A nominal annual interest rate of 9.75% is compounded semiannually. What is the periodic rate of interest applied every half year?

Introduction / Context:
This question examines your understanding of how to convert a nominal annual interest rate into a periodic rate when compounding takes place more than once a year. Financial contracts, such as loans and investments, frequently specify a nominal rate along with the compounding frequency, so being able to move from nominal to periodic rates is a basic but essential skill in financial mathematics.

Given Data / Assumptions:

• Nominal annual rate j = 9.75% per annum
• Compounding frequency is semiannual, which means twice per year
• We must find the interest rate applied every half year (the periodic rate)
• No adjustment for effective annual rate is required, only nominal to periodic conversion

Concept / Approach:
For a nominal annual rate j compounded m times per year, the periodic rate i per compounding period is given by i = j / m. In this problem, j = 9.75% and m = 2 because compounding happens every six months. Therefore, we simply divide 9.75 by 2 to get the rate that is applied at each semiannual compounding date. This step does not yet account for the effect of compounding over a full year; it only tells us the rate used in each period.

Step-by-Step Solution:
Nominal annual rate j = 9.75% Compounding frequency m = 2 (semiannual) Periodic rate i = j / m i = 9.75% / 2 i = 4.875% per half year

Verification / Alternative check:
If we apply 4.875% twice in a year, the nominal rate, by definition, is 2 * 4.875% = 9.75%. If we wanted the effective annual rate, we would compute (1 + 0.04875)^2 - 1, but the question does not ask for that. It only asks for the periodic rate, confirming that the division by 2 is the appropriate operation. Checking the arithmetic shows that 4.875 * 2 = 9.75, so the calculation is consistent.

Why Other Options Are Wrong:
3.785%, 4.865%, and 4.975% per half year do not multiply by 2 to give 9.75%. For example, 3.785% * 2 = 7.57%, which is not the stated nominal rate. Similarly, 4.865% * 2 = 9.73% and 4.975% * 2 = 9.95%, both inconsistent with 9.75%. Only 4.875% correctly corresponds to half of the nominal rate.

Common Pitfalls:
Typical mistakes include treating the nominal rate as if it were already a periodic rate, or attempting to use more complex formulas involving compounding unnecessarily. Some learners also confuse nominal and effective rates and try to convert to an effective annual rate when the question only needs a simple division. Carefully reading the phrase per half year and identifying the compounding frequency avoids these errors.

Final Answer:
The periodic interest rate is 4.875% per half year when a nominal 9.75 percent rate is compounded semiannually.