Sarmanov Distribution Research Articles

Asymptotics for ruin probabilities of a dependent delayed-claim risk model with general investment returns and diffusion

In this paper, we study a delayed-claim insurance risk model perturbed by diffusion with general investment returns, in which each main claim may induce a delayed claim. Assume that the main claim sizes follow a one-sided linear process with independent and identically distributed step sizes. Furthermore, we assume that the step sizes and the inter-arrival times form a sequence of independent and identically distributed random pairs, with each pair obeying a bivariate Sarmanov distribution, and so do the delayed claim sizes and corresponding delayed times. In the presence of heavy tails, asymptotic upper and lower bounds for ruin probabilities are obtained. Finally, in order to verify the accuracy of our results, we conduct numerical simulations by Crude Monte Carlo (CMC) method.

Compound Poisson–Lindley process with Sarmanov dependence structure and its application for premium-based spectral risk forecasting

One of the fundamental challenges insurance companies face is forecasting a fair premium that covers the cost of claims while maintaining profitability. To comprehend the risk of insurance claims, one needs to construct a collective risk model. In this study, we aim to propose a new collective risk model, namely a dependent compound Poisson–Lindley process. We capture the dependence structure between the claim frequency and severity using a bivariate Sarmanov distribution. We then employ this model to perform premium-based risk measure forecasting such that the insurance company can obtain the premium value for the policyholder. We accomplish this task by proposing a specific risk spectrum to adjust the premium for risk-aversion-type insurance companies. Our main results demonstrate that a collective risk model based on the Poisson–Lindley process is able to capture the overdispersion phenomenon in claim frequency that is common in practice. This ability is confirmed by our simulation conducted using real insurance data. Furthermore, when accounting for the Sarmanov dependence structure, the resulting premium value becomes more appropriate.

Rank-Based Multivariate Sarmanov for Modeling Dependence between Loss Reserves

The interdependence between multiple lines of business has an important impact on determining loss reserves and risk capital, which are crucial for the solvency of a property and casualty (P&C) insurance company. In this work, we introduce the two-stage inference method using the Sarmanov family of multivariate distributions to the actuarial literature. In fact, we study rank-based methods using the Sarmanov distribution to adequately estimate the loss reserves and properly capture the dependence between lines of business. An inadequate choice of the dependence structure may negatively impact the estimation of the marginals and, hence, the reserve. Thus, we propose a two-stage inference strategy in this research to address this, while taking advantage of the flexibility of the Sarmanov distribution. We show that this strategy leads to a more robust estimation, and better captures the dependence between the risks. We also show that it generates smaller risk capital and a better diversification benefit. We extend the model to the multivariate case with more than two lines of business. To illustrate and validate our methods, we use three different sets of real data from both a major US property–casualty insurer and a large Canadian insurance company.

Valuing Equity-Linked Death Benefits on Multiple Life with Time until Death following a K n Distribution

The purpose of this paper is to investigate the valuation of equity-linked death benefit contracts and the multiple life insurance on two heads based on a joint survival model. Using the exponential Wiener process assumption for the stock price process and a K n distribution for the time until death, we provide explicit formulas for the expectation of the discounted payment of the guaranteed minimum death benefit products, and we derive closed expressions for some options and numerical illustrations. We investigate multiple life insurance based on a joint survival using the bivariate Sarmanov distribution with K n (i.e., the Laplace transform of their density function is a ratio of two polynomials of degree at most) marginal distributions. We present analytical results of the joint-life status.

Sarmanov distribution for modeling dependence between the frequency and the average severity of insurance claims

Real data studies emphasized situations where the classical independence assumption between the frequency and the severity of claims does not hold in the collective model. Therefore, there is an increasing interest in defining models that capture this dependence. In this paper, we introduce such a model based on Sarmanov's bivariate distribution, which has the ability of joining different types of marginals in flexible dependence structures. More precisely, we join the claims frequency and the average severity by means of this distribution. We also suggest a maximum likelihood estimation procedure to estimate the parameters and illustrate it both on simulated and real data.

