DashboardNewsSettingsHelp
Get Premium
Dashboard
Dashboard Sign out
View details

Rule

Continuously Compounded Interest

When interest is compounded infinitely many times, it is said to be continuously compounded. Let A be the balance of an account that is continuously compounded, P the initial amount, r the interest rate, and t the time. These values are connected by the following formula.

A=Pe^(rt)

Keep in mind that, in this formula, the value of r must be written as a decimal and the time t must be in years. Also, the initial amount P is usually called principal.

Proof
Recall the compound interest formula. A = P (1+r/n)^(nt) In the formula, there is the account's balance A, the initial amount P, the time in years t, and the number of times n the interest is compounded per year. The formula can be rewritten by using the Power of a Power Property. A = P (1+r/n)^(nt) ⇕ A = P [ (1+r/n)^n ]^t If the interest rate is 100 %, or r=1, and the interest is continuously compounded, the formula can be written in terms of P, t, and e. This is because as n goes to infinity, the value of (1+ 1n)^n approaches e. A = P [ (1 + 1/n)^n ]^t [n→ ∞] A = P e^t The value of the expression (1+ rn )^n, on the other hand, approaches e^r. Examine the following interactive graph to better grasp how this is possible.

graphs of $y =e^r$ and $y=(1+r/n)^n$

Using this last approximation, the final form of the formula can be obtained. A = P [ (1 + r/n)^n ]^t [ n→ ∞ ] A = P e^(r t)

Exercises

Recommended exercises

To get personalized recommended exercises, please login first.
Return to lesson.
Edit Lesson