Investments

• An annuity is an account into which, or out of which, a sequence of scheduled payments are made.
• We will deal with fixed annuities, where the payments, period, compounding periods, and other details are fixed.
• A ordinary annuity or fixed annuity is used to save for something
  • retirement, college education, ...
• The money at the end of the annuity is the accumulated amount or future value.
• We compute this with A = p[(1+r/n)^(nt)-1] / (r/n)
  • A is the accumulated amount
  • p is the periodic payment
  • r is the interest rate
  • n is the number of compounding per year
  • t is the number of years
• $10 is invested semiannually for 15 years at 4.5% compounded semiannually. What will the accumulated value be.

         p = $10
         n = 2
         r = 4.5%     A = 10((1+.045/2)^(2x15)-1)/(0.045/2)
         t = 15         = $421.95
      

• We might want to work the other way.
• Semiannual payments of 6% interest are compounded semiannually for 4 years to accumulate 70,000. What is the monthly payment to achieve this?
• We can solve for p. p = A(r/n)/((1+r/n)^(nt)-1)
• This is called a sinking fund

•     p = ?
      n = 2
      r = 6%         p = 70,000(.06/2)/((1+.06/2)^(2x4)-1)
      t = 4 years      = 7,871.95
      A = 70,000
      

• To save for retirement, a student invests $30 each month in an ordinary annuity with 3% interest compounded monthly. Determine the amount of money in the account after 20 years.

  p = 30
  r = 3%    A = 30((1+.03/12)^(12*20)-1)/(.03/12)
  n = 12      = 9,849.06
  t = 20