On a family of counting distributions and recursions for related compound distributions

Abstract

This paper considers a family of counting distributions whose densities satisfy certain second order difference equations. Recursions for the evaluation of related compound distributions are developed in the case of severity distributions which are concentrated on the non-negative integers. From these a characterization of the considered counting distributions is obtained, and it is shown that most of these are compound Poisson distributions.     

Two-Sided Bounds for Tails of Compound Negative Binomial Distributions in the Exponential and Heavy-Tailed Cases

This paper derives two-sided bounds for tails of compound negative binomial distributions, both in the exponential and heavy-tailed cases. Two approaches are employed to derive the two-sided bounds in the case of exponential tails. One is the convolution technique, as in Willmot & Lin (1997). The other is based on an identity of compound negative binomial distributions; they can be represented as a compound Poisson distribution with a compound logarithmic distribution as the underlying claims distribution. This connection between the compound negative binomial, Poisson and logarithmic distributions results in two-sided bounds for the tails of the compound negative binomial distribution, which also generalize and improve a result of Willmot & Lin (1997). For the heavy-tailed case, we use the method developed by Cai & Garrido (1999b). In addition, we give two-sided bounds for stop-loss premiums of compound negative binomial distributions. Furthermore, we derive bounds for the stop-loss premiums of general compound distributions among the classes of HNBUE and HNWUE.     

Comparison of Methods for Evaluation of the Convolution of Two Compound R 1 Distributions

In the present paper we compare four methods for evaluating the convolution of two compound R 1 distributions by counting the numbers of elementary algebraic operations required. Two of the methods are applicable in general, whereas the remaining two are restricted to the case when the two compound distributions have the same severity distribution. This case is discussed separately. We consider in particular the special case when this common severity distribution is concentrated in one, that is, evaluation of the convolution of two R 1 distributions.     

Optimal portfolio positioning within generalized Johnson distributions

Many empirical studies have shown that financial asset returns do not always exhibit Gaussian distributions, for example hedge fund returns. The introduction of the family of Johnson distributions allows a better fit to empirical financial data. Additionally, this class can be extended to a quite general family of distributions by considering all possible regular transformations of the standard Gaussian distribution. In this framework, we consider the portfolio optimal positioning problem, which has been first addressed by Brennan and Solanki [J. Financial Quant. Anal., 1981, 16, 279–300], Leland [J. Finance, 1980, 35, 581–594] and further developed by Carr and Madan [Quant. Finance, 2001, 1, 9–37] and Prigent [Generalized option based portfolio insurance. Working Paper, THEMA, University of Cergy-Pontoise, 2006]. As a by-product, we introduce the notion of Johnson stochastic processes. We determine and analyse the optimal portfolio for log return having Johnson distributions. The solution is characterized for arbitrary utility functions and illustrated in particular for a CRRA utility. Our findings show how the profiles of financial structured products must be selected when taking account of non Gaussian log-returns.     

Unconditional distributions obtained from conditional specification models with applications in risk theory

Bivariate distributions, specified in terms of their conditional distributions, provide a powerful tool to obtain flexible distributions. These distributions play an important role in specifying the conjugate prior in certain multi-parameter Bayesian settings. In this paper, the conditional specification technique is applied to look for more flexible distributions than the traditional ones used in the actuarial literature, as the Poisson, negative binomial and others. The new specification draws inferences about parameters of interest in problems appearing in actuarial statistics. Two unconditional (discrete) distributions obtained are studied and used in the collective risk model to compute the right-tail probability of the aggregate claim size distribution. Comparisons with the compound Poisson and compound negative binomial are made.     

Ruin Problems for Phase-Type(2) Risk Processes

In this paper we consider a risk process in which claim inter-arrival times have a phase-type(2) distribution, a distribution with a density satisfying a second order linear differential equation. We consider some ruin related problems. In particular, we consider the compound geometric representation of the infinite time survival probability, as well as the (defective) distributions of the surplus immediately prior to ruin and of the deficit at ruin. We also consider explicit solutions for the infinite time ruin probability in the case where the individual claim amount distribution is phase-type.     

