Question

You have just won the lottery. You have two payout options. First, you can agree to take 30 equal annual payments, with the first payment to be made today at Year 0 and the last payment to be made at Year 29. Alternatively, using an effective annual rate of 6.0 percent, the lottery is willing to convert the annual payments into an equivalent lump sum payment today of $7,295,360.51. You have no immediate need for this money and plan to simply invest and hold the money until Year 45, and believe that you can earn an effective annual rate of 9.75 percent over each of the next 45 years. Looking at your possible ending values at Year 45, you should be able to determine that you will be better off taking the lump sum value today rather than taking the yearly cash flows. Given this information, determine the difference in dollar values at Year 45 between the two options.

Answer 1

The difference in dollar value between option 2 (taking yearly cash flows) and option 1 (taking a lump sump) is USD 490,959,268.96. Therefore, you are better off with taking yearly cash flows which you can invest in a rate of 9.75% to get a larger sum of money at Year 45

Step-by-step explanation:

Solving this is relatively simply. All you have to do is calculate the dollar amount available to you 45 years later under each of the two options.

So, in option 1, you have 7,295,360.51 available today. Using the future value formula, i.e FV= PV x (1+r)^n.

Where PV is equal to 7,295,360.51

R is equal to 9.75%

N is equal to 45

You will have USD 480,010,302.91 at year 45.

Now the calculation for option 2 is slightly lengthier.

First we calculate the total value of the lottery. We have the present value, the interest rate and the number of periods. Again, we use the FV formula and calculate value at year 30 as 41,900,838.69. We used this calculation because that's how the lottery calculated the figure of 7.3 Mn as todays lump sum amount. All they did was discount the lottery amount 30 years at a rate of 6%. We used the same formula to derived the lottery sum that they used.

Now, we will get equal installments each year up till year 29 which means (just divide the lottery amount by 30 years) we will get USD 1,396,694.62 each year till year 29. For each year's cash flow, we will calculate the future value at year 45. The formula will remain the same. Only the number of periods will change. So for example, at year 0, we will use the number of periods in our FV formula as 45. Whereas at the last installment date i.e. year 29, we will use 16 as the number of periods left till year 45. Once we have values for all periods, simply sum them up which will produce a value of USD 970,969,571.87 which is the the sum available to us in option 2.

As we can see, simply subtract the sum in option 1 from option 2 which indicates that we are better off under option 1 by a whopping USD 490 Mn!