Okay, hi folks, so we are going to look now at another application of models of contagion, and try to look at a little bit of what numbers can tell us about financial contagions. And what I'll do is I'll, I'll base this off of just a simple model I've been working on with Matt Elliott and Ben Golub. And the idea here is that we're going to just explore contagions to try and understand how we might use network analysis to try and understand how the fact that one company or country, has some difficulties, how that might translate to others. So here, we're going to have companies, or countries, et cetera, are going to be linked to each other viri, varig, via various types of contracts. So it could be that they have debts to each other, promised deliveries. It could be, I'm a supplier and I owe you something. Or I'm a buyer, I owe you some money. Or it could be that there's equity, so I actually have stock in a company. And effectively, these kinds of cross-holdings are going to expose one company to others' investments and values. So I'm now exposed to how well some company's doing if, if I owe, if they owe me something and, and there's some value claim I have on their values. So what we're going to do is, is I'll break this into two pieces. So the first part right now, what we'll do is, let's just first take a look at how we might use networks to model exposures, okay? And I'm going to keep this very simple. We are going to work with just sort of straight equity values. But we're just going to first start to see, how it can be that, if I know what percentage of, of the value of some other company that I am exposed to and I have got a whole of series of these things. How can we keep track of the ultimate exposure that we have in the economy? So an organization is going to have direct investments. And it's also going to have some holdings in other organizations. So I'm saying organizations here cause it might be countries, it might be banks, it might be companies, it might be individuals, etc. So here the idea is that there's going to be some fraction. So if we're organization I, some fraction of the value accrues directly to them. And one minus Ci is, is something that's owned, owed to others. They also are going to hold obligations of some other number of organizations, so that might have some degree i and I own, you know, if it's seven, there's seven other organizations that, that I have claim to. Okay so I hold obligations of them, they owe me something. It could be stock that I owe of them and if they make more, if their value goes up then I'm going to get some of it. But we'll model that as some claim. So in particular let's think of, there's this number of organizations, countries, firms, banks. We'll make things really simple. So each organization i has some price of investments. So you just think of this as the cash flow that should be coming in. So what's the value of their investments? It could be, if I am a country, it could be the amount of taxes I am going to able to collect in a given time period. If I am a, a company, it could be the revenue minus the costs that I have access to. So this is sort of the value of the overall price of the investments that I've got in hand. And the cross holdings are going to be where the network comes into play, so the the organizations aren't just in isolation. I has some claim to the, the value of j, and in particular we can think of the fraction of organization i's, value that's owned by organization j's value that's owned by organization i we'll let that be Cij, okay? And you don't own yourself, so you don't own parts claims to yourself. But what we will let is the residual claim so there's a whole series of if I'm organization I, there's a whole series of other organizations that I owe something to. That either own part of the shares of me, and then the remainder of that fraction is owned by private investors. People who don't have claims to each other. So these are the shareholders, the initial shareholders of the company who are private investors and not owned in turn by somebody else, okay? Okay, so now when we look at the value of a given organization i. This should be a subscript here. What's the value? Well, they have, so lets think of the book value, the overall straight book value. Well, they have some direct asset values. The, the revenue that's coming in and then they also have some claim on the values of the other organizations. They have cross holdings, okay? So this is just on the, the value in terms of the assets that they have on the book. So if we look at that, then when we rewrite this equation. Keeping this as a matrix of cross holdings. Then we can say that the vector of v of values is equal to this vector of p. Investment values plus the cross holdings times the next, the values. So if we solve this out. Solve this equation in terms of network calculation what do we get? We get that the actual vector of overall book values looks like i minus c to the minus one. So the inverse of i minus c times its p vector, right? So we just solve this. We bring this so this looks like v times i minus sorry i minus c. Times V equals p and we solve that and we get i minus c to the minus one is equal to V. This is related to what was studied by Leontief, just a straight calculation of, of book values in this case. So when we look at that, what we want to then figure out is the market value, the value to the final private investors who aren't owing something to somebody else; it's not a company that owns, that's then owned by other people. That value is then, what's the amount that's held in private, by private shareholders, so that's C hat ii times Vi. Now we plug in the Vs here, and we get that V is equal to C hat times I minus C inverse times P, okay? So this is a very simple calculation that tells us, as a function of the