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Investments and loans

By now, you should be well equipped to perform calculations with compound interest. This section aims to allow you to use these valuable skills to critically analyse investment and loan options that you will come across in your later life. This way, you will be able to make informed decisions on options presented to you.

At this stage, you should understand the mathematical theory behind compound interest. However, the numerical implications of compound interest are often subtle and far from obvious.

Recall the example `Show me the money' in "Present Values of a series of Payments" . For an extra payment of R29 052 a month, we could have paid off our loan in less than 14 years instead of 20 years. This provides a good illustration of the long term effect of compound interest that is often surprising. In the following section, we'll aim to explain the reason for the drastic reduction in the time it takes to repay the loan.

Loan schedules

So far, we have been working out loan repayment amounts by taking all the payments and discounting them back to the present time. We are not considering the repayments individually. Think about the time you make a repayment to the bank. There are numerous questions that could be raised: how much do you still owe them? Since you are paying off the loan, surely you must owe them less money, but how much less? We know that we'll be paying interest on the money we still owe the bank. When exactly do we pay interest? How much interest are we paying?

The answer to these questions lie in something called the load schedule.

We will continue to use the earlier example. There is a loan amount of R190 000. We are paying it off over 20 years at an interest of 9% per annum payable monthly. We worked out that the repayments should be R1 709,48.

Consider the first payment of R1 709,48 one month into the loan. First, we can work out how much interest we owe the bank at this moment. We borrowed R190 000 a month ago, so we should owe:

I = M × i 12 = R 190 000 × 0 , 75 % = R 1 425

We are paying them R1 425 in interest. We call this the interest component of the repayment. We are only paying off R1 709,48 - R1 425 = R284.48 of what we owe! This is called the capital component. That means we still owe R190 000 - R284,48 = R189 715,52. This is called the capital outstanding. Let's see what happens at the end of the second month. The amount of interest we need to pay is the interest on the capital outstanding.

I = M × i 12 = R 189 715 , 52 × 0 , 75 % = R 1 422 , 87

Since we don't owe the bank as much as we did last time, we also owe a little less interest. The capital component of the repayment is now R1 709,48 - R1 422,87 = R286,61. The capital outstanding will be R189 715,52 - R286,61 = R189 428,91. This way, we can break each of our repayments down into an interest part and the part that goes towards paying off the loan.

This is a simple and repetitive process. [link] is a table showing the breakdown of the first 12 payments. This is called a loan schedule.

A loan schedule with repayments of R1 709,48 per month.
Time Repayment Interest Component Capital Component Capital Outstanding
0 R 190 000 00
1 R 1 709 48 R 1 425 00 R 284 48 R 189 715 52
2 R 1 709 48 R 1 422 87 R 286 61 R 189 428 91
3 R 1 709 48 R 1 420 72 R 288 76 R 189 140 14
4 R 1 709 48 R 1 418 55 R 290 93 R 188 849 21
5 R 1 709 48 R 1 416 37 R 293 11 R 188 556 10
6 R 1 709 48 R 1 414 17 R 295 31 R 188 260 79
7 R 1 709 48 R 1 411 96 R 297 52 R 187 963 27
8 R 1 709 48 R 1 409 72 R 299 76 R 187 663 51
9 R 1 709 48 R 1 407 48 R 302 00 R 187 361 51
10 R 1 709 48 R 1 405 21 R 304 27 R 187 057 24
11 R 1 709 48 R 1 402 93 R 306 55 R 186 750 69
12 R 1 709 48 R 1 400 63 R 308 85 R 186 441 84

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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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