Bobby buys a scratch$-$off ticket every day and today he hits it big with a $$100,000$ winning ticket.

But when he turns in his ticket, he$$s informed of the fine print that states the $$100,000$ is payable

in annual installments of $$10,000$ per year over the next $10$ years. If he wants a lump sum today,

he will only get $$85,000.$ If the interest rate is $5\%,$ is it better for Bobby to take the ten $$10,000$

annual installments or the $$85,000$ lump sum?

C$.$ Compute the present value of the annuity and compare it to the $$85,000$ lump sum. Which

option should Bobby choose? What is Bobby has a preference to receive the money later?

$7$ points

PV $=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

To compare the two options, find the present value of the annual installments:

So the present value of ten annual payments of $$10,000$ is $$77,217.$ Because this is less than the

$$85,000,$ it would be better for Bobby to take the $$85,000.$ If he wants the money later, he can

invest it at $5\%$ and withdraw more than $$10,000$ a year for $10$ years.

Explain the math of getting the $-77,217\mathrm{.}35$