The skewness risk premium in equilibrium and stock return predictability

In this study, we investigate the skewness risk premium in the financial market under a general equilibrium setting. Extending the long-run risks (LRR) model proposed by Bansal and Yaron (J Financ 59:1481–1509, 2004) by introducing a stochastic jump intensity for jumps in the LRR factor and the variance of consumption growth rate, we provide an explicit representation for the skewness risk premium, as well as the volatility risk premium, in equilibrium. On the basis of the representation for the skewness risk premium, we propose a possible reason for the empirical facts of time-varying and negative risk-neutral skewness. Moreover, we also provide an equity risk premium representation of a linear factor pricing model with the variance and skewness risk premiums. The empirical results imply that the skewness risk premium, as well as the variance risk premium, has superior predictive power for future aggregate stock market index returns, which are consistent with the theoretical implication derived by our model. Compared with the variance risk premium, the results show that the skewness risk premium plays an independent and essential role for predicting the market index returns.     

Reinsurance premium principles based on weighted loss functions

ABSTRACT

In this paper, we propose new reinsurance premium principles that minimize the expected weighted loss functions and balance the trade-off between the reinsurer's shortfall risk and the insurer's risk exposure in a reinsurance contract. Random weighting factors are introduced in the weighted loss functions so that weighting factors are based on the underlying insurance risks. The resulting reinsurance premiums depend on both the loss covered by the reinsurer and the loss retained by the insurer. The proposed premiums provide new ways for pricing reinsurance contracts and controlling the risks of both the reinsurer and the insurer. As applications of the proposed principles, the modified expectile reinsurance principle and the modified quantile reinsurance principle are introduced and discussed in details. The properties of the new reinsurance premium principles are investigated. Finally, the comparisons between the new reinsurance premium principles and the classical expectile principle, the classical quantile principle, and the risk-adjusted principle are provided.     

Disruption of a Managed Competition Environment by Low-Ball Premium Bids

Abstract

This paper reviews the movement among multiple health plan options between 1994 and 1998 for Minnesota state employees whose work site was located in the Minneapolis/St. Paul metropolitan area. During this period the employer contribution was based on the lowest family premium bid from a qualified plan in the county of the employee’s work site. In 1995 the largest individual practice association model HMO in the state, Medica, reduced its state premium by 25%, becoming the lowest-priced option. This resulted in massive transfers of enrollees between plans. The point of this study was to estimate the risk changes that resulted from these movements between plan options. We obtained enrollment data by age and gender from Blue Cross Blue Shield of Minnesota (Blue Cross) and applied age/gender risk weight factors derived from actuarial rate tables to the Blue Cross cells. Annual changes in risk weights by 10-20% were common in a number of Blue Cross subpopulations, and in one case, by more than 50%.

The Blue Cross POS plan experienced increases in risk and went into a death spiral, while a second Blue Cross plan with a more restrictive provider network started with low risk, but experienced increases in risk when the Medica plan was withdrawn. Similar demographic data were not available from other plans offered by the state and claim costs were confidential, so the results pertain only to Blue Cross risks. The question is raised as to whether managed competition can work without some means of adjusting premium rates to the expected cost level of the enrollees of a particular health plan. All carriers seemed reluctant to guarantee premium rates after the 1994-1998 experience, and the state soon became self-insured.     

VAR and CTE Criteria for Optimal Quota-Share and Stop-Loss Reinsurance

Abstract

It is well known that reinsurance can be an effective risk management tool for an insurer to minimize its exposure to risk. In this paper we provide further analysis on two optimal reinsurance models recently proposed by Cai and Tan. These models have several appealing features including (1) practicality in that the models could be of interest to insurers and reinsurers, (2) simplicity in that optimal solutions can be derived in many cases, and (3) integration between banks and insurance companies in that the models exploit explicitly some of the popular risk measures such as value-at-risk and conditional tail expectation. The objective of the paper is to study and analyze the optimal reinsurance designs associated with two of the most common reinsurance contracts: the quota share and the stop loss. Furthermore, as many as 17 reinsurance premium principles are investigated. This paper also highlights the critical role of the reinsurance premium principles in the sense that, depending on the chosen principles, optimal quota-share and stop-loss reinsurance may or may not exist. For some cases we formally establish the sufficient and necessary (or just sufficient) conditions for the existence of the nontrivial optimal reinsurance. Numerical examples are presented to illustrate our results.     

Unconventional monetary policy announcements and risk aversion: evidence from the U.S. and European equity markets

ABSTRACT

This paper examines the role of unconventional monetary policy announcements on risk aversion – as proxied by the variance premium – by using panel data analysis. The objective of this empirical analysis is to investigate the risk-taking channel of monetary policy for the major European and U.S. equity markets by studying the impact that the announcements of an unconventional monetary policy has on market uncertainty and risk perception. By measuring the difference between risk-neutral and realised and conditional variance, we estimate the variance premium, which captures the impact that pricing concerns have on the prices of options. The empirical analysis indicates that easing monetary policies can significantly reduce the variance premium. In addition, we examine the risk premium structure across markets to determine the potential differences in investors’ risk aversion.     

Pricing a Heterogeneous Portfolio Based on a Demand Function

Abstract

Consider a portfolio containing number of risk classes. Each class has its own demand function, which determines the number of insureds in this class as a function of the premium. The insurer determines the premiums based on the number of insureds in each class. The “market” reacts by updating the number of the policyholders, then the insurer updates the premium, and so on. We show that this process has an equilibrium point, and then we characterize this point.     

Budget-constrained optimal reinsurance design under coherent risk measures

ABSTRACT

Reinsurance is a versatile risk management strategy commonly employed by insurers to optimize their risk profile. In this paper, we study an optimal reinsurance design problem minimizing a general law-invariant coherent risk measure of the net risk exposure of a generic insurer, in conjunction with a general law-invariant comonotonic additive convex reinsurance premium principle and a premium budget constraint. Due to its intrinsic generality, this contract design problem encompasses a wide body of optimal reinsurance models commonly encountered in practice. A three-step solution scheme is presented. Firstly, the objective and constraint functions are exhibited in the so-called Kusuoka's integral representations. Secondly, the mini-max theorem for infinite dimensional spaces is applied to interchange the infimum on the space of indemnities and the supremum on the space of probability measures. Thirdly, the recently developed Neyman–Pearson methodology due to Lo (2017a) is adopted to solve the resulting infimum problem. Analytic and transparent expressions for the optimal reinsurance policy are provided, followed by illustrative examples.     

Credibilistic risk aversion

In the probabilistic risk aversion approach, risks are presumed as random variables with known probability distributions. However, in some practical cases, for example, due to the absence of historical data, the inherent uncertain characteristic of risks or different subject judgements from the decision-makers, risks may be hard or not appropriate to be estimated with probability distributions. Therefore, the traditional probabilistic risk aversion theory is ineffective. Thus, in order to deal with these cases, we suggest measuring these kinds of risks as fuzzy variables, and accordingly to present an alternative risk aversion approach by employing credibility theory. In the present paper, first, the definition of credibilistic risk premium proposed by Georgescu and Kinnunen [Fuzzy Inf. Eng., 2013, 5, 399–416] is revised by taking the initial wealth into consideration, and then a general method to compute the credibilistic risk premium is provided. Secondly, regarding the risks represented with the commonly used LR fuzzy intervals, a simple calculation formula of the local credibilistic risk premium is put forward. Finally, in a global sense, several equivalent propositions for comparative risk aversion under the credibility measurement are provided. Illustrated examples are presented to show the applicability of the theoretical findings.     

Equity and Credibility

We build on previous work concerned with measuring equity and consider the problem of using observed claim data or other information to calculate premiums which maximize equity. When these optimal premiums are used, we show that gathering more information or refining the risk classification always increases equity. We study the case for which the premium is constrained to be an affine function of the claim data and obtain results analogous to classical credibility theory, including the inhomogeneous and homogeneous cases of the Bu¨hlmann-Straub model. We derive formulas for the credibility weights in certain cases.     

