Convex Risk Measures: Basic Facts, Law-Invariance and Beyond, Asymptotics for Large Portfolios
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Var and Other Risk Measures
What is Risk? Risk Measures Methods of estimating risk measures Bibliography VaR and other Risk Measures Francisco Ramírez Calixto International Actuarial Association November 27th, 2018 Francisco Ramírez Calixto VaR and other Risk Measures What is Risk? Risk Measures Methods of estimating risk measures Bibliography Outline 1 What is Risk? 2 Risk Measures 3 Methods of estimating risk measures Francisco Ramírez Calixto VaR and other Risk Measures What is Risk? Risk Measures Methods of estimating risk measures Bibliography What is Risk? Risk 6= size of loss or size of a cost Risk lies in the unexpected losses. Francisco Ramírez Calixto VaR and other Risk Measures What is Risk? Risk Measures Methods of estimating risk measures Bibliography Types of Financial Risk In Basel III, there are three major broad risk categories: Credit Risk: Francisco Ramírez Calixto VaR and other Risk Measures What is Risk? Risk Measures Methods of estimating risk measures Bibliography Types of Financial Risk Operational Risk: Francisco Ramírez Calixto VaR and other Risk Measures What is Risk? Risk Measures Methods of estimating risk measures Bibliography Types of Financial Risk Market risk: Each one of these risks must be measured in order to allocate economic capital as a buer so that if a catastrophic event happens, the bank won't go bankrupt. Francisco Ramírez Calixto VaR and other Risk Measures What is Risk? Risk Measures Methods of estimating risk measures Bibliography Risk Measures Def. A risk measure is used to determine the amount of an asset or assets (traditionally currency) to be kept in reserve in order to cover for unexpected losses. -
Comparative Analyses of Expected Shortfall and Value-At-Risk Under Market Stress1
Comparative analyses of expected shortfall and value-at-risk under market stress1 Yasuhiro Yamai and Toshinao Yoshiba, Bank of Japan Abstract In this paper, we compare value-at-risk (VaR) and expected shortfall under market stress. Assuming that the multivariate extreme value distribution represents asset returns under market stress, we simulate asset returns with this distribution. With these simulated asset returns, we examine whether market stress affects the properties of VaR and expected shortfall. Our findings are as follows. First, VaR and expected shortfall may underestimate the risk of securities with fat-tailed properties and a high potential for large losses. Second, VaR and expected shortfall may both disregard the tail dependence of asset returns. Third, expected shortfall has less of a problem in disregarding the fat tails and the tail dependence than VaR does. 1. Introduction It is a well known fact that value-at-risk2 (VaR) models do not work under market stress. VaR models are usually based on normal asset returns and do not work under extreme price fluctuations. The case in point is the financial market crisis of autumn 1998. Concerning this crisis, CGFS (1999) notes that “a large majority of interviewees admitted that last autumn’s events were in the “tails” of distributions and that VaR models were useless for measuring and monitoring market risk”. Our question is this: Is this a problem of the estimation methods, or of VaR as a risk measure? The estimation methods used for standard VaR models have problems for measuring extreme price movements. They assume that the asset returns follow a normal distribution. -
A Framework for Dynamic Hedging Under Convex Risk Measures
A Framework for Dynamic Hedging under Convex Risk Measures Antoine Toussaint∗ Ronnie Sircary November 2008; revised August 2009 Abstract We consider the problem of minimizing the risk of a financial position (hedging) in an incomplete market. It is well-known that the industry standard for risk measure, the Value- at-Risk, does not take into account the natural idea that risk should be minimized through diversification. This observation led to the recent theory of coherent and convex risk measures. But, as a theory on bounded financial positions, it is not ideally suited for the problem of hedging because simple strategies such as buy-hold strategies may not be bounded. Therefore, we propose as an alternative to use convex risk measures defined as functionals on L2 (or by simple extension Lp, p > 1). This framework is more suitable for optimal hedging with L2 valued financial markets. A dual representation is given for this minimum risk or market adjusted risk when the risk measure is real-valued. In the general case, we introduce constrained hedging and prove that the market adjusted risk is still a L2 convex risk measure and the existence of the optimal hedge. We illustrate the practical advantage in the shortfall risk measure by showing how minimizing risk in this framework can lead to a HJB equation and we give an example of computation in a stochastic volatility model with the shortfall risk measure 1 Introduction We are interested in the problem of hedging in an incomplete market: an investor decides to buy a contract with a non-replicable payoff X at time T but has the opportunity to invest in a financial market to cover his risk. -
