Welcome back to corporate finance. Last time, we talked about taxes and their impact on our dollar returns. This time, I want to talk about inflation and its impact on our real returns or our consumption. Let's get started. Hey, everybody. Welcome back to Corporate Finance. So last time we introduced taxes, and we explored the impact of taxes on our cost of capital, our discount rate, and our dollar investment returns. What I want to do today is I want to think about the impact of inflation on our returns and ultimately on our decision-making. So, let's get started. Okay, and I want to start with a picture, as I did with taxes, to illustrate the importance of inflation. This picture shows, over the last approximately 110 years, inflation in the United States, and what you can see is a couple things. First of all, over the recent period, inflation has been relatively low, but if you go back beyond, say, the last 30 years, you can see periods of really high inflation, including deflation. And so, what I want to emphasize is that, while inflation doesn't seem particularly important these days, it can be. And if we move outside the US, in some western European countries, inflation becomes incredibly important. So, let's understand it. How does inflation impact our returns? In particular. All right, so let's revisit the example we've done over the last few lectures, right? This example, just to refresh your memory, was how much do we have to deposit or save to day, in an account earning 5%, if we want to withdraw $100 every year over the next four years? And the answer to that question was $354.60. We deposit $354.60. That earns interest at 5%. That increases our balance. We withdraw the money. That reduces our balance, and we continue over the next three years, until we drive the account balance down to zero, exactly. Now, the first lesson is, inflation's not going to affect the money we earn. It doesn't affect the interest over here right? We're still going to earn 5% every year nominal rate of return. What inflation's going to do is it's going to affect what we can buy with the money we're pulling out, and it's going to affect the value of that money. And so we'd like a way to quantify and understand the impact of inflation on that value, so I'm going to introduce the concept of a real discount rate. I'm going to demote it by RR, and 1 plus the real discount rate equals 1 + the nominal discount rate, divided 1 + the expected rate of deflation, which I've denoted by pi. And a commonly used approximation that you'll see, though I'll emphasize this is an approximation, is that the real rate equals the nominal rate, minus the expected rate of inflation. So, in our example, we might have, right, our discount rate was 5%, and if expected inflation is 2.5%, which is approximately what it's been over the recent history in the US, we get a real rate of return on our investment of 2.44%. Substantially lower. Now, that's important, because even though the inflation is not impacting our account balance, how much money, how many dollars we have, it is impacting what we can do with that money, what we can buy with it. And that's ultimately what we care about, so let's try discounting our cash flows now by the real rate of return, RR, which we just showed was equal to 2.44%. Right, and the discounting proceeds mechanically in the exact same way as it was before, only now, I'm using the real rate instead of the nominal rate, okay? And if we do a little bit of arithmetic, we get the present values of all of these future cash flows. We add them up, and we get a value of $376.75. That's the present value of the sum of all of these cash flows. Now. Taxes affect dollars, but inflation does not affect dollars. It affects consumption, so we earn a nominal return, but we can't buy as much with it. Let me illustrate this. So, let's insert, here, into our savings account, the $376.75 that we just computed using the real discount rate, and see what happens when we're withdrawing $100 every year. Well, that money's going to earn interest at 5%, okay? Every year. We pull out $100, and what's going to happen is we're going to be left with this surplus, but that makes sense, right, because inflation doesn't affect the dollars. It affects what we can do with these things that we pull out, okay? See, we have extra money here, so what we really want to do to address inflation is we want to increase how much we pull out every year, right? I don't want to pull out just $100 each year, because prices are going up, so that $100, say, here in year two, can't buy as much food or housing or clothes or whatever we need to buy. So, let's think about what kind of cash flow stream we might want to address inflation. And one way to do that is to simply solve for the cash flows that we want to withdraw each year, given a nominal discount rate, R, of 5%. What is CF? Well, we can solve this. This is just elementary algebra, right? And we want to use the nominal rate here, since that's reflecting the dollars that we're earning. So, solving this for CF, or cash flow, we get $106.25, which is greater than the $100. That makes sense. We're putting in more money at the beginning, right? Remember, originally, I think we were putting in $354.60, if I remember correctly, so that we put in more, which means we can take out more than we used to take out. And let's see what happens now. So, we put in the $376.75. Now, we're going to withdraw $106.25 each year, and we see that we're going to drive the account balance exactly to $0, with nothing left over, but ideally, we want our withdrawals to increase each year to accommodate inflation. Right? I want these withdrawals to go up every year to account for the increase in prices of the goods that I'm going to purchase, goods and services that I'm going to purchase with that money. So, let's think about this. Well, if prices are going up at 2.5% per year, that means what I could buy with $100 dollars today, I'm going to need $102.50 next year, because prices went up by 2.5%. They're going to go up by 2.5% again, so I'll need a little bit more and a little bit more and a little bit more each year. This sequence of withdrawals maintains our purchasing power. We'll be able to buy the same amount of food, the same amount of housing, go on the same vacations, assuming the prices are all going up by 2.5%, the expected rate of inflation. Now, these are all nominal. These are all nominal values corresponding to the real $100 of purchasing power, and so, if we take the present value of these nominal dollars at the nominal discount rate, we get the $376.75. We discount nominal cash flows by the nominal rate. Keep that in mind, it's important to emphasize that. The present value of nominal cash flows at the nominal discount rate, that's going to equal the present value of the real cash flows at the real rate, right? Remember that we're withdrawing $100 each year, but we discounted these at the real rate of return, which I think was 2.44%, right? We got $376.75, and all that's going on is that the inflation term and the numerator and denominator of the real computation is they're canceling one another. Okay, so let's go back to our savings account. We insert the $376.75. We're now going to withdraw money that's growing at a rate of 2.5% per year to keep up with inflation, but our money in the account is earning the nominal rate of 5%, and what happens is we exactly exhaust our funds at the end of four years. And we've been able to do so by increasing our withdrawals each year to keep up with inflation. So, let's summarize this. Inflation does not affect dollar returns. It's not affecting the money in the bank account or the rate at which it's growing. What it's affecting is the purchasing power of that money. So, when I pull it out and go buy something, I can buy less of that good or that service with that same dollar year after a year after a year when we face inflation. So, we introduce the idea of real rate of return that takes into account the effects of inflation. Next time, we're going to to turn to a new topic, interest rates. And we're going to build on what we've already learned to understand how to discount and value cash flow streams that don't happen every year, that are irregular in their timing, and how to deal with different compounding periods as opposed to annual compounding, which is what we've implicitly been doing all along thus far. So, I look forward to seeing you next time. Thanks.