Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claims

Suppose that, over a fixed time interval of interest, an insurance portfolio generates a random number of independent and identically distributed claims. Under the LCR treaty the reinsurance covers the first l largest claims, while under the ECOMOR treaty it covers the first l−1 largest claims in excess of the lth largest one. Assuming that the claim sizes follow an exponential distribution or a distribution with a convolution-equivalent tail, we derive some precise asymptotic estimates for the tail probabilities of the reinsured amounts under both treaties.     

Joint tail of ECOMOR and LCR reinsurance treaties

Researchers in actuarial sciences have investigated the tail behavior of the LCR and ECOMOR reinsurance treaties separately for managing extreme risks in reinsurance business. In practice, a reinsurance company may possess these two treaties simultaneously. Therefore, investigating the joint tail behavior of these two treaties is practically useful in risk management. This paper derives the asymptotic limit of the joint tail of these two reinsurance treaties under the setup of Jiang and Tang (2008).     

Reinsurance of large claims

The large claims reinsurance treaties ECOMOR and LCR are well known not to be very popular. They have been largely neglected by most reinsurers because of their technical complexity. In this paper, we derive new mathematical results connected to asymptotic problems of these reinsurance forms. Perhaps these results can reopen the discussion on the usefulness of including the largest claims in the decision making procedure. Apart from asymptotic estimates for the tail of the distribution of the ECOMOR-quantity, we find its weak laws. We also deal with the weak laws of the LCR-quantity. Finally, we illustrate the outcomes with a number of simulations.     

Dependence and the asymptotic behavior of large claims reinsurance

We consider an extension of the classical compound Poisson risk model, where the waiting time between two consecutive claims and the forthcoming claim are no longer independent. Asymptotic tail probabilities of the reinsurance amount under ECOMOR and LCR treaties are obtained. Simulation results are provided in order to illustrate this.     

Dependence and the asymptotic behavior of large claims reinsurance

We consider an extension of the classical compound Poisson risk model, where the waiting time between two consecutive claims and the forthcoming claim are no longer independent. Asymptotic tail probabilities of the reinsurance amount under ECOMOR and LCR treaties are obtained. Simulation results are provided in order to illustrate this.     

Asymptotics of convolution with the semi-regular-variation tail and its application to risk

In this paper, according to a certain criterion, we divide the exponential distribution class into some subclasses. One of them is closely related to the regular-variation-tailed distribution class, and is called the semi-regular-variation-tailed distribution class. The new class possesses several nice properties, although distributions in it are not convolution equivalent. We give the precise tail asymptotic expression of convolutions of these distributions, and prove that the class is closed under convolution. In addition, we do not need to require the corresponding random variables to be identically distributed. Finally, we apply these results to a discrete time risk model with stochastic returns, and obtain the precise asymptotic estimation of the finite time ruin probability.     

How retention levels influence the variability of the total risk under reinsurance

Recently, Escudero and Ortega (Insur. Math. Econ. 43:255–262, 2008) have considered an extension of the largest claims reinsurance with arbitrary random retention levels. They have analyzed the effect of some dependencies on the Laplace transform of the retained total claim amount. In this note, we study how dependencies influence the variability of the retained and the reinsured total claim amount, under excess-loss and stop-loss reinsurance policies, with stochastic retention levels. Stochastic directional convexity properties, variability orderings, and bounds for the retained and the reinsured total risk are given. Some examples on the calculation of bounds for stop-loss premiums (i.e., the expected value of the reinsured total risk under this treaty) and for net premiums for the cedent company under excess-loss, and complementary results on convex comparisons of discounted values of benefits for the insurer from a portfolio with risks having random policy limits (deductibles) are derived.      

Asymptotics in a time-dependent renewal risk model with stochastic return

In this paper we study the asymptotic tail behavior for a non-standard renewal risk model with a dependence structure and stochastic return. An insurance company is allowed to invest in financial assets such as risk-free bonds and risky stocks, and the price process of its portfolio is described by a geometric Lévy process. By restricting the claim-size distribution to the class of extended regular variation (ERV) and imposing a constraint on the Lévy process in terms of its Laplace exponent, we obtain for the tail probability of the stochastic present value of aggregate claims a precise asymptotic formula, which holds uniformly for all time horizons. We further prove that the corresponding ruin probability also satisfies the same asymptotic formula.     

