Congratulations! You have just won the State Lottery. The lottery prize was advertised as an annuitized $80 million paid out in 20 equal annual payments beginning immediately. The annual payment is determined by dividing the advertised prize by the number of payments. You now have up to 60 days to determine whether to take the cash prize or the annuity A. If you were to choose the annuitized prize, how much would you receive each year? B. The cash prize is the present valuc of the annuity payments. If the interest rate used to calculate the cash prize is 70%, how much will you receive if you choose the cash prize option? C. Now suppose that, as many lotteries do, the annuitized cash flows will grow by 3% per year to keep up. with inflation, but they still add up to $80 million. If you took the annuitized cash prize instead, how much would you receive (before taxes)?

(a) $4,000,000 (b) $45,342,380.97 (c) 2,977,256.60

Explanation:

Solution:

(A) The annual payment is determined by dividing the advertised prize by the number of payments which is stated as follows:

Amount of annuitized prize = $80,000,000/20 = $4,000,000

(B) Since, cash prize is present value of annuity payments discounted at an interest rate of 7%,

Amount of cash prize can be calculated using the formula:

P * [(1-((1+r)^(-n))/r] * (1+r)

= $4000000 * [(1-((1+7%)^(-20))/7%] * (1+7%) = $45,342,380.97

Amount of cash prize = $45,342,380.97

(C) Now, let make an  assumption that the amount received at time 0 is x

The Amount received at time 1 = (1+3%) * x = 1.03 x

Amount received at time 2 = (1+3%)2 * x = 1.032 x

So, on till amount received at time 19 = (1+3%)19 * x = 1.0319 x

Then

The Sum of this series can be find using the formula is shown below:

A(1-rn)/(1-r)

A is the first term i.e. A = x

r is the common ration i.e. r = 1.03

n is number of terms i.e. n = 20

Hence, sum of the payments = x * (1-1.0320)/(1-1.03) = 26.8703745*x

Since sum of series of payments is $80,000,000

Therefore, 26.8703745*x = 80,000,000

x = 80,000,000/26.8703745

x = 2,977,256.60

Therefore as a graduated annuity payment, first amount received is $2,977,256.60