A retirement benefit of 12,000 dollars is to be paid every 6 months for 25 years at 7% interest compounded semiannually. What present value must be set aside now to fund this end of period retirement benefit?

Introduction / Context:
This retirement planning problem is a present value of an ordinary annuity question with semiannual payments and semiannual compounding. The aim is to find how much money must be invested now so that it can fund regular retirement payments over a long period at a fixed interest rate.

Given Data / Assumptions:

• Retirement payment R = 12,000 dollars every 6 months.
• Nominal annual interest rate = 7%, compounded semiannually.
• Semiannual rate i = 7% / 2 = 3.5% = 0.035.
• Payment period = 25 years, so number of payments n = 25 * 2 = 50.
• Payments are made at the end of each 6 month period (ordinary annuity).

Concept / Approach:
The present value of an ordinary annuity is:
PV = R * (1 - (1 + i)^(-n)) / i This formula discounts each future retirement payment back to today at the semiannual rate and sums them into a single equivalent amount now.

Step-by-Step Solution:
Step 1: Set i = 0.035 and n = 50. Step 2: Compute (1 + i)^(-n) = (1.035)^(-50). Step 3: Evaluate the factor (1 - (1.035)^(-50)) / 0.035. Step 4: Multiply this factor by R = 12,000 dollars. Step 5: The resulting present value is approximately 281,468.06 dollars. Thus around 281,468.06 dollars must be invested today.

Verification / Alternative Check:
As a rough check, the total of all payments is 12,000 * 50 = 600,000 dollars. Because the money earns interest, the present value must be significantly less than 600,000 dollars. A value around 280,000 dollars is reasonable for 25 years at 7% nominal interest.

Why Other Options Are Wrong:
Options A, B, and D (245,678; 234,689; 234,578 dollars) are all lower than the computed present value and would not be sufficient to fund the entire stream of payments.
Option E (200,000 dollars) is clearly too low for such a large series of payments.
Only option C closely matches the correct formula based computation.

Common Pitfalls:
Some learners mistakenly use 7% as the semiannual rate instead of dividing by 2, which doubles the interest rate per period and depresses the present value too much. Others may treat the payments as annual rather than semiannual, which halves the number of periods and again gives the wrong answer.

Final Answer:
The present value required to fund the semiannual retirement payments is approximately 281,468.06 dollars.