The effective rate of interest is the amount of money that one unit invested
at the beginning of a period will earn during the period, with interest being
paid at the end of the period; when we speak of the effective rate of interest we mean interest is paid once per measurement period.
An interest rate is called nominal if the frequency of compounding (e.g. a month) is not identical to the basic time unit (normally a year): interest is paid
more than once per measurement period.
When interest is paid (i.e., reinvested) more frequently than once per period,
we say it is payable (convertible, compounded) each fraction of a period,
and this fractional period is called the interest conversion period.
A nominal rate of interest $i^{(m)}$ payable $m$ times per period, where $m$ is a
positive integer, represents $m$ times the effective rate of compound interest
used for each of the $m$-th of a period. In this case, $\frac{i^{(m)}}{m}$ is the effective rate of interest for each $m$-th of a period.
Thus, for a nominal rate of $i^{(12)}=12\%$ compounded monthly, the effective rate of interest per month is $\frac{i^{(12)}}{12}1\%$ since there are twelve months in a year.
Two rates are said to be equivalent if, for the same initial investment and over the
same time interval (one full year, for example), the final value of the investment,
calculated with the two interest rates, is equal.
If $j_q=\frac{i^{(4)}}{4}$ denotes the effective rate of interest per quarter equivalent to the effective rate of interest per month $j_{m}=\frac{i^{(12)}}{12}$ then we can write
$$ \left(1 +\frac{i^{(4)}}{4}\right)^4 =\left(1 +\frac{i^{(12)}}{12}\right)^{12}$$
since each side represents the accumulated value of a principal of $1$ invested
for one year; or equivalently $$ \left(1 +j_q\right) =\left(1 +j_{m}\right)^3$$
since each side represents the accumulated value of a principal of $1$ invested
for one quarter (3 months).
So in your case
- $i^{(12)}=12\%$ is the nominal interest rate compounded monthly;
- $j_m=\frac{i^{(12)}}{12}=1\%$ is the monthly effective rate of interest:
- $j_q=\left(1 +\frac{i^{(12)}}{12}\right)^{3}-1=3.03\%$ is the quarterly effective rate of interest
Let be $P=100$ (for the first 2 years) and $Q=200$ (for the last 2 years) the payments made at the beginning of each period (each quarter) that will produce interest every month, or equivalently a first annuity of $P$ for 4 years and a second annuity of$P$ for 2 years.
The rate of interest is $1\%$ per month. In this annuity-due, there are 48 interest periods and each payment period consists of 3 interest coversion periods. So, the accumulated value is
$$
FV=100\frac{s_{\overline{48}|0.01}+s_{\overline{24}|0.01}}{a_{\overline{3}|0.01}}=100\frac{61.2226 + 26.9735}{2.9410}=2999
$$
