Compound Interest: Simple vs Compound vs Continuous Compounding
Einstein almost certainly never called compound interest the eighth wonder of the world. The mathematics is remarkable enough without the endorsement — and once you see why the number e turns up uninvited, you understand something deep about growth itself.
Here is a question that catches out more students than it should. You put £1,000 into an account paying 10% a year. After two years, how much do you have?
The tempting answer is £1,200. Ten percent of £1,000 is £100, twice is £200, done. That answer is wrong, and the reason it is wrong is the entire subject of this article.
In year two, you are not earning interest on £1,000. You are earning interest on £1,100 — because the £100 you earned in year one is now sitting in the account working for you exactly like the original money does. It does not know it was interest. Money has no memory of where it came from. So year two pays you £110, not £100, and you finish with £1,210.
That extra £10 looks trivial. Over forty years it is the difference between £5,000 and £45,259. That gap — the gap between adding and multiplying — is what we are going to make precise.
Simple interest: the model that almost never applies
Simple interest treats your principal as the only thing that earns. Each period pays the same amount, forever. Nothing compounds.
A = P(1 + rt)
A = final amount · P = principal · r = rate per period as a decimal (7% → 0.07) · t = number of periods
Notice the shape of that expression: it is linear in t. Plot it and you get a straight line. Double the time, double the interest. There is no acceleration anywhere in it.
For example, £2,000 at 5% simple interest for 6 years gives A = 2000(1 + 0.05 × 6) = £2,600. The interest is £600, arriving at exactly £100 per year for six years.
Where does simple interest actually turn up? Short-term commercial paper, some government bonds between coupon dates, certain informal loans — and, importantly for exam questions, as a deliberate contrast to show you what compounding adds. In most of the financial world, simple interest is a teaching device rather than a product.
Compound interest: where money starts working on itself
Let us build the formula rather than hand it to you, because the derivation is three lines and once you have seen it you will never need to memorise the result.
Start with principal P and rate r per period. After one period you have your original money plus r times your original money: P + Pr. Factor out the P and you get P(1 + r).
After two periods, apply the same operation to what you now have. Your new principal is P(1 + r), so you finish with P(1 + r) × (1 + r) = P(1 + r)². After three periods: P(1 + r)³.
The pattern is not a coincidence. Each period you multiply by the same growth factor (1 + r). Multiplication repeated is exponentiation. That is the whole idea.
A = P(1 + r)t
The quantity (1 + r) is the growth factor. Everything about compounding lives inside it.
The core distinction. Simple interest adds a fixed amount each period. Compound interest multiplies by a fixed factor each period. Adding gives you a straight line; multiplying gives you an exponential curve. Over one year they look nearly identical. Over forty, they are not the same species of thing.
Watching the gap open up
£1,000 invested at 8% per year, both models side by side:
| Years | Simple interest | Compound interest | Difference |
|---|---|---|---|
| 1 | £1,080 | £1,080 | £0 |
| 5 | £1,400 | £1,469 | £69 |
| 10 | £1,800 | £2,159 | £359 |
| 20 | £2,600 | £4,661 | £2,061 |
| 30 | £3,400 | £10,063 | £6,663 |
| 40 | £4,200 | £21,725 | £17,525 |
After one year: identical. After forty: compound interest has produced more than five times as much. Nothing changed except that the interest was allowed to earn interest. It is why time in the market matters more than almost anything else you can control.
Common error — percentages do not cancel. A 50% loss followed by a 50% gain does not return you to where you started. £100 falls 50% to £50; a 50% gain on £50 is £25, leaving £75. You need a 100% gain to recover from a 50% loss. Same multiplicative logic: growth factors of 0.5 and 1.5 multiply to 0.75, not 1. Percentages compose by multiplication, never by addition.
Compounding frequency: what if interest arrives more often?
So far we have quietly assumed interest is added once per period. But banks compound monthly. Some accounts compound daily. Does it matter?
Yes — and here the mathematics gets interesting. If a bank quotes a nominal annual rate r but compounds n times per year, each compounding period pays r/n, and over t years there are nt periods.
