A nominal annual interest rate of 8.4% per annum is quoted. If the periodic interest rate is 0.7% per period, what is the compounding frequency per year (that is, the number of compounding periods each year)?

Introduction / Context:
This question explores another situation where you must deduce the compounding frequency per year from the nominal annual interest rate and the periodic interest rate. It directly reinforces the formula that connects these quantities and highlights a common real life case, monthly compounding.

Given Data / Assumptions:

• Nominal annual interest rate r_nom = 8.4% per annum.
• Periodic interest rate r_per = 0.7% per period.
• Compounding frequency per year m is unknown.
• The nominal rate is distributed uniformly across all compounding periods.

Concept / Approach:
The relationship between the nominal annual rate and the periodic rate is r_nom = m * r_per. Given r_nom and r_per, we can solve for m using simple division. Once m is found, we can interpret it. If m equals 12, that directly corresponds to monthly compounding, since there are 12 months in a year.

Step-by-Step Solution:
Step 1: Write r_nom = 8.4% and r_per = 0.7%.Step 2: Use the formula r_nom = m * r_per.Step 3: Rearrange to get m = r_nom / r_per.Step 4: Compute m = 8.4 / 0.7.Step 5: 8.4 / 0.7 = 12.Step 6: Therefore, the compounding frequency is m = 12 times per year, which corresponds to monthly compounding.

Verification / Alternative check:
To verify, suppose interest is compounded monthly so that m = 12. If each period has rate 0.7%, the nominal annual rate would be 12 * 0.7% = 8.4% per annum, which is exactly the rate given. This confirms that our calculation is consistent and that a compounding frequency of 12 is correct for this periodic rate.

Why Other Options Are Wrong:

• 9: If m were 9, the nominal rate would be 9 * 0.7% = 6.3%, less than 8.4%.
• 10: If m were 10, the nominal rate would be 7%, still lower than 8.4%.
• 11: If m were 11, the nominal rate would be 7.7%, which is also less than 8.4%.

Common Pitfalls:
One common mistake is to confuse nominal rates with effective annual rates and attempt to use logarithms or exponential formulas. For this type of question, that is not needed. Another pitfall is incorrect arithmetic when dividing 8.4 by 0.7, especially if students treat the numbers as decimals without careful computation. It is also important to remember that the answer must be an integer since it represents a count of compounding periods per year, so fractional answers are not meaningful in this context.

Final Answer:
The compounding frequency is 12 times per year, corresponding to monthly compounding.