Exponential inequalities for ruin probabilities in the Cox case

Abstract

We consider an insurance model where the underlying point process is a Cox process. Using a martingale approach we obtain extensions of the classical Lundberg inequality.     

Equity and Credibility

We build on previous work concerned with measuring equity and consider the problem of using observed claim data or other information to calculate premiums which maximize equity. When these optimal premiums are used, we show that gathering more information or refining the risk classification always increases equity. We study the case for which the premium is constrained to be an affine function of the claim data and obtain results analogous to classical credibility theory, including the inhomogeneous and homogeneous cases of the Bu¨hlmann-Straub model. We derive formulas for the credibility weights in certain cases.     

Parisian ruin probability with a lower ultimate bankrupt barrier

The paper deals with a ruin problem, where there is a Parisian delay and a lower ultimate bankrupt barrier. In this problem, we will say that a risk process get ruined when it stays below zero longer than a fixed amount of time ζ > 0 or goes below a fixed level ?a. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we identify the Laplace transform of the ruin probability in terms of so-called q-scale functions. We find its Cramér-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples.     

Sequential estimation problems for the scale parameter of a Pareto distribution

Abstract

The Pareto distribution plays a central role in many areas of econometrics. So, we first consider sequential point estimation problems for the scale parameter of a Pareto distribution. Under a very general loss structure, we derive several asymptotic results regarding the associated “risk” and “regret” functions. Then, we consider the problem of constructing a fixed-ratio confidence interval for the scale parameter, and we propose various sampling techniques to achieve the intended goal. Most of our theoretical findings are asymptotic in nature for either problem, and thus we have presented extensive simulation studies to examine moderate sample performances of all the procedures. The findings in the point estimation problem are also supposed to fill many important gaps left in the paper of Wang (1973).     

Minimum Rate of Return Guarantees: The Danish Case

We analyze minimum rate of return guarantees for life-insurance (investment) contracts and pension plans with a smooth surplus distribution mechanism. We specifically model the smoothing mechanism used by most Danish life-insurance companies and pension funds. The annual distribution of bonus will be based on this smoothing mechanism after taking the minimum rate of return guarantee into account. In addition, based on the contribution method the customer will receive a final (non-negative) undistributed surplus when the contract matures. We consider two different methods that the company can use to collect payment for issuing these minimum rate of return guarantee contracts: the direct method where the company gets a fixed (percentage) fee of the customer's savings each year, e.g. 0.5% in Denmark, and the indirect method where the company gets a share of the distributed surplus. In both cases we analyze how to set the terms of the contract in order to have a fair contract between an individual customer and the company. Having analyzed the one-customer case, we turn to analyzing the case with two customers. We consider the consequences of pooling the undistributed surplus over two inhomogeneous customers. This implies setting up different mechanisms for distributing final bonus (undistributed surplus) between the customers.     

Numerical inversion of characteristic functions

Abstract

To anyone working with characteristic functions, or with Laplace transforms of a non-negative random variable, the three papers by Harald Bohman (1971, 1974, 1975) are invaluable. Numerical integration over an infinite interval is extraordinarily beset with pitfalls (vide Davis & Rabinowitz, 1975) and the publication of actual results achieved in actuarial fields is of great value.     

Life Insurance Policies with Statistical Heterogeneous Population

In the paper we consider an endowment insurance contract with a twelve months maturation time. Using the majorization order and Schur-convex functions we derive upper and lower bounds of the premium, the death and survival benefits for a hetrogeneous population of insureds. The bounds are obtained for the exponential, Balducci, and linear approximations.     

On time-heterogeneous credibility estimation

Abstract

Let χ_i be the total claim amount of an insurance policy in calendar year i. We assume that the χ_i's are conditionally independent given an unknown random parameter ø, and that for all i. In the present paper it is under these assumptions shown how to calculate the credibility estimator of m(ø) by recursive updating. We also give estimators for the unknown parameters α_i, β_i, and ?_i based on portfolio data. Finally we mention some related models.     

Stop-loss ordering for scale and power mixtures of distributions

Abstract

It is the custom to introduce mixtures of distributions for explaining non-obvious variations in observed outcomes from a collective of risks. In the present contribution we consider some special kind of mixtures, namely scale and power mixtures. It is shown that the stop-loss criterion for ordering of risks plays an important role for ordering the mixing distribution.     

Aktuareksamen ved Universitetet i Oslo

Abstract

It is a fact that when one is making a decision concerning the probability distribution of a random variable by means of observing this random variable, one is recommended by the statisticians to consider certain functions of the operating characteristic (O.C.) of the decision function as measures of the reliability of the actual decision made. For instance, the confidence coefficient of an interval estimator will as a rule be regarded as a measure of our confidence in the interval.     