A = P(1 + r/n)nt
n = compounding periods per year. n = 1 annually · 2 semi-annually · 4 quarterly · 12 monthly · 365 daily
Watch what happens to £1,000 at a nominal 10% for one year as we crank up n:
| Compounding | n | Final amount |
|---|---|---|
| Annually | 1 | £1,100.00 |
| Semi-annually | 2 | £1,102.50 |
| Quarterly | 4 | £1,103.81 |
| Monthly | 12 | £1,104.71 |
| Daily | 365 | £1,105.16 |
| Every second | 31,536,000 | £1,105.17 |
Look at that final column carefully. More frequent compounding always helps — but the gains are shrinking fast. Annual to semi-annual buys you £2.50. Daily to every-single-second buys you about one penny.
The sequence is converging. It is heading somewhere specific and refusing to go past it. Where?
The number e, and why it appears here of all places
This is where many students first meet e, get told it is “approximately 2.718”, and are expected to accept this. Let us do better, because the reason e exists is exactly this problem.
Take the formula and set r = 1 (a 100% nominal rate, chosen purely to make the algebra clean) and t = 1, giving A = P(1 + 1/n)n. Now push n upward and watch the bracket:
| n | (1 + 1/n)n |
|---|---|
| 1 | 2.000000 |
| 10 | 2.593742 |
| 100 | 2.704814 |
| 1,000 | 2.716924 |
| 10,000 | 2.718146 |
| 1,000,000 | 2.718280 |
| → ∞ | 2.718281828… |
There is a genuine tension inside that expression, and it is worth pausing on. As n rises, the base (1 + 1/n) shrinks toward 1 — and anything to the power of 1 is just 1. But simultaneously the exponent n races to infinity, and a number slightly above 1 raised to an enormous power should explode. These two forces fight each other, and neither wins. They settle on 2.718281828…
That number is e. It was not invented to be mysterious. Jacob Bernoulli found it in 1683 while asking precisely our question: what happens to compound interest if you compound infinitely often? The answer to a banking problem turned out to be one of the most important constants in mathematics.
Continuous compounding: the ceiling on frequency
Generalise from r = 1 back to any rate r, take the limit as n → ∞, and you get the continuous compounding formula:
A = Pert
e ≈ 2.71828. This is the theoretical maximum — no compounding frequency, however extreme, beats it.
Check it against our table. £1,000 at 10% for one year compounded continuously: A = 1000 × e0.10 = 1000 × 1.10517 = £1,105.17. That matches the “every second” row exactly — because compounding every second is, for practical purposes, already continuous.
Why do economists prefer this form? Real economies do not pause to add up interest at midnight on 31 December. Output, population, capital stock and prices all grow smoothly, all the time. Continuous compounding models that honestly. It also makes the calculus enormously easier: the derivative of ert is r·ert, meaning the growth rate of the function is proportional to the function itself. That single property is why ert runs through the Solow model, discounting, Black–Scholes, and every continuous-time growth model you will meet. It is not showing off — it is the only functional form that behaves correctly.
Going backwards: solving for time
Exam questions frequently reverse the problem. Starting from A = P(1 + r)t, divide by P and take logs of both sides:
t = ln(A/P) ÷ ln(1 + r)
For continuous compounding the natural log strips out the e beautifully: t = ln(A/P) ÷ r
How long to double your money at 7% compounded annually? t = ln(2) ÷ ln(1.07) = 0.6931 ÷ 0.0677 = 10.24 years.
The Rule of 72 — and exactly where it fails
Doubling time comes up often enough that there is a mental shortcut: doubling time ≈ 72 ÷ r, where r is the percentage rate. At 7%: 72 ÷ 7 = 10.3 years. The exact answer was 10.24. Close enough to do in your head during an exam or a meeting.
