Spectral Measures of Uncertain Risk
|
|
- Theodora Cobb
- 6 years ago
- Views:
Transcription
1 Spectral Measures of Uncertain Risk Jin Peng, Shengguo Li Institute of Uncertain Systems, Huanggang Normal University, Hubei 438, China Abstract: A key issue in uncertain risk analysis is how to provide risk measures. In this article we present the spectral measure of risk in an uncertain environment which is described by uncertain variable. Firstly, we introduce the concept of spectral measure of risk and harmonious measure of risk based on uncertainty theory. Then, some fundamental properties of the proposed spectral measure of risk are investigated. It is proved that the proposed spectral measure of risk is a harmonious measure of risk. Finally, some numerical examples are illustrated. Keywords: risk analysis, uncertainty theory, uncertain variable, spectral measure of risk, harmonious measure of risk 1 Introduction Risk analysis in an uncertain environment is a very powerful tool for decision making under uncertainty. A key issue in risk analysis is how to provide the risk measures. There has been a great momentum in research on risk measures in recent years. For recent reviews on risk measure we refer the reader to 6, 7, 14, 17, 25, 26. As we know, the simplest way to measure risk in the early era of risk analysis is to calculate the variance and standard deviation of the profit and loss distribution. Value-at-risk (VaR) is the most commonly used measure of market risk 8, 13, 15, 21. However, VaR measure has been subjected to criticism because of mathematical shortcomings (lack of subadditivity and convexity). A related concept, tail value-atrisk (TVaR), also known as expected shortfall (ES) or conditional Value-at-risk (CVaR) in some contexts 22, 24, 28, has proved to be superior in these respects. In 4, 5, Artzner et al. propose a set of axioms to characterize a class of financially meaningful risk measures, called coherent risk measures. A measure of risk is called coherent if it is monotonous, positively homogeneous, translation invariant, and subadditive. Many researchers pay attention to the coherence property of risk measure 1, 12, 16, 27. Acerbi 1 has investigated the coherence of spectral risk measures. Spectral measures of risk are attractive risk measures as they allow the user to obtain risk measures that reflect Proceedings of the First International Conference on Uncertainty Theory, Urumchi, China, August 11-19, 21, pp their weight function or risk-aversion functions. The question of how spectral measures of risk should be produced in decision problems has been extensively discussed in the past few decades, see the articles 2, 3, 9, 11. In economics and finance, risk is customarily measured by means of probabilistic methods. As is well known, probability theory often requires a lot of historical data. However, decision-making problems in real life are not always in these situations. Generally, in risk analysis with little or no relevant historical data, expert judgment is required. Probabilistic risk analysis is subject to a number of severe problems that make its use as measures of risk very questionable. Randomness is only one type of various uncertainties. Liu 18, 2 has found the axiomatic uncertainty theory which identifies general uncertainty as being distinct from probabilistic uncertainty. In some context, risk is a synonym for uncertainty. This motivates uncertain risk analysis a new area of risk analysis which appears to be distinct from the traditional probabilistic risk analysis. Recently, the axiomatic concepts of coherent and convex risk measures have been developed and extended under uncertainty, see 22. We presuppose throughout this paper that risks can be quantified by way of an uncertain variable which can be used to describe uncertain (positive) profit or (negative) loss. In general, we regard a risk measure as a mapping from a set of real valued uncertain variables representing risks to the real line. Given some known risk measures under uncertainty, how to generate a new risk measure? Spectral risk measures is one kind of these methods. More importantly, risk perceptions and rationality in measures of risk is fundamental. We naturally hope the spectral measure of risk is of better properties than the original one in some aspects. The purpose of this paper is to show the spectral measure of risk in an uncertain environment which is described by uncertain variable. The rest of this paper is organized as follows: Section 2 presents some preliminary concepts and results selected from uncertainty theory. In Section 3, the spectral measure of risk of uncertain variable is introduced. Section 4 deals with some properties of the spectral measure of risk. Some computing examples are illustrated in Section 5. The last section contains some concluding remarks.
2 116 JIN PENG AND SHENGGUO LI 2 Preliminaries In this section, we present some definitions and results for risk analysis in uncertain environment within the framework of uncertainty theory. Let Γ be a nonempty set, and L a σ-algebra over Γ. For any Λ L, Liu 18 presented an axiomatic uncertain measure M{Λ} to express the chance that uncertain event Λ occurs. The set function M{ } satisfies the following four axioms: (i) (Normality) M{Γ} = 1; (ii) (Monotonicity) M{Λ 1 } M{Λ 2 }, whenever Λ 1 Λ 2 ; (iii) (Self-Duality) M{Λ} + M{Λ c } = 1 for any Λ L; (iv) (Countable Subadditivity) For every countable sequence of events {Λ i }, we have M { i Λ i } i M{Λ i}. The triplet (Γ, L, M) is called an uncertainty space and an uncertain variable is defined as a function from this space to the set of real numbers (Liu 18). An uncertain variable can be characterized by its uncertainty distribution Φ : R, 1, which is defined by Liu 18 as follows M { γ Γ (γ) x }. (1) Peng and Iwamura 23 have proved that a function Φ : R, 1 is uncertainty distribution if and only if it is an increasing function except Φ(x) and Φ(x) 1. Let be an uncertain variable with uncertainty distribution Φ. Then the inverse function is called the inverse uncertainty distribution of. An uncertainty distribution Φ is called regular if its inverse function (α) exists and is unique for each α (, 1). The expected value of uncertain variable is defined by Liu 18 as E = + M{ r}dr M{ r}dr (2) provided that at least one of the two integrals is finite. As a useful representation of expected value, it has been proved by Liu 19 that E = (α)dα (3) where is the generalized inverse of the uncertainty distribution Φ of uncertain variable. Let be an uncertain variable with finite expected value µ = E. The variance of is defined as V = E ( µ) 2. And the standard deviation of is defined as σ = V which is previously used as a simple risk measure. Liu 18 introduced the independence concept of uncertain variables. The uncertain variables 1, 2,, m are independent if and only if { m } M { i B i } = min 1 i m M { i B i } (4) for any Borel sets B 1, B 2,, B m of R. Consider an uncertain variable which might be seen as the uncertain profit and loss of an investment or portfolio by a fixed time horizon. Positive values of are regarded as profits, while negative values of are regarded as losses. Peng 22 introduced the concepts of VaR and TVaR of uncertain variable in the sense of the following definitions. Definition 1 Let be an uncertain variable and α (, 1 be a chosen risk confidence level. Then the VaR of can be expressed by VaR (α) inf { x M{ x} α } (α) (5) where (α) denotes the generalized inverse of the uncertainty distribution Φ of. Definition 2 Let be an uncertain variable and α (, 1) be the risk confidence level. Then the TVaR of is the function TVaR : (, 1 R such that TVaR (α) = 1 α VaR 1 α 3 Concept of Spectral Measures of Risk. (6) In this section, we deal with spectral measures of risk, a class of measures based on integrals of the inverse distribution of uncertain variable (which may denote portfolio return). Definition 3 Let be an uncertain variable with uncertainty distribution Φ (x), and w :, 1 R be a function. The spectral measure of risk is defined as M w (α)dα (7) where the function w is called the weight function for convenience. Definition 4 A weight function w :, 1 R is called a risk spectrum or risk aversion function if it satisfies the following conditions 1) Nonnegativity: w(y) for all y, 1; 2) Normality: w = 1; 3) Monotonicity: w(y 1 ) w(y 2 ) for all y 1 y 2 1. Remark 1 The spectral measure of risk can be thought of as a particular uncertainty-weighted average of the inverse distribution (α) of uncertain variable.
