Suppose the chances of winning the lottery are 1 in 14 million with the earnings of $20 million to the lucky winner. A ticket costs $1. However, the winnings are paid over 20 years, with
the first $1 million payment occurring immediately. If inflation is 2% per year ,and the winnings are taxable, is the lottery a good investment? (Assume that you are in a 40% marginal income tax bracket and that the appropriate nominal
discount rate is 10% per year.)
To determine if the lottery is a good investment, we need to calculate the net present value (NPV) of the winnings. NPV takes into account the time value of money, inflation, taxes, and discount rate.
To calculate the NPV, we first need to calculate the after-tax winnings for each year. The winnings are taxable, so we need to account for the 40% marginal income tax rate.
To calculate the after-tax winnings for each year, we use the formula:
After-Tax Winnings = (Winnings - Tax) * (1 - Marginal Tax Rate)
In this case, the Winnings are $20 million over 20 years, and the Marginal Tax Rate is 40%.
For the first year, the After-Tax Winnings would be:
After-Tax Winnings Year 1 = ($1 million - $1 million * 40%) = $600,000
For the remaining 19 years, the After-Tax Winnings would be:
After-Tax Winnings Year 2-20 = ($1 million - $1 million * 40%) * (1 - 2% inflation rate)^(Year-2)
Next, we calculate the present value (PV) of each year's after-tax winnings. The present value is the current value of the future cash flows, taking into account the discount rate.
To calculate the present value for each year, we use the formula:
PV = After-Tax Winnings / (1 + Discount Rate)^(Year-1)
In this case, the Discount Rate is 10%.
For the first year, the present value is:
PV Year 1 = $600,000 / (1 + 10%)^(1-1) = $600,000
For the remaining 19 years, the present value is:
PV Year 2-20 = (After-Tax Winnings Year 2-20) / (1 + 10%)^(Year-1)
Finally, we calculate the NPV by summing up the present values of all the years:
NPV = PV Year 1 + PV Year 2-20
If the NPV is positive, then the lottery can be considered a good investment. If it is negative, then it is not a good investment.
By performing these calculations, you should be able to determine whether or not the lottery is a good investment in this scenario.