RESEARCH RADAR
“Capital requirement modeling for market and non-life premium risk in a dynamic insurance portfolio” published in October 2023 in the Annals of Actuarial Science - Cambridge University Press by Nino Savelli (MIB Visiting Faculty and Catholic University of Sacred Heart in Milan) and Stefano Cotticelli (Sapienza, University of Rome)."For some time now, Solvency II requires that insurance companies calculate minimum capital requirements to face the risk of insolvency, either in accordance with the Standard Formula or using a full or partial Internal Model. An Internal Model must be based on a market-consistent valuation of assets and liabilities at a 1-year time span, where a real-world probabilistic structure is used for the first year of projection. In this paper, the major risks of a non-life insurance company are described, i.e. the non-life underwriting risk and market risk and their interactions, focusing in particular on the non-life premium risk, equity risk, and interest rate risk. This analysis is made using some well-known stochastic models in the financial actuarial literature and practical insurance business, i.e. the Collective Risk Model for non-life premium risk, the Geometric Brownian Motion for equity risk, and a real-world version of the G2++ Model for interest rate risk, where parameters are calibrated on current and real market data.
Finally, the modelling results are illustrated by a case study on either a single-line and a multi-line insurance company, in order to see how the risk drivers behave in both stand-alone and aggregate frameworks."
- Nino Savelli -
Conclusions
In this paper, the authors showed that as the cash flows produced by the insurance business are invested, they consequently create a risk, above the underwriting risk (here represented by the premium risk only). This highlights that non-life insurers not only face underwriting risk but also market risk. More specifically, they pointed out that premium risk, equity, and interest rate are all relevant risks for the non-life insurer. Consequently, the authors described an approach to modeling the distributions of the annual rate of return and aggregate total claim amount, in order to calculate the capital requirements for market and premium risk and to illustrate their combined effect. The authors produced a numerical analysis for a single-line and a multi-line insurance company in a multi-annual dynamic perspective, using current and available market data, in order to show a realistic and heterogeneous time-dependent non-life insurance context.In this paper, the authors also showed that capital requirements are clearly more demanding from a methodological point of view when an IM is applied than when using the SF. Moreover, notwithstanding that an IM must be approved by an insurance supervisory authority, the calibration is critical, because it influences the final result of the capital requirements. For this reason, supervisory authorities pay close attention to cases of so-called model change.
They explained the main differences between their IM and the SF according to their numerical analysis, where market and underwriting risks are examined in connection to each other. In particular for their single-line MTPL insurance company, the SF results in higher capital requirements than their IM (35.7% against 28.8% as percentages of the initial GPW, see Table 12). This is caused by three main reasons. Firstly, the authors are given by the higher volatility factor for premium risk (10% for MTPL in the SF), secondly by a more conservative approach regarding the expected profits (not counted as a mitigation of risk in the SF), and thirdly by the fixed multiplier in the SF, irrespective of the relation between the skewness and volatility underlying the Lognormal distribution assumption.
Compared to the single-line insurer, the multi-line insurance company (having the same GPW) has higher market risk for both the SF and their IM, because of larger investment resources (in particular, the initial claims reserve) given by the GTPL characteristic of an extremely high ratio of claims reserve to GPW (more than 400%).
Considering the premium risk, the multi-line insurance company has smaller capital requirements than the single-line insurer if the SF approach is adopted (26.4% against 31.1% as percentages of the initial GPW, see Tables 11 and 20). The significant reduction is mainly given by the diversification benefit among the different LoBs (due to the linear correlation matrix), which is absent for the single-line insurer. By contrast, for their IM, the multi-line capital requirement is higher than in the single-line, because of the high volatility and skewness of GTPL, which is not counterbalanced by either the limited values registered for MOD or the diversification benefit. Consequently, in their case study, the total capital requirement of the SF is lower than their IM when using the Gaussian copula for premium risk (32.5% against 35.7% as percentages of the initial GPW, see again Table 20), while for the single-line insurance company, it is the opposite (35.7% against 28.8% as percentages of the initial GPW, see again Table 12). In addition, in the case of Gumbel copula for the premium risk, which is distinguished by a higher upper tail dependence than the Gaussian copula, the multi-line capital requirement in their IM increases from 35.7% to 38.9%. The authors remind that their IM can only be accounted as partial because a full IM would obviously consider all the sources of risk (e.g. reserve risk, cat risk, counterparty default risk, operational risk, ). Moreover, the analysis was performed with simple assets expressed in the euro currency and issued by Euro Area governments, which means many important sources of market risk (i.e. the spread risk, liquidity risk, and default risk) were not considered. Hence, the authors obtained a market risk much smaller than premium risk, unlike in practice, where market risk is often larger than underwriting risk.
Authors
Nino Savelli, Full Professor of Risk Theory in the Faculty of Banking, Financial and Insurance SciencesCatholic University of Sacred Heart in Milan - Italy and Visting Professor at MIB Trieste School of Management in Risk Management and Insurance Techniques.
Stefano Cotticelli, Fully Qualified Actuary at Zurich Insurance Group - Ph.D. Candidate in Actuarial Sciences at Sapienza, University of Rome.
