Question

Mega Millions has reached a record-breaking jackpot of $1.6 billion. Whoever holds the winning lottery ticket will be given two options: They can collect their winnings as a one-time lump sum that's less than the value of the total jackpot in this case, and the lump sum payment would be $904,900,000, or they can receive the full amount in annual installments stretched out over 29 years. The annuity will pay out the full value of the lottery 1.6 billion dollars (meaning the present value of all future cashflows). The lottery will guarantee a return of 5%.

What would be the value of the annuity payments? If you were to take the lump sum payment what rate of return would make you indifferent between the annuity and the lump sum payment?

Answer 1

The formula for present value of an annuity is

PV = P x {[1 - (1 + r)^-n] / r}

Where P is the annual payment, r is the interest rate and n is the number of years.

P = PV / {[1 - (1 + r)^-n] / r} = 1,600,000,000 / {[1 - (1.05)²⁹] / 0.05} = $105,672,823.8

The future value of these payments would be

FV = P x [(1 + r)ⁿ - 1] / r = 6,585,816,953.6

Suppose the required rate to be indifferent to the lump sum or the annuity is y

904,900,000 x (1 + y)²⁹ should be equal to 6,585,816,953.6 to be indifferent.

904,900,000 x (1 + y)²⁹ = 6,585,816,953.6

(1 + y)²⁹ = 7.278

1 + y = 7.278^1/29

1 + y = 1.07084

y = 0.07084 or 7.084%