Continuously Compounded Interest

Q: Why is it called the 'Pert' or 'shampoo' formula?

The formula $A = Pe^{rt}$ spells out 'P·e·r·t' when written in sequence, which matches the name of the shampoo brand Pert®. This mnemonic helps students remember the formula. Despite the playful name, the formula is used throughout mathematics, science, and finance.

Q: When do you use the continuously compounded interest formula?

You use $A = Pe^{rt}$ whenever a problem states that interest is compounded "continuously." The same formula applies to any exponential growth or decay scenario — such as population growth, radioactive decay, or temperature change — where the rate of change is proportional to the current amount.

Continuously Compounded Interest

Interest that is, hypothetically, computed and added to the balance of an account every instant. This is not actually possible, but continuous compounding is well-defined nevertheless as the upper bound of "regular" compound interest. The formula, given below, is sometimes called the shampoo formula (Pert^®).

Note: This same formula can be used for exponential growth and exponential decay.

Formula: A = Pe^(rt), where A = final amount, P = principle or original amount, r = rate of interest per year, t = time in years
Example: $2000 deposited at 12%/year compounded continuously for 7 years. A = 2000e^(0.12×7) = $4632.73

See also

Half-life, doubling time

Key Formula

A = Pe^{rt}

Where:

$A$ = The final amount (principal plus interest) after time t
$P$ = The principal — the initial amount of money invested or borrowed
$e$ = Euler's number, approximately 2.71828
$r$ = The annual interest rate expressed as a decimal (e.g., 5% = 0.05)
$t$ = The time in years

Worked Example

Problem: You invest $2,000 at an annual interest rate of 6%, compounded continuously. How much will you have after 5 years?

Step 1: Identify the known values: principal, rate, and time.

P = 2000, \quad r = 0.06, \quad t = 5

Step 2: Write the continuously compounded interest formula.

A = Pe^{rt}

Step 3: Substitute the values into the formula.

A = 2000 \cdot e^{(0.06)(5)} = 2000 \cdot e^{0.3}

Step 4: Evaluate

e^{0.3}

using a calculator.

e^{0.3} \approx 1.34986

Step 5: Multiply to find the final amount.

A \approx 2000 \times 1.34986 = 2699.72

Answer: After 5 years, the investment grows to approximately $2,699.72.

Another Example

This example works backward from a target amount, solving for the principal instead of the final balance. It shows how to rearrange the formula, which is a common variation on tests and in real financial planning.

Problem: You want to have $10,000 in an account that earns 4% interest compounded continuously. How much do you need to deposit now if you plan to wait 8 years?

Step 1: Identify the known values. Here the final amount

A

is known, and you need to find

P

A = 10000, \quad r = 0.04, \quad t = 8

Step 2: Rearrange the formula to solve for the principal

P

P = \frac{A}{e^{rt}}

Step 3: Compute the exponent.

rt = (0.04)(8) = 0.32

Step 4: Evaluate

e^{0.32}

and divide.

e^{0.32} \approx 1.37713 \quad \Rightarrow \quad P \approx \frac{10000}{1.37713} \approx 7261.49

Answer: You need to deposit approximately $7,261.49 now to have $10,000 in 8 years.

Frequently Asked Questions

What is the difference between continuously compounded interest and compound interest?

Standard compound interest compounds at fixed intervals — monthly, quarterly, or annually — using the formula

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

. Continuously compounded interest is the theoretical limit of this formula as the number of compounding periods

n

approaches infinity, which simplifies to

A = Pe^{rt}

. Continuous compounding always yields a slightly higher amount than any finite compounding frequency, but the difference is often small.

Why is it called the 'Pert' or 'shampoo' formula?

The formula

A = Pe^{rt}

spells out 'P·e·r·t' when written in sequence, which matches the name of the shampoo brand Pert®. This mnemonic helps students remember the formula. Despite the playful name, the formula is used throughout mathematics, science, and finance.

When do you use the continuously compounded interest formula?

You use

A = Pe^{rt}

whenever a problem states that interest is compounded "continuously." The same formula applies to any exponential growth or decay scenario — such as population growth, radioactive decay, or temperature change — where the rate of change is proportional to the current amount.

Continuously Compounded Interest vs. Compound Interest (n times per year)

	Continuously Compounded Interest	Compound Interest (n times per year)
Formula	$A = Pe^{rt}$	$A = P\left(1 + \frac{r}{n}\right)^{nt}$
Compounding frequency	Infinite (every instant)	Finite: annually ( $n=1$ ), monthly ( $n=12$ ), daily ( $n=365$ ), etc.
Result for same rate and time	Slightly higher — the theoretical maximum	Slightly lower — increases as $n$ increases
Key constant	Uses $e \approx 2.71828$	No special constant needed
Typical use	Finance, calculus, natural growth/decay models	Banking, loans, most real-world savings accounts

Why It Matters

Continuously compounded interest appears in algebra, precalculus, and calculus courses as the primary example of exponential growth. Banks and financial institutions sometimes quote continuously compounded rates, so understanding the formula helps you compare investment options accurately. Beyond finance, the same

A = Pe^{rt}

formula models radioactive decay (with a negative

r

), population growth, and Newton's law of cooling, making it one of the most widely applied formulas in mathematics and science.

Common Mistakes

Mistake: Using the percentage directly instead of converting to a decimal — for example, plugging in

r = 5

instead of

r = 0.05

Correction: Always divide the percentage rate by 100 before substituting into the formula. An interest rate of 5% means

r = 0.05

Mistake: Confusing the continuously compounded formula

A = Pe^{rt}

with the standard compound interest formula

A = P(1 + r/n)^{nt}

and trying to include

n

in the continuous version.

Correction: The continuous formula has no

n

because compounding happens infinitely often —

e^{rt}

already accounts for that. If a problem says "compounded continuously," use

A = Pe^{rt}

with no

n

variable.

Related Terms

Compound Interest — Finite-frequency version that continuous compounding generalizes
Interest — Broad concept covering simple and compound interest
Exponential Growth — Same formula models growth with a positive rate
Exponential Decay — Same formula with a negative rate models decay
Half-Life — Time for a quantity to halve, derived from $A = Pe^{rt}$
Doubling Time — Time to double an investment, found by setting $A = 2P$
Upper Bound of a Set — Continuous compounding is the upper bound of compound interest
e (Euler's Number) — The base of the natural exponential used in the formula