Continuous Compound Interest

Most bank accounts add interest monthly or daily. Now imagine a bank that adds it every second, then every millisecond, then at every instant. You might expect the balance to explode. It doesn’t. It settles at a perfectly finite number, and that number is described by one of the neatest formulas in algebra, continuous compound interest: . It also explains where the mysterious number comes from, which is why it shows up in an algebra course and not just in finance.

What Is Continuously Compounded Interest?

With ordinary compound interest, the bank calculates interest at fixed moments (once a year, once a quarter, once a month) and adds it to the balance. From then on, that interest earns interest of its own. The more often this happens, the sooner each bit of interest starts working.

Continuously compounded interest pushes this to the limit: the number of compounding periods grows without bound, so interest is being added at every instant. The balance no longer climbs in small steps. It rises along a smooth exponential curve.

In practice, very few savings accounts literally compound continuously. The formula matters because it is the ceiling that every other compounding schedule approaches. It is also the model scientists use for anything that grows or decays smoothly, from bacteria cultures to radioactive samples.

The Continuous Compound Interest Formula

The interest earned on its own is the final amount minus what you put in: .

Notice that only the product matters. for years and for years both give , so they multiply your money by exactly the same factor, .

Where e Comes From

Start with the general compound interest formula, where interest is added times a year:

Now run a thought experiment. Put in an account paying a wildly generous a year. Compounded once, you end the year with . Compounded quarterly, each quarter adds to the current balance, and you finish with . Monthly gives . Keep going:

The balance keeps rising but the increases shrink, and the values close in on a ceiling of about . That ceiling is the number :

The same limit works for any rate, and that is how the general formula drops out: as , approaches . That turns into .

How Much Extra Does Continuous Compounding Earn?

Less than most people expect. Here is at for one year under every common schedule:

Going from annual to monthly compounding is worth . Going from daily to continuous is worth less than half a cent. So the real lesson is that how often you compound matters a little, while the rate and the time matter a lot.

Simple vs Compound vs Continuous Interest

These three are easy to blur together, so here they are side by side:

The difference between simple and compound interest is small at first and enormous later. Over three decades it is the difference between a line and a curve:

For the first ten years the three lines hug each other. Simple interest gives , yearly compounding about , continuous about . By year thirty, simple interest has reached while compounding has passed . That gap is interest earning interest, year after year.

Continuous Compound Interest Table

Every continuous-compounding problem boils down to the growth factor . Multiply the principal by the factor in the table to get the final amount:

Some entries repeat across the table, such as for over years and for over years. That is the “only matters” rule at work. And for years gives exactly .

Effective Annual Rate (APY)

Banks often quote a nominal rate together with an effective annual rate, also called the annual percentage yield (APY). That is the plain once-a-year rate that would produce the same growth. For continuous compounding:

A rate compounded continuously has , or about . Here are a few more:

The effective annual rate is the fair way to compare offers that compound on different schedules. Convert each one to an EAR, then pick the biggest.

Solving for Time, Rate or Principal

Sometimes you know the final amount and want something else: how long it takes, what rate you need, or how much to deposit now. The unknown time and rate sit in the exponent, so you bring them down with the natural logarithm, using the fact that .

Finding the principal from the amount is sometimes called reverse compound interest, or present value. It answers questions like “how much do I need to invest today to have in ten years?” The steps for time and rate are the same technique you use for solving exponential equations in general.

Doubling Time

Set and the principal cancels:

Written with the rate as a percentage, that is the rule of 69.3: divide by the rate. At , money compounded continuously doubles in years, whatever the starting amount. People often round to the “rule of 70” or “rule of 72” for mental math.

Worked Example

You deposit in an account paying interest compounded continuously. How much is in the account after years, and how much of that is interest?

Step 1 — list the values, with the rate as a decimal:

Step 2 — multiply the rate by the time:

Step 3 — find the growth factor:

Step 4 — multiply by the principal:

Step 5 — subtract the principal to get the interest:

The account holds about , of which is interest. For comparison, monthly compounding at the same rate gives , just less after twelve years.

Continuously Compounded Interest Word Problems

The last row shows that the formula runs in reverse too. With a negative exponent, is continuous exponential decay, the same model used for depreciation, cooling and radioactive half-life. The exponential growth and decay lesson covers the step-by-step versions of these models.

Continuous Compound Interest in Excel and Google Sheets

Spreadsheets have built in as the EXP function, so the formula translates directly:

With your inputs in cells (principal in A1, rate in B1, years in C1), the continuous version is =A1*EXP(B1*C1). Enter the rate as 0.05 or as 5%. Both work, but typing 5 would mean .

Common Mistakes to Avoid

• Using the percentage instead of the decimal. is . Plugging in gives , which multiplies your money by in a single year.
• Mixing up months and years. The formula assumes is an annual rate, so must be in years. Eighteen months is , not .
• Using the -formula with . You can’t plug infinity into . Continuous compounding has its own formula, .
• Reporting when the question asks for interest. “How much interest is earned?” means , not .
• Rounding too early. Rounding to instead of throws a answer off by about . Keep at least five or six decimal places until the last step.

Practice Problems

Try each one before opening the answer.

Problem 1. is invested at compounded continuously. What is it worth after years?

Problem 2. What is the effective annual rate of compounded continuously?

Problem 3. How long does it take money to double at compounded continuously?

Problem 4. How much must you invest today at compounded continuously to have in years?

Problem 5. An investment compounded continuously grew from to in years. What was the annual rate?

Problem 6. Which is better for over years: compounded monthly, or compounded continuously?

Problem 6 sums up this lesson. Continuous compounding is the mathematical limit of compound interest, and it’s elegant, but the rate and the time you leave the money alone matter far more. To see how this formula fits the wider family of growth models, continue to exponential growth and decay. To see why is the natural partner of , read common and natural logarithms.