If the nominal interest rate is 10% per year compounded semiannually, what is the effective annual return on the principal?

Introduction / Context:
Engineering economy problems routinely compare nominal versus effective interest rates. When compounding occurs more than once per year, the effective annual rate (EAR) exceeds the nominal rate because interest earns interest within the year.

Given Data / Assumptions:

• Nominal annual rate i_nom = 10% per year.
• Compounding frequency m = 2 (semiannual).
• No fees or taxes; standard compounding.

Concept / Approach:
The effective annual rate is computed from the nominal rate and compounding frequency using:
EAR = (1 + i_nom/m) ^ m - 1

Step-by-Step Solution:
1) Substitute i_nom = 0.10 and m = 2.2) Compute periodic rate: i_period = 0.10 / 2 = 0.05.3) Compute EAR: EAR = (1 + 0.05) ^ 2 - 1 = 1.1025 - 1 = 0.1025.4) Convert to percent: 0.1025 * 100 = 10.25%.

Verification / Alternative check:
A quick approximation for small i is i_eff ≈ i_nom + (i_nom^2)/(2m); for i_nom = 10% and m = 2, this also indicates an increase above 10%, confirming 10.25%.

Why Other Options Are Wrong:
10%: ignores compounding effect within the year.
20% or >20%: massively overstates the EAR; semiannual compounding at 10% nominal cannot double the rate.
Exactly 15%: arbitrary and not derived from the compounding formula.

Common Pitfalls:
Confusing nominal with effective rates; forgetting to apply the exponent equal to the number of compounding periods.

Final Answer:
< 20% (specifically, 10.25%)