For a nominal interest rate of 9.75% per annum compounded semi-annually, what is the periodic interest rate applied in each half-year compounding period?

Introduction:
In this problem we are asked to find the periodic interest rate when a nominal annual rate is given and the compounding frequency is specified. The nominal rate is 9.75% per year, and interest is compounded semi annually, meaning twice per year. We simply need to determine the rate per half year.

Given Data / Assumptions:

• Nominal annual rate R = 9.75% per annum.
• Compounding frequency = semi annually, that is 2 periods per year.
• We want the interest rate i per compounding period.

Concept / Approach:
For a nominal rate R compounded m times per year, the periodic rate i is:
i = R / mif R is expressed in percentage terms per year and compounding is simple with equal periods. Here m = 2 because the compounding is semi annual.

Step-by-Step Solution:
Step 1: Identify the nominal rate R as 9.75%.Step 2: Note that semi annual compounding means m = 2 compounding periods in a year.Step 3: Compute the periodic rate: i = 9.75% / 2.Step 4: Perform the division: 9.75 / 2 = 4.875.Step 5: Attach the percent sign: the periodic rate is 4.875% per half year.So each six month period uses an interest rate of 4.875%.

Verification / Alternative check:
If we apply 4.875% twice, the nominal annual rate becomes 9.75% when expressed as a sum of periodic rates, though the effective annual rate would be slightly higher due to compounding. However, the question clearly asks only for the periodic rate, not the effective annual rate, so dividing by 2 is appropriate.

Why Other Options Are Wrong:
3.785%, 4.865%, and 4.975% are arbitrary values that do not match the straightforward division of 9.75 by 2. Option 5.875% is greater than half the nominal rate and cannot be correct. Only 4.875% is exactly half of 9.75% and thus correctly represents the periodic rate for semi annual compounding.

Common Pitfalls:
Some students confuse the periodic rate with the effective annual rate and attempt to compute (1 + i)^2 - 1 instead. Others mistakenly divide by 12 as if the compounding were monthly, or by 4 as if it were quarterly. Always read the compounding frequency carefully and divide the nominal rate by the correct number of periods per year to get the periodic rate.

Final Answer:
The periodic interest rate for a nominal 9.75% per annum compounded semi annually is 4.875% per half year.