JB Crozet and Colum D'Auria report on the latest thinking on correlations between lines of business and the lessons learnt from the World Trade Center and Hurricane Katrina

The World Trade Center attacks and Hurricane Katrina have both transformed the way the reinsurance industry thinks about risks and their interactions during extreme events. Who would have thought previously that a single event could trigger record losses in both the aviation and property markets? Or the energy and property markets? These major catastrophes have connected lines of business which were previously seen as being independent from one another.

Although these "clashes" between otherwise independent risks are low-probability events, they have a significant impact on the day-to-day management of reinsurance companies. They can directly affect the solvency margin, the credit rating and the underwriting decisions of a reinsurer. In short, these events change the business framework.

A new corollary to Murphy's Law could be, "If things can go wrong, they will, and all at the same time". This raises some new questions for reinsurers, specifically:

- How do these connections impact the way they run their business? and

- What methods are being introduced to capture and quantify correlations between lines of businesses in extreme events?

More importantly, before investing time, effort and money in new modelling tools, it is important to understand whether these will be a valuable enhancement or a spurious complication.

Value at risk

The reason why the correlation of risk phenomenon can have such a significant impact is that the insurance and reinsurance industry devotes a lot of attention to value at risk or VaR measures (such as the 1-in-100 or 1-in-250 year events) in its risk assessments.

More and more regulatory frameworks, such as the International Capital Adequacy Standards in the UK or the forthcoming Solvency II regime in Europe, rely on VaR to assess statutory capital requirements of reinsurers. Similarly, rating agencies use such measures for assessing catastrophe risks in their capital adequacy models. Reinsurers also increasingly rely on VaR-type measures in their technical pricing, when using risk-based approaches. For instance, a marginal pricing approach could fail to adequately reflect the interaction between the new business and the existing portfolio. Finally, the definition of a company's risk appetite, as well as its reinsurance purchasing, are often stated in VaR or probability of ruin terms.

VaR measures are extremely useful but they have the drawback of being very sensitive to the underlying assumptions on correlations.

Rethinking correlations

The traditional thinking on correlations is that the interaction between portfolios is captured by a matrix of "correlation coefficients". The correlation coefficient is a statistical measure of the average interaction between two lines of business:

- At one extreme, a positive correlation value of 100% means that two lines will have similar experience, adverse or favourable, at the same time;

- Conversely, a negative correlation value of -100% means that two lines will exhibit opposite experience over time; and

- A correlation value of 0% means that the two lines behave independently from each other.

While correlation coefficients are useful indicators of the average connection between lines of business, it does not capture how the relationship may vary according to the magnitude of the claims experience and, in particular, in extreme situations. For example, two lines like energy and property may be shown to have a very small correlation coefficient, because energy and property losses tend typically to be unrelated on average.

The events of 9/11 and Hurricane Katrina have both highlighted that this relationship is far from a constant one. Under stressed conditions, losses from different lines of business can interact in quite a distinct way. Reinsurers have therefore started to re-examine the way they model correlations, and to look for new approaches.

When considering correlations, it is useful to remember that statistics like correlation coefficients are only proxies for the real connections in the physical world. Their purpose is to infer relationships or dependencies from the observable data.

Where possible, these dependencies can be explicitly modelled. The most obvious example is in relation to catastrophe models, in which simulated events are run through a portfolio to assess the resulting loss. No one would imagine monitoring the aggregations of a property catastrophe book by assigning correlation coefficients between the different treaties. Unfortunately, however, both the number of events and the lines of business for which such an approach is practical are limited.

Nevertheless, this explicit approach to modelling dependencies is expanding as catastrophe models are continually increasing the number of lines of business they can cover. The experience of Hurricane Katrina has reminded us that the extrapolation of these models to rare events does not always capture all the aspects and subtleties of these occurrences.

Scenario testing

Scenario testing, also known as the use of realistic disaster scenarios, offers an intuitive alternative to the dependencies between lines. Interrogating portfolios with "what if" questions frequently gives us some valuable insights into how well-diversified our businesses really are.

It is important nevertheless to recognise that realistic disaster scenarios are most beneficial in qualitative terms, as they help make connections between risks affected by a specific event. Unfortunately, they offer no help when quantifying the likelihood of these events.

In the many instances when there is no physical model to capture dependencies explicitly, one can attempt to capture statistically the altered nature of the dependency between lines of business under such stressed conditions through copulas.

Copulas are statistical functions that have been increasingly used to model the association of distributions of insurance losses in extreme situations. They effectively allow variations in the relationship between one risk and another, according to the intensity of their loss experience. In other words, they enable us to model two lines of business that are only moderately correlated under normal circumstances, but highly positively correlated in extreme events.

Copulas are certainly an improvement on the traditional approach of assuming a static correlation factor across the whole distribution of outcomes. However, there is often insufficient information to parameterise them precisely and some judgement will be required to utilise them effectively.

At a time when many recognise the benefits of diversified portfolios and others seek to obtain this diversification, there is a lot of new thinking and progress being made in the modelling of interactions between lines. The modelling tools are becoming increasingly sophisticated, and recognise that relationships between lines of business do vary with the intensity of the loss experience.

It must not be forgotten, however, that low-probability scenarios, on which VaR measures rely heavily, are by definition rarely observable. As a result, there is a large amount of extrapolation in the models, either from the limited datasets or from the modellers' judgment. We must recognise this uncertainty and not be fooled by a false sense of accuracy.

- JB Crozet is principal consultant and Colum D'Auria is senior consultant in the actuarial and insurance management solutions practice at PricewaterhouseCoopers.