Present Value of Annuity Notes

Present Value of an Ordinary Annuity

  • The formula for the present value of an ordinary annuity is used to calculate the present value of a series of constant cash flows.
  • Formula: PV=C∗1−(1+r)−nrPV = C * \frac{1 - (1 + r)^{-n}}{r}
    • Where:
      • CC = Constant cash flow per period
      • rr = Interest rate per period
      • nn = Number of periods

Lottery Example: Choosing Between Annuity and Upfront Payment

  • Scenario: You win a \$30,000,000 lottery and have two options:
    • Option 1: Receive \$1,000,000 per year for the next 30 years, starting in one year.
    • Option 2: Receive \$12,000,000 upfront.
  • Interest rate: 8% per annum.
  • Question: Which option is preferable?

Analysis

  • To make the best decision, compute the present value of the 30 annual payments and compare it to the \$12,000,000 upfront cash payment.
  • Given:
    • C = $1,000,000
    • r=0.08r = 0.08 (8% per annum)
    • n=30n = 30
  • Calculation:
    • PV = $1,000,000 * \frac{1 - (1 + 0.08)^{-30}}{0.08}
    • PV ≈ $1,000,000 * \frac{1 - (1.08)^{-30}}{0.08} ≈ $11,257,783

Conclusion

  • The present value of receiving \$1,000,000 per year for 30 years is approximately \$11,257,783.
  • Since \$11,257,783 < $12,000,000$$, taking the \$12,000,000 upfront is the better financial choice.