Impact of insurance for operational risk: Is it worthwhile to insure or be insured for severe losses?

Under the Basel II standards, the Operational Risk (OpRisk) advanced measurement approach allows a provision for reduction of capital as a result of insurance mitigation of up to 20%. This paper studies different insurance policies in the context of capital reduction for a range of extreme loss models and insurance policy scenarios in a multi-period, multiple risk setting. A Loss Distributional Approach (LDA) for modeling of the annual loss process, involving homogeneous compound Poisson processes for the annual losses, with heavy-tailed severity models comprised of α-stable severities is considered. There has been little analysis of such models to date and it is believed insurance models will play more of a role in OpRisk mitigation and capital reduction in future. The first question of interest is when would it be equitable for a bank or financial institution to purchase insurance for heavy-tailed OpRisk losses under different insurance policy scenarios? The second question pertains to Solvency II and addresses quantification of insurer capital for such operational risk scenarios. Considering fundamental insurance policies available, in several two risk scenarios, we can provide both analytic results and extensive simulation studies of insurance mitigation for important basic policies, the intention being to address questions related to VaR reduction under Basel II, SCR under Solvency II and fair insurance premiums in OpRisk for different extreme loss scenarios. In the process we provide closed-form solutions for the distribution of loss processes and claims processes in an LDA structure as well as closed-form analytic solutions for the Expected Shortfall, SCR and MCR under Basel II and Solvency II. We also provide closed-form analytic solutions for the annual loss distribution of multiple risks including insurance mitigation.     

Modeling of an insurance system and its large deviations analysis

We model an insurance system consisting of one insurance company and one reinsurance company as a stochastic process in R². The claim sizes {X_i} are an iid sequence with light tails. The interarrival times {τ_i} between claims are also iid and exponentially distributed. There is a fixed premium rate c₁ that the customers pay; c<c₁ of this rate goes to the reinsurance company. If a claim size is greater than R the reinsurance company pays for the claim. We study the bankruptcy of this system before it is able to handle N number of claims. It is assumed that each company has initial reserves that grow linearly in N and that the reinsurance company has a larger reserve than the insurance company. If c and c₁ are chosen appropriately, the probability of bankruptcy decays exponentially in N. We use large deviations (LD) analysis to compute the exponential decay rate and approximate the bankruptcy probability. We find that the LD analysis of the system decouples: the LD decay rate γ of the system is the minimum of the LD decay rates of the companies when they are considered independently and separately. An analytical and numerical study of γ as a function of (c,R) is carried out.     

Discrete -convex extremal distributions: Theory and applications

Given a nondegenerate moment space with s fixed moments, explicit formulas for the discrete s-convex extremal distribution have been derived for s=1,2,3 (see [M. Denuit, Cl. Lefèvre, Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences, Insurance Math. Econom. 20 (1997) 197–214]). If s=4, only the maximal distribution is known (see [M. Denuit, Cl. Lefèvre, M. Mesfioui, On s-convex stochastic extrema for arithmetic risks, Insurance Math. Econom. 25 (1999) 143–155]). This work goes beyond this limitation and proposes a method for deriving explicit expressions for general nonnegative integer s. In particular, we derive explicitly the discrete 4-convex minimal distribution. For illustration, we show how this theory allows one to bound the probability of extinction in a Galton–Watson branching process. The results are also applied to derive bounds for the probability of ruin in the compound binomial and Poisson insurance risk models.     

Risk aggregation in non-life insurance: Standard models vs. internal models

Standard models for capital requirements restrict the correlation between risk factors to the linear measure and disregard undertaking-specific parameters. We consider an alternative framework for risk aggregation in non-life insurance using vine copulas that allow non-linear dependence and are estimated with undertaking-specific parameters. We empirically compare our alternative risk model with three regulatory standard models (Korean risk-based capital, Solvency II, Swiss Solvency Test) and show that the standard models lead to more than 50% higher capital requirements on average. Half of the overestimation results from the uniform parameter selection imposed by regulations and the other half comes from the linear correlation assumption. The differences might distort competition when both standard models and internal risk models are used in a single market.     

Term structure of discount rates for firms in the insurance industry

There is strong evidence in the literature for the hypothesis that interest rates and the market risk premium are not constant during the business cycle. The beta risk of firms in the insurance industry is also time-varying. The major implication of these results is that discount rates for risky cash flows are time varying and must obey a term structure similar to the term structure of interest rates. The purpose of this paper is to estimate discount rates for cash flows with different time horizons for the U.S. insurance industry and for different insurance sectors. We find that the term structure cost of capital takes on different shapes depending on the business cycle. It is therefore meaningful for insurers to evaluate risky projects by selecting a discount rate most appropriate for the nature and the time horizon of each project.     

The Log–Lindley distribution as an alternative to the beta regression model with applications in insurance

In this paper a new probability density function with bounded domain is presented. The new distribution arises from the generalized Lindley distribution proposed by Zakerzadeh and Dolati (2010). This new distribution that depends on two parameters can be considered as an alternative to the classical beta distribution. It presents the advantage of not including any special function in its formulation. After studying its most important properties, some useful results regarding insurance and inventory management applications are obtained. In particular, in insurance, we suggest a special class of distorted premium principles based on this distribution and we compare it with the well-known power dual premium principle. Since the mean of the new distribution can be normalized to give a simple parameter, this new model is appropriate to be used as a regression model when the response is bounded, being therefore an alternative to the beta regression model recently proposed in the statistical literature.     

What are the actual costs of cyber risk events?

Cyber risks are high on the business agenda of every company, but they are difficult to assess due to the absence of reliable data and thorough analyses. This paper is the first to consider a broad range of cyber risk events and actual cost data. For this purpose, we identify cyber losses from an operational risk database and analyze these with methods from statistics and actuarial science. We use the peaks-over-threshold method from extreme value theory to identify “cyber risks of daily life” and “extreme cyber risks”. Human behavior is the main source of cyber risk and cyber risks are very different compared with other risk categories. Our models can be used to yield consistent risk estimates, depending on country, industry, size, and other variables. The findings of the paper are also useful for practitioners, policymakers and regulators in improving the understanding of this new type of risk.     

Supply chain coordination with insurance contract

We propose an insurance contract under which the supplier shares the risk of overstock and understock with the retailer, improving the efficiency of the supply chain with a newsvendor-type product. We first show that the insurance contract could coordinate the supply chain, and obtain bargaining solution in the supply chain model. Then we investigate the effects of agents’ risk aversion on the supply chain model and acquire the Pareto-optimal solution through the mean–variance approach. After that, we compare the insurance contract with the revenue sharing contract, focusing particularly on their differences. Finally, extensive numerical studies are conducted, and managerial implications are proposed.     

影响我国人寿保险保费收入的因素实证分析

本文对影响我国人寿保险需求的因素进行了分析,并建立了非线性回归模型;同时,采用数学软件MATLAB对此模型进行了分析、改进、检验以及预测.     

Optimal time to change premiums

The claim arrival process to an insurance company is modeled by a compound Poisson process whose intensity and/or jump size distribution changes at an unobservable time with a known distribution. It is in the insurance company’s interest to detect the change time as soon as possible in order to re-evaluate a new fair value for premiums to keep its profit level the same. This is equivalent to a problem in which the intensity and the jump size change at the same time but the intensity changes to a random variable with a know distribution. This problem becomes an optimal stopping problem for a Markovian sufficient statistic. Here, a special case of this problem is solved, in which the rate of the arrivals moves up to one of two possible values, and the Markovian sufficient statistic is two-dimensional. This work was partially supported by the US Army Pantheon Project and National Science Foundation under grant DMS-0604491.