Understanding Effective Rate vs. Nominal Rate

The nominal rate ($r$) is the stated interest rate, while the effective rate ($r_e$) is the actual rate earned after accounting for compounding. The question asks for the formula for the effective rate when the nominal rate $r$ is compounded $m$ times per year.

Deriving the Effective Rate Formula

1. Consider a principal amount, say $P = 1$. When interest is compounded $m$ times a year at a nominal rate $r$, the interest rate per compounding period is $\frac{r}{m}$.

2. After one year, the total amount ($A$) will be the principal plus the accumulated interest. The formula for compound interest gives the amount after $m$ periods:

   $ A = P \left(1 + \frac{r}{m}\right)^m $

   Since we assumed $P=1$, the amount after one year is:

   $ A = \left(1 + \frac{r}{m}\right)^m $

3. The effective annual rate ($r_e$) is the simple interest rate that would yield the same amount $A$ over one year. The total interest earned using the effective rate is $A - P$. With $P=1$, the interest is $A - 1$.

   $ \text{Interest} = A - P = \left(1 + \frac{r}{m}\right)^m - 1 $

4. The effective rate ($r_e$) is defined as the total interest earned divided by the principal:

   $ r_e = \frac{\text{Interest}}{P} $

   Substituting the value of interest and $P=1$:

   $ r_e = \frac{\left(1 + \frac{r}{m}\right)^m - 1}{1} $

   $ r_e = \left(1 + \frac{r}{m}\right)^m - 1 $

Final Formula

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

$ r_e = \left(1 + \frac{r}{m}\right)^m - 1 $

This corresponds to Option 4.