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Compound interest
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True or false? A person who has borrowed 100 euro and pays back 110 euro has paid 10 euro in interest.
Michael wants to buy a new moped. Oh, it costs quite a lot of money! Check with the bank, Michael, they might lend you the money! That was fast! What did they say?
Ten percent interest? Okay. And for how long do you plan to borrow the money? Five years. Let's help Michael calculate.
This represents time, and Michael borrows the money here where time is zero. Here, at 'one', one year has passed. Then Michael owes the bank: The principal amount, plus the principal times the interest rate. We can express the same thing like this: the principal times 1.1 That means the debt has increased by 10 percent. Pause the film and make sure you understand that these different ways of describing the debt mean the same thing.
When talking about a loan, it's easiest to use this way of writing it. It is: the principal, times the change factor. The change factor is: one plus the interest rate. This one here is the principal. And after the decimal point comes the interest rate, ten percent.
Then another year passes. How much does Michael owe then, after two years? The debt had increased by 10 percent after one year - does that mean it has grown by 20 percent after two years? Well, actually no. That's not enough, because the bank doesn't use simple interest.
This is how the bank thinks about it: After one year, Michael doesn't just owe them the amount that the moped cost; he also owes them the first year's interest. So now he owes a bit more in total. After two years, the bank will calculate the new interest on the sum of the principal and the first year's interest. Year three, same thing: the bank calculates the new interest on the total debt, including the interest from the previous years. Each year, new interest is added to the debt, and the debt increases.
And since the debt increases, the interest amount increases too, year by year. Four years. Five years. The bank charges interest on the interest. It's called compound interest.
To calculate, we can state it like this: After one year Michael owes: the principal times 1.1 And after two years he owes that amount, again multiplied by 1.1 Use your calculator to see how much the debt is after two years. See what happened? The debt has increased by 21 percent. Pause the film and check the numbers if you think this was weird. Remember that the bank charges interest on the entire original principal, plus the first year's interest.
After three years, the bank calculates the interest from the original principal and two years of interest. So, let's multiply all of it by 1.1 Year four, we do the same thing. And year five. How much does Michael owe the bank at the end of year five? You can key in all of this on your calculator.
But we can simplify it a bit first. Now this bit you might recognise? This is a repeated multiplication; this kind of thing can be stated as an exponential expression. 1.1 multiplied by itself five times equals 1.1^5 First calculate the exponential expression. Here is the change factor for the entire period of the loan: 1.61 In total, the debt after five years is: The principal times 1.1 to the power of five That's the amount Michael has to pay back in five years time.
Ten percent annual interest, for five years, with compound interest makes a debt increase by 61 percent. Had the bank instead used simple interest, the same debt would have only increased by 50 percent. But now Michael pays interest on the interest, and that makes the debt grow faster and faster and faster.