0:18

And what I'll do is I'll, I'll base this off of just

Â a simple model I've been working on with Matt Elliott and Ben Golub.

Â And the idea here is that we're going to just explore contagions

Â to try and understand how we might use network analysis to try and understand how

Â the fact that one company or country, has

Â some difficulties, how that might translate to others.

Â So here, we're going to have companies, or countries, et cetera, are going to

Â be linked to each other viri, varig, via various types of contracts.

Â So it could be that they have debts to each other, promised deliveries.

Â It could be, I'm a supplier and I owe you something.

Â Or I'm a buyer, I owe you some money.

Â Or it could be that there's equity, so I actually have stock in a company.

Â 1:01

And effectively, these kinds of cross-holdings are going

Â to expose one company to others' investments and values.

Â So I'm now exposed to how well some company's doing if, if

Â I owe, if they owe me something and, and there's some value claim

Â I have on their values.

Â So what we're going to do is, is I'll break this into two pieces.

Â So the first part right now, what we'll do is, let's just first

Â take a look at how we might use networks to model exposures, okay?

Â And I'm going to keep this very simple.

Â We are going to work with just sort of straight equity values.

Â But we're just going to first start to see, how

Â it can be that, if I know what percentage

Â of, of the value of some other company that I am exposed to and I have got a

Â whole of series of these things.

Â How can we keep track of the ultimate exposure that we have in the economy?

Â So an organization is going to have direct investments.

Â 1:53

And it's also going to have some holdings in other organizations.

Â So I'm saying organizations here cause it might be countries, it

Â might be banks, it might be companies, it might be individuals, etc.

Â So here the idea is that there's going to be some fraction.

Â So if we're organization

Â I, some fraction of the value accrues directly to them.

Â And one minus Ci is, is something that's owned, owed to others.

Â 2:20

They also are going to hold obligations of some

Â other number of organizations, so that might have some degree

Â i and I own, you know, if it's seven,

Â there's seven other organizations that, that I have claim to.

Â Okay so I hold obligations

Â of them, they owe me something.

Â It could be stock that I owe of them and if they make

Â more, if their value goes up then I'm going to get some of it.

Â But we'll model that as some claim.

Â So in particular let's think of, there's

Â this number of organizations, countries, firms, banks.

Â 3:06

It could be, if I am a country, it could be the amount

Â of taxes I am going to able to collect in a given time period.

Â If I am a, a company, it could be

Â the revenue minus the costs that I have access to.

Â So this is sort of the value of the

Â overall price of the investments that I've got in hand.

Â And the cross holdings are going to be where the network comes

Â into play, so the the organizations aren't just in isolation.

Â I has some claim to the, the value of j,

Â and in particular we can think of the fraction of

Â organization i's, value that's owned by organization j's value that's

Â owned by organization i we'll let that be Cij, okay?

Â And you don't own yourself, so you don't own parts claims

Â to yourself.

Â But what we will let is the residual claim so there's a whole series of

Â if I'm organization I, there's a whole series

Â of other organizations that I owe something to.

Â That either own part of the shares of me, and

Â then the remainder of that fraction is owned by private investors.

Â People who don't have claims to each other.

Â So these are the shareholders, the initial shareholders

Â of the company who are private investors and

Â not owned in turn by somebody else, okay?

Â 4:24

Okay, so now when we look at the value of a given organization i.

Â This should be a subscript here. What's the value?

Â Well, they have, so lets think of the book value, the overall straight book value.

Â Well, they have some direct asset values.

Â The, the revenue that's coming

Â in and then they also have some claim on the values of the other organizations.

Â They have cross holdings, okay?

Â So this is just on the, the value in terms of the assets that they have on the book.

Â So if we look at that, then when we rewrite this equation.

Â Keeping this as a matrix of cross holdings.

Â Then we can say that the vector of v of values is equal to this vector of p.

Â Investment values plus the cross holdings times the next, the values.

Â So if we solve this out.

Â Solve this equation in terms of network calculation what do we get?

Â We get that the actual vector of overall

Â book values looks like i minus c to the minus one.

Â So the inverse of i minus c times its p vector, right?

Â So we just solve this.

Â We bring this so this looks like v times i minus sorry i minus c.

Â Times V equals p and we solve that and we get

Â i minus c to the minus one is equal to V.

Â This is related to what was studied by Leontief, just

Â a straight calculation of, of book values in this case.

Â 5:57

So when we look at that, what we want to then figure out is the market value, the

Â value to the final private investors who aren't owing something

Â to somebody else; it's not a company that owns, that's then owned by other people.

Â That value is then, what's the amount that's held in private, by private

Â shareholders, so that's C hat ii times Vi.

Â Now we plug in the Vs here, and we get that V

Â is equal to C hat times I minus C inverse times P, okay?

Â So this is a very simple calculation that tells us, as a function of the

Â underline asset values, what's the value of different companies?

