Publicity agents for the Detroit Felines announce the signing of a phenomenal new quarterback, Archie Parabola. They say that the contract is worth $1,000,000 and will be paid in 20 installments of $50,000 per year starting one year from now and with one new installment each year for next 20 years. The contract contains a clause that guarantees he will get all of the money even if he is injured and cannot play a single game. Sports writers declare that Archie has become an “instant millionaire.”
(a) Archie’s brother, Fenwick, who majored in economics, explains to Archie that he is not a millionaire. In fact, his contract is worth less than half a million dollars. Explain in words why this is so. _______________. Archie’s college course on “Sports Management” didn’t cover present values. So his brother tried to reason out the calculation for him. Here is how it goes:
(b) Suppose that the interest rate is 10% and is expected to remain at 10% forever. How much would it cost the team to buy Archie a perpetuity that would pay him and his heirs $1 per year forever, starting in 1 year?
(c) How much would it cost to buy a perpetuity that paid $50,000 a year forever, starting in one year?________. In the last part, you found the present value of Archie’s contract if he were going to get $50,000 a year forever. But Archie is not going to get $50, 000 a year forever. The payments stop after 20 years. The present value of Archie’s actual contract is the same as the present value of a contract that pays him $50,000 a year forever, but makes him pay back $50,000 each year, forever, starting 21 years from now. Therefore you can find the present value of Archie’s contract by subtracting the present value of $50,000 a year forever, starting 21 years from now from the present value of $50,000 a year forever.
(d) If the interest rate is and will remain at 10%, a stream of payments of $50,000 a year, starting 21 years from now has the same present value as a lump sum of _________ to be received all at once, exactly 20 years from now.
(e) If the interest rate is and will remain at 10%, what is the present value of $50,000 per year forever, starting 21 years from now?
(f) Now calculate the present value of Archie’s contract.