﻿ Compound interest

# Compound interest

To explain how compound interest works we need to make a little detour to look at repeated percentages.
7.2
PAULA KELLY: A more common way to charge interest is called compound interest. Most loans are charged using compound interest and most savings accounts pay interest using compound interest. If you have a savings account, the interest is added to the amount of money in your account. Future interest is then paid on the total in your account, not just your original investment. To explain how compound interest works, we’ll need to make a little detour to have a look at repeated percentages.
40.8
A rare postage stamp is a square of side length 5 centimetres. To make a larger picture of the stamp, the stamp is photocopied to make a photocopy in which the length of the stamp is increased by 10%. The new photocopy is itself increased by 10%. This process is repeated until the size of the poster required is obtained. What is the size of the stamp on the first five photocopies? So the original size is 5 centimetres. In photocopy one, we have 1.1 multiplied by 5, to give us a side length of 5.5 centimetres. Photocopy two, we have 1.1 times 5.5 to get 6.05 centimetres. Photocopy three is 1.1 times by 6.05 to give us 6.655 centimetres.
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Photocopy four is 1.1 multiplied by 6.655 to give us 7.3205 centimetres. And photocopy five is 1.1 multiplied by 7.3205. That gives us 8.05255 centimetres. This is quite long-winded, so let’s look at what we have done to see if there’s a more efficient method. We started with the length of 5 centimetres and multiplied this by 1.1 five times. So we had 5 multiplied by 1.1 by 1.1, 1.1, 1.1, and 1.1. That gave us 8.05255. A simpler way of writing this is 5 times 1.1 to the power of 5. That gives us 8.05255. This allows us to determine how large the 10th photocopy will be without calculating each intermediate stage.
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So photocopy 10 will be 5 multiplied by 1.1 to the power of 10. This gives the 12.968712, so about 13 centimetres. So compound interest works in the same way.
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MICHAEL ANDERSON: So let’s see how this applies to compound interest. Say we have 200 pounds, and we’re going to invest it for 10 years with a compound interest rate of 3%.
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PAULA KELLY: OK. So this is where you more commonly would see compound interest. We have 200 pounds.
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MICHAEL ANDERSON: Yep.
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PAULA KELLY: Was it 10 years?
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MICHAEL ANDERSON: 10 years, yeah.
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PAULA KELLY: 10 years. OK.
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And then our percentage was 3%?
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MICHAEL ANDERSON: 3%.
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PAULA KELLY: OK. So we could do our 200 multiplied by 0 and add it on, have our very long-winded way. But we’ve seen our quick way of doing it.
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MICHAEL ANDERSON: Yeah.
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PAULA KELLY: So we have our original amount of money, our 200 pounds.
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MICHAEL ANDERSON: Yeah.
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PAULA KELLY: We’re going to multiply this by 1.03.
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MICHAEL ANDERSON: OK.
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PAULA KELLY: So we have our original investment–
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MICHAEL ANDERSON: Yeah.
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PAULA KELLY: –plus our additional 3%.
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MICHAEL ANDERSON: Right. Yeah.
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PAULA KELLY: At the moment, this would calculate how much you’d have in your account at the end of one year.
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MICHAEL ANDERSON: Yeah.
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PAULA KELLY: We’re not looking for one year. You want 10 years.
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MICHAEL ANDERSON: Right. OK.
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PAULA KELLY: So that repeated multiplier, we can write as a power of 10.
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MICHAEL ANDERSON: OK. Because we’re doing it every year for 10 years?
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PAULA KELLY: Exactly. And that same profit is reinvested, the 3% of that amount.
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MICHAEL ANDERSON: Right. OK.
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PAULA KELLY: That’s why this is called compound interest. So on our calculator, again, this would give us 268.78 pounds and lots of other decimals. As we using money, we’d round to two decimal places.
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MICHAEL ANDERSON: OK. So after 10 years, my 200 pounds is now worth nearly 270 pounds?
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PAULA KELLY: Yes, a small increase. A small increase.
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MICHAEL ANDERSON: OK.
The more common way to charge interest is called Compound Interest. Most loans are charged using compound interest and most savings accounts pay interest using compound interest. If you have a savings account the interest is added to the amount of money in your account, future interest is then paid on the total in your account, not just your original investment.
Paula and Michael explain compound interest by first looking at repeated percentages, with an example based upon repeated photocopying.
They then apply the same principles to the compound interest problem.

## Interest annually, monthly or daily?

Using the example in the video, we say in this case that the annual percentage rate, (APR) is 79.58%.
This type of interest is used when you pay off part of the loan each month and interest is calculated each month on the amount of money you still owe.

### Daily interest rates

You need to be careful to see when the interest rate is compounded. With some lending companies the interest is compounded daily.
I borrow £100 and pay back the full amount at the end of one year. The interest rate is 0.2% but this is compounded daily. To calculate the annual percentage rate:
$100 \times 1.002^{365} = 207.3568…$
This means I have to pay back £207.36, of which the interest paid is £107.36. So although an interest rate of 0.2% appears small, when compounded daily the annual percentage rate is over 107%. I will have to pay back over double the amount that I borrowed!
If you hunt around the small print of online loan companies you should be able to find the daily compound interest rate.

## Problem worksheet

Now complete the final questions on this week’s worksheet, questions 16 and 17.