Okay, hi again. So we have just looked at a very simple model of interdependencies among organizations in terms of financial things. And we saw that networks can then allow us to infer with the exposure of one organization to another. And once we do that, we can then use that directed weighted network as a means by trying to figure out, okay, let's suppose that for whatever reason, one of these organizations goes bankrupt and can't suddenly pay out. So we can look at financial contagions. So it says somebody has a failure, and now their asset goes from a value of 1 to say 0. What's the impact on the others? So now others suddenly have lost some value, depending on what that A matrix looked like that we figured out in terms of all these indirect holdings. Some people are going to be very exposed to that, some people might be less exposed. And so the full network of interactions then tells us how indirectly exposed we are, that might cause somebody else to fail if they're very exposed. Okay, so we're going to stick with a simple model. And now what we've got is we've got a situation where we're going to look at some fraction of a value. It's going to accrue directly to a given organization. Some other fraction is owned to others. And now we'll just have some number of organizations that we owe. And in particular, let's just do some simple simulation to set this up. So we'll look at a network with 100 organizations. And what we're going to do is let's start by doing a very simple sort of style network. So what we can do is set the probability that there's a link between organization i and j to be d, which is going to be the expected degree over n-1. Okay, so if we do these with probabilities, the expected number of connections that each organization should have in terms of other organizations that they own should be d. Okay, so this is the expected number. And we can think of that. We'll think of that as a level of diversification. So if I'm not very diversified in terms of the number of other people that I own, or then I just own a couple of companies where I have I'm dealing in terms of the people that owe me things, it's only going to be a couple of companies. As I broaden that out, I'm going to be diversified. I'm going to be getting things from many different other organizations. And so things that are owed to me are coming from a wider range of things, rather than just one. So we'll think of that as diversification. Now the separate thing is that there's some fraction of the organizations that are cross held, some fraction C. And the remainder of that is the part that's held privately. And so we'll think of this as a level of integration. So if C is very high, then it means that most of an organization is actually owed through the economy to other organizations. And less of it goes directly to its private holders. And otherwise, it's more privately held, and not very well integrated into the economy, okay? So if you go through that, then the claim that a given organization i has on some other organization j, the Cij is going to be, okay, how much of organization j is owned outside that C? And then we'll just divide through by the number of organizations that cross-hold. And so Cij is going to be the gij. So whatever does i actually have a connection to j? And then divide it through by the number of holders of j. So this dj now is number of organizations that actually have claims on j. So it's the n degree of j. Okay, so now we have cij. And based on that, we can go ahead and figure out an a matrix. And then we can just do some simple simulations. So each organization will start with some investment. It has at a value of 1. And now what we're going to do is think of bankrupting one of these organizations. And to keep things simple, we'll just run bankruptcy straight to 0. Okay, so we take the 1. Some organization fails, so the value of its assets goes to 0. Okay, now that's just the pi in our previous notation, so the pi goes to 0. So what does that mean? Well, that means that this organization now has less coming in. It still might have a lot coming in from other organizations. And what we would do is see, we have a starting value of different organizations based on what this network is. And if some organizations value falls below some fractions theta of its starting value, it'll fail. So let's suppose it's 80%. So If I'm a company, if I don't have enough value coming in, if I drop below some threshold, I have to close up shop. And then we're assuming that then all my assets are liquidated here for a value of 0. So in the paper, we go through all kinds of other scenarios. This is just a very simple one to do some simulations with. And then we can look at resulting cascade. So the first asset goes to 0. That means that it has caused this company now to fail. So let's suppose that theta, for instance, was 80%. If we went back to our two organization example from the last video, then each one will be exposed two-thirds to itself, and one-third to somebody else. So if that asset went down from 1 to 0, I would lose a third of my value. If I lose a third of my value, then if this is, say, 80%, now I go bankrupt as well, okay? So that's the basic idea. So we'll be varying theta. We vary the number of organizations you're exposed to. We vary the value c. So we get a different scenario. And then we can simulate what happens by dropping one company's asset. Once this drops, now you can re-value what everybody else has. Some of those might be cause to drop below theta. That means that their asset value drops to 0, then see what the ramifications of that are. And you can do this if you have very simple MATLAB code. Okay, so let's look at some just simple simulations. So here this is the d. What's the expected number of cross-holdings we have? And then this is the percent of organizations that fail on the axis here. And this was done with a theta of 93%. So if you drop below 7%, if you lose 7% of your value, then that triggers bankruptcy. And a situation