$5.$ You have just won $$50$ million in the lottery, payable in equal yearly installments over the next $20$ years $($first payment to be made immediately$).$ Instead of taking the annual payments, you also have the option of receiving a lump sum amount immediately. If the interest rate is $6$ percent per year, what is the minimum lump sum amount you would accept in place of the payments? What if the interest rate is $10$ percent per year?