Two-Sided Bounds for Ruin Probabilities when the Adjustment Coefficient does not Exist

In this paper, we derive two-sided bounds for the ruin probability in the compound Poisson risk model when the adjustment coefficient of the individual claim size distribution does not exist. These bounds also apply directly to the tails of compound geometric distributions. The upper bound is tighter than that of Dickson (1994). The corresponding lower bound, which holds under the same conditions, is tighter than that of De Vylder and Goovaerts (1984). Even when the adjustment coefficient exists, the upper bound is, in some cases, tighter than Lundberg's bound. These bounds are applicable for any positive distribution function with a finite mean. Examples are given and numerical comparisons with asymptotic formulae for the ruin probability are also considered.     

On Mixed and Compound Mixed Poisson Distributions

Recursive formulae are derived for the evaluation of the t-th order cumulative distribution function and the t-th order tail probability of compound mixed Poisson distributions in the case where the derivative of the logarithm of the mixing density can be written as a ratio of polynomials. Also, some general results are derived for the evaluation of the t-th order moments of stop-loss transforms. The recursions can be applied for the exact evaluation of the probability function, distribution function, tail probability and stop-loss premium of compound mixed Poisson distributions and the corresponding mixed Poisson distributions. Several examples are also presented.     

Valuación de opciones sobre subyacentes con rendimientos α-estables

In this work, we analyze the log-stable option pricing model, we estimate the parameters of the distribution of the peso-dollar exchange depreciation rate through the methods: 1) maximum likelihood, 2) tabulated quantiles of α-stable distributions and 3) regression on the sample characteristic function; we conducted a qualitative analysis to show the quality of the distribution’s fit and through a quantitative analysis we chose the best α-parameters estimation and we compare the McCulloch (2003) log-stable option pricing model with the Black and Scholes (1973) log-normal model and a MexDer’s prices vector; finally, we show that the log-stable model has advantages over the log-normal model.     

The negative news threshold—An explanation for negative skewness in stock returns

Abstract

A vast literature documents negative skewness in stock index return distributions on several markets. In this paper the issue of negative skewness is approached from a different angle to previous studies by combining the Trueman's 1997 model of management disclosure practices with symmetric market responses in order to explain negative skewness in stock returns. Empirical tests reveal that returns for days when non-scheduled news items are disclosed are the source of negative skewness in stock returns, as predicted. These findings suggest that negative skewness in stock returns is induced by asymmetries in the news disclosure policies of firm management. Furthermore, it is found that the returns are negatively skewed only for non-scheduled firm-specific news disclosures for firms where the management is compensated with stock options.     

Moments of Compound Mixed Poisson Distributions

Recursive formulae are derived for the evaluation of the moments and the descending factorial moments about a point n of mixed Poisson and compound mixed Poisson distributions, in the case where the derivative of the logarithm of the mixing density can be written as a ratio of polynomials. As byproduct, we also obtain recursive formulae for the evaluation of the moments about the origin, central moments, descending and ascending factorial moments of these distributions. Examples are also presented for a number of mixing densities.     

On definition of compound poisson processes

Abstract

Some authors define the (elementary) compound Poisson process in wide sense {χ _t , 0 ? t < ∞} with help of probability distributions where τ is a so-called operational time, a continuous non-decreasing function of t vanishing for t = 0, and V(q, t) is a non-negative distribution function for every t.     

Moment generating functions of compound renewal sums with discounted claims

Léveillé & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered.

In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.     

Seasonal mortality for fractional ages in short term life insurance

Abstract

Fernández-Durán, and Gregorio-Domínguez, Seasonal Mortality for Fractional Ages in Life Insurance. Scandinavian Actuarial Journal. A uniform distribution of deaths between integral ages is a widely used assumption for estimating future-lifetimes; however, this assumption does not necessarily reflect the true distribution of deaths throughout the year. We propose the use of a seasonal mortality assumption for estimating the distribution of future-lifetimes between integral ages: this assumption accounts for the number of deaths that occurs in given months of the year, including the excess mortality that is observed in winter months. The impact of this seasonal mortality assumption on short-term life insurance premium calculations is then examined by applying the proposed assumption to Mexican mortality data.     

