scholarly journals CMPH: a multivariate phase-type aggregate loss distribution

2017 ◽  
Vol 5 (1) ◽  
pp. 304-315 ◽  
Author(s):  
Jiandong Ren ◽  
Ricardas Zitikis
Keyword(s):  
Negative Binomial ◽  
Risk Measures ◽  
Moment Generating Function ◽  
Insurance Companies ◽  
Joint Distributions ◽  
Special Cases ◽  
Risk Sources ◽  
Computational Aspects ◽  
Aggregate Loss ◽  
Aggregate Loss Distribution

AbstractWe introduce a compound multivariate distribution designed for modeling insurance losses arising from different risk sources in insurance companies. The distribution is based on a discrete-time Markov Chain and generalizes the multivariate compound negative binomial distribution, which is widely used for modeling insurance losses.We derive fundamental properties of the distribution and discuss computational aspects facilitating calculations of risk measures of the aggregate loss, as well as allocations of the aggregate loss to individual types of risk sources. Explicit formulas for the joint moment generating function and the joint moments of different loss types are derived, and recursive formulas for calculating the joint distributions given. Several special cases of particular interest are analyzed. An illustrative numerical example is provided.

scholarly journals Pareto-Optimal Risk Exchanges and Related Decision Problems

1978 ◽  
Vol 10 (1) ◽  
pp. 25-33 ◽  
Author(s):  
Hans U. Gerber
Keyword(s):  
Applied Mathematics ◽  
Weighted Average ◽  
Insurance Companies ◽  
Pareto Optimal ◽  
Fixed Amount ◽  
Simple Method ◽  
Optimal Decisions ◽  
Special Cases ◽  
Aggregate Loss ◽  
Stop Loss

In various branches of applied mathematics the problem arises of making decisions to reconcile conflicting criteria. One example is the classical statistical problem, where a type 1 error cannot be arbitrarily reduced without increasing the probability for a type 2 error. Another example, quite familiar to actuaries, is graduation, where a compromise between smoothness and fit has to be reached. This motivates the concept of Pareto-optimal decisions, which is discussed in section 2. There is a simple method, maximizing a weighted average of the scores, to obtain certain Pareto-optimal decisions. In section 3 a condition is given, which is satisfied in most applications, that guarantees that all the Pareto-optimal decisions can be found by this method. This is applied in section 4, where the problem of risk exchange between n insurance companies is considered. The original model of Borch is generalized: it is assumed that some of the companies are not willing to contribute more than a certain fixed amount towards the aggregate loss of the other companies. The theorem in section 4 gives a characterization of all the Pareto-optimal risk exchanges. Because of the restrictions, these risk exchanges do not just depend on the combined surplus (which would amount to pooling) in general, and can be found by an algorithm. One benefit of this generalization of Borch's Theorem is that two seemingly unrelated results (optimality of a stop loss contract, and optimality of certain dividend formulas in group insurance) follow from it as special cases.

scholarly journals Medidas de riesgo para riesgo operacional con un modelo de perdida agregada de Poisson-Lindley

2010 ◽  
pp. 1
Author(s):  
Agustín Hernández Bastida ◽  
Pilar Fernández Sánchez
Keyword(s):  
Risk Measures ◽  
Predictive Distribution ◽  
Loss Distribution ◽  
Gamma Distributions ◽  
Loss Distribution Approach ◽  
Aggregate Loss ◽  
Aggregate Loss Distribution ◽  
High Level ◽  
Measurement Approach

En este trabajo se considera la determinación de medidas de riesgo en riesgo operacional, es decir, la determinación de cuantiles de alto orden. Se considera la aproximación basada en la distribución de la pérdida dentro de la aproximación avanzada. Se calculan, y se comparan entre si, las medidas de riesgo a partir de la distribución de la pérdida agregada y a partir de la distribución predictiva considerando como funciones estructura para los perfiles de riesgo las distribuciones Triangular y Gamma.<br /><br />This paper considers the determination of the risk measures in Operational Risk, i.e. the determination of a high level quantile. The Loss Distribution Approach in the Advanced Measurement Approach is adopted. The risk measures, obtained from the aggregate loss distribution and from the predictive distribution are determined and compared, using the Triangular and Gamma distributions as structure functions of the risk profiles.<br />