Where does 72 come from? The mathematically honest constant is 100 × ln(2) ≈ 69.3. But 72 has more integer divisors (2, 3, 4, 6, 8, 9, 12), which makes mental arithmetic easier, and it happens to be more accurate in the 6–10% range where most realistic rates sit.
| Rate | Rule of 72 | Exact | Error |
|---|---|---|---|
| 2% | 36.0 yrs | 35.0 yrs | +2.9% |
| 6% | 12.0 yrs | 11.9 yrs | +0.8% |
| 8% | 9.0 yrs | 9.0 yrs | 0.0% |
| 15% | 4.8 yrs | 4.96 yrs | −3.2% |
| 30% | 2.4 yrs | 2.64 yrs | −9.1% |
Common error — trusting the Rule of 72 at high rates. It is a first-order approximation and degrades badly above roughly 15%. At 30% it is off by 9%; at 50% it is nearly useless. Fine for a quick sanity check on realistic returns, dangerous for hyperinflation questions where rates run to hundreds of percent. If a question involves extreme rates, use logarithms.
Effective annual rate vs nominal rate
Here is where compounding stops being an academic curiosity and starts being consumer protection law.
Two credit cards. One advertises 18% compounded monthly. The other advertises 18.5% compounded annually. Which is cheaper?
You cannot answer by comparing 18 and 18.5, because those numbers are not measuring the same thing. The nominal rate is a quoted figure with no compounding information baked in. To compare honestly, convert both to an effective annual rate — the rate that, compounded once annually, would produce the same result.
EAR = (1 + r/n)n − 1
And for continuous compounding: EAR = er − 1
Card A: EAR = (1 + 0.18/12)12 − 1 = (1.015)12 − 1 = 19.56%
Card B: already annual, so EAR = 18.50%
Card B: already annual, so EAR = 18.50%
The card advertising the lower number is the more expensive one. This is exactly why regulators in the UK, EU and US mandate APR disclosure — though note that “APR” is used inconsistently across jurisdictions and sometimes still hides fees. The mathematics is the only reliable defence.
Case Study — Payday Lending and the Tyranny of Frequency
A payday lender offers £300 for two weeks at a “fee” of £45. That sounds like 15% — high, but not obviously catastrophic.
Now compound it. There are 26 two-week periods in a year, and the periodic rate is 45/300 = 0.15. So EAR = (1.15)26 − 1 = 4,014%.
The borrower who rolls the loan over for a year owes roughly forty times what they borrowed. Nothing dishonest was said — 15% for two weeks is literally true. The compounding did all the damage, silently.
Following research on rollover behaviour, the UK’s Financial Conduct Authority capped payday costs at 0.8% per day and imposed a 100% total cost cap in January 2015. The cap is fundamentally a statement about compound interest.
Research Spotlight — Do People Actually Understand Compounding?
The finding: no, and the failure is systematic rather than random.
Stango and Zinman (2009), analysing US household survey data, documented what they termed exponential growth bias — a persistent tendency to linearise exponential functions. People do not merely make noisy errors around the right answer; they consistently underestimate compound growth, because the mind defaults to straight-line extrapolation.
Their striking result: the degree of an individual’s exponential growth bias predicts real financial outcomes. Households exhibiting stronger bias borrowed more, saved less, and held higher-cost debt.
Almenberg and Gerdes (2012) tested the direction of the error in Sweden and found people underestimate compound growth far more often than they overestimate it. Levy and Tasoff (2016) then showed the bias operates on both sides of the balance sheet: it makes saving look less attractive than it is, and borrowing look cheaper than it is.
Why this matters for you: the intuition that £1,000 at 10% for two years gives £1,200 is not a careless slip. It is the default setting of the human brain, and it costs households real money. The formula exists precisely because intuition here is unreliable.
Real vs nominal: the return you actually keep
One final complication, and it is the one that separates a competent answer from a strong one.
Your account grows at 6%. Inflation runs at 4%. Your purchasing power has not grown by 6% — it has grown by roughly 2%, and “roughly” is doing real work in that sentence. The exact relationship is the Fisher equation:
(1 + i) = (1 + r)(1 + π)
i = nominal rate · r = real rate · π = inflation. Rearranged: r = (1 + i)/(1 + π) − 1
Exact: r = (1.06 / 1.04) − 1 = 1.92%. Approximation (r ≈ i − π): 6% − 4% = 2%.
