# Beginner's resources on copulas and impact of correlation on loan defaults?

I'm hoping this question is not too banal or off-topic for this forum. Could you please help me understand / point me towards some resources on the impact of correlation on loan defaults?

## First question

I have seen some material that, as correlation goes up, the expected loss of the portfolio remains the same, but the tails become thicker. Does this always hold true or only in some cases? E.g.

+----------------+----------------+-----------------+
|      Item      | No correlation | 20% correlation |
+----------------+----------------+-----------------+
| Expected loss  | 5%             | 5%              |
| 95% percentile | 10%            | 15%             |
+----------------+----------------+-----------------+


## Material

Can you think of any reading / material that could help me understand this? Not necessarily a mathematically rigorous textbook - even just something to understand the basic intuition.

I have found some slides here, but I'll admit I haven't exactly understood everything.

## How to model loan correlation

Let's say I have a portfolio of 10,000 loans and I have calculated a probability of default for each loan. Calculating the expected loss when the loans are not correlated is straightforward. But what when they are correlated? Is the approach something like (and apologies if it's a very banal question):

• You need a distribution for the probability of default of each loan, not just a PD value
• You run Monte Carlo simulations, generating correlated default data (as in the Matlab example below)
• You then calculate expected loss etc on the basis of this simulation

## Code examples

Finally, can you think of some code examples to recommend - ideally in Python? But even in another language, if the material is clear enough to help me understand the key concepts.

I have found this package for Python but I am none the wiser.

This Matlab documentation seems a bit clearer: it starts by generating two uncorrelated random samples, then it correlates them, and compares the difference

Thanks!

• I don't think there's a way to understand correlations without understanding the math involved. Take a look at this book, for example. Definitely more math than you would like, but I see no way around it, sorry. Jul 3 at 13:05
• Can you at least help me understand if it is true that, when correlation increases, the expected loss remains the same, but the tails become thicker? Or this happens only in certain cases? A high-level answer on this shouldn't require complex maths. Thanks! Also, it has been my experience that a) often the most mathematically-able individuals struggle to explain key concepts in a simple way and that b) there are many concepts for which at least the basic intuition can be explained without resorting to detailed maths. Jul 3 at 13:20
• When people start coding without understanding the math (or the product), 2008 ensues. Jul 3 at 13:59
• When correlation increases, both the tendency to default together and the tendency to survive together increases. The first lengthens the tail of the loss distribution making big losses more likely. The second makes the probability of zero and small losses bigger. As a result the expected loss does not change.
– Dom
Jul 6 at 8:43
• I have built a library that does valuation of credit correlation products such as CDS tranches. Here is a link to a notebook. You should be able to extract loss distributions although I do not have a notebook for that currently. github.com/domokane/FinancePy/blob/master/notebooks/products/…
– Dom
Jul 6 at 8:45

At the risk of arming you to create the next quant-apocalypse...

1. The statement that the expected loss does not depend on correlation is typically the result of modelling a portfolio as a sum of individual exposures: X+Y+..., and then using: E[X+Y+...]=E[X]+E[y]+.... This does NOT generalize to statements like "expectations don't depend on correlation", it matters what you are taking the expectation of.

2. When correlation increases do the tails become 'thicker'? Well... there are lots of ways of modelling portfolio losses, and what is meant by 'correlation' is context-dependent. However, I assume that you are thinking of correlation somewhat intuitively as a parameter that, when increased, results in a higher probability of multiple loan defaults (default correlation).

Then intuitively very large losses are more probable and in some sense the tail of the portfolio loss distribution becomes 'thicker', but you have to be very careful with this intuition. Introducing correlation impacts the entire portfolio loss distribution, and you can easily find that while, for example, the 99% loss increases with correlation, lower percentile losses may decrease...

3. Models: the link you attached was a generic copula intro. I suspect that you may find that one of the many many many books/papers that are more specifically about modelling loan/bond portfolio losses would be more helpful for you, most of which would have a section on copula methods. Search something like 'credit risk modelling', 'loan portfolio modelling'...

I find all of them somewhat helpful and somewhat awful at the same time, you have to look around a bit to find one that presents things in a way that makes sense to you. I happen to have a copy of: An introduction to credit risk modelling, by Bluhm Overbeck Wagner which I found explained a few things in a helpfully simple way.

I don't have a good python/matlab link I'm afraid...

Here is a classic 2008 explanation of credit correlation without using any math formulas: • I was looking for something halfway between "correlation can cause more of your loans to default" (thank you very much, I already knew that" and "here's a 500-page book full of advanced maths". Not only is this apparently not possible, but one also gets downvoted for having the audacity to ask a simple question. No wonder that insights.dice.com/2019/04/18/stack-overflow-many-jerks Jul 3 at 16:14