Question

The effective rate, $r_e$ equivalent to the nominal rate $r$ converted $m$ times a year is given by:

• A. $\left(1+\frac{r}{m}\right)$
• B. $\left(1+\frac{r}{m}\right)^m$
• C. $\left(1+\frac{r}{m}\right)^m+1$
• D. $\left(1+\frac{r}{m}\right)^m-1$

Hint:

Logic Tip: Effective annual rate is always greater than nominal rate when compounding occurs more than once a year.

Answer 1

The Correct Option is D

Step 1:
Effective annual rate measures the actual annual return when compounding occurs multiple times in a year.

Step 2:
If nominal annual rate is $r$ compounded $m$ times yearly, then: \[ r_e=\left(1+\frac{r}{m}\right)^m-1 \]

Step 3:

• $\frac{r}{m}$ gives periodic rate
• Raising to power $m$ accounts for compounding
• Subtracting 1 converts amount factor into effective rate

Step 4:
Therefore, the effective rate formula is: \[ \boxed{\left(1+\frac{r}{m}\right)^m-1} \] Hence, the correct answer is: \[ \boxed{\text{(4)}} \]