## Continuous Compounding of Interest

If the interest rate for a period, e.g., one year, is i, then a loan of A dollars should obtain an interest of Ai at the end of the period. The amount of money given back at the end of the period is

If the borrower keeps the money for another year, then he has to pay interest on A(1+i) dollars. Hence, at the end of the second year he owes

If we continue doing this, we obtain the general equation of compounded interest:

Where An is the total of the loan plus interest gained to the end of period n. Effective Rate of Interest

From equation 1.3, we can see that the interest gained over n periods is

Since the original investment was A, the interest rate is .¿o+ir a

This is called the effective interest rate, and equation 1.4 is written as Example 1.1

A local bank announces that a deposit over \$1,000 will receive a monthly interest of 0.5%. If you leave \$10,000 in this account, how much would you have at the end of one year?

According to equation 1.3,

A u - 10,000 (1+0.005)!î ■ lOtOOQ (!.062) = 10,620

This means that over a one-year period, \$620 has been added to our money, which is the same as a 6.2% annual interest rate. We can see that this is not 12 times the monthly interest rate of 0.5% which is 6%. The difference between the 6.2% and 6% rates is the result of compounding monthly rather than annually.