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 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 Optimal Risk Exchanges

1979 ◽  
Vol 10 (3) ◽  
pp. 243-262 ◽  
Author(s):  
Hans Bühlmann ◽  
William S. Jewell
Keyword(s):  
Risk Sharing ◽  
Pareto Optimal ◽  
Individual Utility ◽  
Exponential Form ◽  
Economic Concept ◽  
Side Payments ◽  
Long Run ◽  
Special Cases ◽  
Stop Loss

The determination of optimal rules for sharing risks and constructing reinsurance treaties has important practical and theoretical interest. Medolaghi, de Finetti, and Ottaviani developed the first linear reciprocal reinsurance treaties based upon minimizing individual and aggregate variance of risk. Borch then used the economic concept of utility to justify choosing Pareto-optimal forms of risk exchange; in many cases, this leads to familiar linear quota-sharing of total pooled losses, or to stop-loss arrangements. However, this approach does not give a unique, risk-sharing agreement, and may lead to substantial fixed side payments. Gerber showed how to constrain a Pareto-optimal risk exchange to avoid invasion of reserves.To these ideas, the authors have added the actuarial concept of long-run fairness to each participant in the risk exchange; the result is a unique, Pareto-optimal risk pool, with “quota-sharing-by-layers” of the total losses. There are many interesting special cases, especially when all individual utility functions are of exponential form, giving linear quota-sharing-by-layers. Algorithms and numerical examples are given.

scholarly journals Constraints on the density distribution of type Ia supernovae ejecta inferred from late-time light-curve flattening

2020 ◽  
Vol 493 (4) ◽  
pp. 5617-5624
Author(s):  
Doron Kushnir ◽  
Eli Waxman
Keyword(s):  
Light Curve ◽  
Weighted Average ◽  
Far Infrared ◽  
Average Density ◽  
Light Curves ◽  
Type Ia Supernovae ◽  
Late Time ◽  
Simple Method ◽  
Type Ia ◽  
Ia Supernovae

ABSTRACT The finite time, τdep, over which positrons from β+ decays of 56Co deposit energy in type Ia supernovae ejecta lead, in case the positrons are trapped, to a slower decay of the bolometric luminosity compared to an exponential decline. Significant light-curve flattening is obtained when the ejecta density drops below the value for which τdep equals the 56Co lifetime. We provide a simple method to accurately describe this ‘delayed deposition’ effect, which is straightforward to use for analysis of observed light curves. We find that the ejecta heating is dominated by delayed deposition typically from 600 to 1200 d, and only later by longer lived isotopes 57Co and 55Fe decay (assuming solar abundance). For the relatively narrow 56Ni velocity distributions of commonly studied explosion models, the modification of the light curve depends mainly on the 56Ni mass-weighted average density, 〈ρ〉t3. Accurate late-time bolometric light curves, which may be obtained with JWST far-infrared (far-IR) measurements, will thus enable to discriminate between explosion models by determining 〈ρ〉t3 (and the 57Co and 55Fe abundances). The flattening of light curves inferred from recent observations, which is uncertain due to the lack of far-IR data, is readily explained by delayed deposition in models with $\langle \rho \rangle t^{3} \approx 0.2\, \mathrm{M}_{\odot }\, (10^{4}\, \textrm{km}\, \textrm{s}^{-1})^{-3}$, and does not imply supersolar 57Co and 55Fe abundances.

scholarly journals Generalized Knaster-Kuratowski-Mazurkiewicz type theorems and applications to minimax inequalities

2017 ◽  
Vol 20 (K2) ◽  
pp. 131-140
Author(s):  
Linh Manh Ha
Keyword(s):  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Applied Mathematics ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Minimax Theorems ◽  
Minimax Inequalities ◽  
Special Cases ◽  
Kkm Mappings ◽  
Definition Of

Knaster-Kuratowski-Mazurkiewicz type theorems play an important role in nonlinear analysis, optimization, and applied mathematics. Since the first well-known result, many international efforts have been made to develop sufficient conditions for the existence of points intersection (and their applications) in increasingly general settings: Gconvex spaces [21, 23], L-convex spaces [12], and FCspaces [8, 9]. Applications of Knaster-Kuratowski-Mazurkiewicz type theorems, especially in existence studies for variational inequalities, equilibrium problems and more general settings have been obtained by many authors, see e.g. recent papers [1, 2, 3, 8, 18, 24, 26] and the references therein. In this paper we propose a definition of generalized KnasterKuratowski-Mazurkiewicz mappings to encompass R-KKM mappings [5], L-KKM mappings [11], T-KKM mappings [18, 19], and many recent existing mappings. Knaster-KuratowskiMazurkiewicz type theorems are established in general topological spaces to generalize known results. As applications, we develop in detail general types of minimax theorems. Our results are shown to improve or include as special cases several recent ones in the literature.