ANALISIS RISIKO OPERASIONAL MENGGUNAKAN PENDEKATAN DISTRIBUSI KERUGIAN DENGAN METODE AGREGAT

2011 ◽  
Vol 10 (2) ◽  
pp. 1
Author(s):  
Y. ARBI ◽  
R. BUDIARTI ◽  
I G. P. PURNABA
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Financial Institution ◽  
Expected Value ◽  
Insurance Companies ◽  
Limited Data ◽  
Loss Distribution Approach ◽  
Potential Loss ◽  
The Impact

Operational risk is defined as the risk of loss resulting from inadequate or failed internal processes or external problems. Insurance companies as financial institution that also faced at risk. Recording of operating losses in insurance companies, were not properly conducted so that the impact on the limited data for operational losses. In this work, the data of operational loss observed from the payment of the claim. In general, the number of insurance claims can be modelled using the Poisson distribution, where the expected value of the claims is similar with variance, while the negative binomial distribution, the expected value was bound to be less than the variance.Analysis tools are used in the measurement of the potential loss is the loss distribution approach with the aggregate method. In the aggregate method, loss data grouped in a frequency distribution and severity distribution. After doing 10.000 times simulation are resulted total loss of claim value, which is total from individual claim every simulation. Then from the result was set the value of potential loss (OpVar) at a certain level confidence.

scholarly journals The Stability of the Aggregate Loss Distribution

Risks ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 91 ◽  
Author(s):  
Riccardo Gatto
Keyword(s):  
Stability Analysis ◽  
Survival Function ◽  
Saddlepoint Approximation ◽  
Computational Formula ◽  
Important Indicator ◽  
The Common ◽  
Infinitesimal Perturbation ◽  
Aggregate Loss ◽  
Aggregate Loss Distribution ◽  
The Stability

In this article we introduce the stability analysis of a compound sum: it consists of computing the standardized variation of the survival function of the sum resulting from an infinitesimal perturbation of the common distribution of the summands. Stability analysis is complementary to the classical sensitivity analysis, which consists of computing the derivative of an important indicator of the model, with respect to a model parameter. We obtain a computational formula for this stability from the saddlepoint approximation. We apply the formula to the compound Poisson insurer loss with gamma individual claim amounts and to the compound geometric loss with Weibull individual claim amounts.

scholarly journals Multivariate Tail Moments for Log-Elliptical Dependence Structures as Measures of Risks

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 559
Author(s):  
Zinoviy Landsman ◽  
Tomer Shushi
Keyword(s):  
Risk Measures ◽  
Short Paper ◽  
Dependence Structure ◽  
Elliptical Distributions ◽  
Tail Conditional Expectation ◽  
Special Cases ◽  
Optimal Portfolio Selection ◽  
Multivariate Risk Measures ◽  
Elliptical Models ◽  
Risk Managers

The class of log-elliptical distributions is well used and studied in risk measurement and actuarial science. The reason is that risks are often skewed and positive when they describe pure risks, i.e., risks in which there is no possibility of profit. In practice, risk managers confront a system of mutually dependent risks, not only one risk. Thus, it is important to measure risks while capturing their dependence structure. In this short paper, we compute the multivariate risk measures, multivariate tail conditional expectation, and multivariate tail covariance measure for the family of log-elliptical distributions, which captures the dependence structure of the risks while focusing on the tail of their distributions, i.e., on extreme loss events. We then study our result and examine special cases, as well as the optimal portfolio selection using such measures. Finally, we show how the given multivariate tail moments can also be computed for log-skew elliptical models based on similar approaches given for the log-elliptical case.

scholarly journals A New Discrete Analog of the Continuous Lindley Distribution, with Reliability Applications

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 603
Author(s):  
Abdulhakim A. Al-Babtain ◽  
Abdul Hadi N. Ahmed ◽  
Ahmed Z. Afify
Keyword(s):  
Goodness Of Fit ◽  
Negative Binomial ◽  
Real Data ◽  
Moment Generating Function ◽  
Discrete Analog ◽  
Bias Reduction ◽  
Discrete Distributions ◽  
Likelihood Method ◽  
Probability Mass Function ◽  
Lindley Distribution

In this paper, we propose and study a new probability mass function by creating a natural discrete analog to the continuous Lindley distribution as a mixture of geometric and negative binomial distributions. The new distribution has many interesting properties that make it superior to many other discrete distributions, particularly in analyzing over-dispersed count data. Several statistical properties of the introduced distribution have been established including moments and moment generating function, residual moments, characterization, entropy, estimation of the parameter by the maximum likelihood method. A bias reduction method is applied to the derived estimator; its existence and uniqueness are discussed. Applications of the goodness of fit of the proposed distribution have been examined and compared with other discrete distributions using three real data sets from biological sciences.

