I hope you have 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, I keep saying so, is not the formula It makes so much sense. And once you get a hangup of why we are using an Excel and why we are doing the formula, you'll get a sense of that and it'll be very, 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. And 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 do that later in the class. But two aspects about it most examples of loans or retirement funds are very realistic in this respect, too, 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 the fixed number. But at the same time, see, it's not a simplification of real life. And we'll get to more complicated things. Why am I bringing this up? Because if you didn't have compounding we wouldn'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 relative, it's not controlling you, you control it. Okay, let's move on to the second phase of 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. And by the way, as I'm doing this, I want to tank, instead of just tanking one set, I want to multiple times tanks with my colleagues. I went to University of Chicago for my PhD and before that I have studied a lot. I have to say thanks so many people for showing me the beauty of finance. I also want to thank my colleagues at the University of Michigan Ross School of Business. Ronin Israel is one of them who with me, taught this introductory class many many years ago. And he has been a big influence in how I think and then there are a lot of other colleagues like [INAUDIBLE] who's a great teacher. I want to thank all these guys. For letting me become who I am and 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. It's convention and in this case, remember, because it's present value I'm standing today, I'm doing the opposite of future value. So I'm saying not years to the end, years to discount. And the word discount is coming from a very simple reason. And the r > zero. Because interest rates are positive, future value grows, but because interest rates are positive the future is discounted when you bring it back to date, right? So how much? zero years of discounting, because we're starting today and present value today is zero. Of the present value turns out to be zero. Why? 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 of a, 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. And the reason we are doing present value after future value is in my book, if you understand future value. You're understand compounding. And then when it come backwards you're not turn away by why you're dividing by one plus one, one plus r squared. Let's do it. This is c over one plus r. And the need thing is we run this last time. Only thing is we have to do it three times. See? Why am I putting one plus r each time in parentheses? simply because it's the one plus r is the factor, not one outside r outside anything like that. one plus r is the factor, and the reason it's getting squared and cubed is because pause again, company. So once you recognize this aspect of it, I want you to bear with me for a second and what I'm going to do is I'm going to go to another page, where I actually write out the formula. And 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) cube, right. So, I've just to do three PVs. Now remember, these PVs are not easy to do, right. So that's why I'll at some point, use a calculator. And remember, I can replace this by PNT In my head. In textbooks, we don't use PMTs every, 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 are dealing with right now, okay? Now, let me show you that actually I can take C out and have [NOISE] Right? Pretty straight forward? So what is this? If you think about it a little bit, this is a factor again. And it's a present value of factor of annuity. Of what? What is the annuity? one buck. So if you know the present, if you know this guy for which where do you need to know? R and N. How many years? 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 of $100. That's the beauty of it. How does this formula change if we go to PV of n? Simple, one thing changes. This, there's a bunch of dots, plus 1 over 1 plus R, raise to power N. So, you'll keep going till you arrive at the end, right? I hope this is clear. Again, why am I doing this? I first did the concept, then the formula, but I'm really, really interested now in doing problems. And 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 and 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