Question

The nominal and effective interest rates are equal when the interest is compounded

• A. monthly
• B. continuously
• C. semi annually
• D. annually

Hint:

As compounding frequency \(m\) increases within a single year, the effective interest rate grows increasingly larger than the nominal interest rate: \[ i_{\text{annual}} < i_{\text{semi-annual}} < i_{\text{monthly}} < i_{\text{continuous}} \] They match perfectly only at \(m = 1\) (annual compounding).

Answer 1

The Correct Option is D

Concept: In corporate economic evaluations, interest rates are specified using two definitions:
• Nominal Interest Rate (\(r\)): The stated annual interest rate that does not account for compounding within the year.
• Effective Interest Rate (\(i_{eff}\)): The actual interest rate earned or paid over a full one-year period, taking into account the effects of compounding frequency.

Step 1: Setting up the mathematical conversion equation.
If a nominal annual rate \(r\) is compounded over \(m\) discrete intervals per year, the effective annual interest rate is given by: \[ i_{eff} = \left(1 + \frac{r}{m}\right)^m - 1 \]

Step 2: Evaluating based on the given choices.
Let's analyze the values of \(i_{eff}\) for different compounding frequencies \(m\):
• Monthly (\(m = 12\)): \(i_{eff} = (1 + r/12)^{12} - 1 \neq r\)
• Semi-annually (\(m = 2\)): \(i_{eff} = (1 + r/2)^2 - 1 \neq r\)
• Continuously (\(m \rightarrow \infty\)): \(i_{eff} = e^r - 1 \neq r\)
• Annually (\(m = 1\)): Compounding happens exactly once at the conclusion of the year. Substituting \(m = 1\): \[ i_{eff} = \left(1 + \frac{r}{1}\right)^1 - 1 = 1 + r - 1 = r \] Therefore, the nominal rate equals the effective rate if and only if interest is compounded annually (once per year). This corresponds to option (4).