Annuities and interest conversion

Question

I am having trouble understanding how to find the equivalent rate of interest per payment periods for annuities.

For example, for this question:

Find the accumulated value at the end of four years of an investment fund in which 100 is deposited at the beginning of each quarter for the first two years and 200 is deposited at the beginning of each quarter for the second two years, if the fund earns 12% convertible monthly.

In the solution, it says to let $j$ be the equivalent rate of interest per quarter

$j = (1.01)^3 - 1 = 0.30301$

Why is $(1.01)^3$ put to the power of 3?

Answer 1

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

$$ FV=100\frac{s_{\overline{48}|0.01}+s_{\overline{24}|0.01}}{a_{\overline{3}|0.01}}=100\frac{61.2226 + 26.9735}{2.9410}=2999 $$

Answer 2

The interest is $12\%$ per year, compounded monthly, so it is $1\%$ per month. That means that every month the balance is multiplied by $1.01$. After three months it will be multiplied by $1.01$ three times, so you cube $1.01$ to get $1.01^3$