Â So it's not that a value of different organizations here

Â in terms of value, interms of valuing their, their actual

Â owners, is just dependent directly only on their own asset holdings.

Â But it's going to be dependent on the whole

Â portfolio of asset holdings, in terms of this factor.

Â So we'll call this c hat times i minus c inverse.

Â This is the a matrix. Okay, we'll call it an a matrix.

Â And what's that doing, Aij tells us, what's the

Â fraction of the investments owned by organization j that

Â ultimately accrue to the private shareholders of i, ok?

Â So lets just look at a very simple example to make this clear.

Â So suppose there was just two organizations in the world, and

Â each of them owned half of the shares of the other, okay?

Â And so here, I'm going to do everything in terms of equity.

Â You can do this in terms of debt; it gets a little more complicated.

Â This makes it nice and linear.

Â So, here, I, you know, each owns half of the assets or,

Â or investments of the other, and, and some value.

Â So what's the implied holdings by private investors?

Â Well, half of, of organization two is held by organization one.

Â That means that the remaining half is privately-held.

Â And half of organization one is held by two, and

Â that means that the remaining half is held privately as well.

Â So half of the each of the organization is held by

Â private investors, half is held by the other okay?

Â So if you do the calculation of what the A

Â matrix is, it says that ultimately if you this calculation.

Â That two thirds of the value of the

Â investments of organization one, actually end up going

Â to the owners of organization one, and one

Â third of organization twos' goes to organization ones' owners.

Â And, and vice versa. Okay?

Â So let's try and get an idea of exactly what's going on here.

Â So what this does is that, just on a

Â very simple matrix analysis which is implicitly capturing the network.

Â 8:54

Half is going back to organization two which owns half of,

Â of ones' claims and similarly on the two side, half is

Â going out to private shareholders, half is going back to one,

Â okay so that's the basic structure in terms of the ownership.

Â And, now, let's figure out where this two third, one third came from.

Â So, let's suppose that there's a dollar that, it's a

Â return to a, one of the investments of organization one.

Â Okay,

Â well, here's our dollar.

Â Now, half of that should be going to organization two.

Â Half of that accrues to the private shareholders.

Â 9:29

So we, half goes out here and half goes over here.

Â Now this part is just going off

Â to the private shareholders; that's owned by them.

Â This half, what happens here?

Â Well, half of that is owed to the shared private shareholders.

Â Half of this again

Â goes back to organization one which now

Â has claims on what's coming into organization two.

Â So when we do that calculation, 0.25 goes out here, 0.25 goes back here.

Â Now this 0.25 is going to get split fifty fifty,

Â and you can see where this is going, right?

Â So, so we go ahead, we split this 0.25

Â up, 0.125 goes out one side, 0.125 goes back that.

Â We split that up and so forth.

Â And when you keep

Â doing this, you can just keep iterating on this.

Â As you keep iterating this, eventually two thirds of that went

Â out one side, and one third came out the other side.

Â Okay?

Â And that was exactly the calculation we got for the A matrix.

Â We got that two thirds of, of the ultimate investments of one

Â were going to one's owners, and one third were going to two.

Â Now this is just a very simple calculation with two organizations and so forth.

Â But this general

Â formula would do this for a very complicated network, right?

Â We could have arbitrary sets of, of companies owning different

Â parts of each other and, and having a very complicated network.

Â And it tells you if you want to do the same kind of calculation of injecting some

Â funds and then figuring out where they went

Â in the economy, this would be the same calculation.

Â So here we can see the power of the network as, as figuring this, this out.

Â 11:00

If you through,

Â we did a simple calculation for debt in, among a few countries in Europe, you can

Â actually figure out by looking at how much

Â of Germans' debt is held outside of Germany.

Â How much of French debt is owned outside of France and so forth.

Â You can eventually figure out what the A matrix looks like.

Â And the A matrix here based on some simple calculations.

Â And says that ultimately out

Â of the tax receipts that come into France, 18% of that actually ends up

Â going to the, to the German through German banks and so forth to Germany.

Â 12% of Germany goes to Italy.

Â 11% of Italy goes to, to Germany and so forth.

Â This only lists the numbers above 5% but basically you can do that calculation.

Â You end up with an A matrix, and you can, you

Â know, begin to see what the dependencies of different countries are in

Â terms of how exposed they are to the investments of, of each other.

Â And so you see that, you know

Â 11:59

certain countries, you know, Italy's exposed to Germany.

Â But France is very exposed to Italian.

Â It's relatively exposed to Portugal. To Greece, to Spain, so you begin to

Â see who's, who's at risk in terms of whose investments are, are accruing to which

Â other country.

Â So now what we have is we've put in place a very simple model of cross dependencies,

Â and so forth which allows us to sort

Â of back out the implicit exposure to different actors.

Â In terms of different organizations have in an economy to each other's investments.

Â And then we can begin to use that to, to model contagions

Â and to see how changes in a value in one place might impact

Â