Hi, welcome back to finance for non finance professionals. In this third video of week one, we're going to take the basic ideas of interest rates and compounding that we learned in the first two videos and apply it to discounting future values back into the present. So, we're going to talk about discounting future cash back into the present. Okay, what is cash worth in the future today? It's kind of a funny concept when we think about it. Usually, in the last lecture, we talked about money that I have today, I put it in the bank and earn some rate of return, some interest rate over time, and that's how much I'm going to have in the future. A lot of us have put money into the bank or waited and watched bonds grow or watch the stock market go up and down. And we understand what it means to have stuff today and watch it grow into the future. What we're going to do in this lecture is a little bit more abstract, we're going to think about a promise of cash coming in, in the future. And then ask how much that promise of future cash is worth today? And that basic concept is called discounting. It's a critical component of finance. Because a lot of what we do in finance is, if you think about bonds or stocks or dividend payments or investments, a lot of what we're thinking about is cash that's coming in in the future. Like I'm going to put money in the stock market and I hope to retire in 20 years. What I'm saying is that I hope I'm going to get cash 20 years from now. And I'm making plans, and decisions, and trade-offs today on the expectation of future cash coming in later. And what we're going to do today is think about how to put a present value, a value today, on what that future cash is worth. Okay, from our last lecture we talked about putting present values, growing them into the future, with compound interest rates. And this was the basic formula that we had from the last lecture. Money that I have today, the present value, grown at some interest rate, r, for a number of periods, t. What we're going to do is just very, very simple algebra. We're going to take that basic formula and we're going to flip it over. I'm just going to solve for present value, which means taking the (1+r) to the t, moving it down underneath into the denominator. But what we've done is conceptually very sort of tricky. Instead of growing a cash flow at r into the future, we're taking a future value now, think about what the formula says now. It says the present value is equal to the future value. And what's now over the future value, look at the denominator, what's in there? (1 + r) to the t, the same thing, the same interest rate. But instead of growing into the future, we're taking that future value and what are we doing? We're sort of smashing it down, we're beating it down with that 1 + r, r is bigger than 0, so 1 + r is something bigger than 1. 1+r to the t is something, going to be even bigger than 1 which means we're going to take that future value, we're going to smash it down into the present, that's called discounting. Discounting future values into the present. All we've done is flipped that formula, but conceptually we've done something very important. Okay, so now we can think about some questions, if I offered you $175 payable 5 years from today, like a simple bond or a simple loan, how much would you pay for that offer if interest rates were 4% today? Let's walk through that very simple example and we'll try to get a feel for this idea of discounting. Okay, let's look at this table as I've got it laid out here for 5 years. When is that cash coming in? It's coming in 5 years from now. Okay, how much is that worth to me today? What do I need to do? I need to smash it down once. I need to smash it down twice, three times, four times, five times into the present today. So we're going to do that using the same formula that we've been using for the last two lectures. What am I going to do? I'm going to take that 175, the future value, smash it down one period, 1 + 4% raised to the 1. And that's what I've got right here in this formula, 175 divided by (1 + 4%) to the 1 power. That's smashed down one period. That takes that 5 year cash flow of 175 and moves it back in time one period. That's worth 168.27, that's smashing it down one time into year 4. Now I'm going to smash it down again. Divided by, if I take that 175 and smash it down for two periods. What do I do? 1 + 4%, but this time two periods squared. That smashes it down to 161.80. If I smash that 175 down three periods, you're getting the idea now. 175 divided by 1 + 4%, the interest rate cubed, 3 periods from 5, one, two, three periods down. $175 over (1+4%)3 gives me 155.57. Smashing it down 4 periods gives me 149.59. Smashing it down 5 periods, $175 over (1+4%) raised to the 5th period, 143.84. And that's the answer, 143.84. What have I done? I've taken a 5 year cash flow, something coming in 5 years from today. And smashed it down 5 times at 4% into the present. We've taken a future value, something that's coming in in the future. Something that's coming in way out there, 5 years from now. And said how much is it worth to me today? If somebody offered you a promise of 175 5 years from now, how much would you pay for it? If interest rates were 4%, we'd pay about $143.84 for it today. This is a really important concept because it means now we can think about anything, right? Real estate, mortgages, commercial, industrial loans. We can think about sovereign debt from Argentina or Greece. We can think about stocks and bonds. We can think about anything that's really got a cash flow attached to it that's coming in in the future. How much are you willing to pay for it today, ultimately that's going to depend on how much those promises of future cash are worth to me when I