Simple and Compound Interest

Interest is the price of time. When you lend money you give up the use of it today in exchange for more of it later; when you borrow, you pay for the privilege of spending now. Almost every financial decision you will ever make — a savings account, a car loan, a mortgage, a credit card balance, a retirement fund — is governed by how that price is calculated. There are two fundamentally different ways to calculate it, simple and compound, and the gap between them is the single most important idea in personal finance. Understanding it is the difference between money quietly working for you and quietly working against you.

This page builds the concept from the ground up: what interest is, how simple and compound growth diverge, why the frequency of compounding matters, how "infinitely often" gives birth to the number $e$, and a mental shortcut — the Rule of 72 — that lets you estimate doubling times in your head.

Learning Objectives

• Distinguish simple interest (linear growth on the principal) from compound interest (exponential growth on principal plus accumulated interest).
• Apply the formulas $A = P(1 + rt)$ and $A = P(1 + r/n)^{nt}$ correctly.
• Explain why compounding frequency changes the outcome, and compute effective annual rates.
• Derive continuous compounding and see how the number $e$ emerges as a natural limit.
• Use the Rule of 72 to estimate doubling time, and know when it is accurate.
• Recognize where these ideas appear in real loans, savings, and investments.

Quick Answer

Simple interest pays a fixed amount each period based only on the original principal: $A = P(1 + rt)$. It grows in a straight line. Compound interest pays interest on interest — each period's interest is added to the balance and itself earns interest afterward: $A = P(1 + r/n)^{nt}$, where $n$ is the number of compounding periods per year. More frequent compounding yields more money, but with diminishing returns; the limit as compounding becomes continuous is $A = Pe^{rt}$, where $e \approx 2.71828$. The Rule of 72 estimates how long money takes to double: divide 72 by the annual percentage rate. At 8%, money doubles in roughly $72/8 = 9$ years.

Where It Came From

Interest is older than coinage. On clay tablets from ancient Mesopotamia (Sumer and Babylon), around 3000–2000 BCE, scribes recorded loans of grain and silver at fixed rates. The very words reveal the idea's origin: the Sumerian word for interest, máš, also meant "calf" or "young goat." A loan of livestock literally reproduced — you returned the animal plus its offspring — so charging interest on a loan of silver was understood by analogy as money that "gives birth." The Babylonians already distinguished customary rates (about 20% on silver, 33% on grain) and, remarkably, some tablets show compound interest calculations, accumulating the debt period after period. The great difficulty was that debts compounding at 20% could double in under four years, ruining borrowers — which is why Mesopotamian kings periodically declared debt amnesties, and why later religious codes restricted or forbade interest (usury) altogether.

For millennia, simple interest dominated everyday practice because it is easy to compute by hand. Compound interest returned to prominence with medieval and Renaissance banking in Italy, where merchants reinvesting profits needed to value money across many periods. This is where the mathematics of exponential growth truly began.

The final piece — continuous compounding — arrived through a purely mathematical question. In 1683 the Swiss mathematician Jacob Bernoulli was studying exactly the interest problem: if a bank offers 100% annual interest, you clearly end with twice your money. But what if it compounds semi-annually, giving 50% twice? You get $(1 + 1/2)^2 = 2.25$ times your money. Quarterly gives $(1 + 1/4)^4 \approx 2.44$. Bernoulli asked what happens as you compound more and more often, and proved that the result does not blow up to infinity — it converges to a number between 2 and 3. That limit, $\lim_{n \to \infty}(1 + 1/n)^n \approx 2.71828$, is the constant later named $e$ by Leonhard Euler in the 1720s. The number underpinning half of modern mathematics was first glimpsed inside a compound interest calculation.

Simple Interest: Linear Growth

Simple interest is charged only on the original principal $P$, never on interest already earned. With annual rate $r$ (as a decimal) over $t$ years, the interest is $I = Prt$ and the final amount is:

$A = P(1 + rt)$

Because the interest each year is the same fixed amount $Pr$, the balance grows in a straight line.

Worked example. You deposit $2000 at 5% simple annual interest for 4 years.

• Annual interest: $2000 \times 0.05 = 100$ every year.
• Total interest over 4 years: $100 \times 4 = 400$.
• Final amount: $A = 2000(1 + 0.05 \times 4) = 2000 \times 1.20 = 2400$.

Each year adds exactly $100 — no more, no less — no matter how large the balance grows. Simple interest is common in short-term loans, some bonds, and many auto loans.