Divergence-Based Risk Measures: a Discussion on Sensitivities and Extensions
entropy Article Divergence-Based Risk Measures: A Discussion on Sensitivities and Extensions Meng Xu 1 and José M. Angulo 2,* 1 School of Economics, Sichuan University, Chengdu 610065, China 2 Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain * Correspondence: [email protected]; Tel.: +34-958-240492 Received: 13 June 2019; Accepted: 24 June 2019; Published: 27 June 2019 Abstract: This paper introduces a new family of the convex divergence-based risk measure by specifying (h, f)-divergence, corresponding with the dual representation. First, the sensitivity characteristics of the modified divergence risk measure with respect to profit and loss (P&L) and the reference probability in the penalty term are discussed, in view of the certainty equivalent and robust statistics. Secondly, a similar sensitivity property of (h, f)-divergence risk measure with respect to P&L is shown, and boundedness by the analytic risk measure is proved. Numerical studies designed for Rényi- and Tsallis-divergence risk measure are provided. This new family integrates a wide spectrum of divergence risk measures and relates to divergence preferences. Keywords: convex risk measure; preference; sensitivity analysis; ambiguity; f-divergence 1. Introduction In the last two decades, there has been a substantial development of a well-founded risk measure theory, particularly propelled since the axiomatic approach introduced by [1] in relation to the concept of coherency. While, to a large extent, the theory has been fundamentally inspired and motivated with financial risk assessment objectives in perspective, many other areas of application are currently or potentially benefited by the formal mathematical construction of the discipline. -
G-Expectations with Application to Risk Measures
g-Expectations with application to risk measures Sonja C. Offwood Programme in Advanced Mathematics of Finance, University of the Witwatersrand, Johannesburg. 5 August 2012 Abstract Peng introduced a typical filtration consistent nonlinear expectation, called a g-expectation in [40]. It satisfies all properties of the classical mathematical ex- pectation besides the linearity. Peng's conditional g-expectation is a solution to a backward stochastic differential equation (BSDE) within the classical framework of It^o'scalculus, with terminal condition given at some fixed time T . In addition, this g-expectation is uniquely specified by a real function g satisfying certain properties. Many properties of the g-expectation, which will be presented, follow from the spec- ification of this function. Martingales, super- and submartingales have been defined in the nonlinear setting of g-expectations. Consequently, a nonlinear Doob-Meyer decomposition theorem was proved. Applications of g-expectations in the mathematical financial world have also been of great interest. g-Expectations have been applied to the pricing of contin- gent claims in the financial market, as well as to risk measures. Risk measures were introduced to quantify the riskiness of any financial position. They also give an indi- cation as to which positions carry an acceptable amount of risk and which positions do not. Coherent risk measures and convex risk measures will be examined. These risk measures were extended into a nonlinear setting using the g-expectation. In many cases due to intermediate cashflows, we want to work with a multi-period, dy- namic risk measure. Conditional g-expectations were then used to extend dynamic risk measures into the nonlinear setting. -
Towards a Topological Representation of Risks and Their Measures
risks Article Towards a Topological Representation of Risks and Their Measures Tomer Shushi ID Department of Business Administration, Guilford Glazer Faculty of Business and Management, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 8410501, Israel; [email protected] Received: 8 October 2018; Accepted: 15 November 2018; Published: 19 November 2018 Abstract: In risk theory, risks are often modeled by risk measures which allow quantifying the risks and estimating their possible outcomes. Risk measures rely on measure theory, where the risks are assumed to be random variables with some distribution function. In this work, we derive a novel topological-based representation of risks. Using this representation, we show the differences between diversifiable and non-diversifiable. We show that topological risks should be modeled using two quantities, the risk measure that quantifies the predicted amount of risk, and a distance metric which quantifies the uncertainty of the risk. Keywords: diversification; portfolio theory; risk measurement; risk measures; set-valued measures; topology; topological spaces 1. Introduction The mathematical formulation of risks is based purely on probability. Let Y be a random variable on the probability space (W, F, P), where W represents the space of all possible outcomes, F is the s-algebra, and P is the probability measure. For a single outcome w 2 W, Y(w) is the realization of Y. The theory of risk measures aims to quantify the amount of loss for each investigated risk where such a loss occurs with some level of uncertainty. Thus, risk measures are a powerful tool for any risk management analysis. -
