripling — the rule of 114, since $\ln 3 \approx 1.0986$ and $1.0986/r$ is the analogous formula. Tripling time at 6% is roughly $114/6 = 19$ years. The same logic generates a rule for any target multiplier.
7. Nominal rate vs effective annual rate
Banks quote a nominal annual rate (also called APR for borrowing, stated rate for saving). But if interest is credited more than once a year, what you actually earn after a year is not that nominal number — it's the effective annual rate (EAR), also called APY (annual percentage yield) on the savings side.
For nominal annual rate $r$ compounded $n$ times per year,
$$ \text{EAR} = \left(1 + \frac{r}{n}\right)^{n} - 1. $$For continuous compounding, $\text{EAR} = e^{r} - 1$.
Take 12% nominal compounded monthly: $(1 + 0.12/12)^{12} - 1 = 1.01^{12} - 1 \approx 0.1268$, so EAR ≈ 12.68%. That's the number you'd compare against a competing 12.5%-annual-compounding product to decide which is actually better.
Comparing two rates with different compounding frequencies is meaningless until you convert both to EAR. US regulation requires deposit accounts to disclose APY for exactly this reason — and credit cards to disclose APR. Look for the effective number, not the nominal one.
8. Partnership: capital × time
Several people pool money in a venture. At year-end there's a profit. How do they split it?
If everyone contributed at the same time and stayed in for the same duration, the answer is intuitive: split in proportion to capital. A invests \$3{,}000, B invests \$4{,}500, both for the full year. Ratio $3 : 4.5 = 2 : 3$. A profit of \$1{,}500 splits \$600 / \$900.
The more interesting case is when partners join at different times or pull money out partway through. Each dollar has only been working for part of the year, and a fair split weights it by how long. The principle:
Each partner's share is proportional to their capital × time.
If A contributes $x_1$ for $t_1$ months and B contributes $x_2$ for $t_2$ months, then
$$ \frac{A\text{'s share}}{B\text{'s share}} = \frac{x_1\, t_1}{x_2\, t_2}. $$The same logic extends to any number of partners and to capital that changes mid-year — just split each interval into "capital × duration" chunks and add them up.
A worked split
Ishan invests \$10{,}000 for the full 12 months. Jaya invests \$15{,}000 but only joins 4 months in — so she's in for 8 months. The business profits \$3{,}900.
- Ishan's capital-months: $10{,}000 \times 12 = 120{,}000$.
- Jaya's capital-months: $15{,}000 \times 8 = 120{,}000$.
- Ratio $1 : 1$. Each gets \$1{,}950.
Coincidence, not a rule: Jaya had more money but less time, and the products happened to match. Shift her entry by two months either direction and the split tilts.
A partner who invests \$10{,}000 and then withdraws \$2{,}000 after 4 months has capital-months $10{,}000 \times 4 + 8{,}000 \times 8 = 104{,}000$. Treat every change in capital as the boundary between two intervals and sum the contributions piecewise.