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Can you see what is happened? Making regular payments of R2 000 instead of the required R1,709,48, you will have saved R76 675,20 (= R410 275,20 - R333 600) in interest, and yet you have only paid an additional amount of R290,52 for 166,8 months, or R48 458,74. You surely know by now that the difference between the additional R48 458,74 that you have paid and the R76 675,20 interest that you have saved is attributable to, yes, you have got it, compound interest!
In the same way that when we have a single payment, we can calculate a present value or a future value - we can also do that when we have a series of payments.
In the above section, we had a few payments, and we wanted to know what they are worth now - so we calculated present values. But the other possible situation is that we want to look at the future value of a series of payments.
Maybe you want to save up for a car, which will cost R45 000 - and you would like to buy it in 2 years time. You have a savings account which pays interest of 12% per annum. You need to work out how much to put into your bank account now, and then again each month for 2 years, until you are ready to buy the car.
Can you see the difference between this example and the ones at the start of the chapter where we were only making a single payment into the bank account - whereas now we are making a series of payments into the same account? This is a sinking fund.
So, using our usual notation, let us write out the answer. Make sure you agree how we come up with this. Because we are making monthly payments, everything needs to be in months. So let $A$ be the closing balance you need to buy a car, $P$ is how much you need to pay into the bank account each month, and $i12$ is the monthly interest rate. (Careful - because 12% is the annual interest rate, so we will need to work out later what the monthly interest rate is!)
Here are some important points to remember when deriving this formula:
So, now that we have the right starting point, let us simplify this equation:
Note that this time X has a positive exponent not a negative exponent, because we are doing future values. This is not a rule you have to memorise - you can see from the equation what the obvious choice of X should be.
Let us re-order the terms:
This is just another sum of a geometric sequence, which as you know can be simplified as:
So if we want to use our numbers, we know that $A$ = R45 000, $n$ =24 (because we are looking at monthly payments, so there are 24 months involved) and $i=12\%$ per annum.
BUT (and it is a big but) we need a monthly interest rate. Do not forget that the trick is to keep the time periods and the interest rates in the same units - so if we have monthly payments, make sure you use a monthly interest rate! Using the formula from Grade 11, we know that $(1+i)={(1+i12)}^{12}$ . So we can show that $i12=0,0094888=0,94888\%$ .
Therefore,
This means you need to invest R166 267 each month into that bank account to be able to pay for your car in 2 years time.
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