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. Last time we introduced taxes and we explored the impact of taxes on our cost of capital or 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. Let's get started. 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. What you can see is a couple of 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 and including deflation. What I want to emphasize is that while inflation doesn't seem particularly important these days, it can be, and if we move outside of the US and some Western European countries, inflation becomes incredibly important. Let's understand it. How does inflation impact our returns in particular? Alright, so let's revisit the example we've done over the last few lectures. This example, just to refresh your memory was, how much do we have to deposit or save today in an account earning 5 percent if we want to withdraw $100 every year over the next four years. The answer to that question was 354.60. We deposit 354.60, that earns interest at 5 percent. 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 0. Exactly. Now, the first lesson is, inflation is not going to affect the money we earned. It doesn't affect the interest over here. We're still going to earn 5% every year a nominal rate of return. What inflation is going to do is it's going to affect what we can buy with the money we're pulling out. It's going to affect the value of that money. We'd like a way to quantify and understand the impact of inflation on that value. I'm going to introduce the concept of a real discount rate. I'm going to denote it by RR. One plus the real discount rate equals one plus the nominal discount rate plus, sorry, divided by one plus the expected rate of inflation, which I've denoted by Pi. 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. In our example, we might have our discount rate was 5 percent and if the expected inflation is 2.5 percent, 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 percent, substantially lower. Now, that's important because even though the inflation is not impacting our account balance, how much money, I mean 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 it. Let's try discounting our cash flows now by the real rate of return, RR, which we just showed was equal to 2.44 percent. 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. 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 these cash flows. Now, taxes affect dollars, but inflation does not affect dollars, it affects consumption. We earn a nominal return, but we can't buy as much with it. Let me illustrate this. 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 is going to earn interest at 5 percent every year. We pull out a $100, and what's going to happen is we're going to be left with this surplus. But that makes sense. Because, inflation doesn't affect the dollar, it affects what we can do with these things that we pull out. We have extra money here. What we really want to do to address inflation is we want to increase how much we pull out every year. I don't want to pull out just $100 each year because prices are going up so that a $100, say here in Year 2, can't buy as much food or housing or clothes or whatever we need to buy. Let's think about what cash flow stream we might want to address inflation. 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 five percent. What is CF? We can solve this. This is just elementary algebra. We want to use the nominal rate here since that's reflecting the dollars that we're earning. 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. Remember, originally I think we were putting in $354.60 if I remember correctly. That we put in more which means we can take out more than we used to take out. Let's see what happens now. We put in the $376.75. Now we're going to withdraw $106.25 each year. We see that we're going to drive the account balance exactly to zero with nothing left over. But ideally, we want our withdrawals to increase each year to accommodate inflation. I want these withdrawals to go up every year to account for the increase in prices of the goods and services I'm going to purchase with that money. Let's think about this. Well, if prices are going up at 2.5 percent per year, that means what I could buy with $100 today, I'm going to need a $102.50 next year because prices went up by 2.5 percent, they're going to go up by 2.5 percent again. I'll need a little bit more and a little bit more and a little bit more each year. What this sequence of withdrawals, maintains,, our purchasing power. Will be able to buy the same amount of food, the same amount of housing, go in the same vacations, assuming the prices are all going up by 2.5 percent, the expected rate of inflation. Now, these are all nominal values corresponding to the real $100 of purchasing power. 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. Remember that when we were withdrawing $100 each year. But we discounted these at the real rate of return, which I think was 2.44 percent, we got $376.75. All that's going on is that the inflation term in the numerator and denominator of the real computations they're canceling one another. 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 percent per year to keep up with inflation. But our money in the account is earning the nominal rate of return of 5 percent. 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. 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. 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 year when we face inflation. We introduced the idea of a real rate of return that takes into account the effects of inflation. Next time, we're going 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 this far. I look forward to seeing you next time. Thanks.