How to calculate compound interest n times and continuously.

Question

a total of $12,000 is invested at an annual rate of 9%. Find the balance after 5 years if it is compounded

a. quarterly b. continuously

Answer 1

Quarterly:

A = P(1 + r/n)^nt

A = 12000(1 + 0.09/4)^(4*5)

A = 12000(1.0225)^20

A = 18,726.11

Continuous:

A = Pe^rt

A = 12000*e^(0.09*5)

A = 18,919.75

meaning that you end up with a slightly higher end value when you apply continuous compounding

Answer 2

A = P(1 + r/n)^nt

A = Accrued amount (principal + interest)

P = Principal amount

r = Annual nominal interest rate as a decimal

R = Annual nominal interest rate as a percent

r = R/100

n = number of compounding periods per unit of time

t = time in decimal years; e.g., 6 months is calculated as 0.5 years. Divide your partial year number of months by 12 to get the decimal years.

I = Interest amount

A total of $12,000 is invested at an annual rate of 9%. Find the balance after 5 years if it is compounded

a) quarterly

A = 12000( 1+ .09/4)^4*5

A = 12,000(1 + 0.0225)⁽²⁰⁾

A = $18,726.11

A = P + I where

P (principal) = $12,000.00

I (interest) = $6,726.11

b) continuously

A = Pe^rt

A = 12,000(2.71828)^(0.09)(5)

A = $18,819.75

A = P + I where

P (principal) = $12,000.00

I (interest) = $6,819.75