On the Distribution of the Deficit at Ruin when Claims are Phase-type

Drekic, Steve; Dickson, David; Stanford, David A.; Willmot, Gordon E.

doi:10.1080/03461230110106471

Cited by 48 publications

(28 citation statements)

References 11 publications

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“…In the general case, φ and F may not be available analytically; however, analytic expressions can be found if K or P belong to the class of phase-type distributions (Asmussen (2000), see also Section 11.2 of and Drekic et al (2004)). Note that (1) is a renewal equation for G(x, y) when y is viewed as fixed, but x is allowed to vary.…”

Section: Introductionmentioning

confidence: 98%

Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model

Chadjiconstantinidis¹,

Politis²

2007

Insurance: Mathematics and Economics

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Section: Introductionmentioning

confidence: 98%

Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model

Chadjiconstantinidis¹,

Politis²

2007

Insurance: Mathematics and Economics

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“…Based on a numerical procedure described in Asmussen (2000), the following expression for the infinite-time ruin probabilities in the above risk model is derived in Drekic et al (2004) ψ(u) = 0.23381e −3.6926u − 0.26698e −3.5987u + 0.04517e −1.6967u + 0.46507e −0.5566u .…”

Section: Approximation Of ψ Cmentioning

confidence: 99%

Ruin probabilities in the discrete time renewal risk model

Cossette

Landriault

Marceau

2006

Insurance: Mathematics and Economics

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“…It is known, [29], that when the individual claim amount follows a phase-type distribution P H (v, S), the deficit at ruin if ruin occurs, Y , is also phase-type distributed with representation P H (Π G , S), where…”

Section: Influence Of (Threshold) Proportional Reinsurance On the Defmentioning

confidence: 99%

Some Optimization and Decision Problems in Proportional Reinsurance

2014

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RESUMEN: El reaseguro es uno de los instrumentos que un asegurador puede utilizar para mitigar el riesgo de suscripción y por lo tanto controlar su solvencia. En este artículo, nos centramos en los contratos de reaseguro proporcional y examinamos diversos problemas de optimización y decisión del asegurador respecto de la estrategia de reaseguro. Con este fin, utilizamos como instrumentos de decisión no sólo la probabilidad de ruina sino también la variable aleatoria déficit en el momento de ruina si la ruina ocurre. La función de penalización descontada se utiliza para calcular como casos particulares la probabilidad de ruina, y los momentos y la función de distribución del déficit en el momento de ruina si la ruina ocurre.Consideramos el modelo de la teoría clásica del riesgo asumiendo un proceso de Poisson y una cuantía individual de los siniestros distribuida según una phase-type, modificado con un reaseguro proporcional con un nivel de retención que no es constante y que depende del nivel de las reservas. La función de penalización descontada se comporta diferente según si el nivel inicial de las reservas está por encima o por debajo de un determinado umbral. Se obtienen expresiones generales para esta función de penalización descontada, así como resultados teóricos interesantes y expresiones explícitas para el caso phase-type 2. Estos resultados se aplican en ejemplos numéricos de problemas de decisión basados en la probabilidad de ruina y en diferentes medidas de riesgo del déficit en el momento de ruina si la ruina ocurre (esperanza, Value at risk y Tail Value at Risk.Palabras Clave: Optimización, Distribución phase-type, Reaseguro.ABSTRACT: Reinsurance is one of the tools that an insurer can use to mitigate the underwriting risk and then to control its solvency. In this paper, we focus on the proportional reinsurance arrangements and we examine several optimization and decision problems of the insurer with respect to the reinsurance strategy. To this end, we use as decision tools not only the probability of ruin but also the random variable deficit at ruin if ruin occurs. The discounted penalty function is employed to calculate as particular cases the probability of ruin and the moments and the distribution function of the deficit at ruin if ruin occurs.We consider the classical risk theory model assuming a Poisson process and an individual claim amount phase-type distributed, modified with a proportional reinsurance with a retention level that is not constant and depends on the level of the surplus. Depending on whether the initial surplus is below or above a threshold level, the discounted penalty function behaves differently. General expressions for this discounted penalty function are obtained, as well as interesting theoretical results and explicit expressions for phasetype 2 distribution. These results are applied in numerical examples of decision problems based on the probability of ruin and on different risk measures of the deficit at ruin if ruin occurs (the expectation, the Value at Risk and th...

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On the Distribution of the Deficit at Ruin when Claims are Phase-type

Cited by 48 publications

References 11 publications

Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model

Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model

Ruin probabilities in the discrete time renewal risk model

Some Optimization and Decision Problems in Proportional Reinsurance

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