Vladimir K. Kaishev

An Improved Finite-Time Ruin Probability Formula and Its Mathematica Implementation

An improved version of a ruin probability formula due to Ignatov and Kaishev [Scand. Actu. J. 1 (2000) 46], allowing for the exact evaluation of the finite-time survival probability for discrete, dependent, individual claims, Poisson claim arrivals and arbitrary, increasing premium income function is derived. Its numerical efficiency is studied, using the Mathematica system. Numerical results are provided and computational aspects are discussed. A Mathematica module, realizing the Picard and Lefevre [Scand. Actu. J. 1 (1997) 58] formula has also been developed and used for numerical investigations.

Optimal retention levels, given the joint survival of cedent and reinsurer

A certain volume of risks is insured and there is a reinsurance contract, according to which claims and total premium income are shared between a direct insurer and a reinsurer in such a way, that the finite horizon probability of their joint survival is maximized. An explicit expression for the latter probability, under an excess of loss (XL) treaty is derived, using the improved version of the Ignatov and Kaishevs ruin probability formula (see Ignatov, Kaishev & Krachunov. 2001a) and assuming, Poisson claim arrivals, any discrete joint distribution of the claims, and any increasing real premium income function. An explicit expression for the probability of survival of the cedent only, under an XL contract is also derived and used to determine the probability of survival of the reinsurer, given survival of the cedent. The absolute value of the difference between the probability of survival of the cedent and the probability of survival of the reinsurer, given survival of the cedent is used for the choice of optimal retention level. We derive formulae for the expected profit of the cedent and of the reinsurer, given their joint survival up to the finite time horizon. We illustrate how optimal retention levels can be set, using an optimality criterion based on the expected profit formulae. The quota share contract is also considered under the same model. It is shown that the probability of joint survival of the cedent and the reinsurer coincides with the probability of survival of solely the insurer. Extensive, numerical comparisons, illustrating the performance of the proposed reinsurance optimality criteria are presented.

Two-Sided Bounds for the Finite Time Probability of Ruin

Explicit, two-sided bounds are derived for the probability of ruin of an insurance company, whose premium income is represented by an arbitrary, increasing real function, the claims are dependent, integer valued r.v.s and their inter-occurrence times are exponentially, non-identically distributed. It is shown, that the two bounds coincide when the moments of the claims form a Poisson point process. An expression for the survival probability is further derived in this special case, assuming that the claims are integer valued, i.i.d. r.v.s. This expression is compared with a different formula, obtained recently by Picard & Lefevre (1997) in terms of generalized Appell polynomials. The particular case of constant rate premium income and non-zero initial capital is considered. A connection of the survival probability to multivariate B -splines is also established.

Optimal customer selection for cross-selling of financial services products

A new methodology, for optimal customer selection in cross-selling of financial services products, such as mortgage loans and non life insurance contracts, is presented. The optimal cross-sales selection of prospects is such that the expected profit is maximized, while at the same time the risk of suffering future losses is minimized. Expected profit maximization and mean-variance optimization are considered as alternative optimality criteria. In order to solve these optimality problems a stochastic model of the profit, expected to emerge from a single cross-sales prospect and from a selection of prospects, is developed. The related probability distributions of the profit are derived, both for small and large portfolio sizes and in the latter case, asymptotic normality is established. The proposed, profit optimization methodology is thoroughly tested, based on a real data set from a large Swedish insurance company and is shown to achieve considerable profit gains, compared to traditional cross-selling methods, which use only the estimated sales probabilities.

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 %

Optimal experimental designs for the B -spline regression

Abstract The present interest is to investigate continuous D -optimal designs for polynomial spline regression. Such a design is obtained in the special case of a quadratic spline with a single distinct knot. Two conjectures, concerning the number and position of the supports in a D -optimal design for the general case of spline-regression of arbitrary degree and multiplicities of the knots are given.

Adsorption of dodecylhexaoxyethylene glycol monoether at a stationary mercury electrode: A spline regression model of differential capacity

The differential capacity C of a stationary mercury electrode as a function of electrode potential E° and concentration c of dodecylhexaoxyethylene glycol monoether has been measured by the impedance bridge method. A 0.05-M aqueous solution of Na2SO4 was used as a supporting electrolyte. A spline regression model of the experimentally investigated capacity dependence on E° and c has been constructed, giving the possibility to predict C for any potential and surfactant concentration in the interval investigated. The critical micelle concentration for the system studied was found on the basis of the spline model to be 7×10−5 mod dm−3 C12E6. The free energy of adsorption and some other parameters characterising the adsorption of the surface-active agent have been determined.

On finite-time ruin probabilities in a generalized dual risk model with dependence

In this paper, we study the finite-time ruin probability in a reasonably generalized dual risk model, where we assume any non-negative non-decreasing cumulative operational cost function and arbitrary capital gains arrival process. Establishing an enlightening link between this dual risk model and its corresponding insurance risk model, explicit expressions for the finite-time survival probability in the dual risk model are obtained under various general assumptions for the distribution of the capital gains. In order to make the model more realistic and general, different dependence structures among capital gains and inter-arrival times and between both are also introduced and corresponding ruin probability expressions are also given. The concept of alarm time, as introduced in Das and Kratz (2012), is applied to the dual risk model within the context of risk capital allocation. Extensive numerical illustrations are provided.

Lévy Processes Induced by Dirichlet (B‐)Splines: Modeling Multivariate Asset Price Dynamics

We consider a new class of processes, called LG processes, defined as linear combinations of independent gamma processes. Their distributional and path‐wise properties are explored by following their relation to polynomial and Dirichlet (B‐)splines. In particular, it is shown that the density of an LG process can be expressed in terms of Dirichlet (B‐)splines, introduced independently by Ignatov and Kaishev and Karlin, Micchelli, and Rinott. We further show that the well‐known variance gamma (VG) process, introduced by Madan and Seneta, and the bilateral gamma (BG) process, recently considered by Kuchler and Tappe are special cases of an LG process. Following this LG interpretation, we derive new (alternative) expressions for the VG and BG densities and consider their numerical properties. The LG process has two sets of parameters, the B‐spline knots and their multiplicities, and offers further flexibility in controlling the shape of the Levy density, compared to the VG and the BG processes. Such flexibility is often desirable in practice, which makes LG processes interesting for financial and insurance applications. Multivariate LG processes are also introduced and their relation to multivariate Dirichlet and simplex splines is established. Expressions for their joint density, the underlying LG‐copula, the characteristic, moment and cumulant generating functions are given. A method for simulating LG sample paths is also proposed, based on the Dirichlet bridge sampling of gamma processes, due to Kaishev and Dimitriva. A method of moments for estimation of the LG parameters is also developed. Multivariate LG processes are shown to provide a competitive alternative in modeling dependence, compared to the various multivariate generalizations of the VG process, proposed in the literature. Application of multivariate LG processes in modeling the joint dynamics of multiple exchange rates is also considered.

On the evaluation of finite-time ruin probabilities in a dependent risk model

This paper establishes some enlightening connections between the explicit formulas of the finite-time ruin probability obtained by Ignatovand Kaishev (2000, 2004) and Ignatov et?al. (2001) for a risk model allowing dependence. The numerical properties of these formulas are investigated and efficient algorithms for computing ruin probability with prescribed accuracy are presented. Extensive numerical comparisons and examples are provided.