Nanyang Business School Forum on Risk Management and Insurance
Risk aggregation in non-life insurance: Standard models vs. internal models
The regulatory standard models require aggregating all possible risks from the assets and liabilities to protect the insurer against simultaneous losses in a stressed situation. The distributions of such possible risks vary, especially between different risk factors in the underwriting portfolio as well as the asset portfolio, including significant tail risks. However, the regulatory standard models (e.g. SII and the Korean Risk-based Capital) require insurers to aggregate them under the linear dependence assumption with predefined correlation parameters and do not allow undertakings to replace the correlation parameters by undertaking-specific parameters. In addition, the standard models either calibrate the parameters from different undertakings (SII and the K-RBC) or require a certain distributional assumption on asset and underwriting risks (Swiss Solvency Test). However, the parameters predefined by the regulations do not fully reflect the empirical data for an individual undertaking and distributional properties vary between different assets and underwriting risks in different datasets. It might be inappropriate to apply homogeneous parameters into datasets with heterogeneous properties and to aggregate different risks with the same statistical tools. All these limitations motivate us to study alternatives to the regulatory standard models.
To overcome such limitations, insurers have the opportunity to construct and use internal risk models. The approval of an internal model by the regulator for calculating capital requirements can give a positive signal to a company’s investors and analysts in that the company can optimize its economic capital. However, establishing an internal risk model requires many resources and is thus not affordable for small and mid-sized insurance companies, which might lead to significant competition distortions and consolidation in the insurance business. Developing appropriate internal risk models and quantifying potential deviations to regulatory standard models are thus not only of interest for academics in the field of actuarial science, but also highly relevant for insurance managers, regulators and public policymakers.
In this sense, this paper is comparing regulatory standard models with internal risk models and proposing an improved methodology to aggregate the risks from assets and liabilities. Using
two empirical datasets from Korean and German markets, we construct internal models by considering two specific elements that are not taken into account in the current regulations: one is undertaking-specific risk profile and the other is data-driven correlation matrix. Unlike the literature, the study builds a comprehensive two-step aggregation approach consisting of the base-level aggregation and the top-level aggregation using the regular vine model, which is the up-to-date high-dimensional dependence model. We apply the constructed models to estimate the risk capitals and compare them with the estimates derived based on three distinct regulatory models: Korean Risk-based Capital (RBC), Solvency II and Swiss Solvency Test (SST). In order to prove the superiority of our model, we apply the model further into the diversification effect and the asset allocation strategies and compare them with the outcomes from three regulatory models.
The key findings from our study highlight the following;
Firstly, the regular vine model turns out to be the best fit model for both asset and underwriting portfolios at the base-level, meaning that high-dimensional risk aggregation can be effectively carried out by this pair-wise dependence modeling. The top-level aggregation is conducted most appropriately under the independence assumption.
Secondly, three regulatory models over-estimate the required capital by 35% on average and 54% as the maximum in the Korean RBC case. Specifically, we find that the average size of the overestimation results from two elements, data-driven dependence structure and undertaking-specific risk profile, which address 18 percentage points and 17 percentage points respectively out of 35%.
Thirdly, we identify that the Korean RBC model needs more improvements than other two regulatory models in that the Korean RBC does not count the dependence structure of the market risk module possibly leading to no diversification benefit and does not appropriately categorize the underwriting risk module.
Lastly, our study provides the evidence that the standard models over-estimate the required capital with different asset allocation strategies and the best fit model gives the most diversification benefit. We test robustness of our findings with other risk measures such as ruin probability and expected policyholder deficit.
Our findings provide several economic implications for insurance practitioners and regulators. One of the important implications is that market competition can be distorted if standard models
and internal models are used in one market. This is especially a disadvantage for small and medium size companies that do not have the resources to build internal models. The misestimation of the required capital under the standard models can lead to several challenges for insurance companies. The overestimation requires insurers to hold more capital, thereby taking away an investment opportunity for an insurer, who diversifies the investment basket in the current low interest rate era. On the other hand, a possible underestimation of the required capital can potentially increase the ruin probability of an insurer due to insufficient capital size. Thus, more appropriate design for the regulation considering undertaking-specific parameters can help insurers optimize their capital allocation. Besides, the interaction between risk factors needs to be more sophisticatedly designed to estimate the risk capital because the interaction might result in simultaneous effect from the tails on the estimation. The two-step aggregation approach developed in this study can be utilized in an actuarial pricing scheme, especially when aggregating a range of risk factors in a risk pool.
To sum up, the contribution of this study to the academic literature and insurance practice is twofold. Firstly, we develop an integrated framework for the economic capital calculation that combines asset and insurance portfolios with undertaking-specific parameters and that considers all possible multivariate dependence models in the copula field. This allows us to identify the internal risk model that best fits the data using an up-to-date copula methodology considering potential non-linear dependency. Secondly, we empirically compare the estimated risk model with regulatory standard models and document significant differences between the two approaches. Showing these massive differences yields important policy implications for insurance managers and regulators, since they might create an uneven playing field and disadvantage especially for small and medium sized insurers.
The complete paper is available for download https://www.ivw.unisg.ch/wp-content/uploads/2018/08/WP215.pdf