A retirement benefit of $12,000 is to be paid at the end of every 6 months for 25 years, with interest at 7% per annum compounded semi-annually. What is the present value that must be available today to fund this series of payments?

Introduction:
This question focuses on the present value of a long term semi-annual retirement annuity. A benefit of 12,000 dollars is paid at the end of every 6 months for 25 years, with an interest rate of 7% per year compounded semi-annually. We must find how much money is required today to fund all of these future payments.

Given Data / Assumptions:

• Periodic retirement payment, R = 12,000 dollars
• Nominal annual interest rate = 7% per annum
• Compounding frequency: semi-annually, so 2 periods per year
• Interest rate per 6 month period, i = 7% / 2 = 3.5% = 0.035
• Time = 25 years
• Number of semi-annual payments, n = 25 * 2 = 50
• Payments occur at the end of each period (ordinary annuity)

Concept / Approach:
The present value P of an ordinary annuity is given by: P = R * (1 - (1 + i)^(-n)) / i This formula discounts each payment of amount R back to the present using the periodic interest rate i and the number of periods n. The total present value is the sum of all these discounted payments.

Step-by-Step Solution:
Step 1: Identify R, i, and n. R = 12000, i = 0.035, n = 50 Step 2: Use the present value formula. P = 12000 * (1 - (1 + 0.035)^(-50)) / 0.035 Step 3: Compute (1 + 0.035)^50. (1.035)^50 is approximately 5.516 (using accurate calculation) Step 4: Compute (1.035)^(-50). (1.035)^(-50) ≈ 1 / 5.516 ≈ 0.1813 Step 5: Substitute into the formula. P = 12000 * (1 - 0.1813) / 0.035 P = 12000 * 0.8187 / 0.035 P ≈ 12000 * 23.392 ≈ 280704 Using more precise computation yields a present value of about 281468.06 dollars, which matches the given answer choice.

Verification / Alternative check:
The total of all benefits over 25 years is: Total nominal payments = 12000 * 50 = 600000 dollars The present value of 281,468.06 dollars is much less than 600,000 dollars because the payments are spread over a long period and each future payment is discounted at 3.5% per half year. This order of magnitude is reasonable for an interest rate of 7% per year over 25 years.

Why Other Options Are Wrong:
245678.00 and 234689.00: These are lower than the correct present value and would be insufficient to fund 12,000 dollars every 6 months for 25 years at the stated interest rate. 234578.00 and 201468.06: These figures are even smaller and would run out long before the last payment is due. 281468.06: This matches the precise calculation of the present value using the standard annuity formula.

Common Pitfalls:
Some learners mistakenly use the annual rate directly as i = 0.07 instead of halving it for semi-annual compounding. Others incorrectly multiply the total of all payments by a discount factor that does not reflect the timing of each payment. It is also common to confuse present value and future value formulas, which leads to answers of the wrong magnitude. Careful interpretation of the compounding frequency and payment timing is essential for solving such problems.

Final Answer:
The present value required today to fund 12,000 dollar payments every 6 months for 25 years at 7% per annum compounded semi-annually is approximately 281468.06 dollars.