The Number e & Continuous Compounding

Continuous Compounding

The more often interest is compounded within a given time period, the more interest will be earned. Surprisingly, however, there is a limit on the amount of interest, no matter how often it is compounded.

Example

Suppose that $1 is invested at 100% interest per year, compounded n times per year. Then the interest rate (in decimal form) is 1.00 and the interest rate per period is LaTeX: \frac{1}{n}.1n.

According to the formula (with P = 1 ), the compound amount at the end of 1 yr will be LaTeX: A=\left(1+\frac{1}{n}\right)^nA=(1+1n)n.

A calculator gives the results shown for various values of n. The table suggests that as n increases, the value of LaTeX: \left(1+\frac{1}{n}\right)^n(1+1n)ngets closer and closer to some fixed number. This is indeed the case. This fixed irrational number is called e. (Note that in mathematics, e is a real number and not a variable.)
Table showing the definition of e.PNG

Value of e

LaTeX: e\approx2.718281828359045e2.718281828359045

Formula for Continuous Compounding

If P dollars are deposited at a rate of interest r compounded continuously for t years, the compound amount A in dollars on deposit is given by the following formula.

LaTeX: A=Pe^{rt}A=Pert

Example

Suppose $5000 is deposited in an account paying 3% interest compounded continuously for 5 yr. Find the total amount on deposit at the end of 5 yr.

Solution

LaTeX: A=Pe^{rt}A=Pert

LaTeX: A=5000e^{0.03\left(5\right)}=5000e^{0.15}\approx5809.17\:or\:\text{ }A=5000e0.03(5)=5000e0.155809.17or $5809.17