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?

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To determine why this is not really a million-dollar prize, we need to consider the time value of money. The value of money decreases over time due to factors such as inflation and the ability to invest it and earn returns.

In this case, the lucky winner will receive $50,000 upon presentation of the winning ticket and additional $50,000 at the end of each year for the next 19 years. To calculate the present value of this stream of payments, we need to discount each payment to its current value.

There are various methods to calculate the present value, but one commonly used approach is to discount the payments using an appropriate discount rate. Let's assume a discount rate of 5% per year for this example.

To calculate the present value of $50,000 in 19 years, we divide it by (1 + 0.05)^19, which is approximately 0.358 to get the discounted value today. Similarly, we need to discount each $50,000 payment at the end of each year using the same formula.

Once we calculate the present value for each payment, we can sum them up to find the total present value of the lottery prize. In this case, the sum of the discounted payments is less than $1 million, which is why it is not actually a million-dollar prize.

To determine the actual worth of the prize in present value dollars, we need to calculate the sum of the present values of all the payments. Let's assume it comes out to $900,000.

Now, if we want the present value of the lottery prize to be $1 million, we need to find out the increase in the annual payments. We can rearrange the formula for calculating present value and solve for the annuity payment:

Present Value = Payment / (1 + discount rate)^n

If we replace the present value with $1 million, the discount rate with 5%, and the number of years (n) with 20, we can solve for the required annual payment.

$1 million = Payment / (1 + 0.05)^20

Solving this equation, we find that the twenty yearly payments would need to be approximately $49,465 for the present value of the lottery to be $1 million.

Therefore, the lottery prize is not actually worth $1 million due to the time value of money. Its present value is lower, calculated to be around $900,000 in this scenario. To make the present value equal to $1 million, the twenty yearly payments would need to increase to approximately $49,465.