 Hi everyone, it's MJ the fellow actuary and in this video I want to introduce you to the idea of copulas now copulas essentially are Mathematical objects and you can almost think of them as a model in the sense that we have some inputs the inputs are normally two or more Marginal distribution something that kind of looks like that And then also some of the copulas tend to have parameters that are based on correlations Some of the parameters will even have a function which is sometimes known as the generator function as well as its inverse But all copulas will have the same output which is something known as a joint distribution And this is a quick little preview of what they look like so very very quick introduction on copulas They're mathematical objects their inputs are marginal distributions Some of them have parameters based on crap on correlation Some of them have functions with generator functions and their inverse, but all of them have this output which is the joint Distribution, let's maybe explore this idea in a little bit more detail So why do we need copulas with fancy functions in order that well? If we come back to say probability states, we know that probabilities live between Zero and one a probability cannot be negative and it cannot be greater than one because then it is nonsensical The problem is what happens when we want to add up a whole bunch of probabilities together It's like what's the probability that both a and b occur? We know that if we just had to add these probabilities Directly then there is a possibility that we might exceed the required state space of zero to one And then we no longer have a probability So to maybe put this in like a bit of a numerical example If we have the probability of a happening with sixty percent and the probability of b happening is seventy percent And we want to say well, what's the probability of these two things happening together? Well, you can't just directly add the probabilities because zero point six plus zero point seven is going to give you one point three and That is not a probability. So what do we do? Well, this is the idea behind Copulars Basically, what copy is the saying is that if the probability is of both a and b live on the state space from zero to one Then why don't we transform them so that they live on the space state space of zero to Infinity and once we've transformed our probabilities into this higher space We can then add them together and how when we start adding to them together They're also going to be living on this zero to infinite space Which means we can add as many probabilities as we want and we're not gonna, you know exit the state space Then once we've done that once we've added all of our probabilities together we then simply take the inverse transformation of the sum and Because we take an inverse inverse Transformation we're gonna take the sum and it's now gonna live back on zero to one and it's therefore going to be a Probability again now that is a very very advanced idea. So let's maybe see it in action with our example So what we're going to be doing is our Generator function in this case is we're going to be using a negative log Transformation this will take any number between zero and one and put on a state space of zero to infinity and then what we do is we Take the inverse of this transformation Which will be the negative Exponential transformation and that'll take anything from the state space of zero to infinity back to zero to one So in our example if we take probability a which was zero point six and probability B which was zero point seven and We transform them into this higher space. We get zero point five one and zero point three six Now we can add them together and we can get zero point eight seven now We could have added C D E a whole bunch of other probabilities and it could have even broken You could have been greater than one But what happens is when we do the inverse transformation? So we take the exponential of this amount the negative of it then no matter what that amount is It's gonna come back and be a value between zero and one and that's what we're seeing here We're seeing zero point four two now zero point four two is an interesting number because what essentially it's saying is Well, basically what our calculation has shown is that the probability of a and B are Independent because if we had simply multiplied the two probabilities together, we would have got zero point four two This is something that we learned from statistics back in maybe first and second year that Probability of a occurring and the probability of B occurring if they're independent You simply multiply the two together to get the probability of them both occurring and we see that this is a special type of Copula it's something known as the independent copula over here. So essentially This independent copula which is given by the function the copula of f of x and f of y is equal to f of x times f of y You can see f of x and f of y are our two Marginal distributions that are being inputted into the copula and the function in this case is simply Multiplication or if you want to be fancy. It's the exponential of the negative lins added together And essentially yeah, we're getting this independent copula now copy this We're gonna see they don't have to work on cumulative distributions or marginal distributions They can also work with probabilities But most of the time that we're gonna be looking at them we are going to be looking at these marginal cumulative distributions and We're looking to find a joint cumulative distribution now this independent copula. It's a it's a famous one I'm gonna struggle to say this name because the one name is French and the other name is Finnish But essentially it is this boundary copula And we're gonna be exploring a few of the other ones as well But we can see that even when we have an independence of multiplying two probabilities together They can be constructed in a format