Relation Between Nominal and Real Returns and Inflation

Inflation

Real and nominal return

Real and Nominal Return
Calculating Real Return in Last Year Dollars
Relation Between Nominal and Real Returns and Inflation

Relation Between Nominal and Real Returns and Inflation Relation between nominal and real returns and inflation

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Relation Between Nominal and Real Returns and Inflation

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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 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 we're writing as a decimal
so in the example we've been using
it was 10%, 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 past. So in our example
this was $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 the rate of inflation is 2%, so that would be 0.02
So this, 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, we get, in the numerator
we get 1 + N - 1 + i , I'll write it like that still
all of that over 1 + i , is = 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 + i over a 1+ i
this is completely identical, cause 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 + i + 1 + N - 1 + 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 + the nominal interest rate
in the denominator, we just have a 1 + the rate of inflation
is equal to 1 + the real interest rate
and then we can multiply both sides times the 1 + i
multiply both sides times the 1 + i
and we get an interesting result
and to some degree this is a common sense result
and I want to show you this is completely consistent
with everything so far, these guys cancel out
and that you get when you compound by the nominal interest rate
that's the same thing as growing the real growth
and then that compounded by the rate of inflation

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