Nominal Interest, Real Interest, and the Fisher Equation

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Alright now let's see how the inflation rate affects the interest rate on loans or savings. Let's see that relationship. Okay so we've talked about inflation a bit and remember that inflation, we're just talking about the increase in prices over time. Right? So maybe last year something cost $102. Right? That change in the price, that's the inflation. So so far we've been using the CPI I remember that's the consumer price index to calculate inflation. So were the consumer price index takes the value of goods in different years and then it sees how have those values changed. So we're calculating inflation with this formula here where we take the C. P. I. In the current year minus the C. P. I. In the previous year, divide by the C. P. I. In the previous year, times 100 to make it a percentage. Now this is just a percentage change formula, right? We're just seeing seeing the percentage change in prices and we always when we talk about percentage change, we're just taking the new number minus the old number divided by the old number and that's exactly what we did here. Right? The new being the current year, the old being the previous year, new minus old divided by old. So let's see the relationship that inflation has with interest rates. So when we go to the bank and we deposit money in a savings account or we take out a loan, there's an interest rate, they tell us what the interest rate is and that is the nominal interest rate. That is the stated rate of interest on a loan or on a deposit. Right? So if you go to the bank and they say, hey, we're gonna pay you 5% interest on this bank account. I wish. Right, Well that 5% that they tell you that is the stated interest rate. The nominal interest rate, compare that to the real interest rate. Well, the real interest rate is when we take into account inflation. So we're gonna take the nominal rate and adjust the nominal rate adjusted for inflation. Okay, because remember that inflation affects your purchasing power. Right? So even though they tell you you're earning 5% on your money, well, when you leave it in there, the prices are also going up over that time you're earning interest. So the purchasing power isn't necessarily 5% greater. Let's go ahead and see how this works in an example and will, will make a conclusion here at the end of the example. So you only spend your savings on one essential good. Many porcelain figurines at the beginning of the year. The price of a figurine is $20. And if you were to use your entire $2000 on figurines, how many figurines would you be able to purchase? Well, this is just some standard arithmetic here, Right. You have $2000 And we divided by the $20 price per figurine. So 2000 divided by 20. That gives us 100 figurines we could purchase at the beginning of the year. Okay. So that's pretty straightforward. We would be able to buy 100 figurines right now. But however, let's take interest into account here. So if we go down here, however, suppose you had saved the $2,000 earning 5% interest throughout the year, if the rate of inflation is 2%, how many figurines could you buy at the end of the year? So notice now we've got two moving parts. We're gonna have $2,000 earning 5% interest and then we're gonna have the price. We're gonna have the price of the figurines growing by 2%. Okay so let's go ahead and calculate how much money we would have at the end of the year. The new price of the figurines. And then we'll see how many you can afford at the end of the year. So let's start here with You're ending ending cash. So we'll say the ending cash once you earn some interest, well you would have the $2,000 and we're gonna multiply it times one plus the interest rate. Right? Because we're gonna take one which is the original amount you had plus the interest rate of 5%. We're gonna multiply by 1.5. Right? The +05 is the interest that you're earning, the one is the money you already had. So 2000 times one. That's $2000 right? What you already had? 2000 times 20000.5. That's the interest you're earning. So 2000 times 1.5 that will get us to the ending amount of money. You have 2000 times 1.5. It tells you you'll have $2100 at the end of the year, right? And you could have done this two ways you could have calculated how much interest you get. 2000 times 5%. That would tell you, okay, you're earning $100 in interest. Either way we end with a balance of $2100 and let's see what the ending price is of the figurine price. And I'll put a figurine. So we know what we're talking about here. So the ending price of the figurine. Well we're gonna do the same thing here. The price was $20 at the beginning but it grew By 2%. So we're gonna do the same thing 1.02 to represent the growth of 2%, the one being the original $20 price. The 0 to being the growth of the price. So 20 times 1.02 will tell us the price at the end of the year And that comes out to $20.40. So the price has gone up by $0.40 here and that is that 2% of inflation. So to find out how many figurines can you buy? We're gonna take our amount of money that we have which would now be $2100 and divided by the new price of $20.40. So let's see how many we can actually purchase after saving for a year. 2100, divided by 20.40. It gives us 100 2.94 figurines. So let's imagine that we could buy parts of a figurine. We would have 100 2.94 figurines. So how many more figurines are you able to purchase? Well, you're able to purchase you were able to purchase 100 before now you can purchase 100 2.94. Well you're able to purchase 2.94% more figurines, right? You're able to buy an extra 2.94 which is 2.94% of 100. Right? So the real interest rate, what was the real interest rate that you got? Remember when we think about the purchasing power at the beginning of the year? You were only able to buy 100. Now you're able to buy one oh 2.94. So you didn't really get a 5% interest rate right? Because if you had earned 5% interest, well you should have been able to buy 5% more figurines. However, you're only able to purchase 2.94% more figurines, right? You're not able to buy 100 5. If you had gotten 5% more figurines, you have gone from 100 to 105. Right? But the change in price affected that. So what we can do is we can use this formula to approximate the real interest rate. Uh this is called the Fisher effect, named for the guy who figured it out. The Fisher effect tells us that the nominal rate. So the real the real interest rate is gonna equal in approximates the nominal interest rate minus the inflation rate. So let's see how this works approximately in our problem in our problem. So in our example we had a real interest rate of 2.94, right? And what was our nominal interest rate? We had five and our inflation rate was 2%. Right, so five minus two is three and that's approximately. So we'll put approximately here. Uh 2.94. So at low levels of inflation notice we say that it's at low levels of inflation here. Let me do it in red at low levels of inflation. Uh We can use this formula to approximate the real interest rate. So it's pretty easy. There will be a lot of times on a test where they'll just tell you the nominal rate, is this the inflation rate, is that what is the real interest rate? And you just use this formula nominal minus inflation rate. Cool. Alright, so that's about it here, let's go ahead and look at a graph just to see the nominal interest rate over time. And let's see what we got here. So we've got a graph showing uh the nominal rate and the real interest rate in the US over time. And what do we see the difference between the two is what we would call the inflation rate. Right. The difference between that nominal and the real interest rate is the inflation rate. So if at any point we wanted to approximate what the the inflation rate is, it's going to be the distance between these two lines. So from here to here we would say that is the inflation rate at that point, right? The inflation At that point. But notice what happened at in in about 2000 and 2009 during this recession? What happened? The real interest rate was above the nominal interest rate. Notice the nominal interest rate was basically zero. It was like 0.1 and the real interest rate was above that. How could that happen? Well, that would be if we had negative inflation. Right. So we actually had a short period where we went through deflation where the real interest rate was greater than the nominal interest rate. So what does that mean that prices were going down over time? Right. So you actually had more purchasing power if you waited to spend your money even though that the nominal interest rate was so low, the prices were decreasing. So you have more purchasing power in the future. So we had short lived deflation there during the recession. And then you can see uh in the past few years the nominal interest rate didn't really climb. And we've had actually a negative real interest rate. That means purchasing power has been decreasing over time. If you were to just sit on money. Well, you could buy less stuff because those prices are still going up right at this point, there's still inflation of this amount, right? This is still inflation right here, causing if you're just sitting on a stack of cash at home, well that stack of cash is going to be losing value because of the the negative real interest rate. Okay, so that's about it. The big hitter here is this formula that we have at the top of the screen right now. Um make sure you remember this one because it's easy points on the test when they give you some some question like that, where you calculate the real interest rate or calculate the inflation rate and they give you the the nominal and the real. Cool. Alright, let's pause here and then we'll move on to the next video