$5.$ Congratulations, you and your family have won a lottery that pays you $$100,000$ a year forever. Ignore taxes in answering the following question and the risk of a default by the entity making the lottery payments.

$(4$ points$)$ a$.$ You are offered a lump sum of $$1\mathrm{.}6$ million today to forego the $$100,000$ a year forever. Suppose that the interest rate on risk$-$free investments is $5\%.$ Would you accept the $$1\mathrm{.}6$ million?

$(4$ points$)$ b$.$ Suppose instead that the interest rate on risk$-$free investments is $10\%$ instead of $5\%.$ Would you accept the $$1\mathrm{.}6$ million?