Compound Interest

• Calculate A for P=1000, r = 10%, t = 5 years.
• Do this again, but if we take the money out after 1 year, ...
• This is the idea of compound interest.
• Compounding annually, monthly, weekly, daily, ...
• Compounding continuously.
• A = p(1+r/n)^nt
  • APrt are all as before
  • n is the number of annual compounding periods.
• What happens when n = 1?
• annually - 1 time per year
• Semiannually - twice per year
• Quarterly - four times per year.
• monthly, daily ...
• Examples

  Problem 2, P = $8000, r = 6%, T = 3, n = 1
     
     A = 8000*(1+.06/1)(1*3)  = $9,528.13
  
  Problem 4, P = $4000, r = 4%, t = 2, n = 5
     A = 4000*(1+.04/2)(2*5)  = $4,875.98
  
  Problem 6, P = 2500, r = 8%, t = 6 years, n = 4
     A = 2500*(1+.08/4)(4*6)  = $4,021.09
  
  Problem 8, P = 2500, r = 6.5%, t = 4 years, n = 12
     A = 2500*(1+.065/12)(12*4)  = $3,240.05
      

• We can also compute compounded continuously
• A = Pe^rt
• e = 2.71828182845904... a number like pi.
• It should be on your calculator (e^x)
• Example:

  Repeat the last problem, but compounded continuously
  
     A = 2500 e.065*4 = 3242.33
      

• If we know A, nrt and want to find P it is just algebra.
• P = A/(1+r/n)^nt
• Example:

  Problem 18
       P = ?, A = 12,000, t = 4, n = 2, r = 7%
  
       P = 12,000/ (1+.07/2)2*4 = 9,112.94
      

• Effective Annual Yield is the simple interest rate that produces the same amount of money at the end of one year as when the account is subjected to compound interest at a stated rate.
• The stated rate is called the nominal rate
• Y= (1+r/n)ⁿ -1
• Example

  Problem 24
      r = 6%, n = 4
      Y = (1+.06/4)4-1 = 6.14%
      

• If you are selecting between two investments, the best choice is the one with the greatest effective annual yield.