Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times

We investigate an insurance risk model that consists of two reserves which receive income at fixed rates. Claims are being requested at random epochs from each reserve and the interclaim times are generally distributed. The two reserves are coupled in the sense that at a claim arrival epoch, claims are being requested from both reserves and the amounts requested are correlated. In addition, the claim amounts are correlated with the time elapsed since the previous claim arrival.We focus on the probability that this bivariate reserve process survives indefinitely. The infinite-horizon survival problem is shown to be related to the problem of determining the equilibrium distribution of a random walk with vector-valued increments with ‘reflecting’ boundary. This reflected random walk is actually the waiting time process in a queueing system dual to the bivariate ruin process.Under assumptions on the arrival process and the claim amounts, and using Wiener–Hopf factorization with one parameter, we explicitly determine the Laplace–Stieltjes transform of the survival function, c.q., the two-dimensional equilibrium waiting time distribution.Finally, the bivariate transforms are evaluated for some examples, including for proportional reinsurance, and the bivariate ruin functions are numerically calculated using an efficient inversion scheme.     

Prediction in a non-homogeneous Poisson cluster model

A non-homogeneous Poisson cluster model is studied, motivated by insurance applications. The Poisson center process which expresses arrival times of claims, triggers off cluster member processes which correspond to number or amount of payments. The cluster member process is an additive process. Given the past observations of the process we consider expected values of future increments and their mean squared errors, aiming at application in claims reserving problems. Our proposed process can cope with non-homogeneous observations such as the seasonality of claims arrival or the reducing property of payment processes, which are unavailable in the former models where both center and member processes are time homogeneous. Hence results presented in this paper are significant extensions toward applications.     

Stochastic analysis of life insurance surplus

The aim of the paper is to examine the behavior of insurance surplus over time for a portfolio of homogeneous life policies. We distinguish between stochastic and accounting surpluses and derive their first two moments. A recursive formula is proposed for calculating the distribution function of the accounting surplus. We then examine the probability that the accounting surplus becomes negative in a given insurance year. Numerical examples illustrate the results for portfolios of temporary and endowment life policies assuming a conditional AR(1) process for the rates of return.     

Robust estimation of the Pickands dependence function under random right censoring

We consider robust nonparametric estimation of the Pickands dependence function under random right censoring. The estimator is obtained by applying the minimum density power divergence criterion to properly transformed bivariate observations. The asymptotic properties are investigated by making use of results for Kaplan–Meier integrals. We investigate the finite sample properties of the proposed estimator with a simulation experiment and illustrate its practical applicability on a dataset of insurance indemnity losses.     

Monotone and cash-invariant convex functions and hulls

This paper provides some useful results for convex risk measures. In fact, we consider convex functions on a locally convex vector space E which are monotone with respect to the preference relation implied by some convex cone and invariant with respect to some numeraire (‘cash’). As a main result, for any function f, we find the greatest closed convex monotone and cash-invariant function majorized by f. We then apply our results to some well-known risk measures and problems arising in connection with insurance regulation.     

A new uncertain insurance model with variational lower limit

Uncertainty theory provides a new tool to deal with uncertainty. The paper employs it to propose a new uncertain insurance model with variational lower limit, and gives a ruin index and uncertainty distribution for the uncertain insurance risk process that claim process is a renewal reward process. The model extends and improves uncertain insurance model presented by Liu. Finally, it also provides examples to illustrate the effectiveness of the model.     

Solvency II reporting: How to interpret funds’ aggregate solvency capital requirement figures

Depending on the current risk exposure of an insurance company, the impact of buying an additional unit of a fund on an insurer’s overall Solvency II capital charges, i.e., the Solvency Capital Requirement (SCR), will differ. We call this impact the fund’s SCR contribution and show in which boundaries it lies if only the fund’s aggregate sub-SCR figures are known but not the risk exposures of the insurance company buying the fund. The upper bound of this range, the worst-case SCR contribution, can be used as a conservative measure to assess funds’ Solvency II risk contributions or to assign them to different Solvency II risk categories. We believe that providing funds’ worst-case SCR contributions can be useful information to insurance companies when screening from a broad investment universe.     

Optimal insurance design with a bonus

This paper investigates an insurance design problem, in which a bonus will be given to the insured if no claim has been made during the whole lifetime of the contract, for an expected utility insured. In this problem, the insured has to consider the so-called optimal action rather than the contracted compensation (or indemnity) due to the existence of the bonus. For any pre-agreed bonus, the optimal insurance contract is given explicitly and shown to be either the full coverage contract when the insured pays high enough premium, or a deductible one otherwise. The optimal contract and bonus are also derived explicitly if the insured is allowed to choose both of them. The contract turns out to be of either zero reward or zero deductible. In all cases, the optimal contracts are universal, that is, they do not depend on the specific form of the utility of the insured. A numerical example is also provided to illustrate the main theoretical results of the paper.     

A stochastic programming model for funding single premium deferred annuities

Single Premium Deferred Annuities (SPDAs) are investment vehicles, offered to investors by insurance companies as a means of providing income past their retirement age. They are mirror images of insurance policies. However, the propensity of individuals to shift part, or all, of their investment into different annuities creates substantial uncertainties for the insurance company. In this paper we develop amultiperiod, dynamic stochastic program that deals with the problem of funding SPDA liabilities. The model recognizes explicitly the uncertainties inherent in this problem due to both interest rate volatility and the behavior of individual investors. Empirical results are presented with the use of the model for the funding of an SPDA liability stream using government bonds, mortgage-backed securities and derivative products. Research partially supported by NSF grants CCR-9104042 and SES-91-00216, and AFOSR grant 91-0168. Computing resources were made available by AHPCRC at the University of Minnesota, by NPAC at Syracuse University, New York, and by the GRASP Laboratory at Computer Science Department at University of Pennsylvania.     

MIN-MAD life: A multi-period optimization model for life insurance company investment decisions

This paper presents a model, called the MIN-MAD Life Model, for managing the investments of a life insurance company over a multiperiod planning horizon. The MIN-MAD Life Model is a linear programming under uncertainty model based on Markowitz portfolio theory. Given the insurance company's current position and its forecasts of possible future developments with their associated probabilities, the model helps determine the set of efficient investment decisions over the planning horizon subject to market constraints and to the insurance company's legal and policy constraints. The senior executives of the life insurance company need examine only the set of efficient investment decisions to determine their optimal investment decisions.