Copula

[ChristofferLoov]

Hi David,

I am having trouble understanding the purpose of the copula. As I understand it, we map a uniform distribution to a normal since we can correlate multiple marginal normals to create a multivariate normal distributions to see how loans default simultaneously. Isn’t this what we do in the simulation in structured credit risk by Maltz?

However, when calculating the portfolio UL in Schroeck we can sum the individual UL and the pairwise correlations to get the portfolio UL. If the portfolio is to large this creates a problem with dimensionality and we can assume constant correlation.

Why can we simply sum the pairwise correlation in one case and we have to use the copula in the other? If the problem is the number of pairwise correlations I don’t see how the copula solves that problem since we still need a correlation matrix. What am I missing?

 

[David Harper CFA FRM]

Hi @ChristofferLoov Right, well, I am not myself sure that any multivariate normal application is a showcase for the advantage of copulas; if i think in terms of code (I use r #stats) and if the goal is to simulate (eg) multivariate normal, I am not sure why I would need a "copula." I put quotes because I tend to think of a copula as a cluster of things. I am not a copula expert, so i am not certain about the following statement, but: I also do not think the sharp advantage of a copula is to avoid specifying the correlation matrix, or even to avoid the "curse of dimensionality" of needing to specifying even a large number of pairwise correlations.

Instead, I perceive the advantage of copulas is when the marginal distributions are non-normal and/or the dependence "structure" is non-linear; i.e., when pairwise (Pearson) correlations are inadequate perhaps especially in the tail.

So, if i want to generate several (a vector) of correlated, say, random beta (or whatever non-normal) variables, I can generate random multivariate normal variables (yes informed by a correlation matrix) and then map (inverse transform) them to their (marginal) beta distributions. This is easy and shows the first advantage: because the correlation (copula) function is separated from the marginal distributions, we are free to specify any marginal distributions on the way to our multivariate distribution. This is candidly all I've ever done in code ... as I haven't used the non Gaussian copulas .... again, this is using a correlation matrix (and Gaussian copula) to inform correlated variables that are marginally non-normal; e.g., multiple random beta/whatever variables that are informed by our familiar correlation matrix.

But we can pull off the shelf any number of other copula functions that are non-normal, including copulas with heavy tail dependence. In this case, we are seeking to avoid the correlation matrix because we want something more accurate that a linear relationship, and we can get these non-linear relationships in the form of a convenient functions. (true, in practice, we often assume all pairwise correlations in the matrix are the same so that we can conveniently reduce the matrix to a simple function, but I'm not sure big data fears the correlation matrix. I think the more relevant issue is that linear dependence is not realistic such that we might prefer coupla correlation functions because they reflect heavier tail dependence. And they happen to be conveniently expressed in a function.). So, I think that is the second advantage: going beyond linear correlations by retrieving a more exotic dependency, often to reflect greater likelihood of extreme (tail) joint outcomes than would be anticipated by linear correlation.

Then of course, we can combine both advantages: non-normal marginal distributions that are jointly dependent in a non-linear way. If you ask me to pull that off without copulas, I think i will be stuck!! I think that's really the advantage .... I hope that's helpful!

 

[ChristofferLoov]

Thanks David! To see if I got this right.

So going back to the example of UL, if we think of the individual ULs as distributions. All ULs may have different distributions and by mapping them to normals we can correlate them more accurately. This may however not be necessary when we only are looking at specific ULs for each loan, as in Schroek. In those cases we can rely on the pairwise correlations we have measured.

But if we want a more thorough view of the different possible outcomes we have to simulate. This means random draws from the distributions. In those cases we can’t rely on the ”special case” of the measured pairwise correlation but instead have to know the whole correlation structure for the distributions, therefore we use copulas.

As for the other advantage, by using some copula functions (but not the Gaussian as I understand it), which have high tail dependence we can allow the dependence to go up in extreme cases (the tail of the distribution). This can be useful since correlation tends to increase in, for instance, recessions.

Does this sound somewhat correct or am I trying to fit you explanation into an example that doesn’t work?

 

[David Harper CFA FRM]

Hi @ChristofferLoov I do agree with the gist of your second and third paragraphs above; i.e., if we want a more thorough view of the different possible outcomes, we probably need to simulate; and copulas allows us to model high tail dependence.

The analytical unexpected loss (UL) in Schroeck (and throughout the FRM where its history owes to the original author Ong https://amzn.to/2YhHby9) is actually a standard deviation: it is the standard deviation around the expected future value of the credit asset. Actually, this UL makes no assumption about the underlying distribution. However, as we know about risk, while it helps to know the variance/sigma, when the distribution is non-normal (presumably the case in credit risk), we also want to know the skew and especially the kurtosis (and something about the shape of the extreme tail).

As only a single standard deviation, 1.0σ, UL is often misunderstood to itself equal economic capital (EC). But for most distributions 1.0σ is associated with low confidence; e.g., if the distribution is normal, then 1.0σ is only =NORM.S.DIST(1.0, true) = 84.1% 1-tailed confidence. Our realistic (i.e., tail seeking) confidence levels are therefore a multiple of the analytical UL. Hence, EC[@ realistically high confidence level] = capital_multiplier * UL. So I would say, with respect to the analytical (i.e., formulaic) UL that we use is a convenient function but, to the extent we are interested in anything more (e.g., fitting the tail of the unknown non-normal distribution, modeling non-linear tail dependencies, even estimation of the capital multipliers), we are decidedly encouraged to employ (Monte carlo) simulation and this includes, but does not require, the use of copulas. I hope that's helpful!

 

[ChristofferLoov]

Hi David,

Thank you! Things are much clearer now. As I suspected, the UL may not have been a good example.

I think my original question came down to why we use copulas instead of just relying on pairwise correlation. But you answered that with the thoroughness argument.

Once again thank you! This has been very helpful!