Solvable local and stochastic volatility models: supersymmetric methods in option pricing

In this paper we provide an extensive classification of one- and two-dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black–Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions. These solutions are given by a sum of hypergeometric functions, generalizing the results obtained in a paper by Albanese et al. (Albanese, C., Campolieti, G., Carr, P. and Lipton, A., Black–Scholes goes hypergeometric. Risk Mag., 2001, 14, 99–103). For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3?/?2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class.     

Distribution of linear functions of ordered rectangular variates

Abstract

The derivations of the distributions of functions of ordered statistics are complicated due to the fact that the integrals occurring in the derivation theory are to be evaluated over ordered ranges of variables of integration. The manipulations in such cases are tedious and involved. This difficulty is sometimes partially or completely obviated by transforming the ordered variates to the unordered ones as done by Kabe [2, 3], amongst several authors. Also the given distribution problem may be transformed to an equivalent one in some other variates which are easy to handle, see e.g., Laurent [5]. Since we are adopting Laurent's procedure in this paper we outline it briefly.     

Application of Risk Theory to Interpretation of Stochastic Cash-Flow-Testing Results

Abstract

In this paper we consider the Sparre Andersen insurance risk model. Three cases are discussed: the ordinary renewal risk process, stationary renewal risk process, and s-delayed renewal risk process. In the first part of the paper we study the joint distribution of surplus immediately before and at ruin under the renewal insurance risk model. By constructing an exponential martingale, we obtain Lundberg-type upper bounds for the joint distribution. Consequently we obtain bounds for the distribution of the deficit at ruin and ruin probability. In the second part of the paper, we consider the special case of phase-type claims and rederive the closed-form expression for the distribution of the severity of ruin, obtained by Drekic et al. (2003, 2004). Finally, we present some numerical results to illustrate the tightness of the bounds obtained in this paper.     

Some results on the compound Markov binomial model

Abstract

Some years ago, in the course of an analysis of upper and lower limits for incomplete moments of statistical distributions I established an elementary summation formula¹ which proved rather useful for the purpose I had in view. Subsequently the formula was generalized by professor Steffensen, who showed² that the formula in question could be looked upon as giving the first term of an expansion in a certain type of series. Professor Steffensen established recurrence formulae for the coefficients of the series and computed the second, third and fourth term and the corresponding remainders¹, but did not arrive at a general, explicite expression for the coefficient of the n-th term and the corresponding remainder. A year later I found these expressions accidentally while I was working on some other problem. I also discovered the real nature of the procedure in question which proved to be a certain kind of least square fitted polynomial approximation. I did not, however, at the time publish the result. Taking the question up again later I found that the whole problem could be considerably generalized. The type of generalization in question is analogous to the generalization from polynomials to arbitrary functions.     

From the Minimal Entropy Martingale Measures to the Optimal Strategies for the Exponential Utility Maximization: the Case of Geometric Lévy Processes

In this article, we will consider a multi-dimensional geometric L'evy process as a financial market model. We will first determine the minimal entropy martingale measure (MEMM); we will next derive the optimal strategy for the exponential utility maximization of terminal wealth concretely from the representation of the MEMM. JEL Classification: D46, D52, G12 AMS (2000) Subject Classification: 60G44, 60G51, 60G52,60H20, 60J75, 91B16, 91B28, 94A17     

A unified approach to ruin probabilities with delays for spectrally negative Lévy processes

ABSTRACT

In this paper, we unify two popular approaches for the definition of actuarial ruin with implementation delays, also known as Parisian ruin. Our new definition of ruin includes both deterministic delays and exponentially distributed delays: ruin is declared the first time an excursion in the red zone lasts longer than an implementation delay with a deterministic and a stochastic component. For this Parisian ruin with mixed delays, we identify the joint distribution of the time of ruin and the deficit at ruin, therefore providing generalizations of many results previously obtained, such as in Baurdoux et al. (2016) and Loeffen et al. (in press) for the case of an exponential delay and that of a deterministic delay, respectively.     

Non-exponential Bounds for Ruin Probability with Interest Effect Included

In this paper, we consider a discrete time risk model. First we discuss the classical model, both exponential and non-exponential upper bounds for ruin probabilities are obtained by using martingale inequalities. Then similar results are obtained for the model with investment income.     

Sequential Quasi-Credibility for Scale Dispersion Models

The sequential approach to credibility, developed by Landsman and Makov [(1999a) On stochastic approximation and credibility. Scand. Actuarial J. 1, 15-31; (1999b) Sequential credibility evaluation for symmetric location claim distributions. Insurance: Math. Econ. 24, 291-300] is extended to the scale dispersion family, which contains distributions often used in actuarial science: log-normal, Weibull, Half normal, Stable, Pareto, to mention only a few. For members of this family a sequential quasi-credibility formula is devised, which can also be used for heavy tailed claims. The results are illustrated by a study of log-normal claims.