In some lotteries, you are given the choice of either taking a lump sum of the 1st prize or annual payments for a fixed period of time. Suppose that the 1st. prize was $100,000,000 and you were given the choice of either $100,000,000 now or annual payments of $1,750,000 plus 10% increase every year for a total of 30 years. Which choice is the better deal given that the interest rate for 30 years is 5% compounded annually. Thanks for any help.
This is very easy to determine. All one has to do is to compare the PV of all 30 payments @ 5% to the value of $100 million of today's money:
Now, there are at least 2 methods of doing this very thing:
1) Is to use a specific TVM formula to find the PV directly. This formula looks this:
PV =P x [(1+10%) / (1+5%)]^N - 1 / [10% - 5%]
PV=$1,750,000 x [3.037398177../ .05]
PV=$1,750,000 x 60.747963550
PV=$106,308,936.21.
This is the PV of all 30 payments of $1,750,000 plus 10% annual increase. So, it is quite obvious that this is a better deal than $100,000,000 up front.
2) The second method is to sum up all these 30 payments plus 10% annual increase on a good calculator or an engine such as Wolfram/Alpha as follows:
∑[ (1750000*1.1^n) / 1.05^(n+1)], n=0 to 29, which gives the exact same result as above. Here is the calculation: http://www.wolframalpha.com/input/?i=%E2%88%91%5B+(1750000*1.1%5En)+%2F+1.05%5E(n%2B1)%5D,+n%3D0+to+29
Have fun dreaming of winning a BIG lottery!!.
