Continuous Compounding

In one sentence: Continuous compounding says money grows as if interest were added every instant, and the number e, about 2.718, sets the size of that growth.

$A = P e^{r t}$

Read it out loud: “A equals P times e to the power r times t. A is the final amount, P is the starting amount, e is the number about 2.718, r is the yearly rate as a decimal, and t is the number of years.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Euler’s number e: The number e is about 2.718281828. It is a fixed number, like pi, whose digits never end or repeat. It is the natural base for growth: whenever a quantity grows by a fixed share at every instant, e appears in the formula. The explanation below shows where it comes from.
Exponents, or powers: An exponent is the small raised number. A whole-number power such as $2^{3}$ means $2 \times 2 \times 2$ . A power that is not a whole number, such as $e^{0.5}$ , is worked out with a calculator, and it is the square root of e, about 1.648721. Also $e^{0}$ is 1, and $e^{1}$ is e itself.
Limit: A limit is the value a pattern gets closer and closer to as one number grows without bound. The values 2, 2.25, 2.613 and 2.7146 come from $\left(1 + \frac{1}{n}\right)^{n}$ with larger and larger $n$ , and they approach 2.71828. That target is the limit.
Percent and decimal: A percent is a number out of 100, so 5% is $\frac{5}{100} = 0.05$ . The formula needs the decimal form, so every yearly rate is divided by 100 before it is used. More practice with powers and percentages is on the math basics page.

What each symbol means

Symbol	Meaning
$A$	Final amount, the balance after t years, measured in dollars. It includes the starting amount and all the interest, and the interest alone is A minus P.
$P$	Principal, the starting amount, measured in dollars. It is the sum put in on day one, before any interest starts to build up.
$e$	Euler’s number, a fixed number about 2.718281828 with no unit. It is the base of natural growth, the number that appears when interest is added at every instant. It is defined as the limit of $\left(1 + \frac{1}{n}\right)^{n}$ as $n$ grows without bound, as the explanation shows. Scientific calculators have an e^x key.
$r$	Annual interest rate, written as a decimal per year, so 5% is 0.05. The rate is a yearly figure, and the formula applies it to every moment of the year.
$t$	Time, the number of years, measured in years like the rate. Half a year is t = 0.5, and ten years is t = 10.

The formula piece by piece

$r t$: Rate times time: the yearly rate multiplied by the number of years. For 5% over 10 years, $0.05 \times 10 = 0.5$ . This single number is the power that e is raised to, so it is the total growth in the exponent.
$e$: The base of natural growth, about 2.718. It is the limit that compound interest reaches when the number of compounding steps grows without bound. Raising it to a power gives the growth factor for continuous growth.
$e^{r t}$: The growth factor over the whole time. For $r t = 0.5$ it is about 1.648721, so the amount grows by about 65 percent. Raising e to the power $r t$ gives the multiplier for continuous compounding.
$P e^{r t}$: The starting amount times the growth factor, which gives the final balance $A$ with the starting amount included. Subtract $P$ to find the interest earned.
$A = P e^{r t}$: The whole formula in one sentence: the final balance equals the starting amount times e raised to the power of the rate times the time.

What the formula is doing, and why

Continuous compounding is the idea that interest is added at every instant, not once a year, once a month or once a day. Real accounts add interest in steps, so continuous compounding is an ideal limit. It gives the largest balance you can reach from a given yearly rate, and it is the form most used in mathematics and in finance theory. The formula $A = P e^{r t}$ gives that balance.

What is $e$ ? Take a rate of 1, which is 100%, for one year and split it into $n$ equal steps. Compounding with those steps gives $\left(1 + \frac{1}{n}\right)^{n}$ . With $n = 1$ it is 2. With $n = 2$ it is 2.25. With $n = 12$ it is about 2.6130, and with $n = 365$ it is about 2.7146. The values keep climbing but get closer and closer to 2.718281828. That fixed target is the number $e$ , the limit of this pattern as $n$ grows without bound.

Why $e^{r t}$ ? When the yearly rate $r$ is split into $n$ steps a year for $t$ years, the balance is multiplied by $\left(1 + \frac{r}{n}\right)^{n t}$ . Letting $n$ grow without bound turns that pattern into $e^{r t}$ . The rate and the time multiply inside the power, because the growth is applied over and over, once for each tiny step, so the total growth factor is e raised to the number $r t$ .

Why $e$ appears so often. Because $e$ is the limit of steady compounding, it shows up in any process that grows, or shrinks, by a fixed share at every instant. Population growth and radioactive decay are two examples, which is why $e$ is called the natural base for growth.

Real accounts. Banks usually add interest daily or monthly, in steps. Continuous compounding gives a slightly higher balance than daily compounding, so it is a close upper estimate rather than an exact match for most accounts.

