Compound Interest: Interactive Visualization

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Compound interest in this visualization

Compound interest is interest paid on interest: each payment is added to the balance and earns interest from then on. At an annual rate $r$ paid $n$ times a year, each payment adds $r/n$ of the balance, so after $t$ years a principal $P$ has grown to $P\,(1 + r/n)^{\lfloor nt \rfloor}$ , where $\lfloor nt \rfloor$ is the number of payments made so far (just $P\,(1 + r/n)^{nt}$ at the payment times). Compounding per period is also written $y = a(1 + r)^{x}$ , with $a$ the principal, $r$ the rate per period and $x$ the number of periods; here the rate per period is $r/n$ and a year has $n$ periods. This page follows a principal of 1 for one year and lets $n$ run from 1 to a million.

The yellow staircase is the balance: flat between payments, a step up at each one. The red ticks along the bottom mark the payment times, 1/n of a year apart. At the end of the year the staircase has reached

Balance after one year

\left(1 + \frac{r}{n}\right)^{n}.

As $n$ grows the steps get smaller and more frequent, and the year-end balance increases — but it does not grow without bound. It approaches the blue curve, continuous compounding:

The limit

\lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^{n} = e^{r}, \qquad \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} = e = 2.718281828\ldots

At $r = 100\%$ the limit is the number $e$ itself; this is one of its standard definitions. The blue curve is $e^{rt}$ , the balance when interest is credited continuously.

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Things to notice

The year-end balance always rises with $n$ and always stays below $e^{r}$ . For large $n$ the gap is about $e^{r} r^{2} / (2n)$ , so ten times as many payments leave about a tenth of the gap (not at first: from 1 to 10 payments at 100%, it shrinks to 0.17 of itself).
Most of the gain comes early. At 100%, yearly to monthly adds 0.61; monthly to daily only 0.10; daily to every minute 0.004.
At ordinary rates frequency matters little: at 5%, yearly gives 1.05 and continuous compounding 1.0513.
The window that opens at large $n$ magnifies the last few payments by the same factor in both directions, so the steps and the gap to $e^{r}$ keep their true proportions.

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Reading it correctly

$r$ is the nominal annual rate. The rate actually earned in a year, the effective annual rate (APY), is $(1 + r/n)^{n} - 1$ , which is higher whenever $n > 1$ .
Daily, hourly and minute payments assume a 365-day year ( $n = 365$ , 8760, 525 600). Continuous compounding is a limit, not a very large $n$ : the page states $e^{r}$ exactly instead of reading it off the largest $n$ .

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Where it comes from and where it is used

Jacob Bernoulli studied this question in 1683: how much does 1 earn at 100% if interest is compounded more and more often? He showed that the answer settles at a limit between 2 and 3: the number now called $e$ .
Growth in proportion to the current amount, compounded continuously, is $e^{kt}$ : continuous compounding in finance, and the same exponential law in population growth and radioactive decay.

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Controls

The Rate knob and keys set $r$ . Under Compounding, the keys pick a named frequency; the fader sets the time between payments, $1/n$ , on a log scale, so pushing it right means more payments. Click a value to type it.
n → ∞ moves $n$ up to the next power of ten, then through each power of ten to a million, where it stops — still short of the limit. Compare frequencies lists the year-end balance for each. Presentation mode opens a full-screen view for projectors (Space runs n → ∞, ← → divide or multiply n by 10, H hides $e^{r}$ ).

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