Are copulas passé for capital models?

Copulas are a popular tool in insurance capital modeling, but they are unwieldy, opaque, and don’t offer granularity. Are copulas on their way out for capital modeling? Maybe. Partially. And for good reasons.

Current approach

A typical approach to modeling property & casualty (P&C) insurance risk in capital modeling is to quantify uncertainty (e.g., lognormal parameters) for different lines of business and then execute many simulations – with each simulation varying the loss generated.

To reflect the relationship between lines of business, copulas/correlation matrices are introduced that relate the random variables together. The random variable for line of business X will have some relationship with the random variable for line of business Y, and the outcome is a somewhat correlated set of results that companies use to set capital loads.

However, very few have faith in the copulas/correlation matrices – and not just because they may have played a role in the 2008 financial crisis. Or at least few have faith in the exact numbers being used.

Hopefully, at a minimum, those in charge believe the values inside the matrix are reasonable representations, even if they can’t mathematically show based on data why those values are correct. After all, capital modeling is not an exact science. It’s an exercise in minimizing how wrong the quantitative measurement can be of an unknowable value.

Why copulas fall short

Ignoring the difficulty of quantifying the specific values inside the correlation matrix, there are other problems.

• The values don’t tell a story. It’s very difficult to go to an underwriter and mention there is X% correlation with another line of business and therefore they need to write less business for the good of the overall company’s capital position. It’s a story with a black box in the background and it doesn’t hold up well to disagreement, adjustment or probing.
• The matrices aren’t scalable. Everyone wants to go more granular in their risk analysis. Every new item to include creates a whole row of correlation values that need to be quantified and justified. If you want to get down to policy level, then good luck. Welcome to correlation matrices that are too big to reasonably use in your capital modeling engine.

So copulas/correlation matrices aren’t great. But are they the best we have? Maybe, but I don’t think so.

Beyond copulas: The risk factor approach

There is an approach that tells a better story, enhancing the socialization of the capital model. There is an approach that allows for scalability where going more granular or adding new lines of business is possible. The answer: use a combination of risk factors to influence the random draw.

For example, identify drivers of risk for a line of business. Perhaps it’s the nebulous underwriting cycle. Perhaps it’s changes in the regulatory environment. Really, it can be anything, as long as it can be supported by experts in the lines of business who would agree and say “yes, that does affect my simulated loss ratio for next year.”

Take that collection of risk factors and use those when simulating the random variable for a line of business. How do we relate different lines of business together without using copulas? If they have common risk factors, then each business line’s random draw will be impacted by the same risk factor. The result is a relationship between the lines of business in an explainable way.

Risk factors offer granularity down to the policy level

The risk factors share a similar problem with the copulas – they’re difficult/impossible to truly quantify. However, they do encourage feedback and discussion because they are described, not nebulous like correlation factors. That discussion is key for buy-in. Buy-in is necessary for socialization.

Using the risk factor approach we can now go as granular as we want, including to the policy level. Each policy will have its own distribution of losses, but the distribution itself is impacted by risk factors that are common throughout the line of business. You can simulate on a per policy basis while still having overall correlation throughout the book of business.

If the end analysis is that you know two lines of business are correlated but you can’t identify the risk factor that is causing the correlation, that’s not ideal, but it’s OK. It now puts on paper which part you feel confident about (the risk factors) and which parts are uncertain. That’s a much better story than a generic catchall correlation factor. I’d argue that it’s an invitation to dive deeper with the lines of business experts and try to understand what is causing the relationship.

Industry adoption

I realize this isn’t a new idea and we have seen some clients adopt this approach already. I tend to gravitate to approaches that are based in first principles and create a good story. I’ve never gravitated to copulas because of that preference. I think the industry, in its search of more granularity and socialization of the capital model, will move away from copulas and toward risk factor approaches.

If anyone truly loves copulas, then I have good news. The risk factors themselves will likely be related through copulas, but that’s a story for another time.