Compound Interest Explorer

watch exponential growth outrun linear intuition

Starting values — Compound Interest Explorer

Principal (P)	1000
Annual rate (r)	7
Time (t)	30
Target amount (X)	10000
Compounding mode	monthly (n = 12)
Custom compounds per year (n)	12
Show continuous-compounding limit	off
Show currency symbol	off

Change any control above and the results recompute from these.

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Compounding more often buys less each time 🖖

Compounding more often does help, and it helps less every time. Take the Stock market preset — $10,000 at 7% for thirty years — and step the frequency up: annually gives $76,122.55, quarterly $80,191.83, monthly $81,164.97, daily $81,645.26. The first move up is worth $4,069. The next is worth $973, then $480. Going all the way from hourly to compounding infinitely often, which is what A = Pe^rt means, is worth 69 cents. Almost all of the value is in the first step away from annual.

Interest that earns its own interest 🖖

The magic of compounding is that each period's interest is added to your balance, so next period you earn interest on a bigger amount. That feedback loop is why savings accelerate instead of growing in a straight line. A handy shortcut is the Rule of 72: divide 72 by the annual percent rate to estimate how many years your money takes to double. At 6% that's about 12 years.

The number e was born here 🖖

Try compounding $1 at 100% for one year. Compounded annually it becomes $2, monthly about $2.61, and daily roughly $2.71. The more often you compound, the closer the result creeps to e ≈ 2.71828. Jacob Bernoulli stumbled onto this limit in 1683 while studying compound interest — which is literally where the constant e comes from.

COMPOUND INTEREST — WHICH QUESTION, AND WHICH WAY IS IT POINTING?

Which Compounding Case Are You In?

Interest on the interest is the whole idea, and it changes the arithmetic from a straight line into a curve: A = P(1 + r/n)^(nt) instead of P(1 + rt). Which case you are in depends on what you are asking. How much does compounding actually add? How long until it doubles? What happens when the same curve is pointed at a debt you owe? And how much does the compounding frequency really matter?

A modest rate over a long time — the gap against simple interest A = P(1 + r/n)^nt

How long until it doubles — and why 72 works t_2× ≈ 72 / r%

The same curve pointed at you — debt compounds identically (1 + r/n)ⁿ − 1

How often it compounds — and why the answer stops mattering n → ∞ ⇒ A = Pe^rt

A modest rate over a long time — the gap against simple interest

What you know: A rate you can actually get on savings, left alone for decades. Compare A = P(1 + r/n)^(nt) with the simple-interest line P(1 + rt) to see what compounding is worth.

Formula: A = P(1 + r/n)^nt

Worked example: 5 000 at 4% compounded monthly for 20 years → 11 112.91, against 9 000 with simple interest — the compounding is worth 2 112.91

Open this case: safe savings

A modest rate over a long time — the gap against simple interest. The compound curve pulls away from the straight simple-interest line, slowly at first. A rate you can actually get on savings, left alone for decades. Compare A = P(1 + r/n)^(nt) with the simple-interest line P(1 + rt) to see what compounding is worth. — The compound curve pulls away from the straight simple-interest line, slowly at first.

How long until it doubles — and why 72 works

What you know: A higher rate over a long horizon. What matters is not the yearly gain but the doubling time, which depends on the rate alone and not on how much you started with.

Formula: t_2× ≈ 72 / r%

Worked example: 10 000 at 7% compounded monthly doubles every 9.93 years, so 30 years gives just over three doublings: 81 164.97, or 8.12 times the deposit

Open this case: stock market avg

How long until it doubles — and why 72 works. Each doubling takes the same 9.93 years, so the curve steepens without changing its rule. A higher rate over a long horizon. What matters is not the yearly gain but the doubling time, which depends on the rate alone and not on how much you started with. — Each doubling takes the same 9.93 years, so the curve steepens without changing its rule.

The same curve pointed at you — debt compounds identically

What you know: Nothing in the formula knows whether the money is yours. At a high rate the doubling time is short, and unpaid balances grow the same way savings do.

Formula: (1 + r/n)ⁿ − 1

Worked example: 3 000 at 20% compounded monthly for 5 years → 8 087.91. The doubling time is 3.49 years, and the effective annual rate is 21.94%, not 20%.

