Risk pooling and solvency regulation: A policyholder's perspective

• Published date: 01 December 2022
• Author: Markus Huggenberger,Peter Albrecht
• Date: 01 December 2022
• DOI: http://doi.org/10.1111/jori.12392

Received: 21 July 2020

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Revised: 11 May 2022

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Accepted: 24 May 2022

DOI: 10.1111/jori.12392

ORIGINAL ARTICLE

Risk pooling and solvency regulation:

A policyholder's perspective

Markus Huggenberger|Peter Albrecht

University of Mannheim, Business

School, Mannheim, Germany

Correspondence

Markus Huggenberger, Institute of

Insurance Economics, University of St.

Gallen, Girtannerstrasse 6, 9010 St.

Gallen, Switzerland.

Email: markus.huggenberger@unisg.ch

Abstract

We investigate the benefits of risk pooling for the

policyholders of stock insurance companies under differ-

ent solvency standards. Using second‐degree stochastic

dominance, we document that the utility of risk‐averse

policyholders is increasing in the pool size if the equity

capital is proportional to the premiums written. To the

contrary, an increase in the pool size can reduce the

policyholders' utility if the equity capital is determined

using the Value‐at‐Risk (VaR). We show that pooling with

a larger number of risks is also beneficial for all risk‐averse

policyholders under a VaR‐based regulation if the pool

satisfies an excess tail risk restriction. Our analysis

provides new insights for the design of solvency standards

and reveals a potential disadvantage of risk‐based capital

requirements for policyholders.

KEYWORDS

excess wealth order, exchangeable risks, risk pooling,

solvency regulation, value‐at‐risk

1|INTRODUCTION

Risk reduction through pooling can be viewed as a defining characteristic of the insurance

mechanism from the insurer's perspective. For example, Houston (1964, p. 538) argues that

while individuals see insurance as a risk transfer, insurance companies consider it as a pooling

process that reduces risk by increasing the number of policies. Following this view, the benefits

J Risk Insur. 2022;89:907–950. wileyonlinelibrary.com/journal/JORI

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This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and

reproduction in any medium, provided the original work is properly cited.

© 2022 The Authors. Journal of Risk and Insurance published by Wiley Periodicals LLC on behalf of American Risk and Insurance

Association.

of larger risk pools are typically studied by quantifying the reduction of the insurer's risk as

measured by its default probability or the relative capital buffer (see, e.g., Cummins, 1974,1991).

However, if we take potential losses from a default into account, then the size of the insurance

risk pool can also affect the payoffs to policyholders and thus their overall utility from the

above‐mentioned risk transfer.

1

In this paper, we reinvestigate the benefits of risk pooling from the policyholders'

perspective under different solvency frameworks. We focus on the case of a stock insurer,

which is of special interest because the benefits of reducing the risk per policy have to be shared

between the policyholders and the owners of the company. This risk allocation can drive a

wedge between the occurrence of pooling benefits for the insurer's total position and the

policyholders' wealth. Given the limited liability of equity holders, the default risk that the

policyholders have to bear for a given amount of total risk depends on the equity capital that

the owners of the company provide. We assume that this equity contribution is exogenously

determined according to solvency rules and we consider two cases: minimum capital

requirements that are proportional to the total premiums written and a capital regulation that

is based on the Value‐at‐Risk (VaR). Capital requirements that are proportional to (net)

premiums are an important example for volume‐based systems such as the capital charges for

underwriting risk in the United States and the former European Solvency I framework. The

VaR‐based rule has become the main component of probabilistic solvency systems around the

world, for example, in the European Solvency II framework.

2

Our baseline analysis relies on the following main assumptions: For the risks being insured, we

only require homogeneity and finite expectations. Homogeneity is formalized by assuming that the

random losses are exchangeable, which includes independent and identically distributed losses as

a special case. The finiteness of expectations is necessary to evaluate the resulting wealth positions

within an expected utility framework. We apply a second‐degree stochastic dominance (SSD)

criterion to obtain utility comparisons that are consistent with the preferences of risk‐averse agents

across a wide range of decision models. Default losses for policyholders are modeled endogenously

by comparing the total claim amount to the level of the available reserves (equity capital and

premiums) following ideas developed by Merton (1974) and Doherty and Garven (1986).

Furthermore, we do not apply a specific pricing model but take the insurance premium as

exogenously given. Finally, we assume that the policyholders are offered full coverage and that the

total default loss from a given pool is shared equally among the policyholders.

