Nominal interest, real interest, and inflation calculations | AP Macroeconomics | Khan Academy

Let's say that you agree to lend me some money. Say you're agreed to lend me 100, and I ask you, "All right, do I just have to pay you back 100?" And you say, "No, no, you want some interest."

I say, "How much interest?" And you say that you are going to charge me five percent per year interest. So one way to think about it is if I borrow 100 today, so 100 today— in a year, I'm going to have to pay you back 100 times. I'm going to have to grow it by 5. So that's the same thing as multiplying it by 1.05. This is how much I'm going to have to pay back.

Let me write this down: this is borrow; this is what I'm going to have to pay back. And so this interest rate, that is just the face value of how much more I'm going to have to pay back, this is known as the nominal interest rate. Nominal interest rate, and we can compare this to the real interest rate.

You might say, "Why do we need some other type of interest rate?" Well, even though on the face value I'm paying you back five percent more, that doesn't necessarily mean that you're going to be able to buy five percent more with the money that you get paid back.

You might guess why that is the case: because of inflation. A hundred and five dollars will not necessarily buy you in a year what it might buy you today. And so that's what the real interest rate is trying to get at.

To do that, to calculate our real interest rate, we are going to have to think about inflation. So let me put inflation right over here. And so let's say that we are in a world that has two percent inflation. So an indicative basket of goods that costs a hundred dollars today, if this is the inflation rate, would cost a hundred two dollars in a year.

So there are two ways that folks will calculate the real interest rate given the nominal interest rate and the inflation rate. The first way is an approximation, but it's very simple, and you can do it in your head. And that's why it's often the first way that it's taught, but it's not exactly mathematically correct.

So the first way you'd say, "Well, this could approximately be equal to the nominal interest rate minus the inflation rate." So you could say this could be approximately equal to five percent minus two percent, which would be equal to three percent. And this is a decent approximation, but the actual way that you would want to calculate this, if you wanted to be more mathematically precise, is that your nominal interest rate multiplies things by 1.05—so 1.05—but then things are getting more expensive at a rate of 2 per year, or another way to think about it, costs are being multiplied by 1.02 every year.

So we divide by that amount, 1.02, every year. And so this was going to give us 1.05 divided by 1.02, which is equal to 1.0294—1.0294. And, or another way to think about it, we just got a much better sense of what the real interest rate is: it's actually much closer to 2.94 interest.

And this is a very small difference, and so that's why people like this method; you can do it in your head and it got pretty close. But keep in mind, even very small changes in interest can make a big deal when we compound over many years. In other videos, we've talked about compounding.