Open this case: credit card debt

The same curve pointed at you — debt compounds identically. A high rate makes the same curve steepen fast, and here it is growing against you. Nothing in the formula knows whether the money is yours. At a high rate the doubling time is short, and unpaid balances grow the same way savings do. — A high rate makes the same curve steepen fast, and here it is growing against you.

How often it compounds — and why the answer stops mattering

What you know: Increasing n raises the effective rate, but with rapidly diminishing returns. Beyond daily compounding you are within a rounding error of the continuous limit, A = Pe^(rt).

Formula: n → ∞ ⇒ A = Pe^rt

Worked example: 1 000 at 7% for 30 years: 7 612.26 compounded yearly, 8 116.50 monthly, 8 164.53 daily, and 8 166.17 continuously

Open this case: daily compounding

How often it compounds — and why the answer stops mattering. Effective rate against compounding frequency: almost all of the gain arrives by monthly. Increasing n raises the effective rate, but with rapidly diminishing returns. Beyond daily compounding you are within a rounding error of the continuous limit, A = Pe^(rt). — Effective rate against compounding frequency: almost all of the gain arrives by monthly.

References (1)

Insight block 3 — where the limit turned up: J. Bernoulli, "Quaestiones nonnullae de usuris, cum solutione problematis de sorte alearum." Acta Eruditorum, May 1690, 219–223 — the compound-interest question whose limit is e.

Problems solved in full

The worth of one percentage point of interest rate over thirty years 5 steps
What is one percentage point of interest rate worth over thirty years, and how does that compare with compounding more often? The tool holds P = 1,000 at a nominal 7% compounded monthly for 30 years, with the continuous limit drawn alongside.
1. Monthly compounding cuts the annual rate into twelve equal parts and applies each part to whatever the balance has already become. Thirty years is 360 of those slices.
  
  $i = \frac{r}{n} = \frac{0.07}{12} = 0.0058 \overline{3}, N = n t = 12 \times 30 = 360$
2. Every slice multiplies the balance by the same factor, so the whole thirty years is that one factor raised to the number of slices.
  
  $A = P (1 + i)^{N} = 1000 (1.0058 \overline{3})^{360} = 8116.50, I = A - P = 7116.50$
3. The continuous model replaces the 360 steps with a limit, and it is the version worth working with, because there the rate and the time appear only as the product rt. The gap between the two models is the check that swapping one for the other does not move the answer.
  
  $A_{c} = P e^{r t} = 1000 e^{0.07 \times 30} = 8166.17, Δ = A_{c} - A = 49.67 (0.61%)$
4. If r changes by δ then rt changes by tδ, so the final balance is multiplied by e^tδ — a factor set by the horizon alone, with no P in it. Recomputing the exact monthly balance at 6% tests whether that shortcut can be trusted.
  
  $\frac{A ( r + δ )}{A ( r )} \approx e^{t δ} : e^{- 0.30} = 0.7408 vs \frac{1000 ( 1.005 ) ^{360}}{8116.50} = \frac{6022.58}{8116.50} = 0.7420$
5. Now turn the question round: what change in rate would have bought the same money that compounding infinitely often bought?
  
  $e^{t δ} = 1 + \frac{Δ}{A} ⟹ δ = \frac{ln ( 1.006120 )}{30} = \frac{0.006101}{30} = 2.03 \times 1 0^{- 4}$
Answer

The balance reaches 8,116.50, of which 7,116.50 is interest, and the continuous limit sits just above it at 8,166.17 — a gap of 49.67. That 49.67 is the entire prize for compounding infinitely often instead of monthly, and it is worth about two hundredths of a percentage point on the rate. Dropping the rate to 6% costs 2,093.92 instead, which is 42 times as much, because the horizon multiplies every rate change: over t years a change δ scales the balance by e^tδ, so thirty years turns a small difference in r into a large difference in money. Of the four numbers you can set here, the compounding frequency is the one that matters least — three hundredths of a point on the rate already outweighs the whole jump from monthly to continuous.

