Question

Your plans for the future have finally materialized because you have won the lottery.
Congratulations! The lottery marketing material says you have won $20,000,000 but a more
careful examination of the terms and conditions means that you have won twenty $1,000,000
beginning-of-the-year cash flows with the first cash flow today. Further, the lottery contract
says that instead of waiting for so many years to collect your winnings, you could accept a
lump sum check today in the amount of $12,000,000. If you determine that the appropriate
interest rate to compare these two alternatives is 6%, which alternative is preferred? To answer
this question, solve for the present value of each alternative and make your decision based
strictly on the values. You may ignore taxes in your decision.

Answer 1

Lets consider the first scenario, twenty $1,000,000 beginning of the year cashflows with first cashflow today. With interest rate of 6%, present value of cashflows will be 1000000+1000000/1.06+1000000/1.06^2+.......1000000/1.06^19.

Using present value of annuity formula from 2nd payment to 20th payment P*((1-(1+r)^-n)/r), where P is the periodic payment, r is the interest rate and n is the number of periods. On substituting, 1000000*((1-1.06^-19)/0.06)= 11158116.49. So, including today's payment of 1000000 total value is 12158116.49

So, present value of first alternative is $12.158 Million

Lets consider second scenario, Lumpsum check today for an amount of $12000000 which is $12 Million.

As present value of first scenario ($12.158M) is greater than present value of second scenario ($12M), twenty payments of $1 Million should be accepted.