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.