Welcome to corporate finance and our first lecture. Today we're going to talk about the time value of money. We're going to start off with some intuition. Then I want to introduce the tools associated with the time value of money, namely, the discount factor and the timeline. Finally I want to apply those tools to move money back in time via a process called discounting. Let's get started. Hey, everybody, welcome to corporate finance and our first lecture on the time value of money, and we're going to start off with some intuition and discounting. In particular, I want to give everybody a good sense of what the time value of money means in a context with which they're already familiar. Then I want to introduce some tools, and then I want to apply those tools to our first financial problems. So some intuition. Hopefully in a setting with which everybody's always comfortable and familiar with. That's foreign currency. Imagine I've got 100 euros and $100, and I ask the question, how much money do I have? Well, we can't answer that, at least not yet, because we can't add euros to dollars, right? What we need to do is what? We need to convert the euros to dollars to find out how much we have in dollars, or convert the dollars to euros to find out how much we have in euros. And we do this using an exchange rate. And there's nothing special in this example that there's only two currencies. If I had three currencies, euros, dollars and yuan, I would have to convert two of the currencies into one currency, one base currency, whether it's in the yuan, dollars or euros, using the appropriate exchange rates, in order to answer the question how much money I have or in order to add the different currencies. So what's the message? Well, look, we can't add or subtract different currencies. We have to convert the currencies to a common base currency using an exchange rate. So what does this have to do with the time value of money? Well, the time value of money refers to the fact that money received or paid at different times is like different currencies. You can't add it. Money has a time unit. What you have to do is you have to convert to a common base unit in order to aggregate it, and to do that we need an exchange rate for time. So let's talk about some of the tools that we're going to use to accomplish this task. So first, the time line, which is exactly what it sounds like. I've got a time line here that lays out different time periods, and these periods can be anything. They could be years, they could be months, they could be days, they could be decades. Typically we refer to time period 0 as today or now or the point at which we're answering the question. Underneath that we lay out our cash flows, denoted by CF. The subscript denotes the time period, and this is nothing more than a visual representation of when money is coming or going. That's it. Get in the habit of placing cash flows on a time line. When you work in Excel, it almost automatically does that for you. But it's a great tool to emphasize the point that money arrived at different points in time, has a different time unit and cannot be added, right? Don't add money at different points in time. Let me say it one more time. Never add money received at different points in time. Now I put a little asterisk there because some people will say, well, what if the money arrives very close in time. Then it's not such a sin, but get in the habit of just not doing it. And I say it multiple times because you'll do it, your friends will do it, people in finance do it all the time. But it's really not a good thing to do, and you'll see why in just a moment. What we need is we need rate for time, to convert to a common time unit, and that exchange rate is called the discount factor. Our discount factor is given by 1+R raised to the power t. t, that's just the number of the time periods into the future, if t is greater than 0, or past, if t is less than 0, that we want to move the cash flows. R, that gets its own slide. R is the rate of return offered by investment alternatives in the capital markets of equivalent risk. And that's a mouthful. R goes by several other names. It goes by a discount rate, a hurdle rate, and opportunity cost of capital. The way to think about R is just to ask yourself, what are the risks, or how risky are the cash flows that I'm going to be discounting here? And then think about how that relates to investments in the capital markets. So what I've done here is I've presented average annual returns to six different investments. And what do you notice? You notice that riskier investments, as I move down the columns, riskier investments are met with higher returns. Riskier investment, higher return. So how do we use the tools? Well, we're first going to focus on bringing cash flows back in time and when we move cash flows back in time, that's called discounting. And so here's my time line that we started out with, and I'm simply illustrating that if I want to convert all of these cash flows, from period 1 forward, into time 0 units, what I do is I apply my discount factor to each one. So to move cash flow 1 back to period 0, I'm going to multiply by 1+R to the minus 1. Because I'm moving it back in time, hence it's negative, one period, hence 1. And similarly for cash flows 2, 3 and 4. Again, notice that all of the exponents are negative, because we're moving all of these cash flows back in time to today's units. Once I've done that, once I've discounted each cash flow back to today, I can add all of these numbers. They're all in the same time units, namely, date 0 units. Let's get rid of this mess. These values of the future cash flows, as of today, are called present values. Present values of cash flows is the discounted value of the cash flow as of period t, or in this case period 0. And the notation is just that. Present value as of period 0 of cash flow 1. Present value as of period 0 of cash flow 2. It's just some notation that'll be useful as we move throughout the course. Let's do an example. How much money do you have to save today to withdraw $100 at the end of each of the next four years if you can earn 5% per annum? The first step in tackling any problem is to just put the cash flows on a time line. Let's get our bearings straight, right? The question is how much do I need today, so how much do I need today, period 0, if I'm going to withdraw $100 each year over the next four years thereafter. Well, the naive thing to do would to be to just add these and say $400. But we know that's wrong because that's a no-no. These cash flows, or these dollar amounts, have different time units. We can't add them. What we need to do is move them back in time today by discounting. And since we've assumed a 5% discount rate or 5% rate of return on our investment, that's our R. And you'll notice I'm dividing each one by the discount raised to the appropriate power. We can do a little arithmetic, and now I can add all of these numbers, they're all in the same time units, to answer the question. We're going to need $354.60. More precisely, we need $354.60 today in an account earning 5% each year so that we can withdraw $100 at the end of each of the next four years. Alternatively, the present value of $100 received at the end of the next four years is $354.60 when the discount rate is 5%. Now I know that's just another way of saying what I said in one, but get in the habit of different words meaning the same thing. Finance is just laden with jargon, and the sooner you get used to it and the more exposure you have to it, the more comfortable you become with it. With that segue, Interpretation 3 is today's price for a contract that pays $100 at the end of the next four years is $354.60 when the discount rate is 5%. Three different ways to say the same thing. Now I want to mention we're assuming that the discount rate, R, is constant over time. See, when I pull back each cash flow, I discount each one by the same 5%. That's a common assumption, but I want to emphasize it's still an assumption, one we're going to look at when we talk about the term structure of interest rates and yield curves later on. But what's going on in the background? How is this working? What's actually happening on a dynamic basis? Well, at time period 0 we're inserting or depositing $354.60. Then we're going to earn interest at 5% on that money, which is going to amount to $17.73 cents. And I'm assuming all of the activity happens at the end of the period. It just makes thing a little bit easier conceptually. There's nothing special about that. We can make the period shorter, which we will later on. Okay, so I earn my interest, which is the 354.60 x 0.05. That's going to give me a pre-withdrawal balance of the initial balance plus the interest. But notice that the present value of that amount, the present value of this 372.32, is just our initial withdrawal balance, that's our 354.60. I'm going to pull out $100. That's going to leave me with 272.32 after the first year. And if we continue this process for the next three years, two, three and four, what we see is we're going to exactly offset or exactly exhaust the funds in the account. Okay, so let's summarize what we learned today. Rule number one, never add or subtract cash flows from different time periods. You'll want to do it. Don't do it. What we need to do is recognize that money has time unit, and in order to add money at different points in time, we need to convert that money to a common base unit using an exchange rate for time called our discount factor. We're also going to get into a habit of using a time line to get our bearings straight, a visual representation of when money is moving in or out. And we learned about the present value of a cash flow. It tells us the value of future cash flow as of some earlier time period. It also tells us the price of a claim to those cash flows. And what's coming up next? We're going to go to the other direction, moving cash flows forward, in other words, compounding. Thanks for listening, and I look forward to seeing you in the next lecture.