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 1n.
According to the formula (with P = 1 ), the compound amount at the end of 1 yr will be A=(1+1n)n.
A calculator gives the results shown for various values of n. The table suggests that as n increases, the value of (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.)
Value of e
e≈2.718281828359045
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.
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
A=Pert
A=5000e0.03(5)=5000e0.15≈5809.17or $5809.17