Christian, Ethan, and Yifei enter into a financial arrangement. Christian agrees to pay Ethan 3000 today. Christian also agrees to pay Yifei 1000 at the end of one year. At the end of three years, Yifei will pay Christian 4000. At the end of two years, Ethan will pay X
 to Yifei and 1000 to Christian. 

Using the bottom line approach, the annual yield rate or interest rate is the same for Christian and Yifei.

Calculate X
.

(Round your answer to 2 decimal places.)

To solve this problem using the **bottom line approach** (which treats each party's cash flow as equalized via an annual yield rate), we need to set up a financial equation based on the principle that the net present value (NPV) of inflows and outflows for Christian, Ethan, and Yifei must be balanced with respect to the same annual yield rate, $ i $.

Let's break down the cash flows:

### Christian’s cash flow:
- **Today (t = 0):** Christian pays Ethan \$3000.
- **At the end of 1 year (t = 1):** Christian pays Yifei \$1000.
- **At the end of 2 years (t = 2):** Christian receives \$1000 from Ethan.
- **At the end of 3 years (t = 3):** Christian receives \$4000 from Yifei.

### Yifei’s cash flow:
- **At the end of 1 year (t = 1):** Yifei receives \$1000 from Christian.
- **At the end of 2 years (t = 2):** Yifei receives $ X $ from Ethan.
- **At the end of 3 years (t = 3):** Yifei pays \$4000 to Christian.

### Ethan’s cash flow:
- **Today (t = 0):** Ethan receives \$3000 from Christian.
- **At the end of 2 years (t = 2):** Ethan pays $ X $ to Yifei and \$1000 to Christian.

We need to calculate the value of $ X $ that ensures Christian and Yifei face the same annual yield rate. Using the **present value formula**, we can discount future payments to their present values at the unknown yield rate $ i $.

\[
PV = \frac{F}{(1+i)^t}
\]
where $ F $ is the future value, $ i $ is the interest rate, and $ t $ is the time in years.

### Christian’s Net Present Value (NPV):

Christian’s net present value (NPV) at time $ t = 0 $ in terms of $ i $:

\[
NPV_{\text{Christian}} = -3000 - \frac{1000}{(1+i)} + \frac{1000}{(1+i)^2} + \frac{4000}{(1+i)^3}
\]

### Yifei’s Net Present Value (NPV):

Yifei’s net present value (NPV) at time $ t = 0 $ in terms of $ i $:

\[
NPV_{\text{Yifei}} = \frac{1000}{(1+i)} + \frac{X}{(1+i)^2} - \frac{4000}{(1+i)^3}
\]

Since Christian and Yifei face the same annual yield rate, their net present values must be equal. Setting the NPV of Christian’s and Yifei’s cash flows equal gives us the equation:

\[
-3000 - \frac{1000}{(1+i)} + \frac{1000}{(1+i)^2} + \frac{4000}{(1+i)^3} = \frac{1000}{(1+i)} + \frac{X}{(1+i)^2} - \frac{4000}{(1+i)^3}
\]

Simplifying:

\[
-3000 + \frac{1000}{(1+i)^2} + \frac{4000}{(1+i)^3} = \frac{1000}{(1+i)} + \frac{X}{(1+i)^2} - \frac{4000}{(1+i)^3}
\]

Now, let's collect the terms by powers of $ (1+i) $:

1. Multiply both sides by $ (1+i)^3 $ to eliminate the denominators.

2. Solve for $ X $.

I will solve this equation step by step to find the value of $ X $.The value of $ X $, in terms of the annual yield rate $ i $, is given by:

\[
X = 1000 \times \frac{-3i^3 - 10i^2 - 10i + 5}{i + 1}
\]

To find $ X $, we need the value of the yield rate $ i $. Since the problem does not provide $ i $, we would either need additional information to solve for $ i $, or assumptions about the interest rate could be made.

Do you want to assume a specific value for the interest rate $ i $, or do you have more details about it? Feel free to ask any questions for clarification.

Here are 5 questions to consider:
1. What happens if the interest rate changes annually?
2. Can you specify how to compute if Yifei had a different payment structure?
3. How does Ethan's payment influence the overall balance?
4. What is the impact of early repayments on the yield rate?
5. Could a different financial arrangement affect the outcome?

**Tip**: When working with future cash flows, it's essential to discount each payment to its present value using the appropriate interest rate.

Christian, Ethan, and Yifei enter into a financial arrangement. Christian agrees to pay Ethan 3000 today. Christian also agrees to pay Yifei 1000 at the end of one year. At the end of three years, Yifei will pay Christian 4000. At the end of two years, Ethan will pay X to Yifei and 1000 to Christian. Using the bottom line approach, the annual yield rate or interest rate is the same for Christian and Yifei. Calculate X. (Round your answer to 2 decimal places.)

Explore how to calculate X in a complex financial arrangement involving Christian, Ethan, and Yifei using the bottom line approach and Net Present Value (NPV) formulas. This detailed solution involves discounting cash flows at a common annual yield rate. Suitable for university-level financial mathematics students.

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