The compound Poisson risk model with multiple thresholds

In this paper we consider a multi-threshold compound Poisson risk model. A piecewise integro-differential equation is derived for the Gerber-Shiu discounted penalty function. We then provide a recursive approach to obtain general solutions to the integro-differential equation and its generalizations. Finally, we use the probability of ruin to illustrate the applicability of the approach.     

Lévy risk model with two-sided jumps and a barrier dividend strategy

In this paper, we consider a general Lévy risk model with two-sided jumps and a constant dividend barrier. We connect the ruin problem of the ex-dividend risk process with the first passage problem of the Lévy process reflected at its running maximum. We prove that if the positive jumps of the risk model form a compound Poisson process and the remaining part is a spectrally negative Lévy process with unbounded variation, the Laplace transform (as a function of the initial surplus) of the upward entrance time of the reflected (at the running infimum) Lévy process exhibits the smooth pasting property at the reflecting barrier. When the surplus process is described by a double exponential jump diffusion in the absence of dividend payment, we derive some explicit expressions for the Laplace transform of the ruin time, the distribution of the deficit at ruin, and the total expected discounted dividends. Numerical experiments concerning the optimal barrier strategy are performed and new empirical findings are presented.     

The influence of individual claims on the chain-ladder estimates: Analysis and diagnostic tool

The chain-ladder method is a widely used technique to forecast the reserves that have to be kept regarding claims that are known to exist, but for which the actual size is unknown at the time the reserves have to be set. In practice it can be easily seen that even one outlier can lead to a huge over- or underestimation of the overall reserve when using the chain-ladder method. This indicates that individual claims can be very influential when determining the chain-ladder estimates. In this paper the effect of contamination is mathematically analyzed by calculating influence functions in the generalized linear model framework corresponding to the chain-ladder method. It is proven that the influence functions are unbounded, confirming the sensitivity of the chain-ladder method to outliers. A robust alternative is introduced to estimate the generalized linear model parameters in a more outlier resistant way. Finally, based on the influence functions and the robust estimators, a diagnostic tool is presented highlighting the influence of every individual claim on the classical chain-ladder estimates. With this tool it is possible to detect immediately which claims have an abnormally positive or negative influence on the reserve estimates. Further examination of these influential points is then advisable. A study of artificial and real run-off triangles shows the good performance of the robust chain-ladder method and the diagnostic tool.     

The Gerber-Shiu discounted penalty function for a compound binomial risk model with by-claims

A recursive formula of the Gerber-Shiu discounted penalty function for a compound binomial risk model with by-claims is obtained. In the discount-free case, an explicit formula is given. Utilizing such an explicit expression, we derive some useful insurance quantities, including the ruin probability, the density of the deficit at ruin, the joint density of the surplus immediately before ruin and the deficit at ruin, and the density of the claim causing ruin.     

Optimal reinsurance with general risk measures

This paper studies the optimal reinsurance problem when risk is measured by a general risk measure. Necessary and sufficient optimality conditions are given for a wide family of risk measures, including deviation measures, expectation bounded risk measures and coherent measures of risk. The optimality conditions are used to verify whether the classical reinsurance contracts (quota-share, stop-loss) are optimal essentially, regardless of the risk measure used. The paper ends by particularizing the findings, so as to study in detail two deviation measures and the conditional value at risk.     

Pricing catastrophe swaps: A contingent claims approach

In this paper, we comprehensively analyze the catastrophe (cat) swap, a financial instrument which has attracted little scholarly attention to date. We begin with a discussion of the typical contract design, the current state of the market, as well as major areas of application. Subsequently, a two-stage contingent claims pricing approach is proposed, which distinguishes between the main risk drivers ex-ante as well as during the loss reestimation phase and additionally incorporates counterparty default risk. Catastrophe occurrence is modeled as a doubly stochastic Poisson process (Cox process) with mean-reverting Ornstein-Uhlenbeck intensity. In addition, we fit various parametric distributions to normalized historical loss data for hurricanes and earthquakes in the US and find the heavy-tailed Burr distribution to be the most adequate representation for loss severities. Applying our pricing model to market quotes for hurricane and earthquake contracts, we derive implied Poisson intensities which are subsequently condensed into a common factor for each peril by means of exploratory factor analysis. Further examining the resulting factor scores, we show that a first order autoregressive process provides a good fit. Hence, its continuous-time limit, the Ornstein-Uhlenbeck process should be well suited to represent the dynamics of the Poisson intensity in a cat swap pricing model.     

On a compound Poisson risk model with delayed claims and random incomes

The main focus of this paper is to analyze the Gerber-Shiu penalty function of a compound Poisson risk model with delayed claims and random incomes. It is assumed that every main claim will produce a by-claim which can be delayed with a certain probability. We derive the integral equation satisfied by the Gerber-Shiu penalty function. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the Gerber-Shiu penalty function is derived. Finally, when the premium sizes have rational Laplace transforms, we also obtain the Laplace transform of the Gerber-Shiu penalty function.     

