A lottery offers a prize of $1,000,000. If the winner is paid in annual instalments of $25,000 for forty years, what is the actual value of the prize if money is worth 4.5% compounded annually. If you wish to sell the payment stream half way through the payment period, to a finance company for cash, how much in cash would you accept? Money can earn 5.6% at that time?

To find the actual value of the prize and the cash amount you would accept if selling the payment stream halfway through, we need to perform two calculations:

1. **Calculate the present value of the prize** if paid in annual installments.
2. **Calculate the present value of the remaining payment stream** halfway through the payment period.

### Step 1: Calculate the Present Value of the Prize

The prize is $1,000,000 paid in annual installments of $25,000 for 40 years. We need to find the present value (PV) of these payments, assuming an interest rate of 4.5% compounded annually.

The present value $ PV $ of an annuity is calculated using the formula:
\[ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \]

Where:
- $ PMT $ is the annual payment ($25,000)
- $ r $ is the annual interest rate (0.045)
- $ n $ is the total number of payments (40)

Plugging in the values:
\[ PV = 25000 \times \frac{1 - (1 + 0.045)^{-40}}{0.045} \]

Calculating the values inside the parentheses first:
\[ (1 + 0.045)^{-40} \approx 0.2085 \]

Now:
\[ PV = 25000 \times \frac{1 - 0.2085}{0.045} \]
\[ PV = 25000 \times \frac{0.7915}{0.045} \]
\[ PV = 25000 \times 17.5889 \]
\[ PV \approx 439,722.50 \]

The actual value of the prize, considering the time value of money at 4.5% interest, is approximately **$439,722.50**.

### Step 2: Calculate the Present Value of the Remaining Payment Stream Halfway Through

Halfway through the payment period means after 20 years. At this point, there will be 20 payments of $25,000 remaining. We need to find the present value of these 20 payments, assuming an interest rate of 5.6%.

The present value $ PV_{20} $ of the remaining payments is calculated using the same formula:
\[ PV_{20} = PMT \times \frac{1 - (1 + r_{new})^{-n_{new}}}{r_{new}} \]

Where:
- $ PMT $ is the annual payment ($25,000)
- $ r_{new} $ is the new annual interest rate (0.056)
- $ n_{new} $ is the total number of remaining payments (20)

Plugging in the values:
\[ PV_{20} = 25000 \times \frac{1 - (1 + 0.056)^{-20}}{0.056} \]

Calculating the values inside the parentheses first:
\[ (1 + 0.056)^{-20} \approx 0.3305 \]

Now:
\[ PV_{20} = 25000 \times \frac{1 - 0.3305}{0.056} \]
\[ PV_{20} = 25000 \times \frac{0.6695}{0.056} \]
\[ PV_{20} = 25000 \times 11.9554 \]
\[ PV_{20} \approx 298,885 \]

The cash amount you would accept if selling the remaining payment stream halfway through the period, assuming a 5.6% interest rate, is approximately **$298,885**.

Would you like any further details or explanations on this solution? Here are five more questions you might find interesting:

1. How would the present value change if the interest rate were different?
2. What if the prize installments were paid monthly instead of annually?
3. How is the present value formula for an annuity derived?
4. What would be the present value if the payments were to increase by a fixed percentage each year?
5. How does changing the compounding frequency affect the present value calculations?

**Tip:** When dealing with financial calculations, always consider the impact of interest rates and compounding periods on the present and future values of money.

Learn how to calculate the present value of a lottery prize paid in annual installments using a 4.5% interest rate. Also, find out the cash amount you would accept halfway through the payment period at a 5.6% interest rate. This problem involves the time value of money and annuity calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation