From ruin theory to solvency in non-life insurance

We start from ruin theory considerations in the classical Cramér–Lundberg model. We modify these considerations step by step so that finally we arrive at today’s solvency assessments for non-life insurance companies. These modifications include discussions about time horizons, risk measures, financial returns, and valuation of insurance liabilities.     

Impact of Underwriting Cycles on the Solvency of an Insurance Company

Abstract

This paper studies the solvency of an insurance firm in the presence of underwriting cycles. A small or medium-size insurance company with a price-taker position in the market is considered. Its premium income is assumed to obey an autoregressive process with cycles. Specifically, the premium income for a specific calendar year is influenced by the market experience for the last couple years. Under this classical AR(2) dynamics governing the premium income, an explicit expression for the ultimate ruin probability is derived, using a martingale approach, in the lighttailed claims case. Furthermore, the logarithmic asymptotic behavior of the ultimate ruin probability as well as the typical path to ruin are investigated. Then a comparison is made with the classical case where the same company operates on a market without such cycles. Asymptotically, the presence of market cycles is shown to increase the risk for the company. Numerical illustrations are performed on Canadian motor insurance market data and support the theoretical analysis.     

New Risk-Based Capital Standards in the European Union: A Proposal Based on Empirical Data

In response to criticism concerning the current solvency system, the European Commission is developing new rules for insurance companies operating in the member states of the European Union (EU). Under this so-called Solvency II concept, an insurer is allowed to verify its solvency by using an internal risk management model previously approved by the regulatory authority. In this article we develop such an internal risk management approach for property-liability insurers that is based on dynamic financial analysis (DFA). The proposed concept uses a simulation technique and models the central risk factors from the investment and underwriting areas of an insurance company. On the basis of the data provided by a German insurer, the ruin probabilities under different scenarios and varying planning horizons are calculated.     

Some properties of the ruin function in the collective theory of risk

Abstract

It is well known that the chief aim of all theory of risk is to attain a sort of objective and somehow confirmed opinion of how and to which extent an insurance company ought to reinsure its risks in order that the probability of ruin by random fluctuations of the risk process shall become so small that it can be overlooked in practice.     

Authors’ Reply: Credit Standing and the Fair Value of Liabilities: A Critique - Discussion by Marsha Wallace

Abstract

As investment plays an increasingly important role in the insurance business, ruin analysis in the presence of stochastic interest (or stochastic return on investments) has become a key issue in modern risk theory, and the related results should be of interest to actuaries. Although the study of insurance risk models with stochastic interest has attracted a fair amount of attention in recent years, many significant ruin problems associated with these models remain to be investigated. In this paper we consider a risk process with stochastic interest in which the basic risk process is the classical risk process and the stochastic interest process (or the stochastic return-on-investmentgenerating process) is a compound Poisson process with positive drift. Within this framework, we first derive an integro-differential equation for the Gerber-Shiu expected discounted penalty function, and then obtain an exact solution to the equation. We also obtain closed-form expressions for the expected discounted penalty function in some special cases. Finally, we examine a lower bound for the ruin probability of the risk process.     

Biometric Solvency Risk for Portfolios of General Life Contracts. I. The Single-Life Multiple Decrement Case

Abstract

Solvency II splits life insurance risk into seven risk classes consisting of three biometric risks (mortality risk, longevity risk, and disability/morbidity risk) and four nonbiometric risks (lapse risk, expense risk, revision risk, and catastrophe risk). The best estimate liabilities for the biometric risks are valued with biometric life tables (mortality and disability tables), while those of the nonbiometric risks require alternative valuation methods. The present study is restricted to biometric risks encountered in traditional single-life insurance contracts with multiple causes of decrement. Based on the results of quantitative impact studies, process risk was deemed to be not significant enough to warrant an explicit calculation. It was therefore assumed to be implicitly included in the systematic/parameter risk, resulting in a less complex standard formula. For the purpose of internal models and improved risk management, it appears important to capture separately or simultaneously all risk components of biometric risks. Besides its being of interest for its own sake, this leads to a better understanding of the standard approach and its application extent. Based on a total balance sheet approach we express the liability risk solvency capital of an insurance portfolio as value-at-risk and conditional value-at-risk of the prospective liability risk understood as random present value of future cash flows at a given time. The proposed approach is then applied to determine the biometric solvency capital for a portfolio of general life contracts. Using the conditional mean and variance of a portfolio’s prospective liability risk and a gamma distribution approximation we obtain simple solvency capital formulas as well as corresponding solvency capital ratios. To account for the possibility of systematic/parameter risk, we propose either to shift the biometric life tables or to apply a stochastic biometric model, which allows for random biometric rates. A numerical illustration for a cohort of immediate life annuities in arrears reveals the importance of process risk in the assessment of longevity risk solvency capital.     

On the probability of ruin in the collective risk theory for insurance enterprises with oly negative risk sums

Abstract

1. The determination of the probability that an insurance company once in the future will be brought to ruin is a problem of great interest in insurance mathematics. If we know this probability, it does not only give us a possibility to estimate the stability of the insurance company, but we may also decide which precautions, in the form of f. ex. reinsurance and loading of the premiums, should be taken in order to make the probability of ruin so small that in practice no ruin is to be feared.     

Interplay of insurance and financial risks in a stochastic environment

Consider an insurer who makes risky investments and hence faces both insurance and financial risks. The insurance business is described by a discrete-time risk model modulated by a stochastic environment that poses systemic and systematic impacts on both the insurance and financial markets. This paper endeavors to quantitatively understand the interplay of the two risks in causing ruin of the insurer. Under the bivariate regular variation framework, we obtain an asymptotic formula to describe the impacts on the insurer's solvency of the two risks and of the stochastic environment.     

On the application of finance theory to the insurance firm

The primary argument set forth in this article is that the theory of finance can and should be rigorously applied to the study of the insurance firm. In order to illustrate this point, we turn our attention to the insurance solvency literature, where the implications of default risk for insurance company decision-making and regulatory policy are widely discussed but not nearly as widely understood. Rather than treat the probability of ruin as an exogenous constraint that is arbitrarily imposed by regulators, the approach taken here is to endogenize the probability of ruin with respect to a complex contracting process undertaken by a variety of self-interested claim holders. This treatment enables us to evaluate regulatory constraints such as minimum capital requirements within a rigorous theoretical framework. Our analysis suggests that even in an unregulated market, insurers would voluntarily limit their premium-capital ratios in an effort to economize on contracting costs. Furthermore, mutual insurers are likely,ceteris paribus, to employ less leverage than insurers organized as stock corporations.     

Rechnungslegung und Aufsicht von Versicherungsunternehmen — Zur Vereinbarkeit von IFRS und Solvency II

Current discussions of Insurance Accounting and supervisory regulation present some major challenges for insurance companies. The International Accounting Standards Board (IASB) started a project on Insurance Accounting to apply the principles of fair value to insurance liabilities. At the same time ‘Solvency II’ contains a fundamental and wide-ranging review of the insurance solvency regime in the light of adequate risk consideration. The paper discusses the aims and problems of both projects. The separate illustration presents the basis to identify the essential interdependences of ‘Insurance IFRS’ and ‘Solvency II’.The main problem is to create a unique valuation basis for Insurance Liabilities which makes allowance for relevant and reliable accounting rules as well as for solvency margins. On the basis of an actuarial approach an adequate model is shown. The construction of Fair Value contains the deviation of a Market Value Margin (MVM), which reflects the premium that a marketplace participant would demand for bearing the uncertainty inherent in the cash flows. For the purpose of solvency additional risk components must be integrated due to the fact that the Market Value Margin basically does not allow for all parts of volatility and uncertainty risk in insurance liabilities.     