scholarly journals Simple Method for Estimating the Contribution of Neighbouring n-beam Interactions to Two-beam Structure Factors

1988 ◽  
Vol 41 (3) ◽  
pp. 469
Author(s):  
HJ Juretschke ◽  
HK Wagenfeld
Keyword(s):  
Experimental Determination ◽  
Structure Factor ◽  
Beam Structure ◽  
Simple Method ◽  
Experimental Configuration ◽  
Structure Factors ◽  
Special Cases ◽  
Better Than

Unless special precautions are taken, the experimental determination of two-beam structure factors to better than 1 % may include contributions from neighbouring n-beam interactions. In any particular experimental configuration, corrections for such contributions are easily carried out using the modified two-beam structure factor formalism developed recently (Juretschke 1984), once the full indexing of the pertinent n-beam interactions is known. The method is illustrated for both weak and strong primary reflections and its applicability in special cases, as well as for less than perfect crystals, is discussed.

An analysis of the variation between dairy-sire progeny-groups for milk yield and fat content

1958 ◽  
Vol 1958 ◽  
pp. 19-29 ◽  
Author(s):  
Alan Robertson ◽  
S. S. Khishin
Keyword(s):  
Milk Yield ◽  
Comparison Method ◽  
Fat Content ◽  
Fat Percentage ◽  
Simple Method ◽  
Factors Affecting ◽  
Special Cases ◽  
The Mean ◽  
Milk Marketing ◽  
The Relationship

The past few years have seen the development in Great Britain of the ‘contemporary comparison’ method for evaluating progeny tests of dairy sires (Macarthur, 1954; Robertson, Stewart and Ashton 1956). The final overall figure attached to a sire is the mean difference between the yield of his daughters and that of other heifers milking in the same herd in the same year, with due regard for the numbers of animals in the two groups. Although it has some imperfections in special cases, this is probably the most informative simple method of evaluating a sire for yield and, fortunately, one which could be easily integrated with the existing recording system. The method has been turned into a simple routine in the Bureau of Records of the Milk Marketing Board and several thousand bulls have now been evaluated. In this paper, we shall be mostly concerned to use this material to investigate the heritabilities of milk yield and fat content and the relationship between the two in the different breeds. The information that we shall use consists, for each bull, of the mean contemporary comparison, with its effective ‘weight’, and the average fat percentage of the daughters. Before we deal with the observed results, we should go into rather more detail into the nature of these two figures and into the factors affecting them.

Analytical solutions of a spherical nanoinhomogeneity under far-field unidirectional loading based on Steigmann–Ogden surface model

2020 ◽  
Vol 25 (10) ◽  
pp. 1904-1923
Author(s):  
Youxue Ban ◽  
Changwen Mi
Keyword(s):  
Flexural Rigidity ◽  
Elastic Stiffness ◽  
Surface Model ◽  
Far Field ◽  
Stress Concentrations ◽  
Equilibrium Equations ◽  
Legendre Series ◽  
Simple Method ◽  
Shear Moduli ◽  
Special Cases

For a solid surface or interface that is subjected to transverse loading, the influence of its flexural resistibility to bending deformation becomes significant. A spherical inhomogeneity or void embedded in an infinite elastic medium under the application of nonhydrostatic loads represents a typical example. In this work, we consider the most fundamental loading of a far-field unidirectional tension. Analytical displacements and stresses are developed by the coupling of a Steigmann–Ogden surface mechanical model, the simple method of Boussinesq displacement potentials, the semi-inverse method of elasticity, and Legendre series representations of spherical harmonics. The problem is then solved by converting the equilibrium equations of displacement into a linear system with respect to the Legendre series coefficients. The developed solutions are general in the sense that they may reduce to their classical or Gurtin–Murdoch counterparts as special cases. Analytical expressions reveal that the derived solution depends on four dimensionless ratios from among surface material parameters, shear moduli ratio, and inhomogeneity or void radius. In particular, instead of depending on both flexural parameters in the moment–curvature relation, one fixed combination is sufficient to represent the surface flexural rigidity. This is in contrast with the influence of the in-plane elastic stiffness, in which both surface Lamé parameters matter. Parametric studies further demonstrate that, for metallic inhomogeneities or voids with radii between 10 nm and 100 nm, the effects of surface flexural rigidity on stress distributions and stress concentrations are significant.

scholarly journals An optimal multi-layer reinsurance policy under conditional tail expectation