An asymptotic expansion for the tail of compound sums of Burr distributed random variables

In this paper we show that it is possible to write the Laplace transform of the Burr distribution as the sum of four series. This representation is then used to provide a complete asymptotic expansion of the tail of the compound sum of Burr distributed random variables. Furthermore it is shown that if the number of summands is fixed, this asymptotic expansion is actually a series expansion if evaluated at sufficiently large arguments.     

A strong large deviation theorem

We prove a strong large deviation theorem for an arbitrary sequence of random variables, that is, we establish a full asymptotic expansion of large deviation type for the tail probabilities. An Edgeworth expansion is required to derive the result. We illustrate our theorem with two statistical applications: the sample variance and the kernel density estimator.     

A matrix-less and parallel interpolation–extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices

Numerical Algorithms - In the past few years, Bogoya, Böttcher, Grudsky, and Maximenko obtained the precise asymptotic expansion for the eigenvalues of a Toeplitz matrix Tn(f), under suitable...     

Generalized calculus and sdes with non regular drift

In this note we prove a precise asymptotic estimate for Laplace type functionals for a parabolic SPDE. We use a large deviation principle, the stochastic Taylor expansion, some exponential inequalities and support theorems for our stochastic partial differential equation     

Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments

Let F$F$ be a distribution function with negative mean and regularly varying right tail. Under a mild smoothness condition we derive higher order asymptotic expansions for the tail distribution of the maxima of the random walk generated by F$F$. The expansion is based on an expansion for the right Wiener–Hopf factor which we derive first. An application to ruin probabilities is developed.     

Slowly Varying Solitary Wave Solutions of the Perturbed Korteweg-de Vries Equation Revisited

Using a multi-scale perturbation expansion we reconsider the slowly varying solitary wave asymptotic solution of the perturbed Korteweg-de Vries equation. The well-known results for the variation of the solitary-wave amplitude and the accompanying trailing tail are recovered. Here the analysis is carried through to second order so as to determine a general expression for the first-order speed correction. The result obtained here generalizes and improves previous results.     

Quantile Function Expansion Using Regularly Varying Functions

We present a simple result that allows us to evaluate the asymptotic order of the remainder of a partial asymptotic expansion of the quantile function h(u) as u → 0⁺ or 1^?. This is focussed on important univariate distributions when h(?) has no simple closed form, with a view to assessing asymptotic rate of decay to zero of tail dependence in the context of bivariate copulas. Motivation of this study is illustrated by the asymptotic behaviour of the tail dependence of Normal copula. The Normal, Skew-Normal and Gamma are used as initial examples. Finally, we discuss approximation to the lower quantile of the Variance-Gamma and Skew-Slash distributions.     

Precise large deviations for generalized dependent compound renewal risk model with consistent variation

We investigate the precise large deviations of random sums of negatively dependent random variables with consistently varying tails. We find out the asymptotic behavior of precise large deviations of random sums is insensitive to the negative dependence. We also consider the generalized dependent compound renewal risk model with consistent variation, which including premium process and claim process, and obtain the asymptotic behavior of the tail probabilities of the claim surplus process.     

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This note deals with an insurance company with multiple lines of business. In the context of heavy-tailed heterogeneous claim amounts with the 1st upper-orthant tail dependence, based on the so-called k-out-of-n ruin set, we can exhibit the Radon measure mu and derive the asymptotic ruin probability for some of all lines business to ruin in a finite time. One numerical example is also presented to illustrate our main results.     

Formal asymptotic solution of a singularly perturbed nonlinear optimal control problem

A system of equations that arises in a singularly perturbed optimal control problem is studied. We give conditions under which a formal asymptotic solution exists. This formal asymptotic solution consists of an outer expansion and left and right boundary-layer expansions. A feature of our procedure is that we do nota priori eliminate the control function from the problem. In particular, we construct a formal asymptotic expansion for the control directly. We apply our procedure to a Mayer-type problem. The paper concludes with a worked example.