If $r_{eff}$ = effective rate of interest, $r$ = nominal rate of interest and $m$ = number of conversion periods per year, the relationship between nominal rate and effective rate of interest is:

The nominal rate $(r)$ is the stated annual interest rate. The effective rate $(r_{eff})$ is the actual interest earned after accounting for compounding. The variable $m$ represents the number of conversion periods per year.

When interest compounds multiple times per year, interest is earned on previously accumulated interest, resulting in a higher effective rate than the nominal rate.

The rate per conversion period is $\frac{r}{m}$.

The growth factor for one period is $1 + \frac{r}{m}$.

After $m$ periods (one full year), the growth factor becomes $\left(1 + \frac{r}{m}\right)^m$.

The effective interest rate is this growth factor minus the original principal:

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

Consider $r = 12\%$ (or $0.12$) with quarterly compounding $(m = 4)$:

$r_{eff} = \left(1 + \frac{0.12}{4}\right)^4 - 1$

$r_{eff} = (1 + 0.03)^4 - 1$

$r_{eff} = (1.03)^4 - 1$

$r_{eff} = 1.1255 - 1$

$r_{eff} = 0.1255$

The effective rate is $12.55\%$, which exceeds the nominal rate of $12\%$ due to quarterly compounding.

Therefore, the relationship between nominal rate and effective rate of interest is:

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