Compound Interest

• We are going to look at investments for a this section.
• In this case, you put the money into an account, and withdraw it some time later.
• One method would be just simple interest.
• Example: I have $1000, I deposit for two years at 5% interest.

  A = p + prt  (or A = p(1+rt))
    = 1000 (1 + 0.05 × 2)
    = 1100
     

• What if I decided to take the money out once a year and redeposit it
• Example 2:

  Year 1:
     A = 1000(1+0.05 × 1) 
       = 1050
  Year 2:
     A = 1050(1+0.05 × 1) 
       = 1102.50
     

• What happened in the previous example? Compound interest
• There is a formula, which we will investigate in a moment

     A = p(1+ r/n)nt
        r = interest rate
        n = number of compoundings per year
        t = number of years
        p = principal
     

• Example 3:

     In the previous example, n = 1, t = 2
     A = 1000(1+.05/1)(1× 2)
       = 1102.5
     

• Example 4:

        p = 5000
        r = 5%
        t = 9
        n = 1 (annually)
  
        A = 5000(1+.05/1)^(1*9)
          = 7756.64
        i = A-p
          = 5000 -7756.64 = 2756.64
     

• Other compounding periods

  Name	n
  annually	1
  semiannually	2
  quarterly	4
  monthly	12
  daily	360

• Example: $2500 is deposited at 4 1/4 percent interest compounded daily for three years. How much interest is earned?

     p = 2500
     r = 4.24
     n = 360
     t = 3
  
     A = 2500(1+.0425/360)(3× 360)  
       = 2839.94
     i = A -p 
       = 2839.94 - 2500 = 339.94
     

• APY or annual percentage yield, or effective annual yield.
• APY is the simple interest rate that gives the same amount of interested as the compound rate over the same period of time.
• This can be solved in general:

      r = 3.5%, n=2
      Assume p = 1 and t = 1
      A = 1(1+0.035/2)(1× 2) 
        = 1.0353
        or 3.53%
     

• Sometimes, we have a goal.
• We would like to have 100,000 in 20 years, a bank offers 9.5% interest compounded quarterly. How much must we deposit now?

     A = 100,000
     p = ?
     r = 9.5%
     n = 4
     t = 20
  
     A = p(1+r/n)(nt)
     p = A/(1+r/n)nt
     p = 100000/(1+0.05/4)(4*20)
       = 15,292.79
     

• This is called present value.
• To derive the Compound interest formula:

  use A=p(1+rt)
     
  assume n = 2 and t = 1
    The first 1/2 year     The second 1/2 yeaer
       A = p+pr 1/2          A = p(1+r/2)(1+r/2)
         = p(1+r/2)            = p(1+r/2)2
  
  Assume n = 3 and t = 1
     The first 1/3       The second 1/3     The last 1/3
     A = p(1+r 1/3)    A = p(1+r/3)(1+r/3)  A = p(1+r/3)2(1+r/3)
       = p(1+r/3)        = p(1+r/3)2          = p(1+r/3)3
  
  Assume n=3, t = 2
     Each time we will have 2/6 years or 1/3 year 
     The first 1/6        The second 1/6    The third 1/6
     A = p(1+r 1/3)    A = p(1+r/3)(1+r/3)  A = p(1+r/3)2(1+r/3)
       = p(1+r/3)        = p(1+r/3)2          = p(1+r/3)3
  
     The fourth 1/6          The fifth 1/6          The last 1/6
     A = p(1+r/3)3(1+r/3)   A = p(1+r/3)4(1+r/3)   A = p(1+r/3)5(1+r/3)
       = p(1+r/3)4            = p(1+r/3)5            = p(1+r/3)6