Math Problem Statement

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.)

Solution

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, ii.

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 XX 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 XX to Yifei and $1000 to Christian.

We need to calculate the value of XX 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 ii.

PV=F(1+i)tPV = \frac{F}{(1+i)^t} where FF is the future value, ii is the interest rate, and tt is the time in years.

Christian’s Net Present Value (NPV):

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

NPVChristian=30001000(1+i)+1000(1+i)2+4000(1+i)3NPV_{\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=0t = 0 in terms of ii:

NPVYifei=1000(1+i)+X(1+i)24000(1+i)3NPV_{\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:

30001000(1+i)+1000(1+i)2+4000(1+i)3=1000(1+i)+X(1+i)24000(1+i)3-3000 - \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+1000(1+i)2+4000(1+i)3=1000(1+i)+X(1+i)24000(1+i)3-3000 + \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+i):

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

  2. Solve for XX.

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

X=1000×3i310i210i+5i+1X = 1000 \times \frac{-3i^3 - 10i^2 - 10i + 5}{i + 1}

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

Do you want to assume a specific value for the interest rate ii, 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.

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Math Problem Analysis

Mathematical Concepts

Financial Mathematics
Present Value
Interest Rate
Net Present Value (NPV)

Formulas

Present Value formula: PV = F / (1+i)^t
Net Present Value for Christian: NPV_Christian = -3000 - (1000 / (1+i)) + (1000 / (1+i)^2) + (4000 / (1+i)^3)
Net Present Value for Yifei: NPV_Yifei = (1000 / (1+i)) + (X / (1+i)^2) - (4000 / (1+i)^3)

Theorems

Principle of Equating Net Present Values
Bottom Line Approach

Suitable Grade Level

University Level (Financial Mathematics)