Loss-Based Risk Measures Rama Cont, Romain Deguest, Xuedong He
Loss-Based Risk Measures Rama Cont, Romain Deguest, Xuedong He To cite this version: Rama Cont, Romain Deguest, Xuedong He. Loss-Based Risk Measures. 2011. hal-00629929 HAL Id: hal-00629929 https://hal.archives-ouvertes.fr/hal-00629929 Submitted on 7 Oct 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Loss-based risk measures Rama CONT1,3, Romain DEGUEST 2 and Xue Dong HE3 1) Laboratoire de Probabilit´es et Mod`eles Al´eatoires CNRS- Universit´ePierre et Marie Curie, France. 2) EDHEC Risk Institute, Nice (France). 3) IEOR Dept, Columbia University, New York. 2011 Abstract Starting from the requirement that risk measures of financial portfolios should be based on their losses, not their gains, we define the notion of loss-based risk measure and study the properties of this class of risk measures. We charac- terize loss-based risk measures by a representation theorem and give examples of such risk measures. We then discuss the statistical robustness of estimators of loss-based risk measures: we provide a general criterion for qualitative ro- bustness of risk estimators and compare this criterion with sensitivity analysis of estimators based on influence functions. -
A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations
risks Article A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations Matthias Fischer 1,*, Thorsten Moser 2 and Marius Pfeuffer 1 1 Lehrstuhl für Statistik und Ökonometrie, Universität Erlangen-Nürnberg, Lange Gasse 20, 90403 Nürnberg, Germany; [email protected] 2 Risikocontrolling Kapitalanlagen, R+V Lebensverischerung AG, Raiffeisenplatz 1, 65189 Wiesbaden, Germany; [email protected] * Correspondence: matthias.fi[email protected] Received: 11 October 2018; Accepted: 30 November 2018; Published: 7 December 2018 Abstract: In both financial theory and practice, Value-at-risk (VaR) has become the predominant risk measure in the last two decades. Nevertheless, there is a lively and controverse on-going discussion about possible alternatives. Against this background, our first objective is to provide a current overview of related competitors with the focus on credit risk management which includes definition, references, striking properties and classification. The second part is dedicated to the measurement of risk concentrations of credit portfolios. Typically, credit portfolio models are used to calculate the overall risk (measure) of a portfolio. Subsequently, Euler’s allocation scheme is applied to break the portfolio risk down to single counterparties (or different subportfolios) in order to identify risk concentrations. We first carry together the Euler formulae for the risk measures under consideration. In two cases (Median Shortfall and Range-VaR), explicit formulae are presented for the first time. Afterwards, we present a comprehensive study for a benchmark portfolio according to Duellmann and Masschelein (2007) and nine different risk measures in conjunction with the Euler allocation. It is empirically shown that—in principle—all risk measures are capable of identifying both sectoral and single-name concentration. -
An Introduction to Multivariate and Dynamic Risk Measures Arthur Charpentier
An introduction to multivariate and dynamic risk measures Arthur Charpentier To cite this version: Arthur Charpentier. An introduction to multivariate and dynamic risk measures. 2018. hal-01831481 HAL Id: hal-01831481 https://hal.archives-ouvertes.fr/hal-01831481 Preprint submitted on 5 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. An introduction to multivariate and dynamic risk measures Arthur Charpentier May 2014 This document is lectures notes for xfa doctoral course in Louvain-la- Neuve, given in 2014 1 1 Introduction and Notations All (univariate) risk measures - or to be more specific all downside (or upside) risk - are, somehow, related to quantiles. So, in order to derive some general multivariate risk measures, or dynamic ones, we need to understand more deeply what quantile functions are, and why we need them (in this risk measure context). 1.1 Probablistic and Measurable Spaces Consider some topological space S, metrizable, in the sense that there is a metric d on that space. Assume that S is separable, so that the σ-algebra S of S is generated by open d-balls, centered on a contable dense subset of S. -
A Framework for Time-Consistent, Risk-Averse Model Predictive Control: Theory and Algorithms
