Relation between nominal and real returns and inflation

Let's generalize the mathematics that we've been doing in the last few videos to calculate the real return. And maybe we'll come up with some interesting formulas or some simple approximations. So what we've been doing is we've been, at least in the first video, we converted everything to today's dollars. So the actual dollar return in today's dollars is the amount that we got or the net dollar return. And the net dollar return is the amount that we originally invested compounded by the nominal interest rate. And here we're assuming that we're writing it as a decimal. So in the example we've been using it was 10%. And so this is going to be 0.10. Or this whole value is going to be 1.10. And so this is how much we're going to get after a year has passed. So in our example, this was the $110. $100 compounded by 1.1. And then from that, you want to subtract how much we invested in today's dollars. Well, we originally invested P dollars a year ago. And in today's dollars, we just need to grow it by the rate of inflation. And in the examples we've been doing we assume that the rate of inflation is 2%. So that would be 0.02. So this expression right over here is actually the dollar return in today's dollars. It's this value right here that we calculated in the first video. And to calculate the real return we want the dollar return in today's dollars divided by the investment in today's dollars. And once again, this is the investment in today's dollars. It's the amount we invested originally grown by inflation. And this right over here gives us the real return. Now one thing we can do right off the bat to simplify this is that we have everything in the numerator and everything in the denominator is divisible by P. So let's divide the numerator and the denominator by P. Simplify it a little bit. Just like that. And then we get in the numerator, we get 1 plus N minus 1 plus I. I'll write it like that still. All of that over 1 plus I is equal to R. And I'm giving some space here because one simplification I can do here is I can add 1 to both sides of this equation. So if I add a 1 on the right hand side, I have to add a 1 on the left hand side. But a 1 is the same thing as a 1 plus I over a 1 plus I. This is completely identical because this is dividing the same thing by itself. So this is going to be a 1. So we're adding a 1 on the left. We're adding a 1 on the right. And the reason why I did that is it comes up with an interesting simplification. We have the same denominator here. If I add the numerators, 1 plus I plus 1 plus N minus 1 plus I. So this and this are going to cancel out. And we're going to be left with, in the numerator, we're just left with a 1 plus the nominal interest rate. In the denominator, we just have a 1 plus the rate of inflation is equal to 1 plus the real interest rate. And then we can multiply both sides times the 1 plus I. Multiply both sides times 1 plus I. And we get interesting result. And to some degree, this is a common sense result. And I want to show you that it's completely consistent with everything we've been doing so far. These guys cancel out. And when you compounded by the nominal interest rate, that's the same thing as growing the real growth, and then that compounded by the rate of inflation, which actually makes a ton of sense.