Annuities - SS3 Mathematics Lesson Note

Compound interest is computed on a fixed sum deposited in an account that pays interest that is compounded periodically. However, not many people are able to deposit a large sum of money at once. Rather, they invest money by depositing relatively small amounts at a time. This is termed as an Annuity.

Annuity is a series of equal payments made at regular intervals (not necessarily annually), and the interest is paid when each regular payment is made.

The future value of any annuity is the sum of all payments plus all interests earned. If \(R\) is the periodic payment of an annuity, \(i\) the interest rate per period and \(n\) the number of periods, then the future value \(S\) of an annuity is:

\[S = R\left( \frac{{(1 + i)}^{n} - 1}{i} \right)\]

Example 3 Find the future value of an annuity of \(N300\) at \(10\%\) per annum for \(5\ years\)

Solution

\[R = 300,\ i = 0.1,\ n = 5\]

\[S = 300\left( \frac{{(1 + 0.1)}^{5} - 1}{0.1} \right)\]

\[S = 300\left( \frac{(1.1)^{5} - 1}{0.1} \right)\]

\[S = 300(6.11) = 1833\]

\[S = N1,833\]

Example 4 What is the total interest that will accrue on an annuity of N500 paid yearly for \(8\ years\) at \(3\%\) per annum

Solution

\[R = 500,\ i = 0.03,\ n = 8\]

\[S = 500\left( \frac{(1 + 0.03)^{8} - 1}{0.03} \right)\]

\[S = 500(8.8666)\]

\[S = 4433.30\]

\[Total\ interest\ = \ Amount\ –\ total\ money\ paid\]

\[Total\ interest = 4433.30 - 8(500)\]

\[Total\ interest = 4433.30 - 4000 = N433.30\]

Example 5 John deposits \(N1000\) at the end of each quarter for \(15\ years\) in a bank that pays an annual interest rate of \(18\%\) compounded quarterly. Find the worth of the account.

Solution

\[R = 1000,\ i = \frac{0.18}{4} = 0.045,\ n = 4 \times 15 = 60\]

\[S = 1000\left( \frac{(1 + 0.045)^{60} - 1}{0.045} \right)\]

\[S = 1000(288.8889)\]

\[S = N288,888.90\]