Optimal Risk Pools Under Changing Volatility and Heterogeneous Risk Classes

Optimal Risk Pools under Changing Volatility and Heterogeneous Risk Classes Florian Klein, Hato Schmeiser∗ July 5, 2018 Abstract We extend previous research into risk pooling by introducing a default probability driven premium loading as well as a relation between the premium level and demand through a convex price-demand function. Furthermore, we compare the profitability of a single risk pool for heterogeneous risk classes with multiple risk pools from the perspective of an insurer's shareholders using net present value cal- culus. In addition, we contrast multiple risk pools structured as a single legal entity with the case of multiple legal entities. To achieve the net present value maximizing default probability, the insurer adjusts the underlying equity capital. In this context, we demonstrate with our theoretical consid- erations and numerical examples that multiple risk pools with multiple legal entities are optimal if the equity capital must be decreased. An equity capital increase implies that multiple risk pools in a single legal entity are generally optimal. Moreover, a single risk pool for multiple risk classes rela- tively improves in relation to multiple risk pools with multiple legal entities whenever the standard deviation of the underlying claims increases. Keywords: Risk Pooling, Legal Entity, Default Probability, Price-Demand Function, Heterogeneous Risk Classes JEL classification: G22; G28; G32 1 Introduction Risk pooling is widely acknowledged as the core of insurance activity. Previous research analyzes a decreasing loading under constant default probability with an increasing number of insured (see, e.g., Cummins, 1991; Smith and Kane, 1994; Porat and Powers, 1999; Gatzert and Schmeiser, 2012)1 and the reduction of default probability when increasing the number of insured under a constant loading (see, e.g., Smith and Kane, 1994; Porat and Powers, 1999; Powers et al., 2003; Gatzert and Schmeiser, 2012).2 However, as we understand from empirical research into default probability and willingness to pay, an insurer's changing default probability consequently affects the willingness to pay as long as policyholders are aware of such a variation. Wakker et al.(1997), Zimmer et al.(2009), and Zimmer et al.(2018) ∗Florian Klein (fl[email protected]) and Hato Schmeiser ([email protected]) are from the Institute of Insurance Economics, University of St. Gallen, Tannenstrasse 19, CH-9000 St. Gallen. 1For further research into risk pooling see Albrecht and Huggenberger(2017). The authors analyze mutual insurance from policyholders' perspective for different preference functions. 2These findings hold true whenever the risk in the portfolio is not purely systematic. demonstrate with their empirical research the substantial decrease in the policyholders' willingness to pay under default probability (if transparent). Furthermore, even when policyholders are insensitive based on a small default probability and hence do not adapt their willingness to pay, the loading is affected since the expected indemnity payments decrease with a higher default probability, while the willingness to pay remains stable. We identify potential insensitivity concerning small default probabilities from Tversky and Kahneman(1979). The authors argue that individuals either ignore or overweight highly unlikely events. The assumption of a constant loading under a changing default probability, as considered in previous research, only holds when willingness to pay and expected indemnity payments vary by the same amount. In this context, we extend previous research and argue that the potential insensitivity and the substantial decrease in policyholders' willingness to pay under default probability (if transparent) must be considered when analyzing pooling effects. We introduce an insensitivity parameter Ψ, which describes the default probability that is not recognized by the policyholders, and follow the exponential regression of Klein and Schmeiser(2018), which is based on Zimmer et al.(2018), to explain the depen- dency between willingness to pay and recognized default probability. In addition, previous research does not underlie any relation between loading and number of risk units and hence a lower loading can be perfectly compensated by a larger risk pool and vice versa. However, in practice, a price-demand function underlies this relationship and thus should be considered. We extend previous research and underlie a convex price-demand function (see, e.g., Zimmer et al.(2018)), which leads to loading and demand being only imperfectly compensated. To overcome potential circularity induced by the default probability, we consider a capital adaptation by the insurer. Moreover, we analyze different pooling strategies and investigate risk classes. In this paper, we assume that a risk class is characterized by expected indemnity payments (see, e.g., Hoy, 1982; Crocker and Snow, 2013) as well as premiums (see, e.g., Harrington and Doerpinghaus, 1993) for a given distribution of the indemnity payments. According to substandard annuities, Gatzert et al.(2012) use the term risk class, which includes a range of risks concerning the underlying life expectancy, where the annuity prices are affected by the risk class. In our analysis, within each risk class, the risks are based on the costs being homogeneous. In this context, we focus on risk classes with varying expected indemnity payments as well as deviating optimal premiums and analyze whether it is optimal to pool heterogeneous risk classes to a single risk pool or create multiple risk pools. While a single risk pool is structured as a legal entity, multiple risk pools can form a single legal entity or multiple legal entities. The aim of this paper is to analyze the profitability of multiple risk pools with a single legal entity and multiple risk pools with multiple legal entities in relation to a single risk pool that includes multiple, heterogeneous risk classes from the perspective of an insurer's shareholders. Therefore, we consider the net present value optimality of different pooling strategies and derive general findings, which are sup- ported by proofs. Furthermore, we provide additional insights induced by net present value calculations under changing volatility and varying profitability of a risk class. To determine the net present value, we use an option pricing framework for normally distributed claims. In this regard, the following questions 2 arise: How does the policyholders' insensitivity to small default probabilities affect the net present value and which implications can be derived for the insurer? Under which conditions are the different pooling strategies optimal? What is the impact of a change in volatility, the variation of heterogeneity between risk classes, policyholders' insensitivity to small default probabilities, and a capital transfer between pools on the risk pool optimality? Since risk pooling is highly relevant for the insurance company, we analyze all these questions to develop a deep understanding of which factors are key drivers to build optimal risk pools. This paper is organized as follows. In section 2, we define our model framework. In this context, we first consider pooling effects and extend previous research into risk pooling by introducing a default probability driven loading as well as a relation between premium and demand through a convex price- demand function. Furthermore, we thematize heterogeneous risk classes and different pooling strategies to maximize the net present value. Moreover, we derive general insights concerning the optimal pooling strategy. We run numerical examples in section 3 and demonstrate how the optimality of different pooling strategies changes depending on volatility, profitability of risk classes, and policyholders' insensitivity to default probability. Section 4 focuses on the economic implications of our findings. A summary and conclusion are provided in section 5. 2 Model Framework 2.1 Extending Insights from Risk Pooling With respect to risk pooling, previous research focuses on two cases that are derived from the law of large numbers and lead to diversification.3 Concerning Case 1, with a larger number of risks (higher demand), the individual loading can decrease while the default probability remains constant (see, e.g., Cummins, 1991; Smith and Kane, 1994; Porat and Powers, 1999; Gatzert and Schmeiser, 2012). Case 2 denotes the combination where the individual loading is fixed and the default probability drops as a consequence of the larger number of risks (see, e.g., Smith and Kane, 1994; Porat and Powers, 1999; Powers et al., 2003; Gatzert and Schmeiser, 2012). While Gatzert and Schmeiser(2012) focus on a policyholder's perspective, in this paper, we analyze the perspective of an insurer's owners.4 Since the insurer's shareholders do not exhibit an incentive to reduce the policyholder's loading, we analyze the scenario where the default probability decreases with increasing demand. We assume that the demand n describes n risks and focus on a one-period approach. Moreover, we follow Cummins(1974, 1991) and determine the claims of the risks as independent and identically distributed based on a normal distribution.5 The claim size of risk i at t = 1 is determined by ci and the total amount of the stochastic claims C is defined as 3Powers(2006) argues that risk pooling is not necessarily beneficial for insurance companies in practice. For instance, the author demonstrates that the effect of the law of large numbers can be overcompensated

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