Potential measures and expected present value of operating costs until ruin in renewal risk models with general interclaim times

In this paper, a Sparre Andersen risk process with arbitrary interclaim time distribution is considered. We analyze various ruin-related quantities in relation to the expected present value of total operating costs until ruin, which was first proposed by Cai et al. [(2009a). On the expectation of total discounted operating costs up to default and its applications. Advances in Applied Probability 41(2), 495–522] in the piecewise-deterministic compound Poisson risk model. The analysis in this paper is applicable to a wide range of quantities including (i) the insurer's expected total discounted utility until ruin; and (ii) the expected discounted aggregate claim amounts until ruin. On one hand, when claims belong to the class of combinations of exponentials, explicit results are obtained using the ruin theoretic approach of conditioning on the first drop via discounted densities (e.g. Willmot [(2007). On the discounted penalty function in the renewal risk model with general interclaim times. Insurance: Mathematics and Economics 41(1), 17–31]). On the other hand, without any distributional assumption on the claims, we also show that the expected present value of total operating costs until ruin can be expressed in terms of some potential measures, which are common tools in the literature of Lévy processes (e.g. Kyprianou [(2014). Fluctuations of L'evy processes with applications: introductory lectures, 2nd ed. Berlin Heidelberg: Springer-Verlag]). These potential measures are identified in terms of the discounted distributions of ascending and descending ladder heights. We shall demonstrate how the formulas resulting from the two seemingly different methods can be reconciled. The cases of (i) stationary renewal risk model and (ii) surplus-dependent premium are briefly discussed as well. Some interesting invariance properties in the former model are shown to hold true, extending a well-known ruin probability result in the literature. Numerical illustrations concerning the expected total discounted utility until ruin are also provided.     

Capital requirements and claims recovery: A new perspective on solvency regulation

Protection of creditors is a key objective of financial regulation. Where the protection needs are high, that is, in banking and insurance, regulatory solvency requirements are an instrument to prevent that creditors incur losses on their claims. The current regulatory requirements based on value at risk (V@R) and average value at risk (AV@R) limit the probability of default of financial institutions, but they fail to control the size of recovery on creditors' claims in the case of default. We resolve this failure by developing a novel risk measure, recovery V@R. Our conceptual approach is flexible and allows the construction of general recovery risk measures for various risk management purposes. We provide detailed case studies and applications. We show that recovery risk measures can be used for performance-based management of business divisions of firms and discuss how to calibrate recovery risk measures to historical regulatory standards. Finally, we analyze how recovery risk measures react to the joint distributions of assets and liabilities on firms' balance sheets and compare the corresponding capital requirements with the current regulatory benchmarks based on V@R and AV@R.     

Properties of a Risk Measure Derived from Ruin Theory

This paper studies a risk measure inherited from ruin theory and investigates some of its properties. Specifically, we consider a value-at-risk (VaR)-type risk measure defined as the smallest initial capital needed to ensure that the ultimate ruin probability is less than a given level. This VaR-type risk measure turns out to be equivalent to the VaR of the maximal deficit of the ruin process in infinite time. A related Tail-VaR-type risk measure is also discussed.     

Portfolio Optimization under Solvency Constraints: A Dynamical Approach

We develop portfolio optimization problems for a nonlife insurance company seeking to find the minimum capital required that simultaneously satisfies solvency and portfolio performance constraints. Motivated by standard insurance regulations, we consider solvency capital requirements based on three criteria: ruin probability, conditional Value-at-Risk, and expected policyholder deficit ratio. We propose a novel semiparametric formulation for each problem and explore the advantages of implementing this methodology over other potential approaches. When liabilities follow a Lognormal distribution, we provide sufficient conditions for convexity for each problem. Using different expected return on capital target levels, we construct efficient frontiers when portfolio assets are modeled with a special class of multivariate GARCH models. We find that the correlation between asset returns plays an important role in the behavior of the optimal capital required and the portfolio structure. The stability and out-of-sample performance of our optimal solutions are empirically tested with respect to both the solvency requirement and portfolio performance, through a double rolling window estimation exercise.     

