Annuity Due and Present Value Calculation

Chapter 1: Entire Annuity Period

  • The method to solve annuity problems involves splitting the cash flow stream into two parts:

    • The cash flow at time period zero.
    • The remaining cash flows.

  • We can calculate the present value of each component separately.

  • The remaining cash flows often form an ordinary annuity.

  • The value of an N period ordinary annuity is determined one period before the first cash flow.

  • For a four-period annuity due, the present value can be calculated as the sum of:

    • The present value of a three-period ordinary annuity.
    • The cash flow at time period zero.

  • Formula for the present value of an annuity due:

    • PV=PVordinary annuity with N−1 payments+CPV = PV_{\text{ordinary annuity with } N-1 \text{ payments}} + C
    • Where:
      • PVPV is the present value of the annuity due.
      • CC is the constant cash flow.
      • RR is the interest rate per period.
      • NN is the total number of payments.

  • Lottery Example:

    • Option 1: 1,000,000 per year for the next thirty years with the first cash payment today.
    • Option 2: 12,000,000 upfront.
    • Interest rate: 8% per annum.

Chapter 2: Value Of Annuity

  • Annuity due present value formula (revisited):

    • PV=PVordinary annuity with N−1 payments+CPV = PV_{\text{ordinary annuity with } N-1 \text{ payments}} + C
    • Where:
      • C = $1,000,000
      • R=8%=0.08R = 8\% = 0.08
      • N=30N = 30

  • In the lottery example, the present value of the annuity due (calculated using the formula) is a little over 12,000,000.

  • Conclusion: Receiving $1,000,000 per year for 30 years with a cash payment today is preferable to receiving 12,000,000 upfront, based on the present value calculation.