On Sat May 7, 2016, someone won the Powerball jackpot. The winner gets a choice

ID: 2728966 • Letter: O

Question

On Sat May 7, 2016, someone won the Powerball jackpot. The winner gets a choice between two different payouts. (1) $429.6 million paid out in 30 equal yearly installments. (2) A lump sum payment of $284.1 million. Let us make the following assumptions: (a) the inflation rate is zero, (b) the lottery is well managed so it will actually be able to pay off the 30 installments, and (c) the winner has a life expectancy of more than 30 years so that either choice is a viable option.

What is the estimated average long term interest rate r that the lottery is using to calculate the lump sum payment so that it equals the annuity of yearly installments? You cannot do this problem with one equation. Rather you must iterate answers to the two formulas until both formulas give the same answer (or close to the same answer). (see page 149 of Applied Finite Math)

The compound interest formula is:

Ai = P(1 + )nt

The total value of an ordinary annuity formula is

Aa = m [(1 + )nt -1 ] (r/n)

Let n = 1 and t = 30.

What is P? _______

What is m? _______

Fill in the following table where Ai and Aa are in millions of dollars.

Interest rate r

Estimate the interest rate r so that Ai = Ar.

r = < > _________%. Fill in the number and circle one of the three operators (“=”, “<”, or “>”).

Look up on the internet what the current 30 year US treasury bond interest rate is. ________%

How does this bond rate compare with the interest rate r that you estimated from the table? Circle one of the following.

Less than Greater than About the same

Interest rate r

Explanation / Answer

The compound interest formula is:

Ai = P(1 + r/n )nt

The total value of an ordinary annuity formula is

Aa = m [(1 + r/n )nt -1 ] (r/n)

Let n = 1 and t = 30.

What is P? P is the principal amount or lump sum payment = $ 284.1 million

What is m? m is cash flow per period or yearly installment of annuity = $429.6/30 = $14.32 million

Fill in the following table where Ai and Aa are in millions of dollars.

Interest Rate r

Ai (million $)

Aa (million $)

2.70%

631.81

649.12

2.80%

650.53

659.63

2.91%

671.74

671.43

3.00%

689.59

681.28

3.10%

709.96

692.42

3.20%

730.91

703.79