Question

You have been hired as the financial manager to design a new Powerball jackpot. The marketing team, after a thorough market research, has found that offering an infinite annuity would attract a new market segment. This program is expected to attract at least 2,000,000 participants. The company is planning to sell each ticket for $2 each.

The cash flow is as follows:

• During the first year, every 6-months, $100,000 will be given to the winner

• For the second year and up to year 10, this payment will increase by 10%. So, the

  second year payment would be $110,000 every 6-month, the third year payment

  would be $121,000 every 6-months, etc.

• All the cash flow repeats the same process every 10 year infinitely long. (start with

  100,000 at year 11 and increase by 10% every year)
  The company interest rate is 12% compounded semi-annually.
  What is the lowest ticket price that make this lottery attractive for your company?

Answer 1

Cashflows: First year's two payments 6 months apart = $100,000

Growth rate for next year's payment = 10%, which means 2nd Year's two Payments 6 months apart = $110,000

Similarly, 3rd year's two payments 6 months apart = $121,000

To calculate the Present Value of such a payment series, we will consider each payment payments for an year, then for 10 years and then finally for Infinite.

First we will calculate what effectively would be paid out at the end of First year.

Value of First Year's Payments at the end of 1st year = First payment of year * (1 + Rate/2) + Second Payment of year

Rate of return = 12% | Semi-annual year rate = 6%

Effective annual rate = (1+Semi-Annual rate)² - 1 = (1+6%)² - 1 = 12.36%

Now we can put values and calculate Value of First Year's payments

Value of First Year's payment at the end of 1st year = 100,000*(1+6%) + 100,000 = 106,000 + 100,000

Value of First Year's payment at the end of 1st year = 206,000

Now 206,000 is effectively the First Year payment. We have converted the semi-annual payment into annual payment and we have also calculate the EAR of 12.36% for our further calculations.

We know that cashflow will increase by 10% over previous year's payments till Year 10, hence, their Year end effective payments would also increase at same growth. Therefore, we can use Growing Annuity formula to calculate the Present Value of these payments for next 10 years.

PV of Growing Annuity Formula in our context = (CF at Year 1 / (EAR - Growth))*(1 - ((1+Growth)/(1+EAR))¹⁰)

PV of 10 years payments = (206,000 / (12.36% - 10%))*(1 - ((1+10%)/(1+12.36%))¹⁰)

PV of 10 years payments = 8,728,813.559322 * 0.19125884

Present Value of 10 years Payments = $1,669,462.76

As the 10 years payments will go on infinitely, therefore, PV of 10 years payments becomes the Cashflow for our Perpetuity, where Rate of return would our EAR of 12.36%

PV of Perpetuity formula = Cashflow / Rate of return

Value of all the payments for winner = PV of 10 years Payments / EAR

Value of all the payments for winner = 1,669,462.76 / 12.36%

Value of all the payments for winner = $13,506,980.27

As the company is expecting to sell at least 2,000,000 tickets, the minimum ticket price company should sell tickets on would be Value of all payments for winner divided by number of expected ticket sales.

Minimum Price for Ticket = Value of all the payments for winner / Expected number of ticket sales

Minimum Price for Ticket = 13,506,980.27 / 2,000,000

Minimum Price for Ticket = $6.75

The company can set lowest price at $6.75 which breaks-even the winner payments, any price below $6.75 would be unattractive for the company. Therefore, company should set a price above $6.75.