# 2.1 Investments & Loans, loan schedules, capital outstanding,  (Page 3/4)

 Page 3 / 4

Remember that the bank calculates repayment amounts using the same methods as we've been learning. They decide on the correct repayment amounts for a given interest rate and set of terms. Smaller repayment amounts will make the bank more money, because it will take you longer to pay off the loan and more interest will acumulate. Larger repayment amounts mean that you will pay off the loan faster, so you will accumulate less interest i.e. the bank will make less money off of you. It's a simple matter of less money now or more money later. Banks generally use a 20 year repayment period by default.

Learning about financial mathematics enables you to duplicate these calculations for yourself. This way, you can decide what's best for you. You can decide how much you want to repay each month and you'll know of its effects. A bank wouldn't care much either way, so you should pick something that suits you.

Stefan and Marna want to buy a house that costs R 1 200 000. Their parents offer to put down a 20% payment towards the cost of the house. They need to get a moratage for the balance. What are their monthly repayments if the term of the home loan is 30 years and the interest is 7,5%, compounded monthly?

1. $R1\phantom{\rule{3.33333pt}{0ex}}200\phantom{\rule{3.33333pt}{0ex}}00-R240\phantom{\rule{3.33333pt}{0ex}}000=R960\phantom{\rule{3.33333pt}{0ex}}000$

2. Use the formula:

$P=\frac{x\left[1-{\left(1+i\right)}^{-n}\right]}{i}$

Where

$P=960\phantom{\rule{3.33333pt}{0ex}}000$

$n=30×12=360\mathrm{months}$

$i=0,075÷12=0,00625$

3. $\begin{array}{ccc}\hfill R960\phantom{\rule{3.33333pt}{0ex}}000& =& \frac{x\left[1-{\left(1+0,00625\right)}^{-360}\right]}{0,00625}\hfill \\ & =& x\left(143,0176273\right)\hfill \\ \hfill x& =& R6\phantom{\rule{3.33333pt}{0ex}}712,46\hfill \end{array}$
4. The monthly repayments $=R6\phantom{\rule{3.33333pt}{0ex}}712,46$

## Exercises - investments and loans

1. A property costs R1 800 000. Calculate the monthly repayments if the interest rate is 14% p.a. compounded monthly and the loan must be paid off in 20 years time.
2. A loan of R 4 200 is to be returned in two equal annual instalments. If the rate of interest of 10% per annum, compounded annually, calculate the amount of each instalment.

## Calculating capital outstanding

As defined in "Loan Schedules" , Capital outstanding is the amount we still owe the people we borrowed money from at a given moment in time. We also saw how we can calculate this using loan schedules. However, there is a significant disadvantage to this method: it is very time consuming. For example, in order to calculate how much capital is still outstanding at time 12 using the loan schedule, we'll have to first calculate how much capital is outstanding at time 1 through to 11 as well. This is already quite a bit more work than we'd like to do. Can you imagine calculating the amount outstanding after 10 years (time 120)?

Fortunately, there is an easier method. However, it is not immediately clear why this works, so let's take some time to examine the concept.

## Prospective method for capital outstanding

Let's say that after a certain number of years, just after we made a repayment, we still owe amount $Y$ . What do we know about $Y$ ? We know that using the loan schedule, we can calculate what it equals to, but that is a lot of repetitive work. We also know that $Y$ is the amount that we are still going to pay off. In other words, all the repayments we are still going to make in the future will exactly pay off $Y$ . This is true because in the end, after all the repayments, we won't be owing anything.

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