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VIPSolutions · Accepted Answer

# Objective
To compute the projected benefit obligation (PBO), annual retirement payments, service cost, and interest cost for Anna Ang’s defined benefit pension, based on the details given.

---

## Given Data
- Service years at retirement: 35 (2013–2047)
- Years worked by end of 2027: 15
- Ang’s salary at end of 2027: \$94,000
- Projected salary at retirement (2047): \$300,000
- Annual benefit formula: 1.6% × service years × final year’s salary
- Retirement period: 18 years
- Discount rate: 8%

---

## Step 1: **Calculate Projected Annual Retirement Payments**

**Formula:**  
Annual payment = 1.6% × service years × final salary

At retirement (2047), service years = 35; final salary = \$300,000.

$$
\text{Annual payment} = .016 \times 35 \times \$300,000
$$
$$
= .016 \times 35 = .56
$$
$$
.56 \times \$300,000 = \$168,000
$$

---

## Step 2: **Calculate Projected Benefit Obligation (PBO) at End of 2027 (after 15 years of service)**

PBO is the present value (at 2027) of all future retirement payments earned for 14 years' service (2013–2027).

### **a) Find the annual benefit earned as of 2027**

Service years by end of 2027 = 15  
Annual benefit earned as of 2027 = .016 × 15 × \$300,000  
$$
= .016 \times 15 = .24
$$
$$
.24 \times \$300,000 = \$72,000 \text{ per year}
$$

### **b) Present Value of Retirement Payments at Retirement Date (2047)**

PV at retirement = Annual benefit × PV of ordinary annuity (n = 18, i = 8%)

From PV of ordinary annuity table:  
PV factor (n = 18, i = 8%) ≈ 9.6036

$$
\text{PV at retirement} = \$72,000 \times 9.6036 = \$691,459.20
$$

### **c) Present Value at End of 2027 (20 years before retirement)**

Discount this amount back to 2027 (20 periods at 8%):

From PV of \$1 table:  
PV factor (n = 20, i = 8%) ≈ .2145

$$
\text{PBO at end of 2027} = \$691,459.20 \times .2145 = \$148,296.49
$$

---

## Step 3: **Service Cost for 2027**

Service cost = PV at end of 2027 of benefits earned in 2027 (i.e., for 15th year of service)

### **a) Annual benefit for 2027**

Benefit for 15th year = .016 × 1 × \$300,000 = \$4,800

### **b) PV at retirement (2047) of this payment stream**

PV at retirement = \$4,800 × 9.6036 = \$46,097.28

### **c) PV at end of 2027 (discount 20 years at 8%)**

$$
\text{Service cost} = \$46,097.28 \times .2145 = \$9,887.35
$$

---

## Step 4: **Interest Cost for 2027**

Interest cost = PBO at beginning of 2027 × discount rate (8%)

PBO at beginning of 2027 ≈ PBO at end of 2026 (which is PBO at end of 2027 minus service cost for 2027)

$$
\text{Interest cost} = \$148,296.49 \times .08 = \$11,863.72
$$

---

## Step 5: **Projected Benefit Obligation at End of 2027 (after 15 years)**
$$
\text{PBO at end of 2027} = \text{PBO at beginning of 2027} + \text{Service Cost} + \text{Interest Cost}
$$

---

# **Summary Table**

| Requirement     | Calculation                                               | Result          |
|-----------------|----------------------------------------------------------|-----------------|
| 1. PBO (2027)   | \$72,000 × 9.6036 × .2145                               | \$148,296       |
| 2. Annual Payments | .016 × 35 × \$300,000                                 | \$168,000       |
| 3. Service Cost | \$4,800 × 9.6036 × .2145                                | \$9,887         |
| 4. Interest Cost| \$148,296 × .08                                         | \$11,864        |
| 5. PBO (2027, after 15 years) | Sum of above                               | See below       |

---

## **Final Answer**

**1. Projected benefit obligation at the beginning of 2027:**  
$$
\boxed{\$148,296}
$$

**2. Annual retirement payments at retirement:**  
$$
\boxed{\$168,000}
$$

**3. Service cost for 2027:**  
$$
\boxed{\$9,887}
$$

**4. Interest cost for 2027:**  
$$
\boxed{\$11,864}
$$

**5. Projected benefit obligation at the end of 2027 (after 15 years’ service):**  
$$
\boxed{\$148,296 + \$9,887 + \$11,864 = \$170,047}
$$

---
### **Final Answer Summary**
The projected benefit obligation at the end of 2027, after 15 years of service, is **\$170,047**. This is based on the calculated annual payments, service cost, and interest cost as shown step-by-step above.

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