Multivariate INAR(1) Regression Models Based on the Sarmanov Distribution

A multivariate INAR(1) regression model based on the Sarmanov distribution is proposed for modelling claim counts from an automobile insurance contract with different types of coverage. The correlation between claims from different coverage types is considered jointly with the serial correlation between the observations of the same policyholder observed over time. Several models based on the multivariate Sarmanov distribution are analyzed. The new models offer some advantages since they have all the advantages of the MINAR(1) regression model but allow for a more flexible dependence structure by using the Sarmanov distribution. Driven by a real panel data set, these models are considered and fitted to the data to discuss their goodness of fit and computational efficiency.

Bivariate Mixed Poisson and Normal Generalised Linear Models with Sarmanov Dependence—An Application to Model Claim Frequency and Optimal Transformed Average Severity

The aim of this paper is to introduce dependence between the claim frequency and the average severity of a policyholder or of an insurance portfolio using a bivariate Sarmanov distribution, that allows to join variables of different types and with different distributions, thus being a good candidate for modeling the dependence between the two previously mentioned random variables. To model the claim frequency, a generalized linear model based on a mixed Poisson distribution -like for example, the Negative Binomial (NB), usually works. However, finding a distribution for the claim severity is not that easy. In practice, the Lognormal distribution fits well in many cases. Since the natural logarithm of a Lognormal variable is Normal distributed, this relation is generalised using the Box-Cox transformation to model the average claim severity. Therefore, we propose a bivariate Sarmanov model having as marginals a Negative Binomial and a Normal Generalized Linear Models (GLMs), also depending on the parameters of the Box-Cox transformation. We apply this model to the analysis of the frequency-severity bivariate distribution associated to a pay-as-you-drive motor insurance portfolio with explanatory telematic variables.

A Sarmanov Distribution with Beta Marginals: An Application to Motor Insurance Pricing

Background: The Beta distribution is useful for fitting variables that measure a probability or a relative frequency. Methods: We propose a Sarmanov distribution with Beta marginals specified as generalised linear models. We analyse its theoretical properties and its dependence limits. Results: We use a real motor insurance sample of drivers and analyse the percentage of kilometres driven above the posted speed limit and the percentage of kilometres driven at night, together with some additional covariates. We fit a Beta model for the marginals of the bivariate Sarmanov distribution. Conclusions: We find negative dependence in the high quantiles indicating that excess speed and night-time driving are not uniformly correlated.

Frequency and Severity Dependence in the Collective Risk Model: An Approach Based on Sarmanov Distribution

In actuarial mathematics, the claims of an insurance portfolio are often modeled using the collective risk model, which consists of a random number of claims of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent cost per claim. To facilitate computations, there is a classical assumption of independence between the random number of such random variables (i.e., the claims frequency) and the random variables themselves (i.e., the claim severities). However, recent studies showed that, in practice, this assumption does not always hold, hence, introducing dependence in the collective model becomes a necessity. In this sense, one trend consists of assuming dependence between the number of claims and their average severity. Alternatively, we can consider heterogeneity between the individual cost of claims associated with a given number of claims. Using the Sarmanov distribution, in this paper we aim at introducing dependence between the number of claims and the individual claim severities. As marginal models, we use the Poisson and Negative Binomial (NB) distributions for the number of claims, and the Gamma and Lognormal distributions for the cost of claims. The maximum likelihood estimation of the proposed Sarmanov distribution is discussed. We present a numerical study using a real data set from a Spanish insurance portfolio.

On a class of bivariate mixed Sarmanov distributions

SummaryMultivariate distributions are more and more used to model the dependence encountered in many fields. However, classical multivariate distributions can be restrictive by their nature, while Sarmanov's multivariate distribution, by joining different marginals in a flexible and tractable dependence structure, often provides a valuable alternative. In this paper, we introduce some bivariate mixed Sarmanov distributions with the purpose to extend the class of bivariate Sarmanov distributions and to obtain new dependency structures. Special attention is paid to the bivariate mixed Sarmanov distribution with Poisson marginals and, in particular, to the resulting bivariate Sarmanov distributions with negative binomial and with Poisson‐inverse Gaussian marginals; these particular types of mixed distributions have possible applications in, for example modelling bivariate count data. The extension to higher dimensions is also discussed. Moreover, concerning the dependency structure, we also present some correlation formulas.