put them in present value. Okay, two things are driving how hard we sort of smash down those cash flows. The first is, how far out is that cash? The further out it is, the harder we're discounting it. The second is, how big is the interest rate? The bigger the interest rate, the more we're smashing that cash flow down. To see that graphically, let's talk about a $100 discounted at 10%. And I'm going to show you a graphic of how hard that discounting works as we sort of move further out in time. So think about that $100. If there were no discounting at all, that would be the 0 on the x-axis. And of course, that $100 would be worth exactly $100. But as I smash it down for a year, 2 years, 5 years, 10 years, 20 years, the value of that money goes down, down, down. And you can see it doesn't go down straight, it kind of goes down exponentially. Just like when we talked about compound interest, growing exponentially, discount decay exponentially. So we're sort of saying, how hard are we smashing down those cash flows, the amount that we're smashing down those cash flows, grows exponentially. So it's exponential decay in the present value of future cash, right? You can see that that $100, discounted at 10%, if we go out 50 years, is worth close to nothing, right? It's worth a few pennies because we've discounted it 1 plus that percent, 1 plus that percent, 1 plus that. By the time we're raising that to the 50th power, that exponential decay is hitting so hard that that future promise of cash is essentially worthless. Now the best way to see this is to work a couple of examples, so let's again, let's go to the light board and work a couple of practical examples. All right, let's work a simple example together of using the present value, future value relationship and think about what this might mean for us. So let's say my friend comes up to me and he says, hey, I'm going to start a pizza parlor and I'm trying to raise some money to get it off the ground. And he makes me a promise, he says, I'll pay you $1,000, but I can't pay you that $1,000 for another 7 years. So, 7 years is when I figure the pizza parlor will be up, and I'll be able to pay back my debts. I'm trying to raise money today to get it off the ground, I'll pay you back $1,000, 7 years from now. And I think, all right, well, pizza parlor maybe in downtown Houston is not a bad idea. This guy has been in the restaurant business before. What would be a reasonable rate of return to expect? How about 5%? I'd like to earn 5% on that money over time. And I'll say, all right, well, if you're going to pay me back $1, 000, 7 years from now, and I'm thinking 5% is a reasonable return, that's what I'm going to discount that $1,000 at. So let's use the present value, the future value relationship to figure out how much money I will be willing to lend to him today for that promise of a $1,000, 7 years from now, discounting at 5%. I would use my relationship between the present value and the future value. The present value is the future value times 1 over (1 + r) to the t. Okay, so in this case I'm solving for present value. My future value is 1,000 and 1 over 1 + r, what's r here? 5% raised to the 7th power. Okay good, so now I can put this into any financial calculator and solve that. That answer is going to come out to be $710.70. All right, good. So what have we done? I've said, okay, you are willing to pay me $1,000, but not for 7 years, and I want to make sure I earn some kind of rate of return for lending you the money. I'd be willing to give you $710 for that loan, that's how much I'd be willing to finance you at. I'd be willing to give you $710 for that promise of $1,000 coming in 7 years from now, that's 5%. So that's how the straight loan would work. If I gave him $710.70 and he actually paid me the $1000, 7 years from now, I would have earned 5% a year for 7 years. The discounting and the compounding, that r, is the same. Now, what if my friend came to me and said, hey, I've changed the concept for the pizza parlor a little bit. I'm going to make it all about fish pizza. And maybe that's a great idea, and my friend is really excited about fish pizza. But I'm not as comfortable with fish pizza as business concept as my friend is, and so I think, all right, that changes the risk to me a little bit. You're still going to pay me a $1,000, 7 years from now. But to invest in a fish pizza business which I'm a little bit skeptical about, I'm going to charge you a 15% rate of return. So I'm going to discount a lot harder that promise of a future cash flow because I feel less comfortable of that cash is actually going to arrive 7 years from now. Okay, so if I go back to my future value, present value relationship, what's the present value of a promise of $1,000, 7 years from now, if I'm discounting at 15%? That's going to be $1,000 over 1 + 15% raised to the 7th power. So, what have I done? I'm just changing the r here from 5% to 15%. What is that going to do, as I discount much harder at same period of time, same amount of money? What's that going to come out to? That's going to come out to $375.90. Much lower value, right? I was discounting at 5%, that got me to $710. Discounting at 15% really smashes down that cash flow. My friend says to me, I want to invest in the fish pizza. That's riskier to me. I'm only willing to lend you $375 for the same promise of a $1,000, 7 years from now. With the lower discount rate, I felt more comfortable about the original business plan. At a 5% discount rate, I'd be willing to lend $710. At the riskier discount rate, only 375. So we can see that relationship between the present value and future value depends really on two things, how far out in time is the money coming in? And how hard am I discounting those cash flows based on risks?