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# Relation between nominal and real returns and inflation

Video transcript

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.