A payment of $14,000 is due in 1 year and $11,800 is due in 2 years. What two equal payments, one in 3 years and one in 4 years would replace these original payments? Assume that money earns 4.50% compounded quarterly. Use the focal date in 4 years.


To replace the original payments with two equal payments in 3 and 4 years, we need to use the time value of money principles. Since the interest rate is 4.50% compounded quarterly, we'll calculate the present and future values of the original payments at the focal date (4 years), and then find the amount of the two equal payments.

### Step 1: Break down the problem
- The original payments are:
  - $14,000 due in 1 year.
  - $11,800 due in 2 years.
- We need to replace these with two equal payments, one in 3 years and one in 4 years, with a focal date at the 4-year mark.
- Interest rate: 4.50% compounded quarterly, so the quarterly interest rate $ i_q $ is:
  \[
  i_q = \frac{4.50\%}{4} = 1.125\% \text{ per quarter}.
  \]

### Step 2: Calculate the future value of the original payments at the focal date (4 years)

#### For the $14,000 payment due in 1 year:
This payment has 3 years (12 quarters) to grow to the 4-year mark.
The future value $ FV_1 $ at 4 years is:
\[
FV_1 = 14,000 \times (1 + 0.01125)^{12}
\]

#### For the $11,800 payment due in 2 years:
This payment has 2 years (8 quarters) to grow to the 4-year mark.
The future value $ FV_2 $ at 4 years is:
\[
FV_2 = 11,800 \times (1 + 0.01125)^{8}
\]

### Step 3: Set up the equation for equal payments
Let the two equal payments at 3 years and 4 years be denoted as $ X $. The present value of these payments at the focal date (4 years) must equal the sum of the future values calculated in Step 2.

The first payment is made 1 year (4 quarters) before the focal date, so we discount it to the focal date:
\[
PV_3 = X \times (1 + 0.01125)^{-4}
\]
The second payment is made at the focal date, so it doesn't need discounting:
\[
PV_4 = X
\]

### Step 4: Solve for $ X $

Now, we sum the future values from the original payments and set them equal to the sum of the two replacement payments:
\[
FV_1 + FV_2 = PV_3 + PV_4
\]

Let me calculate this.The future values of the original payments at the 4-year mark are:

- Future value of the $14,000 due in 1 year: **$16,011.44**.
- Future value of the $11,800 due in 2 years: **$12,904.77**.

The total amount these future values sum to is **$28,916.21**.

To replace these payments with two equal payments, the amount of each equal payment (one in 3 years and one in 4 years) would be **$14,781.54**.

Would you like any additional details on this, or do you have any questions?

Here are 5 related questions for further practice:
1. How would the equal payments change if the interest rate were 6% compounded monthly?
2. What if the focal date were set at 3 years instead of 4 years?
3. How would you adjust the calculation if only one payment were required at 4 years instead of two equal payments?
4. How do you calculate the present value of a payment due in 5 years with continuous compounding?
5. What happens if the payments are adjusted for inflation?

**Tip**: Always make sure to adjust the compounding period to match the interest rate when dealing with time value of money calculations.

A payment of $14,000 is due in 1 year and $11,800 is due in 2 years. What two equal payments, one in 3 years and one in 4 years would replace these original payments? Assume that money earns 4.50% compounded quarterly. Use the focal date in 4 years.

Learn how to replace two future payments of $14,000 in 1 year and $11,800 in 2 years with two equal payments in 3 and 4 years, using a 4.50% quarterly compounded interest rate. Detailed step-by-step solution for time value of money problems. Suitable for advanced high school and college-level students.

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