Bounds for Ratios of Posterior Expectations: Applications in the Collective Risk Model

This paper considers the collective risk model for the insurance claims process. We will adopt a Bayesian point of view, where uncertainty concerning the specification of the prior distribution is a common question. The robust Bayesian approach uses a class of prior distributions which model uncertainty about the prior, instead of a single distribution. Relatively little research has dealt with robustness with respect to ratios of posterior expectations as occurs with the Esscher and Variance premium principles. Appropriate techniques are developed in this paper to solve this problem using the k -contamination class in the collective risk model.     

Cybersecurity insurance and risk-sharing

In today’s interconnected digital world, cybersecurity risks and resulting breaches are a fundamental concern to organizations and public policy setters. Accounting firms, as well as other firms providing risk advisory services, are concerned about their clients’ potential and actual breaches. Organizations cannot, however, eliminate all cybersecurity risks so as to achieve 100% security. Furthermore, at some point additional cybersecurity measures become more costly than the benefits from the incremental security. Thus, those responsible for preventing cybersecurity breaches within their organizations, as well as those providing risk advisory services to those organizations, need to think in terms of the cost-benefit aspects of cybersecurity investments. Besides investing in activities that prevent or mitigate the negative effects of cybersecurity breaches, organizations can invest in cybersecurity insurance as means of transferring some of the cybersecurity risks associated with potential future breaches.This paper provides a model for selecting the optimal set of cybersecurity insurance policies by a firm, given a finite number of policies being offered by one or more insurance companies. The optimal set of policies for the firm determined by this selection model can (and often does) contain at least three areas of possible losses not covered by the selected policies (called the Non-Coverage areas in this paper). By considering sets of insurance policies with three or more Non-Coverage areas, we show that a firm is often better able to address the frequently cited problems of high deductibles and low ceilings common in today’s cybersecurity insurance marketplace. Our selection model facilitates improved risk-sharing among cybersecurity insurance purchasers and sellers. As such, our model provides a basis for a more efficient cybersecurity insurance marketplace than currently exists. Our model is developed from the perspective of a firm purchasing the insurance policies (or the risk advisors guiding the firm) and assumes the firm’s objective in purchasing cybersecurity insurance is to minimize the sum of the costs of the premiums associated with the cybersecurity insurance policies selected and the sum of the expected losses not covered by the insurance policies.     

Cybersecurity Disclosure by the Companies on the S&P/TSX 60 Index

Cybersecurity has become a topic of great interest since 2010. Accounting issues surrounding cybersecurity governance, management, and disclosure have gained attention from accounting standard setters, large accounting firms, and professional associations, but only a limited number of studies have looked at cybersecurity disclosure. In this study, we examine whether the content of cybersecurity disclosures of Canadian firms comprising the S&P/TSX 60 index is aligned with best practices—that is, financial regulators' guidelines in that matter. A content analysis was performed of documents issued between January 2017 and mid‐2018, consisting of recent annual information forms (AIFs), annual and quarterly management's discussion and analysis (MD&As), proxy circulars, material change reports, and news releases. To assess the nature and extent of cybersecurity disclosure, we developed a scoring grid featuring 40 items based on financial regulators' guidelines. Results show that cybersecurity disclosure levels are low. Companies vary widely in the amount of detail they provide, and the information is often not company‐specific. The variations among industrial sectors involve the categories related to cybersecurity risk, cybersecurity risk mitigation, and other items. Most of the companies provided cybersecurity disclosures in the annual MD&A, and several reiterated some disclosure items in the AIF and proxy circular. The results of this study highlight some areas where cybersecurity disclosures have evolved and others where they could be improved. They suggest that some firms strive to avoid boilerplate language and be more company‐specific. The findings also suggest that financial regulators could issue more stringent requirements.     

