Question

1.As an employee in the Lottery Commission, your job is to design a new prize. Your idea is to create two grand prize choices: (1) receiving $50,000 at the end of each year beginning in one year for 20 consecutive years, or (2) receiving $500,000 today followed by a one-time payment at the end of 20 years. Using an interest rate of 3%, which of the following comes closest to the amount prize (2) needs to pay at the end of year 20 in order that both prizes to have the same present value?

2.Jack and Diane are lottery winners. They hold the ticket to the Grand Prize in the “Set for Life” Prize that makes 20 consecutive annual payments of $50,000 starting immediately. There is one hitch: the eleventh payment (to be received at the end of year 10) is not $50,000 but only $20,000 (that is, this payment is $30,000 LESS than the other 20). Which of the following comes closest to the present value of the prize if interest rates are 3%?

Answer 1

1.PV of option (1)

PV of 20 consecutive $ 50,000 ordinary annuities at the discount rate of 3% can be computed as:

PV = C x PVIFA (i, n)

C = Periodic cash flow = $ 50,000

i = Rate of interest = 3 %

n = No. of periods = 20

PV = $ 50,000 x PVIFA (3%, 20)

   = $ 50,000 x 14.8775

   = $ 743,875

PV of option (2)

PV = $ 500,000 + P x PVIF x (3 %, 20)

$ 743,875 = $ 500,000 + P x PVIF x (3 %, 20)

                 = $ 500,000 + P x 0.5537

P x 0.5537 = $ 743,875 - $ 500,000

P x 0.5537 = $ 243,875

P = $ 243,875 /0.5537

   = $ 440,446.09

$ 440,446.09 need to pay after 20 years in order to mach PV of both the options.

2.PV of 20 consecutive $ 50,000 annuities due at the discount rate of 3% can be computed as:

PV = C x PVIFAD (i, n)

PV = C x PVIFAD (3%, 20)

But there is a less payment of $ 30,000 at the beginning of period 11^th or end of period 10^th which can be deducted from PV of annuity due as:

PV = C x PVIFAD (3%, 20) - $ 30,000 x PVIFA (30%, 10)

= $ 50,000 x 15.3238 – $ 30,000 x 0.7441

   = $ 766,190 - $ 22,323 = $ 743,867

The present value of annuity due with $ 30,000 shortage at the 11^th payment is nearly equal to

$ 743,867