The approximation is fine here. At Weimar or Zimbabwe inflation rates it collapses completely, because the cross-term r × π that it discards is no longer small.
Common error — subtracting inflation and stopping there. r ≈ i − π is an approximation valid only when both rates are small. Examiners award marks for recognising it as an approximation and knowing when it fails. Above roughly 10% inflation, use the exact Fisher equation — and say why.
The family of formulas
| Situation | Formula | Use when |
|---|---|---|
| Simple interest | A = P(1 + rt) | Interest never reinvested |
| Compound, annual | A = P(1 + r)t | Interest added once per year |
| Compound, n times/year | A = P(1 + r/n)nt | Monthly, quarterly, daily |
| Continuous | A = Pert | Growth models, discounting, theory |
| Solve for time | t = ln(A/P) ÷ ln(1 + r) | “How long until…?” |
| Doubling time (quick) | ≈ 72 ÷ r% | Mental estimate, rates under 15% |
| Effective annual rate | EAR = (1 + r/n)n − 1 | Comparing products fairly |
| Real return | r = (1 + i)/(1 + π) − 1 | Adjusting for inflation |
AP & Cambridge A-Level Exam Technique
1. Convert percentages to decimals first. The single most common source of lost marks is putting 8 into a formula that wants 0.08. Write the conversion as a first line — examiners can see you knew, even if the arithmetic later goes astray.
2. State the formula before substituting. Most mark schemes award a method mark for the correct formula independently of the final answer. That mark is free and unconditional.
3. Watch the units on n and t. Monthly compounding over 3 years means the exponent is nt = 36 and the periodic rate is r/12. Halving one without adjusting the other is the classic error.
4. Round only at the very end. Rounding (1.00833)36 mid-calculation will visibly distort your answer. Carry full precision, round the final figure to two decimal places for money.
5. For “which is better” questions, always go to EAR. Comparing nominal rates with different compounding frequencies is not an answer — it is the trap the question is testing.
6. Sanity-check the magnitude. If you calculate that £1,000 at 5% for 10 years becomes £16,000, something is wrong. The Rule of 72 gives a two-second check: at 5%, doubling takes about 14 years, so after 10 years you should be under £2,000. Examiners reward candidates who notice their own absurd answers.
7. In evaluation questions, mention real returns. Any question about long-run saving that ignores inflation is incomplete. One sentence on real vs nominal frequently picks up an evaluation mark.
Practice Questions
Question 1 — Calculation (4 marks)
£5,000 is invested for 4 years at a nominal annual rate of 6%, compounded quarterly. Calculate the final amount and the total interest earned.
Answer.
Formula: A = P(1 + r/n)nt [1]
Periodic rate = 0.06/4 = 0.015; periods = 4 × 4 = 16 [1]
A = 5000(1.015)16 = 5000 × 1.26899 = £6,344.93 [1]
Interest = 6344.93 − 5000 = £1,344.93 [1]
Formula: A = P(1 + r/n)nt [1]
Periodic rate = 0.06/4 = 0.015; periods = 4 × 4 = 16 [1]
A = 5000(1.015)16 = 5000 × 1.26899 = £6,344.93 [1]
Interest = 6344.93 − 5000 = £1,344.93 [1]
Question 2 — Comparison (6 marks)
Bank A offers 7.2% compounded monthly. Bank B offers 7.5% compounded annually. Bank C offers 7.1% compounded continuously. Determine which gives the highest return, showing your reasoning.
Answer.
Nominal rates are not comparable when compounding frequencies differ; convert each to EAR. [1]
Nominal rates are not comparable when compounding frequencies differ; convert each to EAR. [1]
Bank A: (1 + 0.072/12)12 − 1 = (1.006)12 − 1 = 7.44% [2]
Bank B: already annual → 7.50% [1]
Bank C: e0.071 − 1 = 7.36% [1]
Conclusion: Bank B is best, then A, then C. [1]
Note the lesson: Bank C compounds continuously — the theoretical maximum frequency — and still finishes last, because its nominal rate was lower. Frequency helps, but it cannot rescue a bad rate.