Bayesian Premium Rating with Latent Structure

We propose a fully Bayesian approach to non-life risk premium rating, based on hierarchical models with latent variables for both claim frequency and claim size. Inference is based on the joint posterior distribution and is performed by Markov Chain Monte Carlo. Rather than plug-in point estimates of all unknown parameters, we take into account all sources of uncertainty simultaneously when the model is used to predict claims and estimate risk premiums. Several models are fitted to both a simulated dataset and a small portfolio regarding theft from cars. We show that interaction among latent variables can improve predictions significantly. We also investigate when interaction is not necessary. We compare our results with those obtained under a standard generalized linear model and show through numerical simulation that geographically located and spatially interacting latent variables can successfully compensate for missing covariates. However, when applied to the real portfolio data, the proposed models are not better than standard models due to the lack of spatial structure in the data.     

A Model for Coronary Heart Disease and Stroke with Applications to Critical Illness Insurance Underwriting II: Applications

Abstract

In Part I we constructed a model for the development of coronary heart disease (CHD) or stroke that either incorporates, or includes pathways through, the major risk factors of interest when underwriting for critical illness (CI) insurance. In Part II we extend this model to include other critical illnesses, for example, cancers and kidney failure, and describe some applications of the model. In particular, we discuss CI premium ratings for applicants with combinations of some or all of high body mass index, smoking, high blood pressure, high cholesterol, and diabetes. We also consider the possible effect on CI premium ratings of genetic conditions that increase the likelihood of high blood pressure, high cholesterol, diabetes, CHD event, or stroke.     

The Bankruptcy Cost of the Life Insurance Industry Under Regulatory Forbearance

Abstract

In this study the Taiwan Insurance Guaranty Fund (TIGF) is introduced to investigate the ex ante assessment insurance guaranty scheme. We study the bankruptcy cost when a financially troubled life insurer is taken over by TIGF. The pricing formula of the fair premium of TIGF incorporating the regulatory forbearance is derived. The embedded Parisian option due to regulatory forbearance on fair premiums is investigated. The numerical results show that leverage ratio, asset volatility, grace period, and intervention criterion influence the default costs. Asset volatility has a significant effect on the default option, while leverage ratio is shown to aggravate the negative influence from the volatility of risky asset. Furthermore, the numerical analysis concludes that the premium for the insurance guaranty fund is risk sensitive and that a risk-based premium scheme could be implemented, hence, to ease the moral hazard.     

A credibility theory for automobile bonus systems

Summary

This paper deals with the problem of designing experience rating systems of the bonus type, commonly used in automobile insurance. On the basis of a simple model the mean squared deviation between a policy's expected claim amount and its premium in the nth insurance period as n→∞, is taken as a measure of the efficiency of a bonus system. It is shown that to any set of bonus rules (which determines the bonus class transitions of the policies), there is an optimal premium scale, which coincides with the one proposed by Pesonen in 1963. Thus the problem of choosing an efficient bonus system reduces to choosing efficient bonus rules. Examples are given of comparison between different bonus rules. In one example the present Norwegian bonus system is compared to alternative systems. Comments are made on earlier papers on bonus systems. The credibility theoretic foundation is laid in a separate section.     

Public companies' cybersecurity risk disclosures

Though cybersecurity risks are significant and could materially affect business operations and the integrity of financial reporting, there is limited empirical research on the cybersecurity risk disclosure trends and practices of public companies. In this study, we conduct a longitudinal study of the content and linguistic characteristics of public companies' cybersecurity risk disclosure practices as well as factors that may drive disclosure trends. The results show that the two most commonly disclosed cybersecurity risks are risks of service/operation disruption and risks of data breach. Item 1A of the 10-K Report is the most commonly used disclosure location, but some companies also use Items 1 and 7 to disclose regulation risks and cyber incidents, respectively. The length of cybersecurity risk disclosures increases linearly during the period of our study. This increase is associated with the issuance of SEC guidance (2011 and 2018), industry, overall cybersecurity risks in the general environment, company size, and prior cybersecurity breach incidents. Disclosures have also become more difficult to read in general. They are more difficult to read as firm size increases and are easier to read as the proportion of intangible assets increases or after an executive change. Firms have increased their usage of litigious words in their disclosures. Bigger firms, on average, tend to use less litigious language, but companies in industries with high business information technology intensity (e.g., consumer services, software and services, and banking) tend to use more litigious language than other companies.     