Compound Interest: Growth on Growth

Compound interest adds each period's interest to the balance, so future interest is earned on a larger and larger base. If interest compounds $n$ times per year at annual rate $r$ for $t$ years:

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

Here $r/n$ is the rate per period and $nt$ is the total number of periods.

Worked example (same numbers as above). $2000 at 5% compounded annually ($n = 1$) for 4 years:

$A = 2000(1 + 0.05)^4 = 2000 \times 1.21550625 = 2431.01$

Compare this to the simple-interest result of $2400. The extra $31.01 is interest earned on previously earned interest. Over 4 years the gap is small; over decades it becomes enormous.

Why the gap explodes over time. Take $2000 at 5% for 40 years:

• Simple: $2000(1 + 0.05 \times 40) = 2000 \times 3 = 6000$.
• Compound (annual): $2000(1.05)^{40} = 2000 \times 7.0400 = 14,080$.

Simple interest tripled your money; compound interest multiplied it by seven. This is why Einstein is (probably apocryphally) said to have called compound interest "the eighth wonder of the world."

Compounding Frequency and Effective Rate

The stated (nominal) annual rate is not the whole story — how often it compounds matters. Take $1000 at a nominal 12% for one year at different frequencies:

Frequency	$n$	Calculation	Final amount
Annually	1	$1000(1.12)^1$	$1120.00
Semi-annually	2	$1000(1.06)^2$	$1123.60
Quarterly	4	$1000(1.03)^4$	$1125.51
Monthly	12	$1000(1.01)^{12}$	$1126.83
Daily	365	$1000(1 + 0.12/365)^{365}$	$1127.47

The effective annual rate (EAR) converts any compounding scheme into the single annual rate that would give the same result:

$\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1$

For 12% compounded monthly: $\text{EAR} = (1.01)^{12} - 1 = 0.126825 = 12.68\%$. This is why lenders advertise the low nominal rate but are legally required to disclose the higher effective figure (the APY on savings, the APR on loans). Always compare products by their effective rate.

Continuous Compounding and the Number e

Notice the diminishing gains in the table: going from annual to monthly added over $6, but monthly to daily added only $0.64. Compounding "infinitely often" does not give infinite money — it converges to a ceiling. Setting $m = n/r$, the compound factor becomes

$\left(1 + \frac{r}{n}\right)^{n} = \left[\left(1 + \frac{1}{m}\right)^{m}\right]^{r} \longrightarrow e^{r} \quad \text{as } n \to \infty$

because $\lim_{m \to \infty}(1 + 1/m)^m = e$. This gives the elegant continuous compounding formula:

$A = Pe^{rt}$

Worked example. $1000 at 12% compounded continuously for one year:

$A = 1000 \, e^{0.12} = 1000 \times 1.127497 = 1127.50$

That is just three cents above daily compounding — continuous compounding is the theoretical maximum. It is the standard model in advanced finance (bond pricing, the Black–Scholes option formula) because $e^{rt}$ is far easier to manipulate with calculus than $(1 + r/n)^{nt}$.

Real-World Applications

• Retirement savings. Investing $300/month from age 25 versus age 35 can differ by hundreds of thousands of dollars at retirement — the early decade compounds the longest. Compounding rewards time more than amount.
• Credit card debt. Cards typically compound daily at ~20% APR. A $5000 balance at 20\% ignored for a year grows to about$ 5000 , e^{0.20} \approx $6107$ — the same math, working against you.
• Mortgages and auto loans. Loan payments are engineered so that early payments are mostly interest; understanding compounding explains why extra principal payments early on save so much.
• Inflation. Prices compound too. At 3% inflation, prices double in about $72/3 = 24$ years — the Rule of 72 applied to purchasing power.

Common Mistakes

1. Confusing the nominal rate with what you actually earn or pay. Misconception: "12% compounded monthly means I earn 12% a year." Why wrong: Monthly compounding gives an effective 12.68%. Correction: Always convert to EAR before comparing products.

2. Dividing the rate but forgetting to multiply the exponent (or vice versa). Misconception: Writing $P(1 + r/n)^t$ or $P(1 + r)^{nt}$. Why wrong: The per-period rate is $r/n$ and the number of periods is $nt$; both must change together. Correction: Use $(1 + r/n)^{nt}$ as one linked unit.

