Koherens kockázatmérés és tőkeallokáció = Coherent risk measurement and capital allocation

Bármennyire szeretne is egy bank (vállalat, biztosító) csak az üzletre koncentrálni, nem térhet ki a pénzügyi (hitel-, piaci, operációs, egyéb) kockázatok elől, amelyeket mérnie és fedeznie kell. A teljes fedezés vagy nagyon költséges, vagy nem is lehetséges, így a csőd elkerülésre minden gazdálkodó egységnek tartania kell valamennyi kockázatmentes, likvid tőkét. Koherens kockázatmérésre van szükség: az allokált tőkének tükröznie kell a kockázatokat - azonban még akkor is felmerül elosztási probléma, ha jól tudjuk mérni azokat. A diverzifikációs hatásoknak köszönhetően egy portfólió teljes kockázata általában kisebb, mint a portfóliót alkotó alportfóliók kockázatának összege. A koherens tőkeallokáció során azzal a kérdéssel kell foglalkoznunk, hogy mennyi tőkét osszunk az alportfóliókra, vagyis hogyan osszuk el „kor­rekt” módon a diverzifikáció előnyeit. Így megkapjuk az eszközök kockázathoz való hozzájárulását. A tanulmányban játékelmélet alkalmazásával, összetett opciós példákon keresztül bemutatjuk a kockázatok következetes mérését és felosztását, felhívjuk a figyelmet a következetlenségek veszélyeire, valamint megvizsgáljuk, hogy a gyakorlatban alkalmazott kockázatmérési módszerek [különösen a kockáztatott érték (VaR)] mennyire felelnek meg az elmélet által szabott követelményeknek. ____________________ However much a bank (or company or insurance provider) concentrates only on business, it cannot avoid financial (credit, market, operational or other) risks that need to be measured and covered. Total cover is either very expensive or not even possible, so that every business unit has to hold some risk-free liquid capital to avoid insolvency. What it needs is coherent risk measurement: the capital allocated has to match the risks, but even if the risks are measured well, distribution problems can still arise. Thanks to diversification effects, the total risk of a portfolio is less than the sum of the risks of its sub-portfolios. Coherent capital allocation entails addressing the question of how much capital to divide among the sub-portfolios, or how to distribute ‘correctly’ the advantages of diversification. This yields the contribution of the assets to the risk. The study employs game theory and examples of compound options to demonstrate coherent measurement and distribution of risks. Attention is drawn to the dangers of inconsistencies. The authors examine how far the methods of risk measurement applied in practice (notably VaR—value at risk) meet the requirements set in theory.

The Shapley value for shortest path games

In this paper shortest path games are considered. The transportation of a good in a network has costs and benet too. The problem is to divide the prot of the transportation among the players. Fragnelli et al (2000) introduce the class of shortest path games, which coincides with the class of monotone games. They also give a characterization of the Shapley value on this class of games. In this paper we consider further four characterizations of the Shapley value (Shapley (1953)'s, Young (1985)'s, Chun (1989)'s, and van den Brink (2001)'s axiomatizations), and conclude that all the mentioned axiomatizations are valid for shortest path games. Fragnelli et al (2000)'s axioms are based on the graph behind the problem, in this paper we do not consider graph specic axioms, we take TU axioms only, that is, we consider all shortest path problems and we take the view of abstract decision maker who focuses rather on the abstract problem than on the concrete situations.

Non-uniqueness through duality in the von Neumann growth models

Duality can be viewed as the soul of each von Neumann growth model. This is not at all surprising because von Neumann (1955), a mathematical genius, extensively studied quantum mechanics which involves a “dual nature” (electromagnetic waves and discrete corpuscules or light quanta). This may have had some influence on developing his own economic duality concept. The main object of this paper is to restore the spirit of economic duality in the investigations of the multiple von Neumann equilibria. By means of the (ir)reducibility taxonomy in Móczár (1995) the author transforms the primal canonical decomposition given by Bromek (1974) in the von Neumann growth model into the synergistic primal and dual canonical decomposition. This enables us to obtain all the information about the steadily maintainable states of growth sustained by the compatible price-constellations at each distinct expansion factor.

Reducible von Neumann models and uniqueness

This paper can be regarded as a result of basic research on the technological characteristics of the von Neumann models and their consequences. It introduces a new taxonomy of reducible technologies, explores their key distinguishing features, and specifies which ones ensure the uniqueness of von Neumann equilibrium. A comprehensive comparison is also given between the familiar (in)decomposability ideas and the reducibility concepts suggested here. All these are carried out with a modern approach. Simultaneously, the reader may also acquire a complete picture of and guidance on the fundamental von Neumann models here.

Climate change in game theory context

The aim of this paper is to survey the game theory modelling of the behaviour of global players in mitigation and adaptation related to climate change. Three main fields are applied for the specific aspects of temperature rise: behaviour games, CPR problem and negotiation games. The game theory instruments are useful in analyzing strategies in uncertain circumstances, such as the occurrence and impacts of climate change. To analyze the international players’ relations, actions, attitude toward carbon emission, negotiation power and motives, several games are applied for the climate change in this paper. The solution is surveyed, too, for externality problem.

A simplified implementation of the least squares solution for pairwise comparisons matrices

This is a follow up to "Solution of the least squares method problem of pairwise comparisons matrix" by Bozóki published by this journal in 2008. Familiarity with this paper is essential and assumed. For lower inconsistency and decreased accuracy, our proposed solutions run in seconds instead of days. As such, they may be useful for researchers willing to use the least squares method (LSM) instead of the geometric means (GM) method.