3 SPECTRAL MEASURES OF UNCERTAIN RISK 117 Remark 2 It should be noticed that if, then (α) and then M w. That is to say, non-negative uncertain variable means no risk in the sense of spectral measure. Example 1 Different weight functions produce different spectral measures of risk. Several kinds of weight functions are listed as follows. 1) Power weight function: w(y) = λ(1 y) λ 1 with λ > 1; 2) Proportional weight function: w(y) = 1 µ y 1 µ 1 with µ > 1; 3) Exponential weight function: w(y) = c exp{ cy} 1 exp{ c} with coefficient c >. Example 2 Setting w 1 one gets M w E which the negative expected value is trivially a spectral measure of risk. Example 3 The TVaR in Definition 2 is indeed a particular case of spectral measure. In fact, taking the weight function otherwise, we exactly obtain TVaR (α) 1 α. Example 4 The VaR in Definition 1 is also a typical example of spectral measure. In fact, taking the weight function, which is the so called Dirac delta function, = δ(y α), we reproduce that δ(y α) (α) = VaR (α). (,1) α What mathematical properties a meaningful risk should have? Definition 5 Consider a set U of real-valued uncertain variables. A function r : U R is said to be a harmonious measure of risk if it satisfies the following properties (A 1 ) Monotonicity: If, U are two uncertain variables and, then r() r(). (A 2 ) Positive homogeneity: If U is an uncertain variable and c >, then r(c) = cr(). (A 3 ) Translation invariance: If U and b R is a real number, then r( + b) = r() b. (A 4 ) Independent additivity: If,, + U, and and are independent uncertain variables, then r( + ) = r() + r(). A natural question then arises here: is the spectral measures of risk harmonious? Or under what conditions is it a harmonious measure? We will answer these questions in the coming section. 4 Properties of Spectral Measures of Risk The purpose of this section is to show some properties of the spectral measures of risk. First of all, we exhibit two frequently-used lemmas given by Liu 2. Lemma 1 Let be an uncertain variable with uncertainty distribution Φ, and let f be a strictly increasing function. Then f() is an uncertain variable with inverse uncertainty distribution f() (α) = f(φ 1 (α)), α (, 1). (8) Lemma 2 Let 1, 2,, n be independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If f : R n R is a strictly increasing function, then = f( 1, 2,, n ) (9) is an uncertain variable with inverse uncertainty distribution (α) = f(φ 1 1 (α), Φ 1 2 (α),, Φ 1 n (α)), α (, 1). (1) Remark 3 A real-valued function f(x 1, x 2,, x n ) is said to be strictly increasing if f(x 1, x 2,, x n ) < f(x 1, x 2,, x n) (11) whenever x i x i for i = 1, 2,, n and x j < x j for at least one index j. Theorem 3 Let be an uncertain variable with uncertainty distribution Φ(x), and w :, 1 R be a risk spectrum. Then the spectral measure of risk is defined by M w is a harmonious measure of risk. (α)dα (12) Proof. It is needed to prove that M w satisfies the following axioms (A 1 ) Monotonicity: If, are two uncertain variables and, then M w M w. (A 2 ) Independent additivity: If and are independent uncertain variables, then M w + = M w + M w. (A 3 ) Positive homogeneity: If is an uncertain variable and c >, then M w c = cm w. (A 4 ) Translation invariance: If is an uncertain variable and b R is a real number, then M w + b = M w b. Let us travel the above properties step by step. (1) Since w is a nonnegative function and implies (α) Φ 1(α), it is easy to see that M w = M w. (α)dα (α)dα
4 118 JIN PENG AND SHENGGUO LI (2) Since and are independent uncertain variables, we see from Lemma 1 that Thus we have + M w + (α) + Φ 1(α). + (α)dα ( = M w + M w. (3) Because c >, we have M w c = cm w. (α) + Φ 1(α))dα c (α)dα c (α)dα (4) From the definition of the spectral measure of risk, it is clear that M w + b = M w b. The proof is accomplished. +b (α)dα ( (α) + b)dα Besides the above harmonious property, there are still other desirable properties below that are worthwhile discussing. Theorem 4 Let be an uncertain variable with uncertainty distribution Φ(x), and w :, 1 R be a weight function. If f(x) is a strictly increasing function on R, then the spectral measure of risk f() is M w f() f ( (α) ) dα. (13) Proof. Under the hypothesis, it follows from Lemma 1 that ( ) f() (α) = f (α). The result follows immediately. Theorem 5 (Law-Invariance) Let and be uncertain variables with uncertainty distributions Φ (x) and Φ (x) respectively, and w :, 1 R be a weight function. If Φ (x) = Φ (x) on R, then the spectral measure of risk M w = M w. Proof. It is an easy consequence of the Definition 3. Theorem 6 Let be an uncertain variable with uncertainty distribution Φ(x), and w :, 1 R be a weight function. If f(x) is a strictly decreasing function on R, then the spectral measure of risk f() is M w f() f ( (1 α) ) dα. (14) Proof. Under the hypothesis, it is easy to check that ( ) f() (α) = f (1 α). The result follows immediately. Especially, it is important to keep in mind that M w = M w. Theorem 7 Let w :, 1 R be a weight function and λ 1, λ 2,, λ n, 1 be nonnegative numbers with n λ i = 1. Assume that 1, 2,, n be independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. Then spectral measure of risk of the convex combination is = λ i i M w n λ i i = λ i M w i. (15) Proof. Notice that the inverse uncertainty distribution of the convex combination is Thus we have (α) = n M w n λ i i λ i i (α), α (, 1). (16) = The result is proved. λ i λ i M w i. λ i i (α)dα i (α)dα In fact, the convexity property of the spectral measure of risk follows from the positive homogeneity and independent additivity. Theorem 8 Let w :, 1 R be a weight function. Suppose that and are independent uncertain variables with uncertainty distributions Φ (x) and Φ (x), respectively. Then we
