Question

The gaming commission is introducing a new lottery game called Infinite Progresso. The winner of the Infinite Progresso jackpot will receive $1,100 at the end of January, $2,500 at the end of February, $3,900 at the end of March, and so on up to $16,500 at the end of December. At the beginning of the next year, the sequence repeats starting at $1,100 in January and ending at $16,500 in December. This annual sequence of payments repeats indefinitely. If the gaming commission expects to sell a minimum of 1,150,000 tickets, what is the minimum price they can charge for the tickets to break even, assuming the commission earns 12.00 %/year/month on its investments and there is exactly one winning ticket?

Answer 1

The jackpot amount is distributed monthly starting from $1100 in the first month and rising by $1400 each month.
The interest rate is 12% per anum so we will calculate the monthly interest rate.

Effective Monthly Rate = ((1+Nominal Rate) ^ (1/Compounding Frequency)) - 1

(1.12 ^ (1/12)) - 1 = 0.0095

We will have to calculate the PV of the monthly jackpot amount
The base amount is $1100 while the gradient amount is $1400.

1100 * (P/A, 0.95%, 12) + 1400 * (P/G, 0.95%, 12)
= (1100 * 11.2907) + (1400 * 60.8270)
= 12419.79 + 85157.85
= 97577.64

This is the cost which is perpetual so we will calculate its present worth.

PW of Perpetuity = PMT / Interest Rate

97577.64 / 0.12 = 813147.02

Total tickets = 1150000

813147.02 / 1150000 = 0.7071 or 0.71

The minimum price of the ticke should be $0.71