Introduction, sequences & Series, future value of payments  (Page 2/5)

The obvious way of working that out is to work out how much you need now to afford the payments individually and sum them. We'll work out how much is needed now to afford the payment of R1 000 in a year $(=\mathrm{R}1000\times {(1,10)}^{-1}=\mathrm{R}909,09)$ , the amount needed now for the following year's R1 000 $(=\mathrm{R}1000\times {(1,10)}^{-2}=\mathrm{R}826,45)$ and the amount needed now for the R1 000 after 3 years $(=\mathrm{R}1000\times {(1,10)}^{-3}=\mathrm{R}751,31)$ . Add these together gives you the amount needed to afford all three payments and you get R2 486,85.

So, if you put R2 486,85 into a 10% bank account now, you will be able to draw out R1 000 in a year, R1 000 a year after that, and R1 000 a year after that - and your bank account will come down to R0. You would have had exactly the right amount of money to do that (obviously!).

You can check this as follows:

Perfect! Of course, for only three years, that was not too bad. But what if I asked you how much you needed to put into a bank account now, to be able to afford R100 a month for the next 15 years. If you used the above approach you would still get the right answer, but it would take you weeks!

There is - I'm sure you guessed - an easier way! This section will focus on describing how to work with:

• annuities - a fixed sum payable each year or each month, either to provide a pre-determined sum at the end of a number of years or months (referred to as a future value annuity) or a fixed amount paid each year or each month to repay (amortise) a loan (referred to as a present value annuity).
• bond repayments - a fixed sum payable at regular intervals to pay off a loan. This is an example of a present value annuity.
• sinking funds - an accounting term for cash set aside for a particular purpose and invested so that the correct amount of money will be available when it is needed. This is an example of a future value annuity.

Sequences and series

Before we progress, you need to go back and read Chapter  [link] (from page  [link] ) to revise sequences and series.

In summary, if you have a series of $n$ terms in total which looks like this:

this can be simplified as:

Present values of a series of payments

So having reviewed the mathematics of Sequences and Series, you might be wondering how this is meant to have any practical purpose! Given that we are in the finance section, you would be right to guess that there must be some financial use to all this. Here is an example which happens in many people's lives - so you know you are learning something practical.

Let us say you would like to buy a property for R300 000, so you go to the bank to apply for a mortgage bond. The bank wants it to be repaid by annually payments for the next 20 years, starting at end of this year. They will charge you 15% interest per annum. At the end of the 20 years the bank would have received back the total amount you borrowed together with all the interest they have earned from lending you the money. You would obviously want to work out what the annual repayment is going to be!