Compound Interest

• What if you put money into the bank. 1000 at 4% interest for 3 years.
  • If you use simple interest you get 1000×0.04×3 = 120
  • What if you went to the bank once a year, withdrew the money, to collect the interest, and redeposit the entire thing.?

    i = prt   
    Year 1:  t = 1, p = 1000, r = 0.04   i = 1000*1*0.04 = 40
    Year 2:  t = 1, p = 1040, r = 0.04   i = 1040*1*0.04 = 41.6
    Year 3:  t = 1, p = 1081.60  r = 0.04, i = 1081.60 * 1 * 0.04 = 43.26
    
    Total Interest = 124.86  
    a difference of 4.86
    

  • What happened here? You earned interest on the interest.
  • This is called compounding, and we compounded annually here.
  • Would it make a difference if we compounded monthly?
    • We could work it out by hand, but we would have 36 lines.
    • We could use a computer, but that is a different class.
    • We could look for a new formula.
  • A = P(1+r/n)^rt
    • A = future value
    • P = present value
    • r = interest rate, annual
    • n = number of compounding per year
    • t = number of years.
  • Let us try the annual compounding

    P = 1000
    r = 0.04
    t = 3
    n = 1
    
        A = 1000(1 + 0.04/1)1*3
          = 1000(1.04)3
          = 1000(1.12486)
          = 1,124.86
           

  • So if we deposit $1000 at 4% annual interest, compounded monthly

    P = 1000
    r = .04
    t = 3
    n = 12
    
           A = 1000(1+.04/12)3*12 
             = 1000(1.003)36
    	 = 1000(1.12727)
    	 = 1127.27
           

  • Or what if we wanted daily compounding?

    P = 1000
    r = 0.04
    t = 3
    n = 360
    
          A = 1000(1+.04/360)360*3
            = 1000(1.0001)1080
    	= 1000(1.12749)
    	= 1,127.49
           

  • This can be done at finer and finer levels until we get to continuously compounded, but we need calculus for this.
• Sometimes, we are asked, if we want $10,000 on five years, compounded quarterly at an interest rate of 5.3%, how much should we deposit now.

  A = 10,000
  P = ?
  n = 4
  r = .053
  t = 5
  
      A = P(1+r/n)nt
  
      P = A/(1+r/n)nt
  
      P = 10,000(1+0.053/4)4*5
        = 10,000(1.01325)20
        = 10,000/1.30
        = $7,685.22
  

• This is also known as finding the present value.
• If we have $5,000, at 4.25% interest, compounded semiannually for one year, what is the effective annual yield?
  • This is the simple interest rate
  • Find this by computing the compounded interest

    A = ?
    P = $5,000
    r = 0.0425
    t = 1
    n = 2
    
         A = P (1+r/n)rt
         A = 5,000 (1+.0425/2)2
           = 5,000 (102125)2
           = 5,214.76
    
    We earned $214.76
    	 

  • Now compute what the simple interest rate needed to be to earn this.

    i = prt
    p = $5000
    i = $214.76
    t = 1
        r = i/pt
        r = 214.76(1*5000)
        r = .004295 or 4.30%
    	 

• He gives us a formula Y = (1+r/n)ⁿ -1

  Y = (1+.0425/2)2 - 1
    = 1.04295 - 1 
    = 4.30%
       

Homework