Real and nominal return
Relation Between Nominal and Real Returns and Inflation 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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