5 SPECTRAL MEASURES OF UNCERTAIN RISK 119 have the following results related to the spectral measure of risk M w M w M w M w (α) Φ 1 (α) Φ 1 (α) dα, (17) (α) dα, (18) (α)φ 1 (α) dα, (19) Φ 1 (α) (α) dα. (2) Proof. Note that the following inverse uncertainty distribution expressions / Φ 1 (α) = (α) (α). (α) Φ 1(α), (α) Φ 1(α), (α)φ 1 (α), Thus the results follow directly from the definition of spectral measure of risk. 5 Numerical Examples Now we investigate the spectral measure of risk of some common types of uncertain variable. Example 5 Consider a linear uncertain variable = L(a, b), where a and b are real numbers with < a < b. The uncertainty distribution of the profit and loss uncertain variable is, if x a x a b a, if a x b 1, if x b and its inverse uncertainty distribution is (y) = a(1 y) + by. We obtain the spectral measure of risk expressed as M w L(a, b) = a (y 1)w b Taking the weight function otherwise, yw. we get the spectral measure of risk as follows M w L(a, b) = a b 2 α a. If another weight function w = 1 2 is selected, then y it is easy to see that w is a non-negative, non-increasing, right-continuous integrable function defined on, 1 and w = 1. Thus we have M w L(a, b) = 2a b. 3 Example 6 Consider a zigzag uncertain variable = Z(a, b, c), where a, b, c are real numbers with a < b < c. The uncertainty distribution of the profit and loss uncertain variable is, if x a x a 2(b a), if a x b x + c 2b 2(c b), if b x c 1, if x c. It is easily calculated that for any given confidence level α with < α 1, the inverse distribution of the zigzag uncertain variable can be analytically expressed as { a + 2(b a)y, if y.5 (y) = 2b c + 2(c b)y, if y >.5. Taking the weight function otherwise and recalling the definition M w w(y), we get the spectral measure of risk represented as M w Z(a, b, c) (a b)α a, if α.5 = (b c)α + (c 2b) + 2b a c α 1, if α >.5. 4 Example 7 Now we consider the normal uncertain variable = N (e, σ) with uncertainty distribution ( ( )) 1 π(e x) 1 + exp, x R (21) 3σ
6 12 JIN PENG AND SHENGGUO LI where e and σ are real numbers with σ >. The inverse uncertainty distribution of normal uncertain variable = N (e, σ) is (α) = e + σ 3 π ln α 1 α. (22) Setting the weight function otherwise. In this case, we obtain the spectral measure of risk represented as M w N (e, σ) 1 α 1 α w(y) e + σ 3 πα (e + σ 3 π ln y 1 y )dy ln (1 y) ln ydy e + σ 3 (α 1) ln (1 α) α ln α. πα Example 8 Let us now consider the lognormal uncertain variable = LOGN (e, σ) with uncertainty distribution denoted by ( ( )) 1 π(e ln x) 1 + exp, x (23) 3σ where e and σ are real numbers with σ >. The inverse uncertainty distribution of lognormal uncertain variable = LOGN (e, σ) is ( ) 3σ/π α (α) = exp(e) 1 α Setting the weight function otherwise. (24) In this situation, we has the following representation of spectral measure of risk M w LOGN (e, σ) 1 α w(y) exp(e) α 6 Conclusions ( ) 3σ/π y dy. 1 y Risk analysis aims at making systematic use of available information to identify hazards and to estimate the risk to individuals or populations, property or the environment. One key factor in risk analysis in uncertain environment is to introduce the reasonable measure of risk. At the same time, risk measure has a close relationship with premium calculation or insurance price, and it has a great potential applications in actuarial science. Traditionally, probabilistic risk analysis often requires a lot of historical data. However, uncertain risk analysis often requires a lot of experts empirical data. The expert judgment is typically appropriate when data are sparse or difficult to obtain or data are too costly to obtain. The philosophy behind spectral measure of risk is that a spectral risk measure is a new risk measure transformed from a traditional VaR measure via some kind of weight function. It is desired that the spectral risk measure is of better properties than the original one in some aspects. In this article we mainly present the spectral measure of risk under uncertain environment. Firstly, we introduce the concept of spectral measure of risk and harmonious measure of risk from the viewpoint of uncertainty theory. Then, some fundamental properties of the proposed spectral measure of risk are investigated. It is proved that the proposed spectral measure of risk is a harmonious measure of risk. Finally, some numerical examples are illustrated. Various classes of risk measures including spectral risk measures can be used in the framework of risk optimization under risk constraints. For example, portfolio optimization with appropriate spectral measures of risk under risk constraints should be considered in the future research work. Acknowledgements This work is supported by the National Natural Science Foundation (Grant No and No ), the Innovation Team Project of Hubei Provincial Department of Education, China.
