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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 the the net dollar return and the net dollar return is the amount that we initially 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 we're going to so this is going to be zero point one zero or this whole value is going to be one point one zero and so this is how much we're going to get after a year has passed so in our example this was the hundred and ten dollars one hundred dollars compounded by one point one 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 assumed that 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 grown by inflation 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 plus N minus 1 minus 1 plus I I'll write it like that still all of that over all of that over 1 plus I 1 plus I is equal to our and I'm giving some space here because one simplification I can do here is I can add one 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 and so 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 this comes up with the 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 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 1 plus I multiply both sides times 1 plus I and we get an interesting result and to some degree this is a common-sense result and I want to show you that's completely consistent with everything we've been doing 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 which actually makes a ton of sense