A Note on Randomly Weighted Sums of Dependent Subexponential Random Variables

The author obtains that the asymptotic relations $$\mathbb{P}\left( {\sum\limits_{i = 1}^n {{\theta _i}{X_i}} >x} \right) \sim \mathbb{P}\left( {\mathop {\max }\limits_{1 \le m \le n} \sum\limits_{i = 1}^m {{\theta _i}{X_i}} >x} \right) \sim \mathbb{P}\left( {\mathop {\max {\theta _i}{X_i}}\limits_{1 \le i \le n} >x} \right) \sim \sum\limits_{i = 1}^n \mathbb{P}{\left( {{\theta _i}{X_i} >x} \right)}$$ hold as x → ∞, where the random weights θ1,..., θn are bounded away both from 0 and from ∞ with no dependency assumptions, independent of the primary random variables X1,..., Xn which have a certain kind of dependence structure and follow non-identically subexponential distributions. In particular, the asymptotic relations remain true when X1,..., Xn jointly follow a pairwise Sarmanov distribution.

Interplay of financial and insurance risks in dependent discrete-time risk models

Consider a discrete-time insurance risk model with two kinds of risks, i.e., insurance and financial risks. Within period i, the real-valued net insurance loss caused by traditional claims is regarded as the insurance risk, denoted by Xi, and the positive stochastic discount factor over the same time period is the financial risk, denoted by Yi. Assume that {(X,Y),(Xi,Yi),i≥1} form a sequence of independent and identically distributed random vectors. This work investigates the interplay of the insurance and financial risks, allowing a dependence structure exists between these two kinds of risks. Following the work of Li and Tang (2015), we derive some asymptotic and uniformly asymptotic formulas for the finite-time and infinite-time ruin probabilities, under the assumption that the generic random vector (X,Y) follows a bivariate Sarmanov distribution and every convex combination of the distributions of X and Y is of strongly regular variation. As an extension, we further consider a general pair (X,Y), whose dependence structure is more verifiable than the Sarmanov one to some extent. In such a setting, we study the asymptotic tail behavior of the product XY, which forms an analogue of the well-known Breiman’s theorem in a different and dependent situation.

Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution

Starting from the question: What is the accident risk of an insured individual?, we consider that the customer has contracted policies in different insurance lines: motor and home. Three models based on the multivariate Sarmanov distribution are analyzed. Driven by a real data set that takes into account three types of accident risks, two for motor and one for home, three trivariate Sarmanov distributions with generalized linear models (GLMs) for marginals are considered and fitted to the data. To estimate the parameters of these three models, we discuss a method for approaching the maximum likelihood (ML) estimators. Finally, the three models are compared numerically with the simpler trivariate Negative Binomial GLM and with elliptical copula based models.

Uniform asymptotics for ruin probabilities in a two-dimensional nonstandard renewal risk model with stochastic returns

In this paper, we consider a two-dimensional nonstandard renewal risk model with stochastic returns, in which the two lines of claim sizes form a sequence of independent and identically distributed random vectors following a bivariate Sarmanov distribution, and the two claim-number processes satisfy a certain dependence structure. When the two marginal distributions of the claim-size vector belong to the intersection of the dominated-variation class and the class of long-tailed distributions, we obtain uniform asymptotic formulas of finite-time and infinite-time ruin probabilities.

