Problem $1$:

$13$ lucky engineers won a jackpot in a national lottery. The total winnings are $$257$ million $($after taxes paid$).$ The lottery operator offers these lucky individuals $2$ options for winnings payout:

$1.$ Option $1$: One$-$time lump sum payment of $$150$ million

$2.$ Option $2$: Annual payments of $$12\mathrm{.}85$ million for next $20$ years.

Question: You$$re their financial advisor and know that existing risk$-$free rates of return are about $4\%$ annually. What should these lucky individuals do$?$ Write up your response. Show your calculations.