Abstract
This paper examines the impact of Solvency II on the attainability of target returns, the attainability of portfolio efficiency and the asset allocation of European insurers. I start with a brief introduction to the Solvency II Directive, focusing on the rules for calculating solvency capital requirements (SCR) according to the Solvency II standard formula. The subsequent numerical analysis includes several portfolio optimizations focusing on six relevant asset classes for the 1993–2017 time period. I derive optimal portfolios with respect to the Solvency II capital requirements, with respect to conventional risk measures, and I combine both optimization problems. My results show that the capital requirements according to Solvency II are not adequately calibrated. Nevertheless, due to a solid equity base, the majority of European insurers are still able to attain high target returns and mean-variance-efficiency. However, undercapitalized insurers are not able to hold risk-optimal allocations of equities, real estate and hedge funds any longer. In an environment of very low interest rates, these insurers may also face difficulties obtaining their target returns. To the best of my knowledge, this is the first paper to explicitly incorporate the solvency capital requirement as a numerical constraint into the insurers’ portfolio optimization problem. As a result, my approach first provides insights about the attainable target return and the asset weights as a direct function of insurers’ equity.
Zusammenfassung
Der vorliegende Artikel untersucht die Auswirkungen von Solvency II auf die Asset Allocation europäischer Versicherungsunternehmen, insbesondere im Hinblick auf die Erreichbarkeit von Zielrenditen und Portfolioeffizienz. Ich beginne mit einer kurzen Einführung in das Solvency II Framework mit Fokus auf das Marktrisikomodul der Solvency II Standardformel. In der darauffolgenden numerischen Analyse wird die Asset Allocation über die sechs relevantesten Assetklassen für den Zeitraum von 1993 bis 2017 optimiert. Die Portfolien werden hinsichtlich der Solvenzkapitalanforderungen nach Solvency II und hinsichtlich konventioneller Risikomaße optimiert. In einem weiteren Analyseschritt werden die Portfolien im Hinblick auf beide Zielgrößen simultan optimiert. Meine Ergebnisse zeigen, dass die Solvenzkapital-anforderungen nach Solvency II gegenüber konventionellen Risikomaßen fehlparametrisiert sind. Die Mehrzahl der europäischen Versicherer ist aufgrund hoher Eigenkapitalquoten dennoch in der Lage, hohe Zielrenditen und Portfolioeffizienz zu erreichen. Unterkapitalisierte Versicherer sind nach Solvency II hingegen nicht mehr in der Lage, risikooptimale Anteile an Aktien, Immobilien und Hedge Fonds zu halten. Bedingt durch das Niedrigzinsumfeld geraten unterkapitalisierte Versicherer zudem in Gefahr, ihre Zielrenditen nicht mehr zu erreichen. Nach meinem besten Wissen ist dies der erste Artikel, der die Solvenzkapitalanforderungen nach Solvency II explizit als Nebenbedingung in der Portfoliooptimierung berücksichtigt. Hierdurch zeigt sich erstmals die Sensitivität der erreichbaren Zielrendite und der Gewichte der verschiedenen Assetklassen in Abhängigkeit vom Eigenkapital der Versicherer.
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Notes
See ECB (2017).
According to an analysis by Assekurata (2016), for example, the average guaranteed interest rate on existing policies among German life insurance companies amounted to 2.97% in 2016.
Besides the effects on portfolio efficiency, a reallocation of insurers’ assets could lead to fundamental shifts in demand and pricing for several asset classes, as Fitch Ratings (2011) has already pointed out.
Note that the attainability of a certain target return is necessary to fulfill the interest rate guarantees on existing life insurance policies, as already pointed out.
The European Insurance and Occupational Pensions Authority (EIOPA) is part of a European System of Financial Supervisors that comprises three European Supervisory Authorities.
All data is obtained from Thomson Reuters Datastream.
The BofA Merrill Lynch (Code: MLEX-PEE) European corporate bond index only dates back to 1996. The index shows a very similar risk-return profile and correlation patterns. Therefore, my results are unlikely to be affected by the choice of this index.
In accordance with the EIOPA framework, the value at risk was calculated on an annual basis for the 99.5% level.
The Solvency II covariance matrix is calculated as the outer product of the regulatory correlation matrix (\(R_{\mathrm{reg}})\) and the column vector of capital requirements (\(\boldsymbol{SCR})\), both as shown in Table 1 (\(\Sigma_{\mathbf{reg}}=\boldsymbol{SCR}\otimes R_{\mathrm{reg}}\otimes \boldsymbol{SCR'}).\) The resulting matrix is not positive semi-definite, which may cause a discontinuity in the quadratic objective function (Eq. 11). I therefore apply the algorithm of Higham (2002) in order to obtain the nearest positive semi-definite matrix. Furthermore, there are circularity issues: Both the equity SCR and the interest rate SCR are a function of the portfolio weights themselves (i. e., a function of the solution vector of the optimization program). While the equity SCR accounts for diversification within the equity sub-module, the interest rate SCR is determined by the duration gap, which in turn depends on the weights of corporate bonds and government bonds. To overcome these issues, all N permissible combinations of hedge funds, stocks, corporate bonds and government bonds are enumerated up to the fourth decimal place. For any given target return, the original problem is now solved N times. Each of the N optimizations uses the corresponding preset asset weights as additional constraints (i. e., the weights of the four asset classes with circularity issues are held constant). Hence, the covariance matrix no longer exhibits circular references. Finally, the portfolio allocation with the lowest SCR of all the N optimization results is chosen as the global optimum for the respective target return.
The investment limits I use are particularly inspired by the German “Regulation on the Investment of Restricted Assets of Insurance Undertakings” (Investment Regulation; German: Anlageverordnung).
The lowest portfolio target return is determined by the asset class with the lowest expected return, i. e., money market. At the other extreme, the highest portfolio target return is achieved by sequentially increasing the weights of the assets with higher expected returns, until the individual investment limits are reached.
This corresponds to a value at risk of approximately 95%, assuming returns are normally, identically and independently distributed.
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Appendix
Appendix
Optimal Portfolio Allocations. a Optimal Allocations for SCR-optimized Portfolios, b Optimal Allocations for STD-optimized Portfolios. Notes: Fig. 4 shows the resulting asset allocation for both optimization programs as stated in Sect. 4.1
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Heinrich, M., Wurstbauer, D. The impact of risk-based regulation on European insurers’ investment strategy. ZVersWiss 107, 239–258 (2018). https://doi.org/10.1007/s12297-018-0403-8
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DOI: https://doi.org/10.1007/s12297-018-0403-8