3. Using a percentage instead of a decimal. Misconception: Plugging $r = 5$ for 5%. Why wrong: $(1 + 5)^t$ multiplies your money by 6 every period. Correction: Convert: 5% $\to 0.05$. Always sanity-check that the answer is reasonable.

Comparison and Connections

Feature	Simple Interest	Compound Interest
Interest earned on	Principal only	Principal + accumulated interest
Growth pattern	Linear	Exponential
Formula	$A = P(1 + rt)$	$A = P(1 + r/n)^{nt}$
Over long horizons	Falls far behind	Dominates
Typical use	Short-term loans, some bonds	Savings, investments, credit cards

Compound interest is a geometric sequence (each term is the previous times a constant) and, in the continuous limit, an exponential function $Pe^{rt}$ — the same function that describes population growth and radioactive decay. Simple interest, by contrast, is an arithmetic sequence, i.e., a linear function. The two agree for very short periods and diverge more and more thereafter.

Practice Questions

Recall

State the formula for simple interest and for compound interest, defining every symbol. Answer: Simple: $A = P(1 + rt)$. Compound: $A = P(1 + r/n)^{nt}$. Here $A$ = final amount, $P$ = principal, $r$ = annual rate (decimal), $t$ = time in years, $n$ = compounding periods per year.

Understanding

Explain why compounding daily earns only slightly more than compounding monthly, even though "daily" sounds much more frequent. Guidance: The compound factor converges toward the ceiling $e^{rt}$. Each extra split of the year adds interest on an already tiny sliver of interest, so gains shrink rapidly toward the continuous limit.

Application

You invest $5000 at 6% for 10 years. Find the final amount under (a) simple interest, (b) annual compounding, (c) continuous compounding. Answer: (a) $5000(1 + 0.06 \times 10) = 8000$. (b) $5000(1.06)^{10} = 5000 \times 1.79085 = 8954.24$. (c) $5000 \, e^{0.6} = 5000 \times 1.82212 = 9110.59$.

Analysis

A bank offers 8% compounded quarterly; a rival offers 8.1% compounded annually. Which is better? Guidance: Compare EARs. First: $(1 + 0.08/4)^4 - 1 = (1.02)^4 - 1 = 0.08243 = 8.24\%$. Second: 8.10%. The quarterly account wins despite the lower nominal rate.

FAQ

Is compound interest always better than simple interest? For a saver or investor, yes — you earn more. For a borrower, compound interest is worse because your debt grows faster. It depends which side of the loan you are on.

What does "APR" versus "APY" mean? APR (annual percentage rate) is usually the nominal rate quoted on loans. APY (annual percentage yield) is the effective rate including compounding, quoted on savings. APY is the honest comparison figure.

Where does the number $e$ actually come from here? From asking what $(1 + r/n)^{nt}$ approaches as $n$ grows without bound. That limit is $e^{rt}$, and $e = \lim_{m\to\infty}(1 + 1/m)^m \approx 2.71828$. It was discovered inside exactly this interest problem by Jacob Bernoulli.

Does continuous compounding mean my money grows infinitely fast? No. "Continuous" refers to compounding frequency, not rate. It gives the largest result for a fixed nominal rate, but only marginally more than daily compounding — a finite ceiling of $Pe^{rt}$.

How accurate is the Rule of 72? Very good for rates between about 4% and 12%. It comes from $\ln 2 \approx 0.693$, and dividing 69.3 by the percentage rate would be more exact — but 72 has more divisors (2, 3, 4, 6, 8, 9, 12), making mental math easier, and it slightly better matches typical annual compounding.

Quick Revision

• Simple: $A = P(1 + rt)$ — linear, interest on principal only.
• Compound: $A = P(1 + r/n)^{nt}$ — exponential, interest on interest.
• Continuous: $A = Pe^{rt}$, with $e \approx 2.71828$.
• Effective annual rate: $\text{EAR} = (1 + r/n)^n - 1$.
• Rule of 72: doubling time $\approx 72 \div (\text{percent rate})$.
• Always convert percentages to decimals; change $r/n$ and $nt$ together.
• More frequent compounding earns more, with diminishing returns toward $Pe^{rt}$.

Related Topics

Prerequisites

• Financial Mathematics overview
• Percentages and decimals
• Exponents and geometric sequences

Related Topics

• Time value of money and present/future value
• The number $e$ and exponential functions

Next Topics

• Annuities and loan amortization
• Present value and discounting