10. As of this morning, the New Mexico Lottery power ball jackpot was reportedly worth $111 million. The $111 million dollar value is based on an annuity paying $5.55 million per year for 20 years. Using a discount rate of 3%, what is the present value of the grand prize? Explain how you calculated this.
To calculate the present value of the grand prize, we need to use the concept of the time value of money, specifically the formula for the present value of an annuity.
The formula for the present value of an annuity is:
PV = C * [1 - (1 + r)^(-n)] / r
Where:
PV = Present value
C = Cash flow per period
r = Discount rate
n = Number of periods
In this case, the cash flow per period is $5.55 million, the discount rate is 3%, and the number of periods is 20 years. Plugging in these values into the formula, we can calculate:
PV = 5.55 * [1 - (1 + 0.03)^(-20)] / 0.03
Now, let's calculate this using a calculator:
1 + 0.03 = 1.03
-20 = -20
(1.03)^(-20) ≈ 0.557
5.55 * [1 - 0.557] ≈ 2.43
2.43 / 0.03 ≈ 81
Therefore, the present value of the grand prize is approximately $81 million.