7 SPECTRAL MEASURES OF UNCERTAIN RISK 121 References 1 Acerbi C, Spectral measures of risk: a coherent representation of subjective risk, Journal of Banking and Finance, Vol. 26, No. 7, , Acerbi C, Simonetti P, Portfolio optimization with spectral measures of risk, Adam A, Houkari M, Laurent J-P, Spectral risk measures and portfolio selection, Journal of Banking & Finance, Vol. 32, No. 9, , Artzner P, Delbaen F, Eber JM, and Heath D, Thinking Coherently, Risk, Vol. 1, No. 11, 68-71, Artzner P, Delbaen F, Eber JM, and Heath D, Coherent measures of risk, Mathematical Finance, Vol. 9, No. 3, , Brachinger H W, Weber M, Risk as a primitive: A survey of measures of perceived risk, Operations Research Spectrum, Vol. 19, No. 3, , Cheng S, Liu Y, and Wang S, Progress in Risk Measurement, Advanced Modelling and Optimization, Vol. 6, No. 1, 1-2, Choudhry M, An Introduction to Value-at-Risk, 4th Edition, John Wiley, Cotter J, Dowd K, Extreme spectral risk measures: An application to futures clearinghouse margin requirements, Journal of Banking & Finance, Vol. 3, No. 12, , Dhaene J, Laeven RJA, Vanduffel S, Darkiewicz G, and Goovaerts M J, Can a Coherent Risk Measure be too Subadditive? Journal of Risk and Insurance, Vol. 75, , Dowd K, John Cotter J, and Sorwar G, Spectral risk measures: properties and limitations, Journal of Financial Services Research, Vol. 34, No. 1, 61-75, Fabozzi F J, Tunaru R, On risk management problems related to a coherence property, Quantitative Finance, Vol. 6, No. 1, Gourieroux C, Laurent JP, and Scaillet O, Sensitivity analysis of values at risk, Journal of Empirical Finance, Vol. 7, No. 3, , Jaeger C, Renn O, Rosa E, and Webler T, Risk, Uncertainty, and Rational Action, Earhscan Publications, London, Jorion P, Value at Risk: The New Benchmark for Managing Financial Risk, 2nd ed., McGraw- Hill, New York, Kusuoka S, On law invariant coherent risk measures, in: Advances in Mathematical Economics, Vol. 3, 83-95, Springer, Tokyo, Kaplanski G, Kroll Y, VaR risk measures vs traditional risk measures: An analysis and survey, Journal of Risk, Vol. 4, No. 3, 1-28, Liu B, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, Liu B, Some Research Problems in Uncertainty Theory, Journal of Uncertain Systems, Vol.3, No.1, 3-1, Liu B, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, Morgan J P, RiskMetricsTM- -Technical Document, Fourth Edition, Morgan Guaranty Trust Companies, Inc. New York, Peng J, Value at Risk and Tail Value at Risk in Uncertain Environment. Proceedings of the Eighth International Conference on Information and Management Sciences, Kuming & Banna, China, July 2-28, pp , Peng Z, Iwamura K, A sufficient and necessary condition of uncertainty distribution Rockafeller RT, Uryasev S, Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, Vol. 26, No. 7, , Szegö G, Risk Measures for the 21st Century, New York: John Wiley and Sons, Szegö G, Measures of risk, European Journal of Operational Research, Vol. 163, No. 1, 5-19, Weber S, Distribution-invariant risk measures, information, and dynamic consistency. Mathematical Finance, Vol. 16, , Yamai Y, Toshinao Y, Value-at-risk versus expected shortfall: A practical perspective, Journal of Banking & Finance, Vol. 29, No. 4, , 25.
Value at Risk and Tail Value at Risk in Uncertain Environment
Value at Risk and Tail Value at Risk in Uncertain Environment Jin Peng Institute of Uncertain Systems, Huanggang Normal University, Hubei 438000, China pengjin01@tsinghua.org.cn Abstract: Real-life decisions
More informationTail Value-at-Risk in Uncertain Random Environment
Noname manuscript No. (will be inserted by the editor) Tail Value-at-Risk in Uncertain Random Environment Yuhan Liu Dan A. Ralescu Chen Xiao Waichon Lio Abstract Chance theory is a rational tool to be
More informationSpanning Tree Problem of Uncertain Network
Spanning Tree Problem of Uncertain Network Jin Peng Institute of Uncertain Systems Huanggang Normal University Hubei 438000, China Email: pengjin01@tsinghuaorgcn Shengguo Li College of Mathematics & Computer
More informationKnapsack Problem with Uncertain Weights and Values
Noname manuscript No. (will be inserted by the editor) Knapsack Problem with Uncertain Weights and Values Jin Peng Bo Zhang Received: date / Accepted: date Abstract In this paper, the knapsack problem
More informationNested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model
Nested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model Xiaowei Chen International Business School, Nankai University, Tianjin 371, China School of Finance, Nankai
More informationMatching Index of Uncertain Graph: Concept and Algorithm
Matching Index of Uncertain Graph: Concept and Algorithm Bo Zhang, Jin Peng 2, School of Mathematics and Statistics, Huazhong Normal University Hubei 430079, China 2 Institute of Uncertain Systems, Huanggang
More informationFormulas to Calculate the Variance and Pseudo-Variance of Complex Uncertain Variable
1 Formulas to Calculate the Variance and Pseudo-Variance of Complex Uncertain Variable Xiumei Chen 1,, Yufu Ning 1,, Xiao Wang 1, 1 School of Information Engineering, Shandong Youth University of Political
More informationEuler Index in Uncertain Graph
Euler Index in Uncertain Graph Bo Zhang 1, Jin Peng 2, 1 School of Mathematics and Statistics, Huazhong Normal University Hubei 430079, China 2 Institute of Uncertain Systems, Huanggang Normal University
More informationUncertain Structural Reliability Analysis
Uncertain Structural Reliability Analysis Yi Miao School of Civil Engineering, Tongji University, Shanghai 200092, China 474989741@qq.com Abstract: The reliability of structure is already applied in some
More informationStructural Reliability Analysis using Uncertainty Theory
Structural Reliability Analysis using Uncertainty Theory Zhuo Wang Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 00084, China zwang058@sohu.com Abstract:
More informationAsymptotic distribution of the sample average value-at-risk
Asymptotic distribution of the sample average value-at-risk Stoyan V. Stoyanov Svetlozar T. Rachev September 3, 7 Abstract In this paper, we prove a result for the asymptotic distribution of the sample
More informationAn axiomatic characterization of capital allocations of coherent risk measures
An axiomatic characterization of capital allocations of coherent risk measures Michael Kalkbrener Deutsche Bank AG Abstract An axiomatic definition of coherent capital allocations is given. It is shown
More informationOn Kusuoka Representation of Law Invariant Risk Measures
MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 1, February 213, pp. 142 152 ISSN 364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/1.1287/moor.112.563 213 INFORMS On Kusuoka Representation of
More informationUncertain Risk Analysis and Uncertain Reliability Analysis
Journal of Uncertain Systems Vol.4, No.3, pp.63-70, 200 Online at: www.jus.org.uk Uncertain Risk Analysis and Uncertain Reliability Analysis Baoding Liu Uncertainty Theory Laboratory Department of Mathematical
More informationTheoretical Foundation of Uncertain Dominance
Theoretical Foundation of Uncertain Dominance Yang Zuo, Xiaoyu Ji 2 Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 84, China 2 School of Business, Renmin
More informationVariance and Pseudo-Variance of Complex Uncertain Random Variables
Variance and Pseudo-Variance of Complex Uncertain andom Variables ong Gao 1, Hamed Ahmadzade, Habib Naderi 1. Department of Mathematical Sciences, Tsinghua University, Beijing 184, China gaor14@mails.tsinghua.edu.cn.
