Okay, so let's go on to our second discounted cashflow method of project evaluation. This is called Internal Rate of Return method and Internal Rate of Return is universally stated as IRR, so when someone says IR, they mean internal rate of return. Internal rate of return is a discount rate, or an interest rate, and let's see how it works. It's defined to be, the interest rate or discount rate, that makes the NPV of a project, to have an investment equal to zero. All right, what do you do when you see a definition you memorize it okay. When I was a young spinster as an undergrad I remember taking biology and it seemed pretty overwhelming to me. And my complete strategy in biology was just, simply memorize the definitions, don't do anything else, and I got a solid A in the class. So, I've kept that rule has worked pretty well for me. Whenever you see a definition, memorize I said period, no ifs ands or buts, okay? So internal rate of return, discount rate, or interest rate that makes the NPV of a project equal to 0. So what do we mean by that mathematically? IRR is the interest rate or discount rate that makes this equation true. NPV equals 0 equal what are usually tool NPV equation is except now instead of opportunity cost of capital I've put IRR in there. So we're going to play around with that interest rate until we find the one that makes this equation true. That makes this sum of all the project cash flow is for one player time normalized add up Up to zero, okay? So this, gives us another way to evaluate potential projects. As usual, with all DCF methods, you want to establish, first of all, the opportunity cost of capital for the project. Which, as beginners in the real world, that's going to be given to you by your general manger. That you're working for or a consultant like me, of course another gratuitous, self serving plug. And also you need to put together for that project all of its cash flows, project risk adjusted, the stupid manager risk adjusted. All right, so once we've got that then we want to solve the IRR equation for IRR using those cash flows, those project risk adjusted cash flows. And then this is really nice because it's nice and intuitive, we can just say if the internal rate of Return is greater than our cost of capital return, we should go ahead. And if the internal rate of return is less than or equal to our opportunity cost of capital return, we shouldn't go ahead with this project. This is really great because it's very intuitive, a lot of people like this method. That said, there's a serious problem with this method which makes it not appropriate to use for most developers. In fact, most sponsors of any real estate project, whether it's buying a cash flow producing property and managing it for awhile. Or improving it and then In selling it or developing something from scratch. The IRR works pretty well for bank lenders but not so much for sponsors of real-estate projects, and let's see why. Okay, so when we solve this formula. Okay, what we're going to do for those of you who remember your pre-calculus algebra. Once we do a little algebraic simplification on this, solving this is always going to require finding the roots of a polynomial or a real equation. If the exponents here are an integer, we'll have to find the roots of what's called a real equation mission, okay. As you may remember from when you were tortured in pre-calculus algebra. Sometimes you might not have a real number solution to this, there might only be complex number of roots. Okay, also, you're going to have multiple roots and you're not sure a priori before hand which root is the correct root. The NPV never has this problem, the NPV method, so we call it a more robust Method. Okay, so graphically the mathematicians of the world have done some great things for us. Which we can think about these would be the graphics on this particular page If we have projects looking like this or like this where. What are the characteristics of this project except for a cash flow at say, T equals zero, or at the very end of the project all of the cash flows are either positive or negative. Okay, so in this cases where we have for example one negative cashflow equal to 0 all the rest of the cashflows are positive or a bunch of negative cash flows until the last cash flow. Well that's positive, or there are two more case, right, we could flip this over the X axis and get something like this. Okay, so we'd have one positive cash flow at t equals 0, and all the rest are negative. And I'll let you figure out the fourth one from the second one down here. In those cases, the wonderful mathematicians of the world have proven that the roots to the equation that need to be solved, this is with integer, exponents. Are going to be really numbers, and the smallest real root is the IRR, so that's really great, so what do these sorts of things look like? They look like cash flow diagrams for lenders, basically but this here looks more like a cash flow diagram for a developer, or a sponsor, or what we call the equity participant in a project. Where maybe they have to invest in phase one, and then phase one comes, so online and they make some money but now they've got to invest in phase two. And then, phase two comes online and they make more money, so they have positive and negative cash flows throughout the project. And mathematically, we can't say anything about the math required or the numerical methods required to solve these, so in a case like this use NVP, why? NVP always works, NVP is the word, the light, and the way. Okay, so IRR works very well For bond projects and loan projects and in the bond world, when folks are talking to you about yield to maturity, all they mean is IRR. Kind of a funny IRR, IRR of the project buy the bond where they evaluate with non-risk-adjusted cash flows. Anyway, yield to maturity a certain type of IRR that folks in the bond world use. All right, so let's apply the IRR method to our distressed Seller problem. Remember our mission? Invest in very safe stuff, that is short duration, something like treasuries. Based on that, we went out and found that a diversified portfolio of that type of investment, we would expect to return excuse me, 5% annually. So that's going to be our opportunity cost of capital and distressed seller then comes along and says. For 100 bucks I'll give you a treasury bond, and the Treasury will pay you in one year $110. Okay, so how are we going to apply this? We're just going to apply the internal rate of return formula, we want to say that find this number, solve for this. So that everything adds up to 0. So, I'm just going to plug in, like we did before. Here's our cash flow at t=0 and I'm not going to and bother by dividing that by (1+IRR) to the 0 because (1+IRR) to the 0 is 1. But our cash flow we'd get in a year are 110, I do need to To divide now, to discount it, by (1+IRR) to the 1. Then through the magic of algebra I can multiply both sides of this equation by one plus IRR, and do a little more work, and I get, as you can see in grey. 110 is equal to, you should verify this, do the little bit of algebra required to do this. 110 = 100 * (1 + IRR), and that leads us to our IRR is equal to 110 over 100 minus 1 or 10%. Okay, our IRR is 10%, our internal rate of return for this project if we pursue it is 10%. Our opportunity cost of capital is 5% and what's our rule for internal rate of return? If the IRR is greater than our opportunity cost of capital, we should go for it. So let's go for it, and buy the bond, so there you have it, okay. Here's another example, which I'm going to say that you should solve on your own. The solution here, it's not much harder, it's actually not harder at all and what we just did. But you should make sure that you can get the same answer that I got here just to make sure you understand the method.