scholarly journals SOME DISTRIBUTIONAL PROPERTIES OF A CLASS OF COUNTING DISTRIBUTIONS WITH CLAIMS ANALYSIS APPLICATIONS

2013 ◽  
Vol 43 (2) ◽  
pp. 189-212 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jae-Kyung Woo
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Negative Binomial ◽  
Claims Analysis ◽  
Explicit Formulas ◽  
Distributional Properties ◽  
Special Cases ◽  
Aggregate Claims ◽  
Stop Loss

AbstractWe discuss a class of counting distributions motivated by a problem in discrete surplus analysis, and special cases of which have applications in stop-loss, discrete Tail value at risk (TVaR) and claim count modelling. Explicit formulas are developed, and the mixed Poisson case is considered in some detail. Simplifications occur for some underlying negative binomial and related models, where in some cases compound geometric distributions arise naturally. Applications to claim count and aggregate claims models are then given.

Joint distributions of successes, failures and patterns in enumeration problems

2000 ◽  
Vol 32 (3) ◽  
pp. 866-884 ◽  
Author(s):  
S Chadjiconstantinidis ◽  
D. L. Antzoulakos ◽  
M. V. Koutras
Keyword(s):  
Joint Distribution ◽  
Joint Probability ◽  
Fixed Number ◽  
Composite Pattern ◽  
Joint Distributions ◽  
Special Cases ◽  
Enumeration Problems ◽  
Binomial Type ◽  
Probability Mass ◽  
Mass Functions

Let ε be a (single or composite) pattern defined over a sequence of Bernoulli trials. This article presents a unified approach for the study of the joint distribution of the number Sn of successes (and Fn of failures) and the number Xn of occurrences of ε in a fixed number of trials as well as the joint distribution of the waiting time Tr till the rth occurrence of the pattern and the number STr of successes (and FTr of failures) observed at that time. General formulae are developed for the joint probability mass functions and generating functions of (Xn,Sn), (Tr,STr) (and (Xn,Sn,Fn),(Tr,STr,FTr)) when Xn belongs to the family of Markov chain imbeddable variables of binomial type. Specializing to certain success runs, scans and pattern problems several well-known results are delivered as special cases of the general theory along with some new results that have not appeared in the statistical literature before.

scholarly journals The Zeta-G Class: Some Computational and Analytical Aspects

2020 ◽  
Vol 42 ◽  
pp. e111
Author(s):  
Ana Carla Percontini ◽  
Frank Gomes-Silva ◽  
Gauss Moutinho Crdeiro ◽  
Pedro Rafael Marinho
Keyword(s):  
Maximum Likelihood ◽  
Generating Function ◽  
Real Data ◽  
Data Set ◽  
R Language ◽  
New Class ◽  
Mathematical Properties ◽  
Special Cases ◽  
Computational Aspects

We define a new class of distributions with one extra shapeparameter including some special cases. We provide numerical and computational aspects of the new class. We proposefunctions using the \textsf{R} language to fit any distribution in this family to a data set. In addition, such functions are implemented efficientlyusing the library \textsf{Rcpp} that enables the incorporation of the codes \textsf{C++} in \textsf{R} automatically. Some examples are presentedfor using the implemented routines in practice. We derive some mathematical properties of this class including explicit expressionsfor the moments, generating function and mean deviations. We discuss the estimation of the model parametersby maximum likelihood and provide an application to a real data set.

scholarly journals Pairwise Stable Matching in Large Economies

Econometrica ◽  
2021 ◽  
Vol 89 (6) ◽  
pp. 2929-2974 ◽  
Author(s):  
Michael Greinecker ◽  
Christopher Kah
Keyword(s):  
General Class ◽  
Stable Matching ◽  
High Dimensional ◽  
Transferable Utility ◽  
Stable Matchings ◽  
Matching Problems ◽  
Joint Distributions ◽  
Special Cases ◽  
Matching Models ◽  
Stability Notions

We formulate a stability notion for two‐sided pairwise matching problems with individually insignificant agents in distributional form. Matchings are formulated as joint distributions over the characteristics of the populations to be matched. Spaces of characteristics can be high‐dimensional and need not be compact. Stable matchings exist with and without transfers, and stable matchings correspond precisely to limits of stable matchings for finite‐agent models. We can embed existing continuum matching models and stability notions with transferable utility as special cases of our model and stability notion. In contrast to finite‐agent matching models, stable matchings exist under a general class of externalities.

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