An everyday picture

Think of rain falling steadily into a bucket instead of being poured in once a day. With a steady drip, the water level rises smoothly, and at every moment the water already in the bucket is adding its own share to the total. Continuous compounding is that steady drip: the balance gets its interest all the time, so each tiny slice of interest starts earning right away.

Worked example: Savings at 5% for 10 years, compared four ways

A savings account starts with 1000 dollars and pays 5% per year. We compare the balance after 10 years when interest is added once a year, monthly, daily and continuously. In the formula, $P = 1000$ dollars, $r = 0.05$ per year and $t = 10$ years.

Write the continuous formula and fill it in. We use $A = P e^{r t}$ , so $A = 1000 \times e^{0.05 \times 10}$ with the result in dollars. The rate and the time multiply first, because they sit together in the power, and $0.05 \times 10 = 0.5$ is the number in the power.
Work out e to the power 0.5. The number $e$ to the power 0.5 is the square root of $e$ . A calculator’s e^x key gives $e^{0.5} \approx 1.648721$ . This number is the growth factor over 10 years when interest is added continuously.
Multiply by the starting amount. The final balance is $1000 \times 1.648721 = 1648.72$ dollars, rounded to the nearest cent. This is the continuous compounding answer, the largest balance this 5% rate can give over 10 years.
Compare with yearly compounding. Once a year, the growth factor is $1.05$ , and ten years give $1.05^{10} \approx 1.628895$ . Multiplying by 1000 gives about 1628.89 dollars, which is less than the continuous answer.
Compare with monthly and daily compounding. Monthly compounding adds $\frac{0.05}{12}$ twelve times a year, so over 10 years it has $12 \times 10 = 120$ steps and gives about 1647.01 dollars. Daily compounding has 3650 steps and gives about 1648.66 dollars. Both sit between the yearly and continuous results, and the numbers move closer to the continuous answer as the steps get shorter.
Find the gap at each end. The continuous answer minus the yearly answer is $1648.72 - 1628.89 = 19.83$ dollars. The continuous answer minus the daily answer is only about 0.06 dollars, or 6 cents. Most of the extra comes from the first steps, and the rest is tiny.

Answer: With continuous compounding the balance after 10 years is about 1648.72 dollars, which is 19.83 dollars more than yearly compounding and about 6 cents more than daily compounding. Real accounts sit a little below this continuous ideal, because they add interest in steps.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Reading e^{rt} as e times rt. The rate and time go in the power, so e^{rt} means e raised to the number rt. With rt equal to 0.5 that is about 1.6487, not 2.718 times 0.5, which would be about 1.36.
Entering the rate as 5 instead of 0.05. The power then becomes 50 instead of 0.5, and the balance comes out far too large. Write 5% as 0.05 first.
Treating continuous compounding as the exact rule for a bank account. Most accounts add interest daily or monthly, so the continuous answer is a slight overestimate for them.
Thinking that e is a rate. The number e is a fixed value of about 2.718, not a percentage. The rate is r, which you choose, and e is the base that growth is measured against.

Where you will meet it

Pricing and risk models in finance, such as the Black-Scholes option pricing formula, which discounts money using e raised to a negative power.
Finance textbooks, which often state interest rates as continuously compounded rates, so that a rate and a time combine into one factor e to the power rt.
Population growth and radioactive decay models, where a quantity changes by a fixed share at every instant.
Checking how much frequent compounding can add at most, because continuous compounding is the upper limit for a given yearly rate.

Questions people ask

Is e a random number?

No. The number e is about 2.718281828, and its digits never end or repeat, much like pi. It has a precise definition as the limit of $\left(1 + \frac{1}{n}\right)^{n}$ as $n$ grows, and it appears whenever growth is applied continuously.

How do I work out e to a power on a calculator?

Look for a key labeled e^x or exp. Type the power, here 0.5, and press the key to get e to the power 0.5, which is about 1.648721. If your calculator has no such key, the power key with 2.718281828 as the base gives nearly the same result.

Does my bank pay continuous compounding?

Usually not. Banks compound in steps such as daily or monthly. Continuous compounding gives a slightly higher balance for the same yearly rate, so it is best read as a theoretical upper limit or as the model used when continuous growth is the natural assumption.

Compound Interest: Compound interest says a savings balance grows faster over time, because each round of interest is added to the balance and then earns interest of its own.
Effective Annual Rate: The effective annual rate says what a stated yearly rate really adds up to over a year, once interest is added more than once and earns interest of its own.
Rule of 72: The rule of 72 says that dividing 72 by a yearly percentage rate gives a quick estimate of the number of years it takes money to double.