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### Course: Macroeconomics > Unit 4

Lesson 2: Nominal v. real interest rates- Real and nominal return
- Calculating real return in last year dollars
- Nominal interest, real interest, and inflation calculations
- Relation between nominal and real returns and inflation
- Indexing and its limitations
- Lesson summary: nominal vs. real interest rates
- Nominal vs. real interest rates

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# Relation between nominal and real returns and inflation

Relation between nominal and real returns and inflation. Created by Sal Khan.

## Want to join the conversation?

- At the end of the video, Sal says that (1+N)=(1+R)(1+I) makes a ton of sense.

Why is that, from a basic theoretical standpoint?(32 votes)- I think the equation is more logical when you have Real rate is Nominal divided by Inflation. Ie, your Real rate is how much your Nominal rate is beating Inflation by.

The way he wrote it, Real rate times Inflation rate equals the Nominal rate you would need. Which makes sense, too, but sounds weirder to me.(41 votes)

- at3:10- where did the first 1+I (the denominator) goes? He cancels the numerator, but where does the denominator go?(13 votes)
- Ah! Just get it:

(1+I) / (1+I) + [(1+N) - (1+I)] / (1+I) is the same as: [(1+I) + (1+N) - (1+I)] / (1+I)

meaning the dominator does not change, simple rule: 1/2 + 2/2 = 3/2(18 votes)

- is this the Fisher Effect?(7 votes)
- An economic theory proposed by economist Irving Fisher that describes the relationship between inflation and both real and nominal interest rates. The Fisher effect states that the real interest rate equals the nominal interest rate minus the expected inflation rate. Therefore, real interest rates fall as inflation increases, unless nominal rates increase at the same rate as inflation. in other words: The Fisher effect can be seen each time you go to the bank; the interest rate an investor has on a savings account is really the nominal interest rate. For example, if the nominal interest rate on a savings account is 4% and the expected rate of inflation is 3%, then money in the savings account is really growing at 1%. The smaller the real interest rate the longer it will take for savings deposits to grow substantially when observed from a purchasing power perspective.(19 votes)

- I still dont get why Sal is simplifying at the end in green color? any simple explanation? thank you!(3 votes)
- You twist and turn an equation however you like as long as you stick to the correct mathematical principals. The last move was just to make everything that much clearer from a logical viewpoint. He's saying the nominal rate of growth (1+N) equals the real rate of growth (1+R) times the inflation rate (1+I).(6 votes)

- How did he get rid of P? Theres only one P for each? At2:00(2 votes)
- Imagine it like this, in a simpler fashion:

(p*a - p*b)/ p*c

Both of the values on top are divided by the single value on the bottom, so the single P on the bottom can get rid of both Ps on the top.

Does that answer the question?(5 votes)

- why is 1 = (1+I) / (1+I) ??(1 vote)
- Algebra. anything divided by itself is 1

(except 0/0)(4 votes)

- I can see how "( ( 1 + N ) - ( 1 + I ) ) / ( 1 + I ) = Real Rate" is a useful formula given that the principal (P), the nominal rate (N), and the inflation rate (I) are pre-determined and a financial advisor or economist would be tasked to find the real rate of return for a client or an employer, etc.

How is the end game of this video, resulting in "( 1 + N ) = ( 1 + R ) ( 1 + I )", an applicable equation in a real life scenario? For example would it be useful for the same financial advisor or economist to cross-check or provide proof of their work? Just curious, thanks!(2 votes) - throughout the video, Sal says "compounded by nominal/real interest rate". what does that mean?(1 vote)
- compound is a tricky word that has a nuanced meaning. The most useful definition here would be something along the lines of "to make bigger or greater by counting all the parts of the whole"

So you are taking the the original amount and ending up with a new amount (the new "whole") that counted the original amount and the effect of the interest rate (the "parts" of the "whole").

Therefore you a compounding the original to arrive at the new.

If that still is not clear, here is a physical analogy: A compound bow is a bow that uses a pulley system to compound the amount effort spent drawing the bowstring.(2 votes)

- why would you even want to find the relation between nominal and real returns and inflation(1 vote)
- Does anyone know how to calculate CPI?(1 vote)

## 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.