underline asset values, what's the value of different companies? So it's not that a value of different organizations here in terms of value, interms of valuing their, their actual owners, is just dependent directly only on their own asset holdings. But it's going to be dependent on the whole portfolio of asset holdings, in terms of this factor. So we'll call this c hat times i minus c inverse. This is the a matrix. Okay, we'll call it an a matrix. And what's that doing, Aij tells us, what's the fraction of the investments owned by organization j that ultimately accrue to the private shareholders of i, ok? So lets just look at a very simple example to make this clear. So suppose there was just two organizations in the world, and each of them owned half of the shares of the other, okay? And so here, I'm going to do everything in terms of equity. You can do this in terms of debt; it gets a little more complicated. This makes it nice and linear. So, here, I, you know, each owns half of the assets or, or investments of the other, and, and some value. So what's the implied holdings by private investors? Well, half of, of organization two is held by organization one. That means that the remaining half is privately-held. And half of organization one is held by two, and that means that the remaining half is held privately as well. So half of the each of the organization is held by private investors, half is held by the other okay? So if you do the calculation of what the A matrix is, it says that ultimately if you this calculation. That two thirds of the value of the investments of organization one, actually end up going to the owners of organization one, and one third of organization twos' goes to organization ones' owners. And, and vice versa. Okay? So let's try and get an idea of exactly what's going on here. So what this does is that, just on a very simple matrix analysis which is implicitly capturing the network. And gives us an idea of what the exposure of different investors are to the underlying assets in the economy, okay? So, if we think about this, we've got organization one and two. Half of the value of organization one is going out to, to private shareholders. Half is going back to organization two which owns half of, of ones' claims and similarly on the two side, half is going out to private shareholders, half is going back to one, okay so that's the basic structure in terms of the ownership. And, now, let's figure out where this two third, one third came from. So, let's suppose that there's a dollar that, it's a return to a, one of the investments of organization one. Okay, well, here's our dollar. Now, half of that should be going to organization two. Half of that accrues to the private shareholders. So we, half goes out here and half goes over here. Now this part is just going off to the private shareholders; that's owned by them. This half, what happens here? Well, half of that is owed to the shared private shareholders. Half of this again goes back to organization one which now has claims on what's coming into organization two. So when we do that calculation, 0.25 goes out here, 0.25 goes back here. Now this 0.25 is going to get split fifty fifty, and you can see where this is going, right? So, so we go ahead, we split this 0.25 up, 0.125 goes out one side, 0.125 goes back that. We split that up and so forth. And when you keep doing this, you can just keep iterating on this. As you keep iterating this, eventually two thirds of that went out one side, and one third came out the other side. Okay? And that was exactly the calculation we got for the A matrix. We got that two thirds of, of the ultimate investments of one were going to one's owners, and one third were going to two. Now this is just a very simple calculation with two organizations and so forth. But this general formula would do this for a very complicated network, right? We could have arbitrary sets of, of companies owning different parts of each other and, and having a very complicated network. And it tells you if you want to do the same kind of calculation of injecting some funds and then figuring out where they went in the economy, this would be the same calculation. So here we can see the power of the network as, as figuring this, this out. If you through, we did a simple calculation for debt in, among a few countries in Europe, you can actually figure out by looking at how much of Germans' debt is held outside of Germany. How much of French debt is owned outside of France and so forth. You can eventually figure out what the A matrix looks like. And the A matrix here based on some simple calculations. And says that ultimately out of the tax receipts that come into France, 18% of that actually ends up going to the, to the German through German banks and so forth to Germany. 12% of Germany goes to Italy. 11% of Italy goes to, to Germany and so forth. This only lists the numbers above 5% but basically you can do that calculation. You end up with an A matrix, and you can, you know, begin to see what the dependencies of different countries are in terms of how exposed they are to the investments of, of each other. And so you see that, you know certain countries, you know, Italy's exposed to Germany. But France is very exposed to Italian. It's relatively exposed to Portugal. To Greece, to Spain, so you begin to see who's, who's at risk in terms of whose investments are, are accruing to which other country. So now what we have is we've put in place a very simple model of cross dependencies, and so forth which allows us to sort of back out the implicit exposure to different actors. In terms of different organizations have in an economy to each other's investments. And then we can begin to use that to, to model contagions and to see how changes in a value in one place might impact the values of, of another.