$δ = \frac{ln ( 1 + Δ/ A )}{t} = 2.03 \times 1 0^{- 4} vs n : 12 \to \infty$
Turning 10,000 into 81,164.97 over 30 years at 7% compounded monthly 7 steps
The panel turns 10,000 into 81,164.97 over 30 years at 7% compounded monthly. Nobody does that in their head, which is why everyone quotes the Rule of 72 instead: money doubles in about 72/7 ≈ 10.3 years. Where does 72 come from, and when does it lie?
1. Start with the number on the panel, because the doubling time is hiding inside it.
  
  $A = P (1 + \frac{r}{n})^{n t} = 10000 (1 + \frac{0.07}{12})^{360} = 81 164.97, I = A - P = 71 164.97$
2. Divide by the starting amount and ask what power of two that is. Just over three doublings in thirty years, so one doubling takes 9.9310 years — and notice that the principal cancelled, which is why nobody ever asks how much you started with.
  
  $\frac{A}{P} = 8.1165 = 2^{3.0209} ⟹ t_{2} = \frac{30}{3.0209} = 9.9310$
3. Derive it properly rather than reading it off a ratio: set the balance to twice the principal and solve. The P on both sides cancels for the same reason, and the answer agrees to four decimals with the route above.
  
  $2 P = P (1 + \frac{r}{n})^{n t_{2}} ⟹ t_{2} = \frac{ln 2}{n ln ( 1 + r / n )} = \frac{0.693147}{12 ln ( 1.00583 3 )} = 9.9310$
4. Now compare against the folk rule. The Rule of 72 is stated for annual compounding, so use n = 1: the exact doubling time is 10.2448 years and the rule says 10.2857. Fifteen days out, over a decade.
  
  $n = 1 : t_{2} = \frac{ln 2}{ln 1.07} = 10.2448 \frac{72}{7} = 10.2857 Δ = 0.0409 (\approx 15 d)$
5. So where does 72 come from? Multiply the doubling time by the rate as a percentage and the r almost cancels — expand the logarithm to two terms and what is left is not a constant at all, but 69.31 with a small correction that grows with the rate.
  
  $100 r t_{2} = \frac{100 r ln 2}{ln ( 1 + r )} \approx \frac{100 r ln 2}{r ( 1 - \frac{r}{2} )} = \frac{69.31}{1 - \frac{r}{2}} \approx 69.31 (1 + \frac{r}{2})$
6. Evaluate that at three rates. The true constant drifts upward, and 72 is simply the value it passes through at around 8% — a compromise centred on the rates people actually argue about.
  
  $r = 2% : 70.01 r = 8% : 72.05 r = 15% : 74.39$
7. One thing remains unexplained: 69.31 rounds to 69, and 70 is closer to correct across low rates than 72 is. Neither won.
  
  $72 = 2^{3} \cdot 3^{2} \Rightarrow 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 7270 = 2 \cdot 5 \cdot 7 \Rightarrow 1, 2, 5, 7, 10, 14, 35, 70$
Answer

The constant is not a constant: it runs 70.0 at 2%, 72.1 at 8% and 74.5 at 15%, and 72 is where it happens to sit for a typical long-run return. The reason the folk rule chose it over the more accurate 69.3 or 70 is arithmetic, not analysis. 72 = 2³·3², so it divides cleanly by 1, 2, 3, 4, 6, 8, 9, 12, 18, 24 and 36; 70 = 2·5·7 divides cleanly by almost nothing you would meet in a conversation about interest. A rule you do in your head has to be divisible in your head, and that beat accuracy.

$t_{2} r_{%} \approx 69.31 (1 + \frac{r}{2}) : 70.0 (2%), 72.1 (8%), 74.5 (15%)$

By the 42Math team · Updated 2026-08-17

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

safe savings - $5,000 at 4% compounded monthly for twenty years reaches $11,112.91, a little more than double. Compounding continuously instead would add $14.79 to that.
stock market avg - $10,000 at 7% compounded monthly for thirty years reaches $81,164.97 - eight times the principal, and the rate never changed.
credit card debt - $3,000 at 20% compounded monthly for five years becomes $8,087.91. The arithmetic that grows savings works exactly as hard against you.
daily compounding - $1,000 at 7% compounded daily for thirty years gives $8,164.53, which is $1.64 short of the continuous limit of $8,166.17. Daily is already almost infinitely often.