Cybersecurity breaches and investors’ interest in the firm as an investment

Cybersecurity breaches pose a significant risk to firms. To combat these risks, many firms engage in strategic cybersecurity risk management initiatives. While these efforts may reduce the likelihood of a cybersecurity breach, they do not eliminate the risk of a breach. In the event of a cybersecurity breach, firms may issue an apology to investors. This study uses an experiment to examine whether a firm indicates cybersecurity risk management is a strategic initiative and whether a post-cybersecurity breach apology by the CEO impacts nonprofessional investors’ investment interest in the firm. Results show that, in response to a cybersecurity breach, the presence of a CEO apology positively impacts investors’ investment impression and their perceptions of CEO affective and CEO cognitive trust. We find that investors’ investment interest is lowest for a firm that previously indicates cybersecurity risk management is a strategic initiative and where the CEO does not issue an apology. The CEO apology, however, does not significantly impact investment amount, a secondary measure of investor interest. Results from this study have implications for managers, investors, and regulators.     

Equilibrium Pricing of General Insurance Policies

Abstract

A model is developed for determining the price of general insurance policies in a competitive, noncooperative market. This model extends previous single-optimizer pricing models by supposing that each participant chooses an optimal pricing strategy. Specifically, prices are determined by finding a Nash equilibrium of an N-player differential game. In the game, a demand law describes the relationship between policy sales and premium, and each insurer aims to maximize its (expected) utility of wealth at the end of the planning horizon. Two features of the model are investigated in detail: the effect of limited total demand for policies, and the uncertainty in the calculation of the breakeven (or cost price) of an insurance policy.

It is found that if the demand for policies is unlimited, then the equilibrium pricing strategy is identical for all insurers, and it can be found analytically for particular model parameterizations. However, if the demand for policies is limited, then, for entrants to a new line of business, there are additional asymmetric Nash equilibria with insurers alternating between maximal and minimal selling. Consequently it is proposed that the actuarial cycle is a result of price competition, limited demand, and entry of new insurers into the market. If the breakeven premium is highly volatile, then the symmetric equilibrium premium loading tends to a constant, and it is suggested that this will dampen the oscillatory pricing of new entrants.     

Changes in corporate cybersecurity risk disclosures after SEC comment letters

Gao et al. (2020) examined the content and linguistic characteristics of public companies' cybersecurity risk disclosure practices as well as factors that may drive disclosure trends. In this paper, we extend Gao et al. (2020) by exploring SEC comment letter practices related to cybersecurity risk disclosures and investigating how SEC comment letters lead to changes in companies’ cybersecurity risk disclosures. Coinciding with newly issued cybersecurity guidelines, SEC comment letters related to cybersecurity disclosure deficiencies spiked in 2011. On average, it takes about 26 days for a registrant to respond to a comment letter and only 10 percent of registrants respond within the recommended 10-day period. Most comment letters (75 percent) are resolved within one round of communication. Multiple rounds of communication are often required when deficiencies surround disclosure of a cyber breach. Though 81 percent of registrants respond to comment letters related to cybersecurity breaches by claiming that there was no need for disclosure as the breaches were not material, the SEC will likely reject that claim and require the registrant to provide the required detail. We find evidence that the SEC uses comment letters to signal that the staff wish to see an explicit statement in the registrant’s cybersecurity risk disclosures on whether or not the firm suffered security breaches during a reporting period. The SEC scrutinizes cybersecurity risk disclosures to verify they are sufficient subsequent to a published security breach. Firms change their disclosure behavior one year after receiving a comment letter. Specifically, the length of cybersecurity risk disclosures increases, specificity increases, and readability and clarity improve one year after a registrant receives a comment letter that points to deficiencies in the firm’s cybersecurity risk disclosures.