A lottery pays out $10 million over 5 Years. The interest rate is 6% and the first payment is made immediately. What is the grand prize really worth?

This problem requires calculating the **present value (PV)** of the lottery's payout, given that the payments are spread over 5 years with the first payment made immediately, and the interest rate is 6%.

We can break it down into steps:

### Step 1: Identify the cash flow
- Total payout: $10 million over 5 years.
- Assuming equal payments each year, the annual payout will be:  
  \[
  \text{Annual payment} = \frac{10,000,000}{5} = 2,000,000
  \]

### Step 2: Account for the first payment
The first payment of $2,000,000 is made immediately. Since it's already in the present, no discounting is needed for this amount.

### Step 3: Calculate the present value of remaining payments
The remaining 4 payments occur at the end of each of the following 4 years. The present value of each future payment is discounted by the interest rate (6% or 0.06). The formula to calculate the present value of a future cash flow is:

\[
PV = \frac{\text{Payment}}{(1 + r)^t}
\]
Where:
- Payment = $2,000,000
- $r = 6\% = 0.06$ (interest rate)
- $t$ = time in years.

### Step 4: Calculate present value for each payment

1. **Year 1 (already counted)**: $2,000,000
2. **Year 2**: 
\[
PV_2 = \frac{2,000,000}{(1 + 0.06)^1} = \frac{2,000,000}{1.06} \approx 1,886,792.45
\]
3. **Year 3**:
\[
PV_3 = \frac{2,000,000}{(1 + 0.06)^2} = \frac{2,000,000}{1.1236} \approx 1,779,057.98
\]
4. **Year 4**:
\[
PV_4 = \frac{2,000,000}{(1 + 0.06)^3} = \frac{2,000,000}{1.191016} \approx 1,678,356.57
\]
5. **Year 5**:
\[
PV_5 = \frac{2,000,000}{(1 + 0.06)^4} = \frac{2,000,000}{1.262476} \approx 1,584,289.21
\]

### Step 5: Add up the present values
Now, we sum the present values of all 5 payments:

\[
\text{Total Present Value} = 2,000,000 + 1,886,792.45 + 1,779,057.98 + 1,678,356.57 + 1,584,289.21
\]

\[
\text{Total Present Value} \approx 8,928,496.21
\]

### Conclusion:
The real worth of the grand prize (its present value) is approximately **$8,928,496.21**.

Would you like more details or have any questions on how this was calculated?

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Here are 5 related questions to explore:

1. What if the interest rate was 5% instead of 6%? How would the present value change?
2. How does increasing the number of years affect the present value of payments?
3. What if the payments were not equally split over 5 years, but increased each year?
4. How does making payments at the end of each period (instead of immediately) affect the present value?
5. What if there were 10 smaller payments instead of 5?

**Tip**: Present value calculations allow you to understand the true value of future payments today, taking into account the interest rate or discount factor.

A lottery pays out $10 million over 5 years. The interest rate is 6% and the first payment is made immediately. What is the grand prize really worth?

Discover how to calculate the present value of a $10 million lottery payout spread over 5 years, using a 6% interest rate. Learn about annuities, discounting future payments, and how the first immediate payment affects the calculation. Suitable for high school students in Grades 10-12.

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