Ruin probabilities in multivariate risk models with periodic common shock

We propose a multidimensional risk model where the common shock affecting all classes of insurance business is arriving according to a non-homogeneous periodic Poisson process. In this multivariate setting, we derive upper bounds of Lundberg-type for the probability that ruin occurs in all classes simultaneously using the martingale approach via piecewise deterministic Markov processes theory. These results are numerically illustrated in a bivariate risk model, where the beta-shape periodic claim intensity function is considered. Under the assumption of dependent heavy-tailed claims, asymptotic bounds for the finite-time ruin probabilities associated to three types of ruin in this multivariate framework are investigated.     

Economic crisis and determinants of solvency in the insurance sector: new evidence from Spain

This paper analyses the factors that determine the solvency of insurance companies operating in Spain. The selected time span, from 2008 to 2015, encompasses a period of economic instability characterised by record low interest rates and low or even negative economic growth. Using a dynamic panel data model, we conclude that actual solvency margins are positively related to profitability, underwriting risk and a mutual-type organisation but inversely related to size, reinsurance use, longer-tailed business and life insurance specialisation. We also find that less concentrated markets and the context of an economic crisis decrease solvency margins.     

Ruin probabilities and optimal capital allocation for heterogeneous life annuity portfolios

A large part of the actuarial literature is devoted to the derivation of ruin probabilities in various non-life insurance risk models. On the contrary, very few papers deal with ruin probabilities for life insurance portfolios. The difficulties arise from the dependence and non-stationarity of the annual payments made by the insurance company. This paper shows that the ruin probability in case of life annuity portfolios can be computed from algorithms derived by De Pril (1989) and Dhaene & Vandebroek (1995). Approximations for ruin probabilities are discussed. The present article complements the works of Frostig et al. (2003) who considered whole life, endowment, and temporary assurances, and of Denuit & Frostig (2008) who considered homogeneous life annuities portfolios. Here, heterogeneous portfolios (with respect to age and/or face amounts) are studied. Particular attention is paid to the capital allocation problem. The total amount of reserve is shared among the risk classes in order to minimize the ruin probability. It is then fair to charge a higher margin to the risk classes requiring more capital.     

Reversible Jump Markov Chain Monte Carlo Method for Parameter Reduction in Claims Reserving

Abstract

We present an application of the reversible jump Markov chain Monte Carlo (RJMCMC) method to the important problem of setting claims reserves in general insurance business for the outstanding loss liabilities. A measure of the uncertainty in these claims reserves estimates is also needed for solvency purposes. The RJMCMC method described in this paper represents an improvement over the manual processes often employed in practice. In particular, our RJMCMC method describes parameter reduction and tail factor estimation in the claims reserving process, and, moreover, it provides the full predictive distribution of the outstanding loss liabilities.     

Zur Exaktheit der Standardformel

The Solvency II standard formula permits the approximate computation of the solvency capital requirement for an insurance company. The solvency capital requirement is defined as the 99,5 % quantile of the distribution of the basic own funds of the insurance company taking into account all risks of the company. The standard formula decomposes the total risk into risk units and conducts the aggregation of the solvency capital requirements for the single risk units using their correlations. The present paper provides a condition on the joint distribution of all risk units under which the risk aggregation via the standard formula is exact.     

财务再保险与偿付能力最适边界和可解决域研究

偿付能力监管是现代保险监管方法的重要组成部分,更是衡量保险公司经营稳定与安全性的主要指标,如何改善偿付能力是保险业界必须讨论和研究的热点问题。再保险特别是财务再保险,由于其自身所具有的本质特性,将成为改善保险公司偿付能力的主要手段之一。本文基于再保险的角度去探索改善保险公司偿付能力的途径,从财务再保险的基本理论出发,介绍了财务再保险的定义、特征和分类,并结合保险公司偿付能力的有关知识分析了二者之间的关系。通过模型计算,得出了财务再保险与偿付能力最适边界和可解决域,这将极大的方便保险公司在购买财务再保险时的决策。