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.