Suppose the opportunity cost of capital is 10 percent and you have just won a $1 million lottery that entitles you to $100,000 at the end of each of the next ten years.
a. What is the minimum lump sum cash payment you would be willing to take now in lieu of the ten-year annuity?
b. What is the minimum lump sum you would be willing to accept at the end of the ten years in lieu of the annuity?
c. Suppose three years have passed and you have just received the third payment and you have seven left when the lottery promoters approach you with an offer to "settle-up for cash." What is the minimum you would accept (the end of year three)?
d. How would your answer to part (a) change if the first payment came immediately (at t = 0) and the remaining payments were at the beginning instead of at the end of each year?
a. The minimum lump sum you should take is the present value of the cash payments.
PV = $100,000 × Annuity Factor (i = .10, t = 10)
= $100,000 × 6.145
= $614,500
b. This question is essentially (a) in reverse. You are looking for the future value of the cash payments. Looking in the future value in arrears table, the annuity factor is 15.937.
FV = $100,000 × 15.937
= $1,593,700
c. This is similar to (a). This time, t = 7.
PV = $100,000 × Annuity Factor (i = .10, t = 7)
= $100,000 × 4.868
= $486,800
d. To convert an end-of-year payment schedule to a beginning-of-year schedule, we need only multiply by 1 + r. The minimum payment is $614,500 × 1.10 = $675,900.
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