A nominal annual interest rate of 9.0% is compounded quarterly. What is the periodic interest rate applied each quarter?

Introduction:
This question checks the basic concept of converting a nominal annual interest rate into a rate per compounding period. The nominal annual rate is 9.0%, and it is compounded quarterly. We need to find the corresponding interest rate that is applied to the account each quarter.

Given Data / Assumptions:

• Nominal annual interest rate, r_nominal = 9.0% per year
• Compounding frequency: quarterly, that is 4 times per year
• We assume the same periodic rate is used in each quarter

Concept / Approach:
When interest is compounded m times per year at a nominal annual rate r_nominal, the periodic rate per compounding period is: i_periodic = r_nominal / m However, r_nominal must first be expressed as a percentage or decimal consistently. Since the question asks for the periodic interest rate in percent, we can perform the calculation directly with percent values:

Step-by-Step Solution:
Step 1: Note the compounding frequency. Compounding quarterly means m = 4 periods per year. Step 2: Divide the nominal annual rate by the number of periods. Periodic rate = 9.0% / 4 Periodic rate = 2.25% So, each quarter, an interest rate of 2.25% is applied to the balance.

Verification / Alternative check:
We can confirm this by converting to decimal form as well: r_nominal_decimal = 9.0% = 0.09 i_periodic_decimal = 0.09 / 4 = 0.0225 Converting back to a percentage: 0.0225 * 100 = 2.25% This confirms that the periodic rate is indeed 2.25% per quarter.

Why Other Options Are Wrong:
3.45%: This rate is too high for a quarterly rate when the nominal annual rate is only 9%. 5.25% and 6.25%: These are much too large and would imply annual rates far above 9% if used as periodic quarterly rates. 1.25%: This is too small and would correspond to a nominal annual rate of only 5% if compounded quarterly. 2.25%: This is exactly one quarter of the 9% nominal rate and is therefore the correct periodic rate.

Common Pitfalls:
A typical error is confusing the periodic rate with the effective annual rate, which requires using (1 + i_periodic)^m - 1. Here the question asks only for the rate per quarter, not the effective annual rate. Another mistake is dividing by 12 instead of 4, treating the rate as if it were compounded monthly. Care must be taken to match the correct frequency with the nominal annual rate.

Final Answer:
The periodic interest rate applied each quarter when the nominal annual rate is 9.0% compounded quarterly is 2.25% per quarter.