2017 ◽  
Vol 12 (1) ◽  
pp. 130-146
Author(s):  
Amir T. Payandeh Najafabadi ◽  
Ali Panahi Bazaz
Keyword(s):  
Optimality Criterion ◽  
Risk Measure ◽  
Insurance Companies ◽  
Total Risk ◽  
Unknown Parameters ◽  
Practical Applications ◽  
Simulation Based ◽  
Cartesian Coordinate ◽  
Stop Loss ◽  
Conditional Tail Expectation

AbstractAn usual reinsurance policy for insurance companies admits one or two layers of the payment deductions. Under optimality criterion of minimising the Conditional Tail Expectation (CTE) risk measure of the insurer’s total risk, this article generalises an optimal stop-loss reinsurance policy to an optimal multi-layer reinsurance policy. To achieve such optimal multi-layer reinsurance policy, this article starts from a given optimal stop-loss reinsurance policy f(⋅). In the first step, it cuts down the interval [0, ∞) into intervals [0, M1) and [M1, ∞). By shifting the origin of Cartesian coordinate system to (M1, f(M1)), and showing that under the CTE criteria $$f\left( x \right)I_{{[0,M_{{\rm 1}} )}} \left( x \right){\plus}\left( {f\left( {M_{{\rm 1}} } \right){\plus}f\left( {x{\minus}M_{{\rm 1}} } \right)} \right)I_{{[M_{{\rm 1}} ,{\rm }\infty)}} \left( x \right)$$ is, again, an optimal policy. This extension procedure can be repeated to obtain an optimal k-layer reinsurance policy. Finally, unknown parameters of the optimal multi-layer reinsurance policy are estimated using some additional appropriate criteria. Three simulation-based studies have been conducted to demonstrate: (1) the practical applications of our findings and (2) how one may employ other appropriate criteria to estimate unknown parameters of an optimal multi-layer contract. The multi-layer reinsurance policy, similar to the original stop-loss reinsurance policy is optimal, in a same sense. Moreover, it has some other optimal criteria which the original policy does not have. Under optimality criterion of minimising a general translative and monotone risk measure ρ(⋅) of either the insurer’s total risk or both the insurer’s and the reinsurer’s total risks, this article (in its discussion) also extends a given optimal reinsurance contract f(⋅) to a multi-layer and continuous reinsurance policy.

An Approximate Method for Determining Areal Sweep Efficiency and Flow Capacity in Formations with Anisotropic Permeability

10.2118/16-pa ◽  
1961 ◽  
Vol 1 (04) ◽  
pp. 277-286 ◽  
Author(s):  
M. Mortada ◽  
G.W. Nabor
Keyword(s):  
Principal Axis ◽  
Bedding Plane ◽  
Sweep Efficiency ◽  
Simple Method ◽  
Anisotropic Permeability ◽  
Preferred Direction ◽  
Flow Capacity ◽  
Principal Axes ◽  
Special Cases ◽  
Directional Permeability

Abstract The effects of anisotropic or directional permeability on the areal sweep efficiency and the flow capacity are examined. The paper points out the importance of taking directional permeability into consideration in planning a flood. It analyzes the two-dimensional flow pattern associated with the skewed line drive for a unit mobility ratio. The direct and staggered line drives are treated as special cases of the skewed line drive. Analytical expressions are developed for the areal sweep efficiency at breakthrough and the flow capacity. They are related to the spacing between like wells, the distance between a row of injectors and the nearest row of producers, and the degree of skewness of the line drive. The latter quantity is defined such that it is equal to zero for the direct line drive and equals one-half for the staggered line drive. The a real sweep efficiency and the flow capacity depend also on the orientation of the flood pattern with respect to the principal axis of anisotropy. The paper provides a simple method for determining the a real sweep efficiency and the flow capacity for a formation in which the permeability in the bedding plane is anisotropic. Introduction Directional or anisotropic permeability is manifested by the ability of the formation to conduct fluids more readily along certain preferred directions. This situation occurs in many producing formations and is usually attributed to depositional features in which the sand grains are oriented in a preferred direction. In some cases it results from the formation of a major and a minor fracture system. Directional permeability should be taken into account in many phases of the production and exploration activities. Recognizing its existence in the formation of interest and planning accordingly can lead to increased recovery and substantial savings. For instance, the areal sweep efficiency in a water flood depends to a great extent on the orientation of the flood pattern with respect to the principal axis of permeability. Anisotropic permeability is specified by the directions of its three principal axes and the permeability along each axis. The principal axes of permeability are mutually perpendicular. This paper deals with the areal sweep efficiency at breakthrough and the flow capacity for formations with anisotropic permeability. The flood pattern considered consists of alternate rows of injecting and producing wells. The rows of wells are parallel and form a developed, skewed line drive which is illustrated in Fig. 1. The staggered and direct line drives are treated as special cases of the skewed line drive.

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.

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