that represents the copula, which is given here by this copula f of x f of y is equal to the two multiplied together Now the big thing which kind of happens with copulas is we're not that interested in the independent case we're more interested to see and well what happens if there is correlation and This is where we can now improve our generator function by adding in this parameter of Alpha and alpha what we're gonna see it's linked to correlation But the whole idea with copulas is not only if you've got a generator function You also want the inverse of the generator function and we can see this is where sometimes the maths can get a little bit tricky But if you're quite solid with your math You shouldn't have any problem by seeing negative lin f of x to the power of alpha the inverse of that is the exponential of negative f of x to the power of one divided by alpha and Like I said just to bring it back What we were almost looking at with the independent one is we just had the lin and the exponential What we're doing now is we can introduce Correlation with this parameter of alpha and like I say it is linked to correlation in the sense that Kendall's tau Which is one of the the rain correlates measures for correlation is going to be equal to one Minus one divided by alpha So if you can calculate the correlation you can then use this to find your alpha and you can then use that in your copula So that you can calculate the joint distribution of two marginal distributions that are Correlated or connected and this is the idea or this went when we use these generator functions We get something called the Archimedean Gumbel copula now the Archimedean Gumbel copula is saying that we will be exploring in more detail in these videos But essentially this maybe talk about why we actually care about copulas as Actories, Actories love copulas because what they allow us to do is we can now model risk profiles So we've made courses where we can measure market risk where we can measure credit risk And there's other courses we can make where we measure operational and other types of risk now a copula Can be used for each of these so you can see what is the credit portfolio or the market portfolio So you could have a copula that combines all your market risks together But you can also use the copula to combine all of these different risks together based on their dependency and You might be saying well why don't I just use a multivariate Distribution or the whole idea with the copula or why it's a better is it allows this dependency between the risks to have Flexibility specifically let's look quickly at the Gumbel Copula we will see that it is designed in such a way that its dependency increases with extreme Positive values that means that when we enter into a recession we can expect a whole bunch of our credit losses They're all gonna start defaulting at the same time or their dependency is going to increase and the Gumbel copula Will be able to capture this change in dependency Way better than any multivariate Distribution possibly could now look copulas do have a little bit of a bad reputation when it comes to The financial markets because some people do blame them or blame the financial crisis of 2008 on them And what we see what was happening in the crash of 2008 is they weren't using the Gumbel copula In fact, they weren't actually using copula that Incorporated fat tails or had this dependency that increases under market conditions like recessions and all of those like what it was basically designed for no instead they use something called the normal copula and It's fascinating because we know from long-term capital management and the whole crisis that they had in the 90s That they fell apart because of this normal assumption towards market risk It turns out that they didn't learn their lesson Wall Street didn't learn their lesson because they kept using this normal copula Well, it's either that they were stupid and they didn't learn their lesson or Or it's because they thought hold on if we had to use the Gumbel copula to Measure the risk of a mortgage bond and you know the whole bundles for Securitization then the Gumbels copula is actually gonna tell us that these things are very very risky and The more risky something is the higher return that it needs to generate and thus the lower price Whereas if you use the normal copula, it will allow you to Understate risk of these securitizations and these very highly complicated financial instruments And because you've understated risk you can get a higher credit rating because you have a higher credit rating You can offer a lower return and how you get away with offering a lower return is by selling something at a higher price So at the end of the day, I don't think copulas should get all the blame that they kind of got I mean there were some articles written by a wired magazine about how the copula was the formula that brought down Wall Street I think that is unfair I think greed had a much bigger role to play than the copulas because like I said If they'd used the Gumbels copula or even the student T copula, which allows for more fat in the tails Then we were seeing that these mortgage-backed securities that were calculated using copulas Would have actually shown them to be a lot more risky and therefore their price should have been a lot lower But anyway there we're getting into a little bit too much of the application And I want these videos to be very much theoretical so that you can apply to a whole host of different topics And not necessarily only just say financial risks, but anyway at the end of the day that is copulas We are going to be exploring these boundary copulas in a lot more detail We're also going to be looking at these Archimedean copulas in a lot more detail But before we do that, let's focus a little bit more on the theory something known as Sklar's theorem So we're going to be talking about that in the next video. Thanks for watching. Cheers