Question

Titus Tribble wins the big Powerball lottery which pays $17 million at the beginning of each of the next 30 years. The reported prize is $510 million which is just the total of these 30 annuity payments. What is the present value of this prize today at a 5% annual rate? Express your answer in terms of millions of dollars and round the answer to the nearest tenth of a million, for example 17.0 for 17 million.

Answer 1

The present value of the $510 million prize is approximately $229.5 million.

Step-by-step explanation:

To calculate the present value of the prize, we need to determine the value of receiving $17 million at the beginning of each of the next 30 years, discounted back to today. We can use the formula for present value of an annuity: PV = PMT * ((1 - (1 + r)^-n) / r), where PV is the present value, PMT is the annual payment, r is the interest rate, and n is the number of years.

In this case, the annual payment is $17 million, the interest rate is 5% (or 0.05 in decimal form), and the number of years is 30. Plugging in these values, we get PV = 17 * ((1 - (1 + 0.05)^-30) / 0.05), which equals approximately $229.5 million. Rounding to the nearest tenth of a million, the present value of the prize is $229.5 million.

Answer 2

Present value of the prize today rounded to the nearest tenth of a million:

274.4 millions

Step-by-step explanation:

We will calculate the present value of the 30 years annuity-due of 17 millions at 5% discount rate

`C * (1-(1+r)^(-time) )/(rate) * (1+r)= PV\\`

C 17,000,000

time 30 years

rate 5% = 5/100 = 0.05

`17000000 * (1-(1+0.05)^(-30) )/(0.05) times (1+0.05)= PV\\`

PV $274,398,250.83

Rounding: 274.4 millions

The additional term (1+ r) is added because the cash received capitalize for an additional period, so it generates a higher present value.