So let's start off with the Net Present Value method. First of all, what do we mean by Net Present Value? Net Present Value is simply all of the projects cash flows for one player, inflows and outflows, as normalized back to T = 0 or T0. So mathematically we'd say that net present value or NPV is equal to the sum of present values of all the projects' cash flows, including anything we receive or pay out at any time, okay? So getting more mathematical here using summation notation. We'd say MPV, the double equal sign means defined to be, is defined to be the sum of all of the cashflows. But before we sum them up, I just want to emphasize here, each one has to be normalized back to or discounted back to T equals zero. And note that the index i here for the cashflow starts at zero, not one. This includes all the cash flows for one player, or for one project. So, that's our definition of Net Present Value, and now we're going to have to learn what's called the Net Present Value Method, and to get there let's say a few more things about it. So the NPV method says, if we want to put it in words, if all the project's time normalized cash flows are greater than zero, we're going to make money on this project, okay? If not, we won't. So we, we bring them all back to this apples to apples basis, and add them all up, and if that sum is positive, we're going to make money. And if it's not, we're not going to, okay. So the NPV evaluation method is going to work extraordinarily simply, like this. The NPV evaluation method simply says we're going to invest in a project, we're going to go forward with a project, we're going to execute a project if and only if, that's not a typo, that stands for if and only if, the potential project's net present value for us, one player at a time, is greater than zero. So that's it. Pretty straightforward. Okay, so now we need to spend a little bit of time on what Cash Flows and Discount Rates we should be considering for use in the NPV Method okay. We want to consider in general the risks, that include risk-free risk, that sounds like an oxymoron. That's really the risk of US treasuries and there is some risk there right? Inflation and other things like that are built into there even though we call it risk, call it risk-free risk, industry or market risks, And then Project-Specific. And then finally of course time, the longer any project goes on the riskier it gets, because the harder it is to understand what's going to be happening with any precision the further we go out into the future timewise. So we want to consider all four of these risks. And in general with the NPV method, most people work things out this way. They include the project-specific risks in the cash flows. And another good way to think of project specific risk is stupid manager risk. So these are risks that have nothing to do with the industry or the market, or the macro-economy of the US, or anywhere else in the world. These are risks by someone executing the project making bad decisions. So here's just a very simple example. If we're looking at a project, if everything goes perfect, we're going to get $10 out of this project, that'll be cash flow coming in to us. And we believe we've done a lot of work on that and there're 95% probability that that's the way this project is going to go. However, if the manager screws up this project we're only going to get $5 out of this this project and happily, we think there's only a 5% probability of that coming our way. So how do we deal with this project's specific risk? Well, we take a weighted average and we'd say that the cash flow at time one that we're going to use is 95% things going well plus the 5% of things going badly or 9.75. So that's just one very simple example of how we could handle project specific risk or stupid manager risk, okay? The next thing we want to think about is what discount rate or we give the second here and in the discount rates we want to include Are risk free risk. And our industry or market or market And we're going to use for that what we call our Opportunity Cost of Capital discount rate, okay? And we note that as ROCC, and it's also called sometimes a market discount rate and it's also denoted as SRM. It means the same thing. Different folks use different subscripts, but opportunity cost of capital is equal to market discount rate, RM. Okay, so now let's go ahead and plug these quantities into our MPV formula and when we do that, we have something that looks like this. Now the NPV of a project for one player is going to be, again a time, the sum the of individual time-normalized cash flows where we have our project risk taking care of in the cash flows discounted by our opportunity cost of capital which includes our risk-free risk, and our market-risk. All right, so with opportunity cost of capital, it's probably good to get a slightly better handle on what exactly we mean by this. How could we compute it? Where does it come from? So for a given project, we define opportunity cost of capital as the known discount rate or interest rate. Again, same thing mathematically. Interest rate if we're going forward in time, discount rate if we're going back in time. So the known interest rate or discount rate for a diversified set of alternative investments or projects with similar market or industry risks. And again, sometimes we call this RM instead of ROCC. All right, so how should this opportunity class of capital be computed? First big thing to note here is there is no consensus on this. There have been on going wars within finance and economics on how best to compute this since the concept first arose. There are several methods in common usage, two methods are presented in the appendix to this chapter. And of course, warning, gratuitous and self serving plug, in my Introduction to Business Finance text, which is available at appliedfinance.flickrocket.com. Okay. No matter how it's computed, we don't have time here to spend time going over different methods of computing. [COUGH] Excuse me. Opportunity cost of capital. But no matter how it's computed, it has two important properties. Let's remember first what it is, double equal sign, this always means defined to be, to be our opportunity cost and capital. Defined to be the known interest rate for diversified set of alternative investments with similar market or industry risks to the project we're considering at hand, okay? So with that in our pocket, that definition, two things become fairly obvious if you think about it. First, we can make investments with a return of our opportunity cost of capital at any time by adding to this position, this diversified set of alternative investments, with some more industry risks, okay? So that implies that [COUGH] the opportunity cost of capital is going to be the rate of return, the interest rate or discount rate we'd forgo by pursuing a new opportunity in our investment milieu. In this case, our investment milieu is developing multifamily rental properties and selling them. Okay? Second thing about the opportunity cost of capital is that the opportunity cost of capital, of course, increases with the riskiness of our investment mission. So, for example, if someone's mission is to invest in Treasury securities, which are considered the safest debt securities in the world by many people, not everyone, but most people. The opportunity cost of capital for an investor like that is going to be less than the opportunity cost of capital for someone whose mission in life is to invest in sub-prime mortgages, for example. Subprime mortgages are, by definition, very risky mortgages. Okay, so now we're ready to get back to, now we knew everything we need to know to jump back into the NPV project evaluation method. So here's a reminder of the formula. The net present value of a project for one player is simply the algebraic sum of every stinkin' cash flow for this project for that player, which has been before adding up, which has been discounted back to T equals zero, or if you want to think about it in other way, discounted back to T equals zero using what? Using the opportunity cost of capital. And what cash flows are we talking about again? Cash flows that have been risk adjusted for project specific risk or what we call sometimes, what I call sometimes stupid manager risk. Okay, so now we've got two important things about the NPV formula itself. One is that the net present value of opportunity cost investments, that diversified portfolio or observable stuff that we could have invested in alternately that gave us our opportunity cost of capital. The NPV from investing in more of that diversified portfolio of stuff fin our investment milieu, it's always going to equal 0. And what I have in gray here is an explanation and proof, explanation for and proof of that. The information in gray, I'm not going to cover. You're welcome to look at it. It's in my textbook if you're interested and you can certainly check it out, but we're just going to accept this on face value here. Second thing that we're going to accept is that if we find investments that are in our investment milieu, again, for a firm like CAG, if we find projects, remember project, the synonym for investment. If we find development projects for multifamily rental buildings that we can put together that we feel will have an NPV greater than zero after we estimated all the cash quotes for the project, we should go ahead. We should go ahead because that's going to do better for us than putting the money we could invest in this project in our opportunity cost of capital investments. All right, so that gives us a great rule for applying the NPV formula, which is, invest in a potential project if the NPV is greater than zero. Don't invest if the NPV is less than zero.