Compound interest

Question

Suppose you are managing an account in which you deposit  $20,000 at the end of each year for 20 years. How much amount you have accumulated with the assumption that you earn 5% interest compounded annually.

Answer 1

Let's define the variables. 

 

a: amount of deposited money each year

r: annual interest rate

 

You deposit the money at the end of each year for 20 yrs. The accumulated amount is calculated as below. 

 

Year: 1st  2nd 3rd               ......................                 19th  20th       value of money at the end of 20th yr

         

          a    ..........................................................................                    a(1+r)¹⁹

                a   ..................................................................                       a(1+r)¹⁸

                       a   ..............................................................                    a(1+r)¹⁷

                                 .................................................

                                                                                a  .....                     a(1+r)¹

                                                                                         a                   a(1+r)⁰

 

So, the accumulated amount (AA) is the sum of the value of money at the end (20th year).  

That is, 

 

AA = a(1+r)¹⁹ + a(1+r)¹⁸ + a(1+r)¹⁷ + .... + a(1+r)¹ + a(1+r)⁰

     = a[(1+r)²⁰ - 1]/[(1+r) -1]

     = a[(1+r)²⁰ - 1]/r

 

a = 20,000

r = 0.05

 

So,

AA = 661,319

 

Answer 2

Yousaf, hello.  The compound interest formula, when compounding annually, is A=P(1+r)^t.  Here, A is the amount that you have in the account after t years assuming a beginning principle amount of P and an annual interest rate of r.  So, we that A=20000(1+0.05)²⁰ = $53,065.95.