On the Moments of the Time of Ruin with Applications to Phase-Type Claims

Abstract

We describe an approach to the evaluation of the moments of the time of ruin in the classical Poisson risk model. The methodology employed involves the expression of these moments in terms of linear combinations of convolutions involving compound negative binomial distributions. We then adapt the results for use in the practically important case involving phase-type claim size distributions. We present numerical examples to illuminate the influence of claim size variability on the moments of the time of ruin.     

Multivariate affine generalized hyperbolic distributions: An empirical investigation

The aim of this paper is to estimate multivariate affine generalized distributions (MAGH) using market data. We use the Ibovespa, CAC, DAX, FTSE, NIKKEI and S&P500 indexes. We estimate the univariate distributions, bi-variate distributions and six-dimensional distribution. Then we assess their goodness of fit using Kolmogorov distances. As an application we study the efficient frontier.     

On the Joint Distributions of the Time to Ruin,the Surplus Prior to Ruin,and the Deficit at Ruin in the Classical Risk Model

Abstract

The seminal paper by Gerber and Shiu (1998) unified and extended the study of the event of ruin and related quantities, including the time at which the event of ruin occurs, the deficit at the time of ruin, and the surplus immediately prior to ruin. The first two of these quantities are fundamentally important for risk management techniques that utilize the ideas of Value-at-Risk and Tail Value-at-Risk. As is well known, calculation of these and related quantities requires knowledge of the associated probability distributions. In this paper we derive an explicit expression for the joint (defective) distribution of the time to ruin, the surplus immediately prior to ruin, and the deficit at ruin in the classical compound Poisson risk model. As a by-product, we obtain expressions for the three bivariate distributions generated by the time to ruin, the surplus prior to ruin, and the deficit at ruin. Finally, we consider mixed Erlang claim sizes and show how the joint (defective) distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin can be calculated.     

A Direct Approach to the Discounted Penalty Function

Abstract

This paper provides a new and accessible approach to establishing certain results concerning the discounted penalty function. The direct approach consists of two steps. In the first step, closed-form expressions are obtained in the special case in which the claim amount distribution is a combination of exponential distributions. A rational function is useful in this context. For the second step, one observes that the family of combinations of exponential distributions is dense. Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus process has downward and upward jumps, modeled by two independent compound Poisson processes. If the distribution of the upward jumps is exponential, a series of new results can be obtained with ease. Subsequently, certain results of Gerber and Shiu [H. U. Gerber and E. S. W. Shiu, North American Actuarial Journal 2(1): 48–78 (1998)] can be reproduced. The two-step approach is also applied when an independent Wiener process is added to the surplus process. Certain results are related to Zhang et al. [Z. Zhang, H. Yang, and S. Li, Journal of Computational and Applied Mathematics 233: 1773–1 784 (2010)], which uses different methods.     

Extending composite loss models using a general framework of advanced computational tools

ABSTRACT

Composite models have a long history in actuarial science because they provide a flexible method of curve-fitting for heavy-tailed insurance losses. The ongoing research in this area continuously suggests methodological improvements for existing composite models and considers new composite models. A number of different composite models have been previously proposed in the literature to fit the popular data set related to Danish fire losses. This paper provides the most comprehensive analysis of composite loss models on the Danish fire losses data set to date by evaluating 256 composite models derived from 16 parametric distributions that are commonly used in actuarial science. If not suitably addressed, inevitable computational challenges are encountered when estimating these composite models that may lead to sub-optimal solutions. General implementation strategies are developed for parameter estimation in order to arrive at an automatic way to reach a viable solution, regardless of the specific head and/or tail distributions specified. The results lead to an identification of new well-fitting composite models and provide valuable insights into the selection of certain composite models for which the tail-evaluation measures can be useful in making risk management decisions.