General annuities are essentially the opposite of simple annuities. Recall, for simple annuities, the frequency of the payment periods and compounding periods are the same. The defining characteristic of general annuities is that the frequency of the payment periods and compounding periods ARE NOT the same. Note that in this section, we are still only looking at simple annuities. Thus, there is still no payment at the beginning of the annuity.
The steps to solve general annuity problems use knowledge from previous lessons. The main idea is to turn a general annuity problem into a simple annuity problem. There are 2 steps to solving general annuity problems:
For a refresher on simple annuities, click here.
Example 1
Jamie deposits $250 into an account at the end of each month into an account that makes 5.75% compounded quarterly. How much money will be in the account after 5 years?
Solution
This is a simple general annuity because the deposits are made at the end of each month (simple) and the frequency of the payments, monthly, is not the same as the frequency of the compounding periods, quarterly (general). We are asked to find the future value of the annuity
First, we find the interest rate compounded monthly which is equivalent to 5.75% compounded quarterly.
\begin{align} \left(1+\frac{r}{12}\right)^{12} &= \left(1+\frac{0.0575}{4}\right)^4 \\ \left[\left(1+\frac{r}{12}\right)^{12}\right]^{\frac{1}{12}} &= \left[\left(1+\frac{0.0575}{4}\right)^4\right]^{\frac{1}{12}} \\ 1+\frac{r}{12} &= \left(1+\frac{0.0575}{4}\right)^{\frac{4}{12}} \\ \frac{r}{12} &= \left(1+\frac{0.0575}{4}\right)^{\frac{4}{12}} -1 \\ r &= 12\left[\left(1+\frac{0.0575}{4}\right)^{\frac{4}{12}} -1\right] \\ r &= 0.057226659 \end{align}
Thus, the equivalent interest rate is 5.7226659%. Now, we can use this in the simple annuity formula.
\(S_n = ? | R=$250 | i=\frac{5.7226659%}{12} | n=5*12=60\)
\(S_n = \frac{250[(1+\frac{0.057226659}{12})^{60}-1]}{\frac{0.057226659}{12}} = $17,318.75$\)
Thus, the account will have $17,318.75 after 5 years
Example 2
A car was purchased with monthly payments of $325 for 7 years. If interest is 8.5% compounded semi-annually, what is the purchase price of the car?
Solution
This is a general annuity question because the frequency of payments (monthly) is different than the frequency of compounding periods (semi-annually). First, calculate the interest rate compounded monthly which is equivalent to 8.5% compounded semi-annually. Here, we are looking for the present value.
\begin{align} \left(1+\frac{r}{12}\right)^{12} &= \left(1+\frac{0.085}{2}\right)^2 \\ \left[\left(1+\frac{r}{12}\right)^{12}\right]^{\frac{1}{12}} &= \left[\left(1+\frac{0.085}{2}\right)^2\right]^{\frac{1}{12}} \\ 1+\frac{r}{12} &= \left(1+\frac{0.085}{2}\right)^{\frac{2}{12}} \\ \frac{r}{12} &= \left(1+\frac{0.085}{2}\right)^{\frac{2}{12}} -1 \\ r &= 12\left[\left(1+\frac{0.085}{2}\right)^{\frac{2}{12}} -1\right] \\ r &= 0.0835532745 \end{align}
Thus, the equivalent interest rate is 8.3532745%
\(A_n = ? | R=$325 | i = \frac{8.3532745%}{12} | n=7*12=84\)
\(A_n = \frac{325[(1-(1+\frac{0.0835532745}{12})^{-84}]}{\frac{0.0835532745}{12}} = $20,618.16\)
Thus, the purchase price of the car was $20,618.16