On a Class of Bivariate Mixed Sarmanov Distributions

In this paper, we extend the class of bivariate Sarmanov distributions by introducing some bivariate mixed Sarmanov distributions. Special attention is paid to the bivariate mixed Sarmanov distribution with Poisson marginals and, in particular, to the resulting bivariate Sarmanov distribution with negative binomial and Poisson-Inverse Gaussian marginals; these particular types of mixed distributions have possible applications in modelling bivariate count data. The extension to higher dimensions is also discussed. Moreover, since one of the purposes of mixing is to obtain new dependency structures, we also present some correlation formulas.

Asymptotics for the finite-time ruin probability in a discrete-time risk model with dependent insurance and financial risks*

We consider a discrete-time risk model with insurance and financial risks. Within period i ≥ 1, the real-valued net insurance loss caused by claims is the insurance risk, denoted by X i , and the positive stochastic discount factor over the same time period is the financial risk, denoted by Y i . Assume that {(X, Y), (X i , Y i ), i ≥ 1} form a sequence of independent identically distributed random vectors. In this paper, we investigate a discrete-time risk model allowing a dependence structure between the two risks. When (X, Y ) follows a bivariate Sarmanov distribution and the distribution of the insurance risk belongs to the class ℒ(γ) for some γ > 0, we derive the asymptotics for the finite-time ruin probability of this discrete-time risk model.

Multivariate Count Data Generalized Linear Models: Three Approaches Based on the Sarmanov Distribution

Starting from the question: “What is the accident risk of an insured?”, this paper considers a multivariate approach by taking into account three types of accident risks and the possible dependence between them. Driven by a real data set, we propose three trivariate Sarmanov distributions with generalized linear models (GLMs) for marginals and incorporate various individual characteristics of the policyholders by means of explanatory variables. Since the data set was collected over a longer time period (10 years), we also added each individual’s exposure to risk. To estimate the parameters of the three Sarmanov distributions, we analyze a pseudo-maximumlikelihood method. Finally, the three models are compared numerically with the simpler trivariate Negative Binomial GLM.

Asymptotic results for ruin probability in a two-dimensional risk model with stochastic investment returns

This paper considers a two-dimensional time-dependent risk model with stochastic investment returns. In the model, an insurer operates two lines of insurance businesses sharing a common claim number process and can invest its surplus into some risky assets. The claim number process is assumed to be a renewal counting process and the investment return is modeled by a geometric Lévy process. Furthermore, claim sizes of the two insurance businesses and their common inter-arrival times correspondingly follow a three-dimensional Sarmanov distribution. When claim sizes of the two lines of insurance businesses are heavy tailed, we establish some uniform asymptotic formulas for the ruin probability of the model over certain time horizon. Also, we show the accuracy of these asymptotic estimates for the ruin probability under the risk model by numerical studies.

On Some Multivariate Sarmanov Mixed Erlang Reinsurance Risks: Aggregation and Capital Allocation

Following some recent works on risk aggregation and capital allocation for mixed Erlang risks joined by Sarmanov's multivariate distribution, in this paper we present some closed-form formulas for the same topic by considering, however, a different kernel function for Sarmanov's distribution, not previously studied in this context. The risk aggregation and capital allocation formulas are derived and numerically illustrated in the general framework of stop-loss reinsurance, and then in the particular case with no stop-loss reinsurance. A discussion of the dependency structure of the considered distribution, based on Pearson's correlation coefficient, is also presented for different kernel functions and illustrated in the bivariate case.

On the Evaluation of Some Multivariate Compound Distributions with Sarmanov's Counting Distribution

In this paper, we consider Sarmanov's multivariate discrete distribution as counting distribution in two multivariate compound models: the First model assumes different types of independent claim sizes (corresponding to, e.g., different types of insurance policies), while in the second model, we introduce some dependency between the claims (motivated by the events that can simultaneously affect several types of policies). Since Sarmanov's distribution can join different types of marginals, we also assume that these marginals belong to Panjer's class of distributions and discuss the evaluation of the resulting compound distribution based on recursions. Alternatively, the evaluation of the same distribution using the Fast Fourier Transform method is also presented, with the purpose to significantly reduce the computing time, especially in the dependency case. Both methods are numerically illustrated and compared from the point of view of speed and accuracy.