A nominal annual interest rate of 8.4% per annum is quoted. If the periodic interest rate is 2.1% per period, what is the compounding frequency per year (number of compounding periods each year)?

Introduction / Context:
Here we again connect a nominal annual interest rate with a given periodic interest rate and use that to find how many times interest is compounded every year. This type of problem is common in banking and finance, where a lender specifies a nominal rate and an implied compounding frequency.

Given Data / Assumptions:

• Nominal annual interest rate r_nom = 8.4% per annum.
• Periodic interest rate r_per = 2.1% per period.
• Compounding frequency per year m is unknown.
• The nominal rate is evenly distributed across all periods within the year.

Concept / Approach:
The fundamental relationship is r_nom = m * r_per, where r_nom is nominal annual rate, r_per is periodic rate, and m is the number of compounding periods per year. If we know r_nom and r_per, we can solve for m by simple division. Because both rates are given in percent form, we can work directly with the numerical values 8.4 and 2.1.

Step-by-Step Solution:
Step 1: Write r_nom = 8.4% and r_per = 2.1%.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 / 2.1.Step 5: 8.4 / 2.1 = 4.Step 6: Therefore, interest is compounded 4 times per year, which corresponds to quarterly compounding.

Verification / Alternative check:
To verify, suppose interest is compounded quarterly, so m = 4. Then the nominal rate implied by a periodic rate of 2.1% is 4 * 2.1% = 8.4% per annum, which exactly matches the given nominal rate. This confirms that m = 4 is consistent with the information in the question.

Why Other Options Are Wrong:

• 1: If m were 1, the nominal annual rate would be only 2.1%, not 8.4%.
• 2: If m were 2, the nominal rate would be 2 * 2.1% = 4.2%, which is half of 8.4%.
• 3: If m were 3, then 3 * 2.1% = 6.3%, still less than the required 8.4%.

Common Pitfalls:
Learners sometimes try to use compound interest formulas that involve exponents when this simple relationship is enough. Another mistake is not keeping track of units, for example mixing percentages with decimals without proper conversion. It is also common to perform the division in the reverse order, 2.1 / 8.4, which gives 0.25 and then misinterpret that figure as the answer. Remember that the compounding frequency must be a positive whole number of periods per year.

Final Answer:
The compounding frequency is 4 times per year, which corresponds to quarterly compounding.