Question

A lottery offers two options for the prize. 

Option A: $1000 a week for life.

Option B: $1 000 000 in one lump sum.

If you choose Option B, you have the opportunity to place the winnings into an investment that also makes regular payments, at a rate of 3%/a, compounded weekly. The annuity will pay out a specific amount weekly based on how long you want the annuity to last.

Which option would the winner choose if you expect to live for another:

20 years?

50 years?

show full work pleases

Answer 1

For 20 years,

There are roughly 1042.86(approx) weeks in 20 years,

so with option A, the winner would be getting 1042.86 * 1000 = $ 1042860

With option B,

Amount received after investment for 20 years = 1000000 * (1 + 0.03/52.143)^(52.143*20) , as there are 52.143(approx) in a year.

Amount received = $1821804.44625

So after 20 years winner would be getting $778944.44625 more if winner select option B,

For 50 years,

Number of weeks in 50 years is 2607.14 (approx).

So with option A, the winner would be getting 2607.14 * 1000 = $ 2607140

with option B,

Amount received after investment for 50 years = 1000000 * (1 + 0.03/52.143)^(52.143*50) = $4479756.3544

So after 50 years winner would be getting $1872616.3544 more if winner select option B,

But with option B if the person is taking out money every week then is getting $575.340889477 per week

So after 20 years, he will be having 575.340889477 * 52.143 * 20 + 1000000  = $1600000

which is still better by $557140.

And after 50 years, he will be having 575.340889477 * 52.143 * 50 + 1000000 = $2500000

in which case option A is better by 2607140 - 2500000 = $107140