Compound Interest

• The last section dealt with simple interest.
  • This is mostly a type of loan you would take out.
• This section deals with compound interest
  • This is more likely to be an investment you make.
  • A CD -Certificate of Deposit is an example of this type of loan.
  • 
• I got $1000 for my birthday.
  1. I invest this for 3 years at 1.35% interest, on Jan 1, 2017

     On Jan 1 2020
     Simple Interest
     p = 1000
     r = 1.35%
     t = 3 years.
     
     i = 1000 x 0.0135 x 3 = 40.50
     
     For a total of 1040.50
     	

  2. But what if I played a trick on the bank?
     • I deposit the money on Jan 1, 2017
     • I withdraw the money on Jan 1 2018

       Simple Interest
       p = 1000
       r = 1.35%
       t = 1 year
       
       i = 1000 x 0.0135 x 1 = 13.50
       	     

     • Then I deposit all of this and do the same thing in Jan 2019

       p = 1013.50 (Why?)
       r = 1.35%
       t = 1 year.
       
       i = 1013.50 x 0.0135 x 1 = 13.68 (Why is this higher by 18 cents?)
       	     

     • And I do this again one more time.

       p = 1027.18
       r = 1.35%
       t = 1 year.
       
       i = 1027.18 x 0.0135 x 1 = 13.87
       	     

     • In the end I have $1,041.05
     • What was the actual interest rate?

       p = 1000
       i = 41.05
       t = 3 years
       
       r = i/pt
       
              41.05/(1000*3) = 1.37%
       	     

• Where did the extra 55 cents come from?
• Would this change if I did it twice a year?
• How about four times a year? ...
• This is an example of compound interest.
  • There is a formula to compute compound interest
  • A = p(1+r/n)^nt
    • A is the final amount = p + i
    • p is the principal
    • i is the interest
    • r is the compound interest rate
    • t is the time
    • n is the number of compounding per year.
      • Annually, n = 1
      • semiannually, n= 2 see about 2:40 in
      • Quarterly, n = 4
      • Monthly, n = 12
      • Daily, n = 360
  • So for my birthday

    p = 1000
    r = 1.35%
    t = 3 years
    n = 1
    
    A = p(1+r/n)^(nt)
         A = 1000 * (1 + 0.0135/1)^(1*3)
         A = 1041.05 
    	

• Calculator 1000 x (1 + 0.0135/ 1) ^ (1 * 3)
• Do some examples, P 627 1-14
• We might do the reverse of this computation.
  • If I want an $10,000 in 5 years at an investment ration of 1.4% compounded monthly, how much must I invest now?
  • We know A, r, t, and n, and want to find p
  • So we solve for p.
  • p = A/(1+r/n)^(nt)

    A = 10000
    r = 1.4%
    n = 12
    t = 5
    
       p = 10000/(1+0.014/12)^(12*5) = 9,324.31
    	

• Do some problems Page 627-629