Question

Lottery jackpots are often paid out over 20 or more years, but the winner can often choose to take a lump sum instead. For some, the cash option is a lot better. In January 2007, an 84 year old retired electrician and WWII veteran in Missouri won the $254 million Powerball jackpot. He was given the option of receiving the jackpot as $8.467 million immediately and $8.467 million per year for the next 29 years, or $120.504 million immediately. What discount rate does this option imply?

Answer 1

the discount rate for this option implies to be 6.26%

Step-by-step explanation:

From the given information; we are to determine the discount rate for the cash option.

Let r represent the discount rate and
`C_o` represent the cash option

The the discount rate for the cash option is related to the sum of all the Present Value of the cash flows together with the discount rate.

r = discount rate = ???

`C_o = 8.467` for the next 29 years.

Mathematically;

`PV = (1+r)*((C_o)/(r)) *({1 - (1)/((1+r)^(30))) = \$ 120.504 \ million`

If discount rate (r) = 1%; we have:

`PV = (1+0.01)*((8.467)/(0.01)) *({1 - (1)/((1+0.01)^(30))) = \$ 120.504`

`PV = (1.01)*(846.7) *({1 - 0.7419229178}) = \$ 120.504`

`PV =220.699 \\eq \$ 120.504`

If the discount rate r= 2% ;

`PV = (1+0.02)*((8.467)/(0.02)) *({1 - (1)/((1+0.02)^(30))) = \$ 120.504`

`PV = (1.02)*(423.35}) *({1 - 0.552070889}) = \$ 120.504`

`PV = 193.4234049 \\eq \$ 120.504`

If the discount rate r= 4% ;

`PV = (1+0.04)*((8.467)/(0.04)) *({1 - (1)/((1+0.04)^(30))) = \$ 120.504`

`PV = (1.04)*(211.675}) *({1 - 0.308318668}) = \$ 120.504`

`PV = 152.2681118 \\eq \$ 120.504`

If the discount rate r = 6%

`PV = (1+0.06)*((8.467)/(0.06)) *({1 - (1)/((1+0.06)^(30))) = \$ 120.504`

`PV = (1.06)*(141.12}) *({1 - 0.1741101309}) = \$ 120.504`

PV = 123.5396349 ≠ 120.504 (but that was so close)

If the discount rate r = 6.26%

`PV = (1+0.0626)*((8.467)/(0.0626)) *({1 - (1)/((1+0.0626)^(30))) = \$ 120.504`

`PV = (1.626)*(135.26}) *({1 - 0.161772856) = \$ 120.504`

PV = 120.4722 million which is approximately equal to $120.504 million

Thus ,the discount rate for this option implies to be 6.26%