Randomized observation times for the compound Poisson risk model: The discounted penalty function

In the framework of collective risk theory, we consider a compound Poisson risk model for the surplus process where the process (and hence ruin) can only be observed at random observation times. For Erlang(n) distributed inter-observation times, explicit expressions for the discounted penalty function at ruin are derived. The resulting model contains both the usual continuous-time and the discrete-time risk model as limiting cases, and can be used as an effective approximation scheme for the latter. Numerical examples are given that illustrate the effect of random observation times on various ruin-related quantities

On the time to ruin for a dependent delayed capital injection risk model

© 2019 Elsevier Inc. All rights reserved.In this paper, we propose a generalisation to the Cramér–Lundberg risk model, by allowing for a delayed receipt of the required capital injections whenever the surplus of an insurance firm is negative. Delayed capital injections often appear in practice due to the time taken for administrative and processing purposes of the funds from a third party or the shareholders of an insurance firm. The delay time of the capital injection depends on a critical value of the deficit in the following way: if the deficit of the firm is less than the fixed critical value, then it can be covered by available funds and therefore the required capital injection is received instantaneously. On the other hand, if the deficit of the firm exceeds the fixed critical value, then the funds are provided by an alternative source and the required capital injection is received after some time delay. In this modified model, we derive a Fredholm integral equation of the second kind for the ultimate ruin probability and obtain an explicit expression in terms of ruin quantities for the Cramér–Lundberg risk model. In addition, we show that other risk quantities, namely the expected discounted accumulated capital injections and the expected discounted overall time in red, up to the time of ruin, satisfy a similar integral equation, which can also be solved explicitly. Finally, we extend the capital injection delayed risk model, such that the delay of the capital injections depends explicitly on the amount of the deficit. In this generalised risk model, we derive another Fredholm integral equation for the ultimate ruin probability, which is solved in terms of a Neumann series.Peer reviewe

The W,ZW,Z scale functions kit for first passage problems of spectrally negative Levy processes, and applications to the optimization of dividends

First passage problems for spectrally negative L\'evy processes with possible absorbtion or/and reflection at boundaries have been widely applied in mathematical finance, risk, queueing, and inventory/storage theory. Historically, such problems were tackled by taking Laplace transform of the associated Kolmogorov integro-differential equations involving the generator operator. In the last years there appeared an alternative approach based on the solution of two fundamental "two-sided exit" problems from an interval (TSE). A spectrally one-sided process will exit smoothly on one side on an interval, and the solution is simply expressed in terms of a "scale function" $W$ (Bertoin 1997). The non-smooth two-sided exit (or ruin) problem suggests introducing a second scale function $Z$ (Avram, Kyprianou and Pistorius 2004). Since many other problems can be reduced to TSE, researchers produced in the last years a kit of formulas expressed in terms of the "$W,Z$ alphabet" for a great variety of first passage problems. We collect here our favorite recipes from this kit, including a recent one (94) which generalizes the classic De Finetti dividend problem. One interesting use of the kit is for recognizing relationships between apparently unrelated problems -- see Lemma 3. Last but not least, it turned out recently that once the classic $W,Z$ are replaced with appropriate generalizations, the classic formulas for (absorbed/ reflected) L\'evy processes continue to hold for: a) spectrally negative Markov additive processes (Ivanovs and Palmowski 2012), b) spectrally negative L\'evy processes with Poissonian Parisian absorbtion or/and reflection (Avram, Perez and Yamazaki 2017, Avram Zhou 2017), or with Omega killing (Li and Palmowski 2017)

Operational risk and insurance: a ruin probabilistic reserving approach

A new methodology for financial and insurance operational risk capital estimation is proposed. It is based on using the finite time probability of (non-)ruin as an operational risk measure, within a general risk model. It allows for inhomogeneous operational loss frequency (dependent inter-arrival times) and dependent loss severities which may have any joint discrete or continuous distribution. Under the proposed methodology, operational risk capital assessment is viewed not as a one off exercise, performed at some moment of time, but as dynamic reserving, following a certain risk capital accumulation function. The latter describes the accumulation of risk capital with time and may be any nondecreasing, mpositive real function hHtL. Under these reasonably general assumptions, the probability of mnon-ruin is explicitly expressed using closed form expressions, derived by Ignatov and Kaishev (2000, 2004, 2007) and Ignatov, Kaishev and Krachunov (2001) and by setting it to a high enough preassigned mvalue, say 0.99, it is possible to obtain not just a value for the capital charge but a (dynamic) risk capital accumulation strategy, hHtL. In view of its generality, the proposed methodology is capable of accommodating any (heavy tailed) mdistributions, such as the Generalized Pareto Distribution, the Lognormal distribution the g-and-h mdistribution and the GB2 distribution. Applying this methodology on numerical examples, we demonstrate that dependence in the loss severities may have a dramatic effect on the estimated risk capital. In addition, we show also that one and the same high enough survival probability may be achieved by mdifferent risk capital accumulation strategies one of which may possibly be preferable to accumulating capital just linearly, as has been assumed by Embrechts et al. (2004). The proposed methodology takes into account also the effect of insurance on operational losses, in which case it is proposed to take the probability of joint survival of the financial institution and the insurance provider as a joint operational risk measure. The risk capital allocation strategy is then obtained in such a way that the probability of joint survival is equal to a preassigned high enough value, say 99.9

