Okay, so here's what we've done with our tree, including time value in a beautifully formatted version here. And I would like to go over some of the key points here. We get everything out of this tree that we had before, the software computes are expected value, which is now the expected NPV of the project, we should be able to verify that using our sum product, each terminal cash times the probability of getting to that terminal cash. One quick note. What you're going to see in the spreadsheet here, 383 is a little bit different than what we're seeing on this slide. And that simply because I have more significant figures here when I did this previously and made a snapshot of it I put more significant figures into these numbers so 385, is a little more accurate than 383. Two big things have happened okay. The expected value of the project in time value consideration terms which is the only reasonable way to consider a project like this has gone down a lot okay. From 9,000 units to 385 in thousand units. And the optimal decisions to make for this project have changed. So including time value is extremely, extremely important. You can get wildly incorrect answers if you don't include time value. So this is again going over what the decision tree is telling us. As pretty much we have just looked at it. And again, we reiterate the key features here that this is extremely important without considering time value. Your off by millions and millions of dollars okay, and the you may make the wrong decisions when you do that too. Okay, A little typo here. Also as we've seen, decision trees can easily accommodate discount rates. We saw how easy that was. Just get the present value of each cashflow and plug it in. Okay, so now we're ready to think about should HDL go ahead with this project, first project of year. So how are we going to determine that? We're going to go back to that risk reward policy, okay. What's the risk reward policy? We want it in time value terms now. Of course, it was, they want in terms of reward, right. They want to make sure that the expected value is maximized, so that's their reward. And here there are two risk criteria. They want to make sure the expected value of the NPV is greater than 0 which it is okay 385,000. They also want to make sure that the probability o f the NPV being less than 40 million is zero. Okay, and let's go ahead and think about this one piece is at a time. Okay? So for their reward analysis, we've got our NPV. Is greater than 0, so that is excellent, and the expected value of the NPV, 385 million, just a good sort of statistical probabilistic point here, could they ever realize is that. Well, let's go back up to, Our decision tree and think about it. Well, here are all the things that can happen and the amounts of money associated with them. Do we see 385,000 as any of these? No, there's no way they're going to get that exact number. Again, what the expected value means is, if they kept doing this project over and over and over again under the exact same circumstances. Sometimes they get this. They'd never get this because their not going to make that decision, it never get that because their not going to make that decision. Sometimes they get this on average over the long the amount their winners and their losers is going to be 385,000, okay? But that's not possible in any one trial, okay, for this project decision tree. All right. So now let's think about their risk criteria. Okay, their risk criteria is the probability, the key one. Is the probability that the NPV of the project being less than 40 million is 0, so there is no chance of them losing net more than 40 million on this project. How do we determine that? We just look at all of the net cash they're left with at each terminal branch that has a non-zero probability. So we take out these guys and we see up here this is the one more they sale the variance property assuming the variance is approved etc. They get 17, 000 that's It’s clearly greater than minus 40 million, and down here 30% probability if their variance is not approved they’re going to lose 40 million, but they’re not losing more than 40 million, they wanted to put a floor on that with the amount they spent to get the variance and they have succeeded in that, okay? So is there any non-zero-probability end point for which we could lose more than 40 million? No. No. So this project passes the reward criteria of HDL and the risk criteria of the HDL. So HDL should proceed with the project. That yeah. Okay. One other thing that we consider in the real world on the time and I just want to give you a taste of because it's very important is model limitations analysis. As I said at the beginning of this module, the farther we go into finance with complicated models the more simplifying assumptions we have to make. And the more trouble we can get in if we don't understand those assumptions and how those limit the applicability of our model. Okay, so you want to ask. Things like this, what are the key assumptions of our model? How does the model misrepresent reality and what would make it useless. Clearly, this three questions are not independent. In a certain way they're asking the same thing in different ways but that's what you want to do you want to ask the same thing the different ways to make sure you don't forget something obvious. Just to answer some of these questions. Key assumptions of the model here are that we can guesstimate our probabilities. And also I would say we can estimate our future cash flows on all scenarios. That's a pretty big assumption. Okay so just think about, let's just look at this one to finish up. What would make this model useless? Well we did think about a localized market crash, but what about a climate event, we didn't consider anything like this. I meant something like hurricane Sandy we have been talking about, we have been talking about hurricane Sandy, created enormous devastating the station in New York, something like that. So you want to ask these questions. And just make sure that you understand how the answers to these questions are limiting your model's effectiveness and applicability. I hope this has been helpful to you. It's certainly been very enjoyable for me and I hope you enjoyed the course overall. Thank you.