Calculate the effective annual interest rate corresponding to a nominal annual rate of 13% when interest is compounded quarterly.

Introduction:
This question asks for the effective annual interest rate when a nominal rate is compounded quarterly. This is a core idea in finance: nominal rates are quoted, but effective rates show the real growth over a year after accounting for compounding frequency.

Given Data / Assumptions:

• Nominal annual interest rate r_nominal = 13% per annum
• Compounding frequency = quarterly (4 times per year)
• We need the effective annual rate r_effective, which gives the same annual growth with one compounding per year.

Concept / Approach:
The effective annual rate for a nominal rate r_nominal compounded m times per year is given by:
r_effective = (1 + r_nominal / m)^m - 1Here r_nominal must be expressed as a decimal. For quarterly compounding, m = 4. Finally, convert r_effective back to percentage form.

Step-by-Step Solution:
Step 1: Convert nominal rate to decimal.r_nominal = 13% = 0.13Step 2: Identify compounding periods.m = 4 (quarterly)Step 3: Apply the effective rate formula.r_effective = (1 + 0.13 / 4)^4 - 1r_effective = (1 + 0.0325)^4 - 1r_effective = (1.0325)^4 - 1Calculating (1.0325)^4 gives approximately 1.1365.
r_effective ≈ 1.1365 - 1 = 0.1365r_effective ≈ 13.65%

Verification / Alternative check:
A rough approximation method is to note that the effective rate will be slightly higher than 13% due to compounding. Since 0.13 / 4 = 3.25% per quarter, the total simple interest equivalent would be 13%, and compounding this slightly increases it to about 13.6%. This checks well with our more exact computation of 13.65%.

Why Other Options Are Wrong:

• 14.665%, 15.65%, 16.65%: These are significantly higher than the correct effective rate and would correspond to larger growth than 13% nominal quarterly.
• 13.00%: This would be the rate with no compounding effect, which is not the case here.

Common Pitfalls:
A common mistake is to assume that the effective rate is the same as the nominal rate or to multiply 13% by 4, which is completely incorrect. Some also forget to convert the percentage to decimal before applying the formula. Always remember to use the expression (1 + r/m)^m - 1 for effective rate calculations.

Final Answer:
The effective annual interest rate corresponding to 13% nominal compounded quarterly is approximately 13.65%.