A loan has a nominal annual interest rate of 8.4% per annum. If the periodic interest rate is 4.2% per period, what is the compounding frequency per year (number of compounding periods in one year)?

Introduction / Context:
This problem tests the concept of how a nominal annual interest rate is related to a periodic interest rate and the number of compounding periods per year. Understanding this relationship is important in many financial calculations because it helps you interpret how often interest is actually added to the principal.

Given Data / Assumptions:

• Nominal annual interest rate r_nom = 8.4% per annum.
• Periodic interest rate r_per = 4.2% per period.
• Interest is compounded a fixed number of times per year, which we denote by m.
• The nominal rate is assumed to be evenly split across all periods.

Concept / Approach:
The relationship between nominal annual rate, periodic rate, and compounding frequency is given by r_nom = m * r_per, where r_nom is the nominal annual rate in percent, r_per is the periodic rate in percent, and m is the number of compounding periods per year. To find m, we simply divide the nominal annual rate by the periodic rate. Once we obtain m, we match it with the correct option.

Step-by-Step Solution:
Step 1: Write the nominal annual rate as r_nom = 8.4%.Step 2: Write the periodic rate as r_per = 4.2% per period.Step 3: Use the formula r_nom = m * r_per, so m = r_nom / r_per.Step 4: Compute m = 8.4 / 4.2.Step 5: 8.4 / 4.2 = 2.Step 6: Therefore, the compounding frequency is m = 2 times per year, which usually corresponds to semiannual compounding.

Verification / Alternative check:
We can verify the answer by reversing the calculation. If m = 2 and each period has an interest rate of 4.2%, then the nominal annual rate would be 2 * 4.2% = 8.4%, which matches the data given in the question. Since the calculated nominal rate is consistent, the frequency of 2 compounding periods per year is correct.

Why Other Options Are Wrong:

• 1: If m were 1, then the nominal rate would equal the periodic rate, that is 4.2%, which does not match the given 8.4%.
• 3: If m were 3, then the nominal rate would be 3 * 4.2% = 12.6%, which is greater than 8.4%.
• 4: If m were 4, then the nominal rate would be 4 * 4.2% = 16.8%, which is also not equal to 8.4%.

Common Pitfalls:
A common mistake is to confuse nominal and effective rates and try to use exponent formulas unnecessarily. Another error is to divide in the wrong order, such as 4.2 / 8.4, which gives 0.5 and does not correspond to a meaningful compounding frequency. Learners may also forget that the rates in this formula must both be expressed in the same units, here both in percent per year and percent per period, so that they cancel correctly when calculating m.

Final Answer:
The compounding frequency is 2 times per year, corresponding to semiannual compounding.