More informationHamilton Index and Its Algorithm of Uncertain Graph
Hamilton Index and Its Algorithm of Uncertain Graph Bo Zhang 1 Jin Peng 1 School of Mathematics and Statistics Huazhong Normal University Hubei 430079 China Institute of Uncertain Systems Huanggang Normal
More informationMembership Function of a Special Conditional Uncertain Set
Membership Function of a Special Conditional Uncertain Set Kai Yao School of Management, University of Chinese Academy of Sciences, Beijing 100190, China yaokai@ucas.ac.cn Abstract Uncertain set is a set-valued
More informationWhy is There a Need for Uncertainty Theory?
Journal of Uncertain Systems Vol6, No1, pp3-10, 2012 Online at: wwwjusorguk Why is There a Need for Uncertainty Theory? Baoding Liu Uncertainty Theory Laboratory Department of Mathematical Sciences Tsinghua
More informationRunge-Kutta Method for Solving Uncertain Differential Equations
Yang and Shen Journal of Uncertainty Analysis and Applications 215) 3:17 DOI 1.1186/s4467-15-38-4 RESEARCH Runge-Kutta Method for Solving Uncertain Differential Equations Xiangfeng Yang * and Yuanyuan
More informationUncertain Second-order Logic
Uncertain Second-order Logic Zixiong Peng, Samarjit Kar Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China Department of Mathematics, National Institute of Technology, Durgapur
More informationFinanzrisiken. Fachbereich Angewandte Mathematik - Stochastik Introduction to Financial Risk Measurement
ment Convex ment 1 Bergische Universität Wuppertal, Fachbereich Angewandte Mathematik - Stochastik @math.uni-wuppertal.de Inhaltsverzeichnis ment Convex Convex Introduction ment Convex We begin with the
More informationOn the convergence of uncertain random sequences
Fuzzy Optim Decis Making (217) 16:25 22 DOI 1.17/s17-16-9242-z On the convergence of uncertain random sequences H. Ahmadzade 1 Y. Sheng 2 M. Esfahani 3 Published online: 4 June 216 Springer Science+Business
More informationA New Approach for Uncertain Multiobjective Programming Problem Based on P E Principle
INFORMATION Volume xx, Number xx, pp.54-63 ISSN 1343-45 c 21x International Information Institute A New Approach for Uncertain Multiobjective Programming Problem Based on P E Principle Zutong Wang 1, Jiansheng
More informationAn Uncertain Bilevel Newsboy Model with a Budget Constraint
Journal of Uncertain Systems Vol.12, No.2, pp.83-9, 218 Online at: www.jus.org.uk An Uncertain Bilevel Newsboy Model with a Budget Constraint Chunliu Zhu, Faquan Qi, Jinwu Gao School of Information, Renmin
More informationA numerical method for solving uncertain differential equations
Journal of Intelligent & Fuzzy Systems 25 (213 825 832 DOI:1.3233/IFS-12688 IOS Press 825 A numerical method for solving uncertain differential equations Kai Yao a and Xiaowei Chen b, a Department of Mathematical
More informationOperations Research Letters. On a time consistency concept in risk averse multistage stochastic programming
Operations Research Letters 37 2009 143 147 Contents lists available at ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl On a time consistency concept in risk averse
More informationOn Liu s Inference Rule for Uncertain Systems
On Liu s Inference Rule for Uncertain Systems Xin Gao 1,, Dan A. Ralescu 2 1 School of Mathematics Physics, North China Electric Power University, Beijing 102206, P.R. China 2 Department of Mathematical
More informationMultivariate Stress Testing for Solvency
Multivariate Stress Testing for Solvency Alexander J. McNeil 1 1 Heriot-Watt University Edinburgh Vienna April 2012 a.j.mcneil@hw.ac.uk AJM Stress Testing 1 / 50 Regulation General Definition of Stress
More informationA Note on the Swiss Solvency Test Risk Measure
A Note on the Swiss Solvency Test Risk Measure Damir Filipović and Nicolas Vogelpoth Vienna Institute of Finance Nordbergstrasse 15 A-1090 Vienna, Austria first version: 16 August 2006, this version: 16
More informationCoherent risk measures
Coherent risk measures Foivos Xanthos Ryerson University, Department of Mathematics Toµɛας Mαθηµατ ικὼν, E.M.Π, 11 Noɛµβρὶoυ 2015 Research interests Financial Mathematics, Mathematical Economics, Functional
More informationRisk Aggregation with Dependence Uncertainty
Introduction Extreme Scenarios Asymptotic Behavior Challenges Risk Aggregation with Dependence Uncertainty Department of Statistics and Actuarial Science University of Waterloo, Canada Seminar at ETH Zurich
More informationCoherent Risk Measures. Acceptance Sets. L = {X G : X(ω) < 0, ω Ω}.
So far in this course we have used several different mathematical expressions to quantify risk, without a deeper discussion of their properties. Coherent Risk Measures Lecture 11, Optimisation in Finance
More informationA Chance-Constrained Programming Model for Inverse Spanning Tree Problem with Uncertain Edge Weights
A Chance-Constrained Programming Model for Inverse Spanning Tree Problem with Uncertain Edge Weights 1 Xiang Zhang, 2 Qina Wang, 3 Jian Zhou* 1, First Author School of Management, Shanghai University,
More informationThe newsvendor problem with convex risk
UNIVERSIDAD CARLOS III DE MADRID WORKING PAPERS Working Paper Business Economic Series WP. 16-06. December, 12 nd, 2016. ISSN 1989-8843 Instituto para el Desarrollo Empresarial Universidad Carlos III de
More informationUNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.
