Correlation Structure

Integration and Risk Aggregation

ERM is a fully integrated model. ERM does not make any assumptions about a company’s exposure to different risk categories or about correlations between them; on the contrary, these characteristics arise logically from ERM analysis. ERM employs a bottom-up approach to risk aggregation—the only proper way to handle correlations between risks of a dynamic company in a changing environment. The top-down approach is based upon the presumption that the total corporate structure could be broken down to a number of high level business segments and the total risk could be derived from the correlation structure between the segments. While this method is easy to implement, it has some major deficiencies:

1. There is usually no statistically sound way to asses these high level correlations. As a result, companies have to rely on expert opinion to come up with the estimate for the correlations between segments. Expert opinions may be irrelevant in a dynamic market environment.
2. Business segments are not static; their composition may transform over time. While weights of the separate components within the segment change, correlations between the segments change as well and this, in turn, may lead to the risk miscalculation.

In contrast, ERM parameters come from current market data and the correlations are measured starting from the lowest level (such as individual positions or lines of business (LOBs)) and then expanding up to any desired level: for example, to wholly owned subsidiaries or sister companies.

Correlation Structure and Calibration

All ERM risk factors exhibit correlations. It is easy to observe that the yields of different maturities move in tandem, or that the equity returns of different sectors are usually highly correlated. Yet it is often the case that macroeconomic factors drive the premium rates as well as frequency and severity of the insured events, just like they affect the default probabilities in the credit models. For example, an increase in the interest rates will result in larger profits from invested premiums, which, in competitive markets, will apply downward pressure to the premium rates. Recessions lead to increased frequency of Workers’ Compensation claims and increased average age of the insured cars.

ERM risk factors and their correlation structure are estimated from the latest market data and both public and proprietary insurance data sources. The long-term horizon required by the nature of insurance business and the distinct differences between investment instruments and insurance liabilities present a number of unique problems for the proper calibration of the correlation structure. Among them:

• Stability. The random noise inherent to large correlation matrices will, in effect, get amplified over a long forecasting period and, if not filtered out, render the risk statistics unreliable. This is especially true if any optimization of a business structure is attempted while making use of a raw correlation matrix.
• Correlation between assets and liabilities. Large correlation matrices commonly employed on the investment side require long time series for estimation. The large required number of data points does not present a problem for financial time series which are generally available on a daily basis. On the other hand, the natural time unit for insurance data is one year, and it would be practically impossible to collect a sufficient history of premium and loss data without incurring complicated seasonal effects. As a result, there is no feasible way to represent both investment assets and insurance liabilities in the form of a “grand” correlation matrix.

In order to assure stability and the statistical significance of the estimates, Seabury ERM applies various calibration techniques, such as Principal Component Analysis (PCA) and regression analysis. ERM employs the latest techniques of random matrix theory developed in physical science [24] in order to discriminate between significant components and noise. PCA drastically reduces the “effective” dimensionality, which allows establishing the dependence of the insurance risk factors on the macroeconomic environment through regression analysis (Appendix B:).