Question 3 — Solving for time (5 marks)
An investor wants to triple an initial investment of £8,000 in an account paying 9% compounded annually. (a) Calculate the time required exactly. (b) Explain why the Rule of 72 cannot be used directly here.
(a) Target = 3 × 8000 = £24,000, so 3 = (1.09)t [1]
Take natural logs: ln(3) = t · ln(1.09) [1]
t = 1.0986 ÷ 0.08618 = 12.75 years [1]
Take natural logs: ln(3) = t · ln(1.09) [1]
t = 1.0986 ÷ 0.08618 = 12.75 years [1]
(b) The Rule of 72 estimates doubling time only, because 72 approximates 100 × ln(2). Tripling requires ln(3), so the rule does not apply. [1]
It can be used indirectly: doubling takes ≈ 72/9 = 8 years, so tripling must take between 8 and 16 years (two doublings = quadrupling) — consistent with 12.75. [1]
Question 4 — Application and evaluation (8 marks)
A saver deposits £10,000 at 5% compounded annually. Inflation averages 3%. (a) Calculate the nominal value after 20 years. (b) Calculate the real value in today’s purchasing power. (c) Evaluate the claim that “compound interest guarantees you get richer over time.”
(a) A = 10000(1.05)20 = £26,533 [2]
(b) Real rate via Fisher: r = (1.05/1.03) − 1 = 1.9417% [1]
Real value = 10000(1.019417)20 = £14,699 [2]
(Equivalent method: deflate the nominal figure — 26533 ÷ (1.03)20 = £14,691, minor rounding difference.)
(c) Evaluation — credit for any three developed points: [3]
• Misleading as stated. Nominal wealth rose 165%, but real purchasing power rose only 47%. Inflation consumed most of the gain. Compounding grew the number, not necessarily the wellbeing.
• It depends on r exceeding π. Had inflation averaged 5.5%, the real rate would be negative and the saver poorer in real terms despite a growing balance — precisely what many savers faced in 2021–2023 when deposit rates lagged inflation.
• It assumes no withdrawals, taxes or fees. Tax on interest and a 1% annual fee would meaningfully reduce the effective rate; compounding works against you on costs exactly as it works for you on returns.
• Behavioural caveat. Stango and Zinman’s exponential growth bias means savers systematically underestimate compounding — arguably making the claim under-appreciated rather than false, provided the real rate is positive.
• Conclusion. Compound interest is a mechanism, not a guarantee. It amplifies whatever the real rate happens to be, in either direction.
Summary
Compound interest is the mathematics of multiplication repeated. Simple interest adds; compound interest multiplies; and over long horizons those two operations produce results that are not remotely comparable. Increasing compounding frequency helps, but with sharply diminishing returns, and it converges on a hard ceiling defined by e — a constant that exists because Bernoulli asked this exact question in 1683. Nominal rates are not comparable across different compounding frequencies, so EAR is the only honest basis for comparison. And the number that matters at the end is always the real one, after inflation.
If you take one thing away: your intuition about exponential growth is reliably wrong, and it is wrong in a specific direction. The research shows people underestimate compounding, not overestimate it. That is why we write the formula down.
References
- Almenberg, J. and Gerdes, C. (2012) ‘Exponential growth bias and financial literacy’, Applied Economics Letters, 19(17), pp. 1693–1696.
- Bernoulli, J. (1690) ‘Quaestiones nonnullae de usuris, cum solutione problematis de sorte alearum’, Acta Eruditorum.
- Financial Conduct Authority (2014) Detailed rules for the price cap on high-cost short-term credit, Policy Statement PS14/16. London: FCA.
- Fisher, I. (1930) The Theory of Interest. New York: Macmillan.
- Levy, M. and Tasoff, J. (2016) ‘Exponential-growth bias and lifecycle consumption’, Journal of the European Economic Association, 14(3), pp. 545–583.
- Stango, V. and Zinman, J. (2009) ‘Exponential growth bias and household finance’, The Journal of Finance, 64(6), pp. 2807–2849.
- Zinman, J. (2015) ‘Household debt: facts, puzzles, theories, and policies’, Annual Review of Economics, 7, pp. 251–276.