A Framework for Time-Consistent, Risk-Averse Model Predictive Control: Theory and Algorithms Yin-Lam Chow, Marco Pavone Abstract— In this paper we present a framework for risk- planner conservatism and the risk of infeasibility (clearly this averse model predictive control (MPC) of linear systems af- can not be achieved with a worst-case approach). Second, fected by multiplicative uncertainty. Our key innovation is risk-aversion allows the control designer to increase policy to consider time-consistent, dynamic risk metrics as objective functions to be minimized. This framework is axiomatically robustness by limiting confidence in the model. Third, risk- justified in terms of time-consistency of risk preferences, is aversion serves to prevent rare undesirable events. Finally, in amenable to dynamic optimization, and is unifying in the sense a reinforcement learning framework when the world model that it captures a full range of risk assessments from risk- is not accurately known, a risk-averse agent can cautiously neutral to worst case. Within this framework, we propose and balance exploration versus exploitation for fast convergence analyze an online risk-averse MPC algorithm that is provably stabilizing. Furthermore, by exploiting the dual representation and to avoid “bad” states that could potentially lead to a of time-consistent, dynamic risk metrics, we cast the computa- catastrophic failure [9], [10]. tion of the MPC control law as a convex optimization problem Inclusion of risk-aversion in MPC is difficult for two main amenable to implementation on embedded systems. Simulation reasons. First, it appears to be difficult to model risk in results are presented and discussed. -
Dynamic Risk Measurement, with an Application to a Pension Fund Setting
Dynamic Risk Measurement, with an Application to a Pension Fund Setting by Servaas van Bilsen 24.26.44 A thesis submitted in partial fulfillment of the requirements for the degree of Master in Quantitative Finance and Actuarial Sciences Faculty of Economics and Business Administration Tilburg University July, 2010 Supervised by: dr. R.J.A. Laeven prof. dr. J.M. Schumacher Abstract The financial crisis has highlighted that economic agents have not been successful in associating capital requirements with the risks un- dertaken by financial institutions. The connection between capital requirements and risk-taking has to be improved. This thesis focuses on the improvement of risk measurement. First, we provide a litera- ture review on static and dynamic convex measures of risk. An explicit convex measure of risk is given by the entropic risk measure. We study this risk measure and its connection with stochastic differential equa- tions. As an extension to the entropic risk measure, we also investigate a risk measure that allows for some degree of ambiguity in choosing a probabilistic model of some random variable. The literature has often focused on risk measurement for random variables. To take into ac- count the dynamic fluctuation of intermediate cash flows, we develop an entropic risk measure for random processes. Finally, we apply some measures of risk to a pension fund setting. Keywords: Static Measures of Risk, Dynamic Measures of Risk, Time-consistency, Entropic Risk Measure, Ambiguity, Random Pro- cesses, Pension Funds, Backward Stochastic Differential Equations ii Contents 1 Introduction 1 2 Static Risk Measures 6 2.1 Monetary, Convex and Coherent Risk Measures . -
Estimation of Distortion Risk Measures
Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS046) p.5035 Estimation of Distortion Risk Measures Hideatsu Tsukahara Department of Economics, Seijo University, Tokyo e-mail address: [email protected] The concept of coherent risk measure was introduced in Artzner et al. (1999). They listed some properties, called axioms of `coherence', that any good risk measure should possess, and studied the (non-)coherence of widely-used risk measure such as Value-at- Risk (VaR) and expected shortfall (also known as tail conditional expectation or tail VaR). Kusuoka (2001) introduced two additional axioms called law invariance and comonotonic additivity, and proved that the class of coherent risk measures satisfying these two axioms coincides with the class of distortion risk measures with convex distortions. To be more speci¯c, let X be a random variable representing a loss of some ¯nancial position, and let F (x) := P(X · x) be the distribution function (df) of X. We denote ¡1 its quantile function by F (u) := inffx 2 R: FX (x) ¸ ug; 0 < u < 1. A distortion risk measure is then of the following form Z Z ¡1 ½D(X) := F (u) dD(u) = x dD ± F (x); (1) [0;1] R where D is a distortion function, which is simply a df D on [0; 1]; i.e., a right-continuous, increasing function on [0; 1] satisfying D(0) = 0 and D(1) = 1. For ½D(X) to be coherent, D must be convex, which we assume throughout this paper. The celebrated VaR can be written of the form (1), but with non-convex D; this implies that the VaR is not coherent.