José Garrido

On a general class of renewal risk process: analysis of the Gerber-Shiu function

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a K n distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.

On The Expected Discounted Penalty function for Lévy Risk Processes

Abstract Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, the joint distribution of the time of ruin, the surplus before ruin, and the deficit at ruin was studied (Gerber and Shiu 1997, 1998a, 1998b; Gerber and Landry 1998). These works use the classical and the perturbed risk models and hint that the results can be extended to gamma and inverse Gaussian risk processes. In this paper we work out this extension to a generalized risk model driven by a nondecreasing Lévy process. Unlike the classical case that models the individual claim size distribution and obtains from it the aggregate claims distribution, here the aggregate claims distribution is known in closed form. It is simply the one-dimensional distribution of a subordinator. Embedded in this wide family of risk models we find the gamma, inverse Gaussian, and generalized inverse Gaussian processes. Expressions for the Gerber-Shiu function are given in some of these special cases, and numerical illustrations are provided.

Moments of compound renewal sums with discounted claims

Abstract Delbaen and Haezendonck [Ins. Math. Econ. 6 (1987) 85] and Willmot [Scand. Actuarial J. 1 (1989) 1] give an analytical expression for the net premium density of a compound Poisson present value risk (CPPVR) process. Their calculation is based, essentially, on the independence of the increments of the CPPVR process. In this paper, under regularity conditions, we derive the first two moments of a compound renewal present value risk (CRPVR) process using renewal theory arguments. Some examples, extensions and limiting results are also given.

The Gerber–Shiu function in a Sparre Andersen risk process perturbed by diffusion

We consider a Sparre Andersen risk process that is perturbed by an independent diffusion process, in which claim inter-arrival times have a generalized Erlang(n) distribution (i.e. as the sum of n independent exponentials, with possibly different means). This leads to a generalization of the defective renewal equations for the expected discounted penalty function at the time of ruin given by Tsai and Willmot [10,11] and Gerber and Shiu [21,22]. The limiting behavior of the expected discounted penalty function is studied, when the dispersion coefficient goes to zero. Finally, explicit results are given for the case where n=2.

Recursive Moments of Compound Renewal Sums with Discounted Claims

Under regularity conditions, Le´veille´& Garrido [6] gives a derivation of the first two moments (resp. asymptotic) of a Compound Renewal Present Value Risk (CRPVR) process using renewal theory arguments. In this paper, with the same procedure and assuming that all the moments of the claim severity and the claims number process exist, we get recursive formulas for all the moments (resp. asymptotic) of the CRPVR process.

Extending pricing rules with general risk functions

The paper addresses pricing issues in imperfect and/or incomplete markets if the risk level of the hedging strategy is measured by a general risk function. Convex Optimization Theory is used in order to extend pricing rules for a wide family of risk functions, including Deviation Measures, Expectation Bounded Risk Measures and Coherent Measures of Risk. Necessary and sufficient optimality conditions are provided in a very general setting. For imperfect markets the extended pricing rules reduce the bid-ask spread. The findings are particularized so as to study with more detail some concrete examples, including the Conditional Value at Risk and some properties of the Standard Deviation. Applications dealing with the valuation of volatility linked derivatives are discussed.

Moment generating functions of compound renewal sums with discounted claims

Léveillé & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersens (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylors (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered. In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.

Regime-Switching Periodic Models For Claim Counts

Abstract We study a Cox risk model that accounts for both seasonal variations and random fluctuations in the claims intensity. This occurs with natural phenomena that evolve in a seasonal environment and affect insurance claims, such as hurricanes. More precisely, we define an intensity process governed by a periodic function with a random peak level. The periodic intensity function follows a deterministic pattern in each short-term period and is illustrated by a beta-type function. A Markov chain with m states, corresponding to different risk levels, is chosen for the level process, yielding a so-called regime-switching process. The properties of the corresponding claim-counting process are discussed in detail. By properly defining a Lundberg-type coefficient, we derive upper bounds for finite time ruin probabilities in a two-state case.

On the computation of aggregate claims distributions: some new approximations

Abstract This paper proposes a new approximation to the aggregate claims distribution based on the inverse Gaussian (IG) distribution. It is compared to several other approximations in the literature. The IG approximation compares favorably to the well-known gamma approximation. We also propose an IG-gamma mixture that approximates the true distribution extremely accurately, in a large variety of situations.

A review of discrete-time risk models

In this paper, we present a review of results for discrete-time risk models, including the compound binomial risk model and some of its extensions. While most theoretical risk models use the concept of time continuity, the practical reality is discrete. For instance, recursive formulas for discretetime models can be obtained without assuming a claim severity distribution and are readily programmable in practice. Hence themodels, techniques used, and results reviewed here for discrete-time risk models are of independent scientific interest. Yet, results for discrete-time risk models can give, in addition, a simpler understanding of their continuous-time analogue. For example, these results can serve as approximations or bounds for the corresponding results in continuous-time models. This paper will serve as a detailed reference for the study of discrete-time risk models.ResumenEn este artículo hacemos un repaso de los resultados para modelos de riesgo en tiempo discreto, incluyendo el modelo de riesgo binomial-compuesto, así como algunas de sus extensiones. Aunque gran parte de los modelos teóricos de riesgo se basen en el concepto de continuidad del tiempo, la realidad práctica es en sí discreta. Por ejemplo, en la práctica actuarial se programan fórmulas recursivas para modelos en tiempo discreto, sin necesidad de suponer una distribución de pérdidas conocida. Con lo cual estos modelos, las técnicas y los resultados que listamos para modelos de riesgo en tiempo discreto, generan un cierto interés científico propio. Pero más allá de sus aplicaciones directas, estos resultados para modelos en tiempo discreto también proporcionan un camino más simple hacia los modelos de riesgo análogos en tiempo continuo. Por ejemplo, los resultados en tiempo discreto pueden servir de aproximaciones o de cotas para sus resultados correspondientes en tiempo continuo. El propósito de este artículo es que pueda servir de referencia detallada para el estudio de modelos de riesgo en tiempo discreto.