Conditional probability calculations of compound binomial risk model in Markov-chain environment

肖临在Cossette(2004)的基础上改进并建立了马氏链环境中复合二项风险模型,针对Cossette(2004)中所提出的几个命题在肖临的模型框架下给出了详细的证明,得出了有限时间的条件非破产概率递推公式及赔付额的条件概率函数的递推公式.     

Predicting automobile claims bodily injury severity with sequential ordered logit models

Despite the large cost of bodily injury (BI) claims in motor insurance, relatively little research has been done in this area. Many companies estimate (and therefore reserve) bodily injury compensation directly from initial medical reports. This practice may underestimate the final cost, because the severity is often assessed during the recovery period. Since the evaluation of this severity is often only qualitative, in this paper we apply an ordered multiple choice model at different moments in the life of a claim reported to an insurance company. We assume that the information available to the insurer does not flow continuously, because it is obtained at different stages. Using a real data set, we show that the application of sequential ordered logit models leads to a significant improvement in the prediction of the BI severity level, compared to the subjective classification that is used in practice. We also show that these results could improve the insurer’s reserves notably.     

On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals

In this paper, we consider an insurance risk model governed by a Markovian arrival claim process and by phase-type distributed claim amounts, which also allows for claim sizes to be correlated with the inter-claim times. A defective renewal equation of matrix form is derived for the Gerber-Shiu discounted penalty function and solved using matrix analytic methods. The use of the busy period distribution for the canonical fluid flow model is a key factor in our analysis, allowing us to obtain an explicit form of the Gerber-Shiu discounted penalty function avoiding thus the use of Lundberg’s fundamental equation roots. As a special case, we derive the triple Laplace transform of the time to ruin, surplus prior to ruin, and deficit at ruin in explicit form, further obtaining the discounted joint and marginal moments of the surplus prior to ruin and the deficit at ruin.     

Asset management and surplus distribution strategies in life insurance: An examination with respect to risk pricing and risk measurement

In this paper, we investigate the impact of different asset management and surplus distribution strategies in life insurance on risk-neutral pricing and shortfall risk. In general, these feedback mechanisms affect the contract’s payoff and hence directly influence pricing and risk measurement. To isolate the effect of such strategies on shortfall risk, we calibrate contract parameters so that the compared contracts have the same market value and same default-value-to-liability ratio. This way, the fair valuation method is extended since, in addition to the contract’s market value, the default put option value is fixed. We then compare shortfall probability and expected shortfall and show the substantial impact of different management mechanisms acting on the asset and liability side.     

Statistical analysis of model risk concerning temperature residuals and its impact on pricing weather derivatives

In this paper we model the daily average temperature via an extended version of the standard Ornstein Uhlenbeck process driven by a Levy noise with seasonally adjusted asymmetric ARCH process for volatility. More precisely, we model the disturbances with the Normal inverse Gaussian (NIG) and Variance gamma (VG) distribution. Besides modelling the residuals we also compare the prices of January 2010 out of the money call and put options for two of the Slovenian largest cities Ljubljana and Maribor under normally distributed disturbances and NIG and VG distributed disturbances. The results of our numerical analysis demonstrate that the normal model fails to capture adequately tail risk, and consequently significantly misprices out of the money options. On the other hand prices obtained using NIG and VG distributed disturbances fit well to the results obtained by bootstrapping the residuals. Thus one should take extreme care in choosing the appropriate statistical model.     

The compound binomial model with randomly paying dividends to shareholders and policyholders

Considering surplus of a joint stock insurance company based on compound binomial model, set up thresholds a₁, a₂ for shareholders and policyholders respectively. When surplus is no less than the thresholds, the company randomly pays dividends to shareholders and policyholders with probabilities q₁, q₂ respectively. For this model, we have derived the recursive formulas of both the expected discount penalty function and ruin probability, and the distribution function of the deficit at ruin.     

On optimal investment in a reinsurance context with a point process market model

We study an insurance model where the risk can be controlled by reinsurance and investment in the financial market. We consider a finite planning horizon where the timing of the events, namely the arrivals of a claim and the change of the price of the underlying asset(s), corresponds to a Poisson point process. The objective is the maximization of the expected total utility and this leads to a nonstandard stochastic control problem with a possibly unbounded number of discrete random time points over the given finite planning horizon. Exploiting the contraction property of an appropriate dynamic programming operator, we obtain a value-iteration type algorithm to compute the optimal value and strategy and derive its speed of convergence. Following Schäl (2004) we consider also the specific case of exponential utility functions whereby negative values of the risk process are penalized, thus combining features of ruin minimization and utility maximization. For this case we are able to derive an explicit solution. Results of numerical computations are also reported.     