More informationUncertain Systems are Universal Approximators
Uncertain Systems are Universal Approximators Zixiong Peng 1 and Xiaowei Chen 2 1 School of Applied Mathematics, Central University of Finance and Economics, Beijing 100081, China 2 epartment of Risk Management
More informationVaR vs. Expected Shortfall
VaR vs. Expected Shortfall Risk Measures under Solvency II Dietmar Pfeifer (2004) Risk measures and premium principles a comparison VaR vs. Expected Shortfall Dependence and its implications for risk measures
More informationEstimating the Variance of the Square of Canonical Process
Estimating the Variance of the Square of Canonical Process Youlei Xu Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 184, China uyl1@gmail.com Abstract Canonical
More informationThe covariance of uncertain variables: definition and calculation formulae
Fuzzy Optim Decis Making 218 17:211 232 https://doi.org/1.17/s17-17-927-3 The covariance of uncertain variables: definition and calculation formulae Mingxuan Zhao 1 Yuhan Liu 2 Dan A. Ralescu 2 Jian Zhou
More informationUncertain Satisfiability and Uncertain Entailment
Uncertain Satisfiability and Uncertain Entailment Zhuo Wang, Xiang Li Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China zwang0518@sohu.com, xiang-li04@mail.tsinghua.edu.cn
More informationBregman superquantiles. Estimation methods and applications
Bregman superquantiles. Estimation methods and applications Institut de mathématiques de Toulouse 2 juin 2014 Joint work with F. Gamboa, A. Garivier (IMT) and B. Iooss (EDF R&D). 1 Coherent measure of
More informationRepresentation theorem for AVaR under a submodular capacity
3 214 5 ( ) Journal of East China Normal University (Natural Science) No. 3 May 214 Article ID: 1-5641(214)3-23-7 Representation theorem for AVaR under a submodular capacity TIAN De-jian, JIANG Long, JI
More informationSensitivity and Stability Analysis in Uncertain Data Envelopment (DEA)
Sensitivity and Stability Analysis in Uncertain Data Envelopment (DEA) eilin Wen a,b, Zhongfeng Qin c, Rui Kang b a State Key Laboratory of Virtual Reality Technology and Systems b Department of System
More informationOn a relationship between distorted and spectral risk measures
MPRA Munich Personal RePEc Archive On a relationship between distorted and spectral risk measures Gzyl Henryk and Mayoral Silvia November 26 Online at http://mpra.ub.uni-muenchen.de/916/ MPRA Paper No.
More informationMinimum Spanning Tree with Uncertain Random Weights
Minimum Spanning Tree with Uncertain Random Weights Yuhong Sheng 1, Gang Shi 2 1. Department of Mathematical Sciences, Tsinghua University, Beijing 184, China College of Mathematical and System Sciences,
More informationReliability Analysis in Uncertain Random System
Reliability Analysis in Uncertain Random System Meilin Wen a,b, Rui Kang b a State Key Laboratory of Virtual Reality Technology and Systems b School of Reliability and Systems Engineering Beihang University,
More informationSome limit theorems on uncertain random sequences
Journal of Intelligent & Fuzzy Systems 34 (218) 57 515 DOI:1.3233/JIFS-17599 IOS Press 57 Some it theorems on uncertain random sequences Xiaosheng Wang a,, Dan Chen a, Hamed Ahmadzade b and Rong Gao c
More informationElliptic entropy of uncertain random variables
Elliptic entropy of uncertain random variables Lin Chen a, Zhiyong Li a, Isnaini osyida b, a College of Management and Economics, Tianjin University, Tianjin 372, China b Department of Mathematics, Universitas
More informationCOHERENT APPROACHES TO RISK IN OPTIMIZATION UNDER UNCERTAINTY
COHERENT APPROACHES TO RISK IN OPTIMIZATION UNDER UNCERTAINTY Terry Rockafellar University of Washington, Seattle University of Florida, Gainesville Goal: a coordinated view of recent ideas in risk modeling
More informationRisk Aversion and Coherent Risk Measures: a Spectral Representation Theorem
Risk Aversion and Coherent Risk Measures: a Spectral Representation Theorem Carlo Acerbi Abaxbank, Corso Monforte 34, 2122 Milano (Italy) July 1, 21 Abstract We study a space of coherent risk measures
More informationBregman superquantiles. Estimation methods and applications
Bregman superquantiles Estimation methods and applications Institut de mathématiques de Toulouse 17 novembre 2014 Joint work with F Gamboa, A Garivier (IMT) and B Iooss (EDF R&D) Bregman superquantiles
More informationExpected Shortfall is not elicitable so what?
Expected Shortfall is not elicitable so what? Dirk Tasche Bank of England Prudential Regulation Authority 1 dirk.tasche@gmx.net Modern Risk Management of Insurance Firms Hannover, January 23, 2014 1 The
More informationApplications of axiomatic capital allocation and generalized weighted allocation
6 2010 11 ( ) Journal of East China Normal University (Natural Science) No. 6 Nov. 2010 Article ID: 1000-5641(2010)06-0146-10 Applications of axiomatic capital allocation and generalized weighted allocation
More informationPéter Csóka, P. Jean-Jacques Herings, László Á. Kóczy. Coherent Measures of Risk from a General Equilibrium Perspective RM/06/016
Péter Csóka, P. Jean-Jacques Herings, László Á. Kóczy Coherent Measures of Risk from a General Equilibrium Perspective RM/06/016 JEL code: D51, G10, G12 Maastricht research school of Economics of TEchnology
More informationMultivariate Stress Scenarios and Solvency
Multivariate Stress Scenarios and Solvency Alexander J. McNeil 1 1 Heriot-Watt University Edinburgh Croatian Quants Day Zagreb 11th May 2012 a.j.mcneil@hw.ac.uk AJM Stress Testing 1 / 51 Regulation General
More informationUncertain Programming Model for Solid Transportation Problem
INFORMATION Volume 15, Number 12, pp.342-348 ISSN 1343-45 c 212 International Information Institute Uncertain Programming Model for Solid Transportation Problem Qing Cui 1, Yuhong Sheng 2 1. School of
More informationThe α-maximum Flow Model with Uncertain Capacities
International April 25, 2013 Journal7:12 of Uncertainty, WSPC/INSTRUCTION Fuzziness and Knowledge-Based FILE Uncertain*-maximum*Flow*Model Systems c World Scientific Publishing Company The α-maximum Flow
More informationUncertain Quadratic Minimum Spanning Tree Problem
Uncertain Quadratic Minimum Spanning Tree Problem Jian Zhou Xing He Ke Wang School of Management Shanghai University Shanghai 200444 China Email: zhou_jian hexing ke@shu.edu.cn Abstract The quadratic minimum
More informationExpected Shortfall is not elicitable so what?
