A nominal annual interest rate of 13% is compounded quarterly. What is the corresponding effective annual interest rate to the nearest hundredth of a percent?

Introduction:
This problem focuses on converting a nominal annual interest rate that is compounded several times a year into its equivalent effective annual rate. Understanding this conversion is important in comparing different financial products that use different compounding frequencies but advertise nominal yearly rates.

Given Data / Assumptions:

• Nominal annual interest rate, r_nominal = 13% per year
• Compounding frequency: quarterly, that is 4 times per year
• We want the effective annual rate, r_effective, which represents the total percentage growth over one full year
• No deposits or withdrawals during the year; we are only interested in the growth factor due to interest

Concept / Approach:
When a nominal annual rate r_nominal is compounded m times per year, the periodic rate per compounding period is: i = r_nominal / m expressed in decimal form. The effective annual rate r_effective is then: r_effective = (1 + i)^m - 1 This formula accounts for the interest-on-interest effect of multiple compounding periods during the year.

Step-by-Step Solution:
Step 1: Convert the nominal rate from percent to decimal. r_nominal = 13% = 0.13 Step 2: Determine the quarterly rate i. m = 4 (quarterly compounding) i = r_nominal / m = 0.13 / 4 = 0.0325 Step 3: Compute the effective annual rate using the formula. r_effective = (1 + i)^m - 1 r_effective = (1 + 0.0325)^4 - 1 Step 4: Evaluate the power. (1.0325)^4 ≈ 1.1365 (rounded) Step 5: Subtract 1 to get r_effective in decimal form. r_effective ≈ 1.1365 - 1 = 0.1365 Step 6: Convert to a percentage. r_effective ≈ 0.1365 * 100 = 13.65%

Verification / Alternative check:
We can approximate by noting that 13% simple interest would give 13% in one year, but because compounding quarterly adds interest four times, the effective rate must be slightly higher than 13%. The computed value of 13.65% is only a little above 13%, which is consistent with expectations for quarterly compounding at this nominal rate.

Why Other Options Are Wrong:
14.66%: This is too high for quarterly compounding at a nominal 13% rate. It might correspond to a higher nominal rate or to more frequent compounding. 15.65% and 16.65%: These are much too high and do not match the formula for a 13% nominal rate compounded quarterly. 13.25%: This is closer to a simple adjustment that ignores compounding, such as a minor addition, but it does not follow the correct effective interest rate formula. 13.65%: This matches the exact calculation from the standard effective annual rate formula and is therefore correct.

Common Pitfalls:
A typical mistake is to forget that the nominal rate must be divided by the number of compounding periods to get the periodic rate. Another common error is to simply add a small percentage to the nominal rate without using the power function, which leads to inaccurate results. Some learners also confuse effective rate with nominal rate and try to divide or multiply percentages directly without using (1 + i)^m - 1.

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