On optimal completions of incomplete pairwise comparison matrices

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigen-vector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper.

A new proportional procedure for the n−person cake−cutting problem

We present a new cake−cutting procedure which guarantees everybody a proportional share according to his own valuation.

A method for solving LSM problems of small size in the AHP

The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. It provides with ratio-scale measurements of the prioirities of elements on the various leveles of a hierarchy. These priorities are obtained through the pairwise comparisons of elements on one level with reference to each element on the immediate higher level. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM), Logarithmic Least Squares Method (LLSM), Weighted Least Squares Method (WLSM) and Chi Squares Method (X2M) are of the tools for computing the priorities of the alternatives. This paper studies a method for generating all the solutions of the LSM problems for 3 × 3 matrices. We observe non-uniqueness and rank reversals by presenting numerical results.

Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices

The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 × 3 and 4 × 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants.

Súlyok meghatározása páros összehasonlítás mátrixok legkisebb négyzetes közelítése alapján

A páros összehasonlítások módszere a többszempontú döntési feladatok megoldásának egy lehetséges eszköze mind a szempontsúlyok meghatározásában, mind az alternatívák értékelésében. A szempontokat páronként összehasonlítva, fontosságaiknak a döntéshozó által megítélt arányait mátrixba rendezve a feladat a súlyvektor meghatározása úgy, hogy annak komponensei valamilyen értelemben jól illeszkedjenek a döntéshozó által megadott értékekhez. A páros összehasonlítás mátrixból a súlyok kiszámítására leggyakrabban használt sajátvektor módszer (Analytic Hierarchy Process) mellett számos távolságminimalizáló módszer is létezik. Ezek egyike a legkisebb négyzetek módszere, melynek megoldása nemlineáris, nemkonvex függvény feltételes optimalizálását jelenti. A cikkben olyan módszereket mutatunk be a páros összehasonlítás mátrixok legkisebb négyzetes becslésére, amelyek a célfüggvény összes lokális és globális minimumhelyének meghatározására alkalmasak.

Magánnyugdíj-járadékok közötti választás (Choice amongst Hungarian pension funds' annuities )

Ebben a cikkben a magánnyugdíj-járadékok közötti választást vizsgálom. Arra vagyok kíváncsi, hogy a racionális döntéshozó hogyan választ a lehetséges járadékokból. A döntéshozó a várható hasznosságátmaximalizálja. A cikkben megvizsgálom azt az esetet is, ha a döntéshozó számára fontos, hogy gondoskodjék az utódairól, és megvizsgálom a házaspárok optimális döntését is. Felhívom a figyelmet, hogy a nemek közötti differenciálás tilalmának az lehet a következménye, hogy a döntéshozók olyan járadékokat választanak, amely a szolgáltatónak összességében veszteséget okoz. Bemutatom, hogy a halasztott járadékkal kombinált ütemezett pénzkivonás a vizsgált környezetben nem rosszabb, mint a többi lehet}oség, a döntéshozó viszont ebben az esetben a befektetési lehet}oségeket maga választhatja meg. _____________ This paper investigates the decision-maker's choice amongst the pension fund's life annuities in Hungary. Hungarian pension funds have not started providing annuities yet, and the environment requires further regulation. The most significant problems is that the annuity providers must not differentiate between sexes. Pension fund members can choose among different types of life annuities (e.g. they can choose guarantee period or joint life annutity), and the member's choice may cause loss to the annuity provider in general. Since differentation between sexes is not allowed member's opportunities may shrink (e.g. they can not choose guarantee period or joint life annuity).

Solution of the Least Squares Method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima.

On Saaty's and Koczkodaj's inconsistencies of pairwise comparison matrices

The aim of the paper is to obtain some theoretical and numerical properties of Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices (PRM). In the case of 3 × 3 PRM, a differentiable one-to-one correspondence is given between Saaty’s inconsistency ratio and Koczkodaj’s inconsistency index based on the elements of PRM. In order to make a comparison of Saaty’s and Koczkodaj’s inconsistencies for 4 × 4 pairwise comparison matrices, the average value of the maximal eigenvalues of randomly generated n × n PRM is formulated, the elements aij (i < j) of which were randomly chosen from the ratio scale ... ... with equal probability 1/(2M − 1) and a ji is defined as 1/a ij . By statistical analysis, the empirical distributions of the maximal eigenvalues of the PRM depending on the dimension number are obtained. As the dimension number increases, the shape of distributions gets similar to that of the normal ones. Finally, the inconsistency of asymmetry is dealt with, showing a different type of inconsistency.

CVaR számítás SRA algoritmussal (CVaR minimization by the SRA algorithm)

A CV aR kockázati mérték egyre nagyobb jelentőségre tesz szert portfóliók kockázatának megítélésekor. A portfolió egészére a CVaR kockázati mérték minimalizálását meg lehet fogalmazni kétlépcsős sztochasztikus feladatként. Az SRA algoritmus egy mostanában kifejlesztett megoldó algoritmus sztochasztikus programozási feladatok optimalizálására. Ebben a cikkben az SRA algoritmussal oldottam meg CV aR kockázati mérték minimalizálást. ___________ The risk measure CVaR is becoming more and more popular in recent years. In this paper we use CVaR for portfolio optimization. We formulate the problem as a two-stage stochastic programming model. We apply the SRA algorithm, which is a recently developed heuristic algorithm, to minimizing CVaR.