Introduction, sequences & Series, future value of payments  (Page 3/5)

Let $X$ be the annual repayment, $i$ is the interest rate, and $M$ is the amount of the mortgage bond you will be taking out.

Time lines are particularly useful tools for visualizing the series of payments for calculations, and we can represent these payments on a time line as:

The present value of all the payments (which includes interest) must equate to the (present) value of the mortgage loan amount.

Mathematically, you can write this as:

The painful way of solving this problem would be to do the calculation for each of the terms above - which is 20 different calculations. Not only would you probably get bored along the way, but you are also likely to make a mistake.

Naturally, there is a simpler way of doing this! You can rewrite the above equation as follows:

Of course, you do not have to use the method of substitution to solve this. We just find this a useful method because you can get rid of the negative exponents - which can be quite confusing! As an exercise - to show you are a real financial whizz - try to solve this without substitution. It is actually quite easy.

Now, the item in square brackets is the sum of a geometric sequence, as discussion in [link] . This can be re-written as follows, using what we know from Chapter  [link] of this text book:

Note that we took out a common factor of $v$ before using the formula for the geometric sequence.

So we can write:

This can be re-written:

So, this formula is useful if you know the amount of the mortgage bond you need and want to work out the repayment, or if you know how big a repayment you can afford and want to see what property you can buy.

For example, if I want to buy a house for R300 000 over 20 years, and the bank is going to charge me 15% per annum on the outstanding balance, then the annual repayment is:

This means, each year for the next 20 years, I need to pay the bank R47 928,44 per year before I have paid off the mortgage bond.

On the other hand, if I know I will only have R30 000 a year to repay my bond, then how big a house can I buy? That is easy ....

So, for R30 000 a year for 20 years, I can afford to buy a house of R187 800 (rounded to the nearest hundred).

The bad news is that R187 800 does not come close to the R300 000 you wanted to pay! The good news is that you do not have to memorise this formula. In fact , when you answer questions like this in an exam, you will be expected to start from the beginning - writing out the opening equation in full, showing that it is the sum of a geometric sequence, deriving the answer, and then coming up with the correct numerical answer.