Question 1193987
**1. Determine the key parameters:**

* **Payment amount (PMT):** P2,500
* **Interest rate per period:** 12.75% per year compounded semi-annually, so the periodic interest rate is 12.75% / 2 = 6.375% or 0.06375
* **Number of periods:**
    * **Total periods:** 9 years * 2 periods/year = 18 periods
    * **Periods to the start of the annuity:** 3 years * 2 periods/year = 6 periods
    * **Periods within the annuity:** 18 periods - 6 periods = 12 periods

**2. Calculate the present value of the annuity at the start of the annuity:**

We can use the formula for the present value of an ordinary annuity:

* **Present Value (PV) = PMT * [(1 - (1 + r)^-n) / r]**

Where:
* PMT = payment amount
* r = periodic interest rate
* n = number of periods

* PV = 2500 * [(1 - (1 + 0.06375)^-12) / 0.06375] 
* PV ≈ 2500 * 7.0462 
* PV ≈ 17,615.50

**3. Calculate the present value of the annuity at the end of 2 years:**

* **Periods to the end of 2 years:** 2 years * 2 periods/year = 4 periods
* **Present Value at the end of 2 years = PV at the start of the annuity / (1 + r)^number of periods**
* Present Value at the end of 2 years = 17,615.50 / (1 + 0.06375)^4 
* Present Value at the end of 2 years ≈ 17,615.50 / 1.2838 
* Present Value at the end of 2 years ≈ 13,708.68

**4. Calculate the present value of the annuity at the end of 5 years:**

* **Periods to the end of 5 years:** 5 years * 2 periods/year = 10 periods
* **Periods between the end of 5 years and the start of the annuity:** 10 periods - 6 periods = 4 periods
* **Present Value at the end of 5 years = PV at the start of the annuity / (1 + r)^number of periods**
* Present Value at the end of 5 years = 17,615.50 / (1 + 0.06375)^4 
* Present Value at the end of 5 years ≈ 17,615.50 / 1.2838 
* Present Value at the end of 5 years ≈ 13,708.68

**Therefore:**

* **a. At the end of 2 years, the sum of these values is approximately P13,708.68.**
* **b. At the end of 5 years, the sum of these values is also approximately P13,708.68.** 

**Note:** These calculations assume that the interest rate remains constant throughout the entire period.