where half of any company is owned by other companies, and half is owned by private shareholders. If you do that, then what do you get? Well for very low degrees, 0.3 less than 1, you get fewer than 10% of the organizations fail. Once you're above, about 7.5 or so, you're again below 10%, and it drops very slow. And then in this middle range from owning between 1 and 6, 7 or other organizations, now you see a substantial amount of organizations that fail. And in particular, we get is a. So this a world where, as you increase the density of the network, it's not as if more cross-holdings mean that we end up with just larger and larger amounts of contagion. Once you get past a certain point, here in this case, about 3, 3.5, it actually starts to lower a bit. And why is it coming down? What's the aspect of that? So let's just have a look behind the scenes. Well first of all, as you vary this, you can vary the parameters. We did it with .93. You could make it .96 .90 here, .87 and so forth. So you could change those levels, and you would get different curves. What's the intuition? Okay, with very low levels of d, we're not getting much the contagion. Why not? Because not many of the organizations own each other, so there's not much of a chance for contagion. So if this one goes bankrupt, that doesn't affect anybody else. If this one goes bankrupt, it affects a few, but not very many. Once we get to medium diversification, now if somebody hits, it has a chance to actually hit a number of different nodes. Once we get to full diversification, it could spread a lot. But what's different here is now this organization is basically part cross-held, or owned by just two others. And they, in this case, with low diversification, they can be substantially exposed to this. So when you've only got a few partnerships, if you one of your partners goes down, that can be devastating. And so here, the fact that we've got connection, but also low numbers of partners means that everybody is substantially exposed to a few people. And that allows things to spread. Once you get high diversification, you have a chance for things to reach more broadly. But now everybody has lots of partners. And so the failing of any one partner doesn't necessarily represent as much of a change in terms of your own holdings. Because when you look at that a matrix, now I'm not necessarily exposed to as much of that particular organization's assets. Okay, so low diversification, we get a fragmented network, no widespread contagion. Medium diversification, what do we get? We get a connected network, contagion's possible. And you get exposure to only a few others, make it easy to spread, and we actually get things going off the ground. High diversification, well, a contagion's still possible. But now we've got little exposure to anybody else and so the failures don't spread as much, okay? So what does this tell us? It's a very simple point here. But what it says is we can begin to use these simple network models by using the A matrix. By beginning to look at this, we can begin to see how these things spread. And under what conditions are we going to get broader ramifications of some failures in the economy? And what conditions might it be more isolated? So one other thing we can do is, we can vary the c parameter. And if you vary the c parameter, you're also going to get some differences. And here what's happening is, as you increase the c parameter, now more of what I owe, own is in other companies. And now if one of those goes bankrupt, I'm more exposed to things. So as you increase this c parameter, you're getting more exposure. So ultimately, if you increase it even more, if you keep taking it way up, it starts to drop back down again. Why does it drop back down again? Well now everybody's exposed to everybody else, but I'm not exposed as much to myself to my own holdings. And so if I have a failure of my own assets, that doesn't necessarily cause problems for me, because I'm well integrated in terms of the other organizations and economy. So you get 9 minus here in both cases. So low integration, low exposure to others, failures, no triggering others. Middle integration, the substantial enough to trigger contagion, and then high integration. Now it's difficult to get any first failures, because the ownership of your own assets doesn't necessarily trigger failure if things are well integrated in the economy. So you can analyze much richer kinds of networks in this model. So you can begin to look at core periphery kinds of networks where there's few large, say large banks that have investments in and out of many other organizations. You can begin to look at how these diversification and integration interact with size of shocks, correlation of shocks, heterogeneity in networks. So you can begin to do a lot of different things. But the point here is that the network analysis is something which begins to allow us to understand how the economy reacts to shocks to particular organizations. To changes in values of particular organizations, how that ripples through an economy. What the ultimate ramifications of that are, and things like this a matrix can be useful in just keeping track of giving organizations exposures to others, and indirect exposures to others. So instead of just looking at a given organization in isolation, we can begin to say how exposed is one organization to the investments in another place. And that allows us to begin then to understand what the potential dangers are. And for regulators, or for other individuals to begin to assess risk, and to try and manage that risk effectively. So here we just see a situation where simple network models can give us some very basic and substantial insights. And we can do this with the tools that use fairly basic. Now obviously, these are oversimplified models, we've crossed over a lot of things. But it gives us some idea of how to model exposure, what kind of things we might want to start with. And we can enrich the models, and go from there.