Question

rear everT Part D Your dream of becoming rich has just come true. You have won the State of Tranquility's Lottery. The State offers you two payment plans for the $5 million jackpot. You can take annual payments of $250,000 for the next 20 years or $2,867,480 today. a. If your investment rate over the next 20 years is 8%, which payoff will you choose? b. If your investment rate over the next 20 years is 5%, which payoff will you choose? c. At what investment rate will the annuity stream of $250,000 be the same as the lump sum payment of $2,867,480?

Answer 1

Present value of an annuity = Annuity x Present Value Interest Factor of the Annuity.

Present Value Interest Factor = [ { 1 - ( 1 / ( 1 + r ) ⁿ } / r ]

a. If r = 8 %, and n = 20,

Present value of the annuities of $ 250,000 = $ 250,000 x [ { 1 - ( 1 / 1.08 ) ²⁰ } / 0.08 ] = $ 250,000 x 9.8181 = $ 2,454,525

Since the present value of the annuities of $ 250,000 at $ 2,454,525 is less than the lump sum of $ 2,867,480, the payoff of $ 2,867,480 should be accepted today.

b. PVIFA _{r = 5 %, n=20} = [ { 1 - ( 1 / 1.05 ) ²⁰ } / 0.05 ] = 12.4622

Present value of the annuities of $ 250,000 = $ 250,000 x 12.4622 = $ 3,115,550.

As the present value of the annuities exceeds $ 2,867,480 payoff today, the payoff in the form of annual payments of $ 250,000 should be accepted.

c. The required PVIFA = $ 2,867,480 / $ 250,000 = 11.4699

[ { 1 - ( 1 / 1.06 ) ²⁰ } / 0.06 ] = 11.4699

At an investment rate of 6 %, the annuity stream of $ 250,000 will be the same as the lump sum payment of $ 2,867,480.