Expected Shortfall is not elicitable so what? Dirk Tasche Bank of England Prudential Regulation Authority 1 dirk.tasche@gmx.net Finance & Stochastics seminar Imperial College, November 20, 2013 1 The opinions
More informationChance Order of Two Uncertain Random Variables
Journal of Uncertain Systems Vol.12, No.2, pp.105-122, 2018 Online at: www.jus.org.uk Chance Order of Two Uncertain andom Variables. Mehralizade 1, M. Amini 1,, B. Sadeghpour Gildeh 1, H. Ahmadzade 2 1
More informationA Theory for Measures of Tail Risk
A Theory for Measures of Tail Risk Ruodu Wang http://sas.uwaterloo.ca/~wang Department of Statistics and Actuarial Science University of Waterloo, Canada Extreme Value Analysis Conference 2017 TU Delft
More informationDistortion Risk Measures: Coherence and Stochastic Dominance
Distortion Risk Measures: Coherence and Stochastic Dominance Dr. Julia L. Wirch Dr. Mary R. Hardy Dept. of Actuarial Maths Dept. of Statistics and and Statistics Actuarial Science Heriot-Watt University
More informationON LIU S INFERENCE RULE FOR UNCERTAIN SYSTEMS
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 18, No. 1 (2010 1 11 c World Scientific Publishing Company DOI: 10.1142/S0218488510006349 ON LIU S INFERENCE RULE FOR UNCERTAIN
More informationAsymptotic Bounds for the Distribution of the Sum of Dependent Random Variables
Asymptotic Bounds for the Distribution of the Sum of Dependent Random Variables Ruodu Wang November 26, 2013 Abstract Suppose X 1,, X n are random variables with the same known marginal distribution F
More informationAn Analytic Method for Solving Uncertain Differential Equations
Journal of Uncertain Systems Vol.6, No.4, pp.244-249, 212 Online at: www.jus.org.uk An Analytic Method for Solving Uncertain Differential Equations Yuhan Liu Department of Industrial Engineering, Tsinghua
More informationA MODEL FOR THE INVERSE 1-MEDIAN PROBLEM ON TREES UNDER UNCERTAIN COSTS. Kien Trung Nguyen and Nguyen Thi Linh Chi
Opuscula Math. 36, no. 4 (2016), 513 523 http://dx.doi.org/10.7494/opmath.2016.36.4.513 Opuscula Mathematica A MODEL FOR THE INVERSE 1-MEDIAN PROBLEM ON TREES UNDER UNCERTAIN COSTS Kien Trung Nguyen and
More informationMultivariate comonotonicity, stochastic orders and risk measures
Multivariate comonotonicity, stochastic orders and risk measures Alfred Galichon (Ecole polytechnique) Brussels, May 25, 2012 Based on collaborations with: A. Charpentier (Rennes) G. Carlier (Dauphine)
More informationNonlife Actuarial Models. Chapter 4 Risk Measures
Nonlife Actuarial Models Chapter 4 Risk Measures Learning Objectives 1. Risk measures based on premium principles 2. Risk measures based on capital requirements 3. Value-at-Risk and conditional tail expectation
More informationGeneral approach to the optimal portfolio risk management
General approach to the optimal portfolio risk management Zinoviy Landsman, University of Haifa Joint talk with Udi Makov and Tomer Shushi This presentation has been prepared for the Actuaries Institute
More informationDistance-based test for uncertainty hypothesis testing
Sampath and Ramya Journal of Uncertainty Analysis and Applications 03, :4 RESEARCH Open Access Distance-based test for uncertainty hypothesis testing Sundaram Sampath * and Balu Ramya * Correspondence:
More informationOn Backtesting Risk Measurement Models
On Backtesting Risk Measurement Models Hideatsu Tsukahara Department of Economics, Seijo University e-mail address: tsukahar@seijo.ac.jp 1 Introduction In general, the purpose of backtesting is twofold:
More informationCompetitive Equilibria in a Comonotone Market
Competitive Equilibria in a Comonotone Market 1/51 Competitive Equilibria in a Comonotone Market Ruodu Wang http://sas.uwaterloo.ca/ wang Department of Statistics and Actuarial Science University of Waterloo
More informationUNCERTAIN OPTIMAL CONTROL WITH JUMP. Received December 2011; accepted March 2012
ICIC Express Letters Part B: Applications ICIC International c 2012 ISSN 2185-2766 Volume 3, Number 2, April 2012 pp. 19 2 UNCERTAIN OPTIMAL CONTROL WITH JUMP Liubao Deng and Yuanguo Zhu Department of
More informationInclusion Relationship of Uncertain Sets
Yao Journal of Uncertainty Analysis Applications (2015) 3:13 DOI 10.1186/s40467-015-0037-5 RESEARCH Open Access Inclusion Relationship of Uncertain Sets Kai Yao Correspondence: yaokai@ucas.ac.cn School
More informationTranslation invariant and positive homogeneous risk measures and portfolio management. Zinoviy Landsman Department of Statistics, University of Haifa
ranslation invariant and positive homogeneous risk measures and portfolio management 1 Zinoviy Landsman Department of Statistics, University of Haifa -It is wide-spread opinion the use of any positive
More informationUncertain risk aversion
J Intell Manuf (7) 8:65 64 DOI.7/s845-4-3-5 Uncertain risk aversion Jian Zhou Yuanyuan Liu Xiaoxia Zhang Xin Gu Di Wang Received: 5 August 4 / Accepted: 8 November 4 / Published online: 7 December 4 Springer
More informationCharacterization of Upper Comonotonicity via Tail Convex Order
Characterization of Upper Comonotonicity via Tail Convex Order Hee Seok Nam a,, Qihe Tang a, Fan Yang b a Department of Statistics and Actuarial Science, University of Iowa, 241 Schaeffer Hall, Iowa City,
More informationStability and attractivity in optimistic value for dynamical systems with uncertainty
International Journal of General Systems ISSN: 38-179 (Print 1563-514 (Online Journal homepage: http://www.tandfonline.com/loi/ggen2 Stability and attractivity in optimistic value for dynamical systems
More informationMFM Practitioner Module: Risk & Asset Allocation. John Dodson. February 4, 2015
& & MFM Practitioner Module: Risk & Asset Allocation February 4, 2015 & Meucci s Program for Asset Allocation detect market invariance select the invariants estimate the market specify the distribution