These assumptions are sufficient to generate monotonically increasing benefits of risk

pooling on the pool level. More specifically, the riskiness of the average claim per policyholder

is nonincreasing in the pool size under the given assumptions, in line with the general notion

that diversification reduces risk. However, we demonstrate that the resulting effect on the

policyholders' utility depends on the form of capital regulation due to the asymmetric risk

sharing between policyholders and equity holders.

1

Several studies have argued that policyholders are highly sensitive to nonperformance risks of insurance contracts.

Compare the discussion in Froot (2007) and the literature on “probabilistic insurance”(see, e.g., Wakker et al., 1997;

Zimmer et al., 2018) for empirical results. This also applies to markets with guaranty funds whose protection is often

only incomplete and associated with additional transaction costs (see Cummins & Sommer, 1996, p. 1075 or Cummins

& Weiss, 2016, p. 130).

2

Holzmüller (2009) and Cummins and Phillips (2009) provide reviews of the solvency regulation in the United States

and in Europe. See Geneva Association (2016) for the results of a recent global survey. For empirical results on

insurance insolvencies, we refer to De Bandt and Overton (2022) and the references therein.

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Under a simple volume‐based solvency framework in which the equity capital is

proportional to the premiums, the pooling benefits for policyholders are consistent with the

overall risk reduction. In particular, we show that the policyholders' utility level is

nondecreasing in the pool size so that all risk‐averse policyholders at least weakly prefer

insurance in larger risk pools.

In contrast, the occurrence of a risk reduction on the pool level does not necessarily

translate into utility gains for policyholders if the amount of equity capital is determined using

the VaR. Although, by construction, a VaR‐based equity capital limits the probability of default,

the relationship between the pool size and the policyholders' utility level depends on the

distribution of the risks that are pooled. Varying the distributional assumption on the losses, we

illustrate that the policyholders' utility level can be (i) globally increasing, (ii) locally

decreasing, or even (iii) globally decreasing in the size of the risk pool. We then derive a

condition that is necessary and sufficient for nonnegative pooling benefits under a VaR‐based

regulation. In particular, our results relate a preference for larger risk pools to a decrease in the

excess tail risk of the average claim as measured by the difference between Average Value‐at‐

Risk (AVaR) and VaR. In addition, we provide sufficient conditions on the joint distribution of

individual risks, which imply that the excess tail risk condition for the average claim of the pool

is satisfied.

Finally, we investigate a case in which the policyholders also own an equity stake in the

insurance company. In this case, the effect of risk pooling on the policyholders' utility is always

nonnegative—independent of the form of the minimum capital requirements.

We then discuss several extensions of our baseline analysis: First, we take a variable

expense loading into account and confirm the intuition that cost benefits can reinforce risk‐

related pooling benefits or compensate pooling‐related utility losses resulting from increases in

excess tail risk. Moreover, we demonstrate that the benefits of risk pooling are robust to

introducing independent investment risk if the equity capital is proportional to the premiums.

To obtain a corresponding result under VaR‐based capital requirements, we have to impose an

additional shape restriction on the distribution of the investment return. We then study the

special case of independent risks, which allows us to relax our full coverage and equal loss‐

sharing assumptions and to derive sufficient conditions for utility gains from risk pooling with

more general contract types and with alternative sharing rules for the total default loss. Next,

we extend our analysis to heterogeneous risk pools and derive extensions of our results for risks

with differences in expected losses and different levels of dispersion. Finally, we show that a

risk‐based premium which reflects the default risk of the company can at least partly resolve

the adverse effects documented for VaR‐based capital requirements and distributions that do

not satisfy our excess tail risk condition.

Our analysis is related to the literature on the benefits of risk pooling and diversification.

As mentioned above, several authors have studied the relationship between the size of the

risk pool and the insurer's risk, often applying asymptotic arguments that build on the law of

large numbers or the central limit theorem (see, Cummins, 1974,1991;Houston,1964;Smith

&Kane,1994, Chap. 1, among others). However, the impact of risk pooling on the

policyholders' utility has so far mainly been investigated in the mutual insurance case. In

particular, Gatzert and Schmeiser (2012) and Albrecht and Huggenberger (2017) show that

policyholders benefit from risk pooling in this setting by exploiting that mutual insurance

companies attain a complete sharing of profits and losses independent of premiums or capital

reserves. Due to this insight, the analysis of the mutual insurance case is related to general

results on diversification benefits for risk‐averse decision makers (Rothschild & Stiglitz, 1971;

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