The state lottery claims that its grand prize is $1 million. The lucky winner will receive $50,000 upon presentation of the winning ticket plus $50,000 at the end of each year for the next 19 years. Why isn't this really a million-dollar prize? What would it actually be worth in dollars to you? What would the twenty yearly payments need to be for the present value of the lottery to be $1 million?
To determine why the state lottery's claim of a $1 million grand prize is not accurate, we need to understand the concept of present value. The present value refers to the current worth of a future sum of money, accounting for the time value of money and potential interest or investment earnings.
In this case, the winner of the lottery receives $50,000 upon presentation of the winning ticket. However, the remaining $50,000 is distributed annually for the next 19 years. To calculate the present value, we need to discount each cash flow to its present value and sum them up.
To find the present value, we need to know the discount rate or the expected rate of return on an alternate investment. Let's assume a discount rate of 5%. We can now calculate the present value of the twenty yearly payments:
PV = (50,000 * (1 - (1 + 0.05)^-19)) / 0.05
Using this formula, the present value of the twenty yearly payments would be $615,310.13. Hence, the actual worth of the prize in present dollars is $615,310.13 rather than $1 million.
To determine the twenty yearly payments needed for the present value of the lottery to be $1 million, we can rearrange the formula. Assuming a discount rate of 5%, we have:
1,000,000 = (X * (1 - (1 + 0.05)^-19)) / 0.05
Solving this equation, the twenty yearly payments would need to be approximately $72,859.52 for the present value of the lottery to amount to $1 million.