If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00×1.52 = $2.25 at the end of the year. Compounding quarterly yields $1.00×1.254 = $2.4414…, and compounding monthly yields $1.00×(1+1/12)12 = $2.613035… If there are ncompounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be $1.00×(1 + 1/n)n.
Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals. Compounding weekly (n = 52) yields $2.692597…, while compounding daily (n = 365) yields $2.714567…, just two cents more. The limit as n grows large is the number that came to be known as e; with continuous compounding, the account value will reach $2.7182818…. More generally, an account that starts at $1 and offers an annual interest rate of R will, after t years, yield eRt dollars with continuous compounding. (Here R is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, R = 5/100 = 0.05)
Sitworld Analytics Team