Question

You have recently won the super jackpot in the Washington State Lottery. On reading the fine print, you discover that you have the following two options: a. You will receive 31 annual payments of $340,000, with the first payment being delivered today. The income will be taxed at a rate of 28 percent. Taxes will be withheld when the checks are issued. b. You will receive $620,000 now, and you will not have to pay taxes on this amount. In addition, beginning one year from today, you will receive $290,000 each year for 30 years. The cash flows from this annuity will be taxed at 28 percent. Using a discount rate of 7 percent, what is the present value of your winnings? (Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 32.16.)

Answer 1

a. You will receive 31 annual payments of $340,000, with the first payment being delivered today. It is like the present value (PV) of annuity due

Formula of the present value (PV) of annuity due

PV of annuity due = X * [1- (1+i) ^-n / i] * (1+i)

Where,

Present value (PV) of annuity due =?

Annual receipts = $340,000 but tax is deducted at the tax rate of 28%; therefore net annual receipts X = $340,000 * (1- 28%) = $244,800

And i= I/Y = 7% is the discount rate per annum

The time period n = 31 years

Therefore,

PV of annuity due = $244,800 * [1- (1+7%) ^-31/ 7%] * (1+7%)

Or PV of annuity due = $3,282,533.28

The present value of your winnings is $3,282,533.28

b. This part has two components -

$620,000 now (it is already a present value)

And you will receive $290,000 each year for 30 years, beginning one year from today

Here we can use PV of an Annuity formula in following manner

PV = PMT * [1-(1+i) ^-n)]/i

Where PV =?

PMT = Annual payment = $290,000 but tax is deducted at the tax rate of 28%; therefore net annual receipts X = $290,000 * (1- 28%) = $208,800

n = N = number of payments = 30 years

i = I/Y = discount rate per year = 7%

Therefore,

PV = $208,800* [1- (1+7%) ^-30]/7%

PV = $2,591,007.80

Therefore total present value of option b. = $2,591,007.80 + $620,000

= $3,211,007.80

As the present value of option a is more than the present value of option b. therefore you should choose option a.