Question

4. The state lottery claims that its grand prize is $1 million. The lucky winner will receive $100,000 upon presentation of the winning ticket plus $100,000 at the end of each year for the next 9 years. Assume a 8% discount rate.

a-Why isn't this really a million-dollar prize?

b. What would it actually be worth in dollars to you?

c. What would the 10 yearly payments need to be for the present value of the lottery to be $1 million? Hint: The first step is to determine what type of an annuity the lottery represents.

Answer 1

• a) As we will get the amount in the coming years, we will be losing out on interest that we would have earned had we got the lump sum at the time of presentation. Therefore we need to discount the future amounts to bring it back to time 0 before adding it up. This will reduce the amount of each payment. And this is the reason that this is not a 1 million dollar prize
• b) First of all we should understand that this is an annuity due as the first payment is made today ie at the time of presentation of the winning ticket
  • Next to find the PV we need to follow the below steps
  • We are given the following information

  • PMT	100000
    r	8.00%
    n	10
    frequency	1

  • We need to solve the following equation to arrive at the required PV
  • 
    PV = PMT X- 1-(1+r)-n - X (1+r)
    
  • 
    PV = 100000 x 1- (1 +0.08) -10 0,08 X (1+0.08)
    
  • 
    PV = 724688.79
    
  • So the PV of the annual installments is is $724688.79
• c) To find the PMT that will amount to 1 million we need to resolve the above equation for PMT by substituting 1 million as PV
  • We are given the following information

  • PV	1,000,000
    r	8.00%
    n	10
    frequency	1

  • We need to solve the following equation to arrive at the required PV
  • 
    PV = PMT X- 1-(1+r)-n - X (1+r)
    
  • 
    1000000 = PMT X- 1-(1 + 0.08) -10 - 0,08 X (1+0.08)
    
  • 
    PMT = 137990.27
    
  • So the annual installment needs to be 137990.27 for the PV to be 1 million