Assessing the cost of capital for longevity risk

The cost of capital is a key element of the embedded value methodology for the valuation of a life business. Further, under some solvency approaches (in particular, the Swiss Solvency Test and the developing Solvency 2 project) assessing the cost of capital constitutes a step in determining the required capital allocation.Whilst the cost of capital is usually meant as a reward for the risks encumbering a given life portfolio, in actuarial practice the relevant parameter has been traditionally chosen, at least to some extent, inconsistently with such risks. The adoption of market-consistent valuations has then been advocated to reach a common standard.A market-consistent value usually acknowledges a reward to shareholders’ capital as long as the market does, namely if the risk is systematic or undiversifiable. When dealing with a life annuity portfolio (or a pension plan), an important example of systematic risk is provided by the longevity risk, i.e. the risk of systematic deviations from the forecasted mortality trend. Hence, a market-consistent approach should provide appropriate valuation tools.In this paper we refer to a portfolio of immediate life annuities and we focus on longevity risk. Our purpose is to design a framework for a valuation of the portfolio which is market-consistent, and therefore based on a risk-neutral argument, while involving some of the basic items of a traditional valuation, viz best estimate future flows and allocated capital. This way, we try to reconcile the traditional with a market-consistent (or risk-neutral) approach. This allows us, in particular, to translate the results obtained under the risk-neutral approach in terms of a properly redefined embedded value.     

Gerber–Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy

This paper studies a Sparre Andersen model in which the inter-claim times are generalized Erlang(n) distributed. We assume that the premium rate is a step function depending on the current surplus level. A piecewise integro-differential equation for the Gerber–Shiu discounted penalty function is derived and solved. Finally, to illustrate the solution procedure, explicit expression for the Laplace transform of the time to ruin is given when the inter-claim times are generalized Erlang(2) distributed and the claim amounts are exponentially distributed.     

On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution

In this paper, we consider the Sparre Andersen risk model with an arbitrary interclaim time distribution and a fairly general class of distributions for the claim sizes. Via a two-step procedure which involves a combination of a probabilitic and an analytic argument, an explicit expression is derived for the Gerber-Shiu discounted penalty function, subject to some restrictions on its form. A special case of Sparre Andersen risk models is then further analyzed, whereby the claim sizes’ distribution is assumed to be a mixture of exponentials. Finally, a numerical example follows to determine the impact on various ruin related quantities of assuming a heavy-tail distribution for the interclaim times.     

On the expected discounted penalty function for the compound Poisson risk model with delayed claims

In this paper, we consider an extension to the compound Poisson risk model for which the occurrence of the claim may be delayed. Two kinds of dependent claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed with a certain probability. Both the expected discounted penalty functions with zero initial surplus and the Laplace transforms of the expected discounted penalty functions are obtained from an integro-differential equations system. We prove that the expected discounted penalty function satisfies a defective renewal equation. An exact representation for the solution of this equation is derived through an associated compound geometric distribution, and an analytic expression for this quantity is given for when the claim amounts from both classes are exponentially distributed. Moreover, the closed form expressions for the ruin probability and the distribution function of the surplus before ruin are obtained. We prove that the ruin probability for this risk model decreases as the probability of the delay of by-claims increases. Finally, numerical results are also provided to illustrate the applicability of our main result and the impact of the delay of by-claims on the expected discounted penalty functions.     

The influence of non-linear dependencies on the basis risk of industry loss warranties

Index-linked catastrophic loss instruments represent an alternative to traditional reinsurance to hedge against catastrophic losses. The use of these instruments comes with benefits, such as a reduction of moral hazard and higher transparency. However, at the same time, it introduces basis risk as a crucial key risk factor, since the index and the company’s losses are usually not fully dependent. The aim of this paper is to examine the impact of basis risk on an insurer’s solvency situation when an industry loss warranty contract is used for hedging. Since previous literature has consistently stressed the importance of a high degree of dependence between the company’s losses and the industry index, we extend previous studies by allowing for non-linear dependencies between relevant processes (high-risk and low-risk assets, insurance company’s loss and industry index). The analysis shows that both the type and degree of dependence play a considerable role with regard to basis risk and solvency capital requirements and that other factors, such as relevant contract parameters of index-linked catastrophic loss instruments, should not be neglected to obtain a comprehensive and holistic view of their effect upon risk reduction.     

A Bayesian approach to pricing longevity risk based on risk-neutral predictive distributions

We present a Bayesian approach to pricing longevity risk under the framework of the Lee-Carter methodology. Specifically, we propose a Bayesian method for pricing the survivor bond and the related survivor swap designed by Denuit et al. (2007). Our method is based on the risk neutralization of the predictive distribution of future survival rates using the entropy maximization principle discussed by Stutzer (1996). The method is illustrated by applying it to Japanese mortality rates.