Stochastic Risk Processes Applied to Insurance Capital Recovery Methods

Over recent decades, insurance and financial industries have been affected by the volatility of economic cycles. A severe financial crisis struck the market in the year 2000 and subsequently between 2007 and 2012. During these economic downturns, financial businesses (including insurance companies) experienced technical bankruptcy due to insufficient capital holdings. Therefore, the private sector and, in some cases, national governments were called upon to provide a means of recovery, in terms of capital, since their bankruptcy would cause a serious threat to the economy and community as a whole. In response to this adverse environment, governments and regulators have since drawn up stringent rules and regulations, within the insurance industry, to provide a more prudent risk assessment and, in turn, minimise the possibility of future bankruptcy. These regulations are usually known as `directives' and have been implemented across the EU, USA, Australia and China, among others. One of the most efficiently employed capital recovery methods, used in practice, is the provision of capital injections. This injection of capital is usually sourced from a companies shareholders (as long as it is profitable for them to do so) or, in some extreme cases, by the national government. Throughout the majority of this thesis, we employ the classical continuous-time risk model to analyse the financial impact of capital injections under the regulatory constraints of Solvency II and, further, by capturing the realistic procedure of financial and administrative processing linked to raising such funds, consider the risk exposure during the delay between requesting and receiving a capital injection. In the final chapter, we move to a discrete-time setting and discuss alternative capital recovery methods for a different line of business. In this case, where we consider pharmaceutical and petroleum businesses, the classic insurance risk model of the previous chapters is unsuitable and the so-called dual risk model is analysed. Moreover, it is believed that the fall into deficit (bankruptcy) can be recovered within a given time period from normal trading strategies. That is, capital injections are not required and the company can recover from deficit without financial assistance

From ruin to bankruptcy for compound Poisson surplus processes

In classical risk theory, the infinite-time ruin probability of a surplus process Ct is calculated as the probability of the process becoming negative at some point in time. In this paper, we consider a relaxation of the ruin concept to the concept of bankruptcy, according to which one has a positive surplus-dependent probability to continue despite temporary negative surplus. We study the resulting bankruptcy probability for the compound Poisson risk model with exponential claim sizes for different bankruptcy rate functions, deriving analytical results, upper and lower bounds as well as an efficient simulation method. Numerical examples are given and the results are compared with the classical ruin probabilities. Finally, it is illustrated how the analysis can be extended to study the discounted penalty function under this relaxed ruin criterion

On Gerber-Shiu functions and optimal dividend distribution for a L\'{e}vy risk process in the presence of a penalty function

This paper concerns an optimal dividend distribution problem for an insurance company whose risk process evolves as a spectrally negative L\'{e}vy process (in the absence of dividend payments). The management of the company is assumed to control timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividend payments received until the moment of ruin and a penalty payment at the moment of ruin, which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. A complete solution is presented to the corresponding stochastic control problem. It is established that the value-function is the unique stochastic solution and the pointwise smallest stochastic supersolution of the associated HJB equation. Furthermore, a necessary and sufficient condition is identified for optimality of a single dividend-band strategy, in terms of a particular Gerber-Shiu function. A number of concrete examples are analyzed.Comment: Published at http://dx.doi.org/10.1214/14-AAP1038 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities

This paper is concerned with the compound Poisson risk model and two generalized models with still Poisson claim arrivals. One extension incorporates inhomogeneity in the premium input and in the claim arrival process, while the other takes into account possible dependence between the successive claim amounts. The problem under study for these risk models is the evaluation of the probabilities of (non-)ruin over any horizon of finite length. The main recent methods, exact or approximate, used to compute the ruin probabilities are reviewed and discussed in a unified way. Special attention is then paid to an analysis of the qualitative impact of dependence between claim amounts.compound Poisson model; ruin probability; finite-time horizon; recursive methods; (generalized) Appell polynomials; non-constant premium; non-stationary claim arrivals; interdependent claim amounts; impact of dependence; comonotonic risks; heavy-tailed distributions