If r per annum is the rate at which the principal A is compounded annually, then at the end of k years, the money due is

         Q = A (1 + r)^k

Suppose compounding is done continuously. i.e. at every instant the principal A is compounded at R per annum. Then,

         Q = A e^Rk

The relationship between R and r is given by the following reasoning:

         A (1 + r)^k = A e^Rk

Example 

If R   = 5.25%, then ln(1 + r) = 5.25% or r = 5.39%

Example 

Suppose Rs.100 is being compounded annually at the rate of 10% per annum. What is the future value of Rs.100 at the end of the third year? What is the effective continuously compounded rate of interest? What is the future value of Rs.100 at the end of the third year, using this interest rate?

FV(Rs.100) = 100 x (1.10)³  = 133.1

If r = 0.1, then the continuously compounded rate of interest R is given by

R = ln(1 + 0.1) = 0.0953

FV(Rs.100) = 100 e^{0.0953 x 3} = 100 x 1.331 = 133.1