More informationResearch Article Continuous Time Portfolio Selection under Conditional Capital at Risk
Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2010, Article ID 976371, 26 pages doi:10.1155/2010/976371 Research Article Continuous Time Portfolio Selection under Conditional
More informationStochastic Optimization with Risk Measures
Stochastic Optimization with Risk Measures IMA New Directions Short Course on Mathematical Optimization Jim Luedtke Department of Industrial and Systems Engineering University of Wisconsin-Madison August
More informationInsurance: Mathematics and Economics
Contents lists available at ScienceDirect Insurance: Mathematics Economics journal homepage: www.elsevier.com/locate/ime Optimal reinsurance with general risk measures Alejro Balbás a, Beatriz Balbás a,
More informationGeneralized quantiles as risk measures
Generalized quantiles as risk measures F. Bellini 1, B. Klar 2, A. Müller 3, E. Rosazza Gianin 1 1 Dipartimento di Statistica e Metodi Quantitativi, Università di Milano Bicocca 2 Institut für Stochastik,
More informationA Note of the Expected Value and Variance of Fuzzy Variables
ISSN 79-3889 (print, 79-3897 (online International Journal of Nonlinear Science Vol.9( No.,pp.86-9 A Note of the Expected Value and Variance of Fuzzy Variables Zhigang Wang, Fanji Tian Department of Applied
More informationA new approach for stochastic ordering of risks
A new approach for stochastic ordering of risks Liang Hong, PhD, FSA Department of Mathematics Robert Morris University Presented at 2014 Actuarial Research Conference UC Santa Barbara July 16, 2014 Liang
More informationThe Canonical Model Space for Law-invariant Convex Risk Measures is L 1
The Canonical Model Space for Law-invariant Convex Risk Measures is L 1 Damir Filipović Gregor Svindland 3 November 2008 Abstract In this paper we establish a one-to-one correspondence between lawinvariant
More informationMarket Risk. MFM Practitioner Module: Quantitiative Risk Management. John Dodson. February 8, Market Risk. John Dodson.
MFM Practitioner Module: Quantitiative Risk Management February 8, 2017 This week s material ties together our discussion going back to the beginning of the fall term about risk measures based on the (single-period)
More informationInverse Stochastic Dominance Constraints Duality and Methods
Duality and Methods Darinka Dentcheva 1 Andrzej Ruszczyński 2 1 Stevens Institute of Technology Hoboken, New Jersey, USA 2 Rutgers University Piscataway, New Jersey, USA Research supported by NSF awards
More informationA SECOND ORDER STOCHASTIC DOMINANCE PORTFOLIO EFFICIENCY MEASURE
K Y B E R N E I K A V O L U M E 4 4 ( 2 0 0 8 ), N U M B E R 2, P A G E S 2 4 3 2 5 8 A SECOND ORDER SOCHASIC DOMINANCE PORFOLIO EFFICIENCY MEASURE Miloš Kopa and Petr Chovanec In this paper, we introduce
More informationA Note on Robust Representations of Law-Invariant Quasiconvex Functions
A Note on Robust Representations of Law-Invariant Quasiconvex Functions Samuel Drapeau Michael Kupper Ranja Reda October 6, 21 We give robust representations of law-invariant monotone quasiconvex functions.
More informationHybrid Logic and Uncertain Logic
Journal of Uncertain Systems Vol.3, No.2, pp.83-94, 2009 Online at: www.jus.org.uk Hybrid Logic and Uncertain Logic Xiang Li, Baoding Liu Department of Mathematical Sciences, Tsinghua University, Beijing,
More informationOptimizing Project Time-Cost Trade-off Based on Uncertain Measure
INFORMATION Volume xx, Number xx, pp.1-9 ISSN 1343-45 c 21x International Information Institute Optimizing Project Time-Cost Trade-off Based on Uncertain Measure Hua Ke 1, Huimin Liu 1, Guangdong Tian
More informationMonetary Risk Measures and Generalized Prices Relevant to Set-Valued Risk Measures
Applied Mathematical Sciences, Vol. 8, 2014, no. 109, 5439-5447 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43176 Monetary Risk Measures and Generalized Prices Relevant to Set-Valued
More informationИЗМЕРЕНИЕ РИСКА MEASURING RISK Arcady Novosyolov Institute of computational modeling SB RAS Krasnoyarsk, Russia,
ИЗМЕРЕНИЕ РИСКА EASURING RISK Arcady Novosyolov Institute of computational modeling SB RAS Krasnoyarsk, Russia, anov@ksckrasnru Abstract Problem of representation of human preferences among uncertain outcomes
More informationStochastic dominance and risk measure: A decision-theoretic foundation for VaR and C-VaR
Hong Kong Baptist University HKBU Institutional Repository Department of Economics Journal Articles Department of Economics 2010 Stochastic dominance and risk measure: A decision-theoretic foundation for
More informationAn Uncertain Control Model with Application to. Production-Inventory System
An Uncertain Control Model with Application to Production-Inventory System Kai Yao 1, Zhongfeng Qin 2 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Economics
More informationMotivation General concept of CVaR Optimization Comparison. VaR and CVaR. Přemysl Bejda.
VaR and CVaR Přemysl Bejda premyslbejda@gmail.com 2014 Contents 1 Motivation 2 General concept of CVaR 3 Optimization 4 Comparison Contents 1 Motivation 2 General concept of CVaR 3 Optimization 4 Comparison
More informationGeneralized quantiles as risk measures
Generalized quantiles as risk measures Bellini, Klar, Muller, Rosazza Gianin December 1, 2014 Vorisek Jan Introduction Quantiles q α of a random variable X can be defined as the minimizers of a piecewise
More informationCORVINUS ECONOMICS WORKING PAPERS. On the impossibility of fair risk allocation. by Péter Csóka Miklós Pintér CEWP 12/2014
CORVINUS ECONOMICS WORKING PAPERS CEWP 12/2014 On the impossibility of fair risk allocation by Péter Csóka Miklós Pintér http://unipub.lib.uni-corvinus.hu/1658 On the impossibility of fair risk allocation
More information