Some lotteries claim to pay the winner $1 million. The lottery actually pays the winner $100,000 per year for the next 10 years. The payments arrive at the beginning of every year. What is the present value for the lottery ticket? More or less than $1 million? Why? 
 

To determine the present value (PV) of the lottery payments, we need to discount the future cash flows (the $100,000 annual payments) back to the present value using a discount rate (denoted as $ r $). The present value will depend on the discount rate because money today is worth more than money in the future due to the time value of money.

### Step-by-Step Calculation of Present Value:

Given:
- The lottery pays $100,000 annually for 10 years.
- Payments are made at the **beginning** of each year (this is an annuity due).

To compute the present value for an annuity due, we use the following formula:

\[
PV_{\text{annuity due}} = C \times \left( 1 + \frac{1 - (1 + r)^{-n}}{r} \right)
\]

Where:
- $ C $ is the annual payment ($100,000),
- $ r $ is the annual discount rate,
- $ n $ is the number of periods (10 years).

### Present Value Less than $1 Million?

The present value will be **less than $1 million** if the discount rate is positive because future payments lose value as they are discounted back to the present.

#### Why is the present value less than $1 million?

This is due to the **time value of money**. A dollar today is worth more than a dollar in the future because you can invest today's dollar and earn interest. Therefore, even though the total nominal payments equal $1 million, the present value is lower because future payments are discounted.

Would you like me to compute the exact present value using a specific discount rate (such as 5%)?

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Here are five related questions:
1. What happens to the present value if the discount rate increases?
2. How does an annuity due differ from an ordinary annuity in terms of present value?
3. What is the impact of inflation on the real value of lottery payments over time?
4. How would the present value change if the lottery paid over 20 years instead of 10?
5. What is the future value of the lottery payments at the end of the 10 years?

**Tip**: The time value of money principle helps in comparing cash flows occurring at different points in time effectively.

Some lotteries claim to pay the winner $1 million. The lottery actually pays the winner $100,000 per year for the next 10 years. The payments arrive at the beginning of every year. What is the present value for the lottery ticket? More or less than $1 million? Why?

Learn how to calculate the present value of a lottery that pays $100,000 annually for 10 years. Understand key financial concepts such as the time value of money, annuity due, and discount rates. Discover why the present value is less than $1 million, even though the total payments sum up to $1 million.

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