I hope you've taken a break and gotten familiar with what we are doing. As I said, today's going to be intense because I want you very early on in this class to really get a feel for finance. To me, the beauty of finance, as I keep saying so is not the formula. It makes so much sense and once you get a hang up of why we're are using an Excel and why we are doing the formula, you'll get a sense of that and it'll be very easy for you to recognize the value of each. I have nothing against Excel, but here's a simple reason we use Excel is because we don't have time to calculate all these numbers. We know how to do it, but we don't have the time. It would be a pretty bizarre thing to calculate numbers for the heck of it. The second reason is these cash flows in this problem are fixed. They could change. That's what life is, and we'll do that later in the class. But two aspects about it, most examples of loans or retirement funds are very realistic in this respect to IE, the C is fixed. People choose to operate like that, whether it's convenience, whatever and loans many times are fixed rate loans. So what I would encourage you to do is recognize that these are simplifications at some level, the C being a fixed number, but at the same time C is not a simplification of real life. We'll get to more complicated things. Why am I bringing this up? Because if we didn't have compounding, we won't need Excel and if we didn't have things changing over time, we wouldn't need Excel. So Excel is awesome, but keep it where it is. It's not controlling you. You control it. Let's move on to the second phase of an annuity, which is present value. So again, there's this chart, I would like you to look at it for a second. Same thing just to make life simple. What I'm going to do is, I'm going to stick with the same example of three years. By the way, as I'm doing this, I want to thank, instead of just thanking once, I want to multiple times, thanks to my colleagues. I went to University of Chicago for my PhD and before that I've studied a lot. I have to say thank so many people for showing me the beauty of finance. I also want to thank my colleagues at University of Michigan, Ross School of Business. Ronen Israel is one of them who with me taught this introductory class many years ago and he has been a big influence in how I think. Then there are a lot of other colleagues like MP Narayanan, who's a great teacher. I want to thank all of these guys for letting me become who I am. If I'm worth anything, it's due to other people not due to me. So let's get started. The first cash flow at year zero, zero and not because it's forced to be, its convention. In this case, remember, because its present value, I'm standing today, I'm doing the opposite of future values. So I'm saying not years to the end, years to discount. The word discount is coming from a very simple reason and the R is greater than zero. Because real interest rates are positive, future value grows. But because interest rates are positive, the future is discounted when you bring it back today. So how much? zero years of discounting, because we're standing today in present value today zero. What if the present value turns out to be zero? That is simply because there's no cash flow here. If there were, it'll be exactly the same number because of no years to discounting. So in some sense, years to discounting is a key variable. Now here, things are a little bit simpler. One, the first C is one year away, two, three. So because we have done this table before, I'm not going to spend too much time on it. However, recognize that we are doing the exact opposite of what we did last time. The reason we are doing present value after future value is in my book, if you understand future value, you understand compounding. Then when you come backwards, you're not thrown away by why are you dividing by one plus R, one plus R squared. So let's do it. This is C over one plus R. The neat thing is we have done this last time. Only thing is we have to do it three times. Why am I putting one plus R each time in parenthesis? Simply because one plus R is the factor, not one outside or R outside or anything like that. One plus R is the factor and the reason it's getting squared and cubed is because, again, compounding. So once you recognize this aspect of it, I want you to bear with me for a second. What I'm going to do is, I'm going to go to another page where I actually write out the formula. Again, I'm not going to try to simplify it. Simplifications can be done very easily by you and in some sense, it's useful to do it. So let me write out the present value formula. The present value formula for a three year annuity, and I use this three just to remind me that it's just three years, is what? C over one plus R plus C over one plus R squared plus C over one plus R cubed. So I've just to do three PVs. Now, remember, these PVs are not easy to do. So that's why I'll at some point use a calculator. Remember, I can replace this by PMT, in my head. In textbooks we don't use PMTs. We say C because C is a generic word for cash. The other thing is Cs are fixed, and that's because of the nature of the beast we're dealing with right now. Now let me show you that actually I can take C out and have, pretty straightforward. So what is this? If you think about it a little bit, this is a factor again and it's a present value factor of an annuity. What is the annuity? One buck. So if you know the present, if you know this guy, for which what do you need to know? R and N. How many years and what's the interest rate? If you know the value of one buck paid three times, you can know the value of C bucks, whether it's half a buck or it's $100. That's the beauty of it. How does this formula change if you go to PV of n? Simple, one thing changes. There's a bunch of dots plus one over one plus R raised to power n. So you'll keep going until you arrive at n. I hope this is clear. Again, why am I doing this? I first did the concept, then the formula, but I'm really interested now in doing problems. Again, I repeat, if you need to pause now, it's good to do it because I do not want you to get overwhelmed by formulas and so on. So let's go on. I'm going to even pause for a second to remind you, you don't have to keep watching the video. Take a break and let's do one problem at a time.