 Hi everyone, it's MJ the fellow actuary and in this video. I want to talk about the Gumbel Copula Now Gumbel was a German statistician who was very much anti Hitler And he kind of got kicked out of Germany because he was very much against the Nazi party Anyway history aside You'll see Gumbel comes up quite a lot with extreme value theory and other statistic things But in this video we want to talk about His copula because Gumbel's copula is part of this Archimedean copula Which we discussed in the previous video Essentially we have these Generator functions where we add the marginal distributions together and then we take the inverse of the sum of them Essentially the big idea is taking space space zero to one to zero to infinity Adding there and then bringing it back to zero to one so that we have a probability Gumbel's generator function is given by the following Negative lin f of x to the power of alpha where candle's tau is equal to one minus one divided by alpha So we're seeing that this alpha is the correlation parameter Alpha lives between the values of one and infinity We see that when it is equal to one we get the independent copula We see that if we had to take it all the way to infinity we'll actually get the minimum copula and What this essentially does or how this Gumbel copula flexes dependency is through the following It'll have a upper tail dependency But it won't have a lower tail dependency and this is something that we can use when it comes to say Credit portfolios specifically with credit losses We'll see that their dependency increases during times of recession. So this is an application of the Gumbel's copula and What we can do? I mean, I'm just explaining that in the bivariate case You can take it all the way to the n-dimensional multivariate case and that is essentially your formula It's not that scary when you kind of break it down and you look at the components Anyway, let's look at some pictures of the Gumbel copula So what I'm gonna be doing is I'm gonna be playing with the parameter alpha and I drew these in a statistical program known as R So you can see here when we have it as one it is the Independent copula and it's kind of just this big block. Okay, nothing too too special But as soon as we start adding some sort of correlation You can see how the graph starts to kind of take this the shape It almost looks a little bit like a stingray the big idea or something that I want you to focus on is The upper tail so the tail of the stingray over there You can see one of the ways to to read These these type of graphs is to also pay attention to the axis. So you can see here is from one to one Here we're going from two to one here's five to one fifteen to one sixty to one And that's just showing how much more weight is getting put in the upper tail So we call this the stingray's tail Well, it looks like a stingray's got its little wings little head there I don't think anybody else causes the stingray Don't refer to it as the stingray in the exam It's just how I kind of see this but the big idea here as you can see that as we're Changing this parameter and you can see the minimum copula was full positive Dependency so as we make alpha bigger, we're gonna see more Dependency between our marginal distributions and what the Gumbel does is it allocates a lot of the weight in the tail And that is something that we're seeing when we start increasing The parameter you can see sixty that is a lot of weight over there Compared to to these other ones and the reason why I say focus on these axis is here is because otherwise the picture looks Like very much the same but when you realize that they're being scaled you can actually see the the effect of it Anyway, this is very much a quick introduction to Gumbels copula