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Incorporating Extreme Events Into Risk Measurement
Lecture notes on risk management, public policy, and the financial system Incorporating extreme events into risk measurement Allan M. Malz Columbia University Incorporating extreme events into risk measurement Outline Stress testing and scenario analysis Expected shortfall Extreme value theory © 2021 Allan M. Malz Last updated: July 25, 2021 2/24 Incorporating extreme events into risk measurement Stress testing and scenario analysis Stress testing and scenario analysis Stress testing and scenario analysis Expected shortfall Extreme value theory 3/24 Incorporating extreme events into risk measurement Stress testing and scenario analysis Stress testing and scenario analysis What are stress tests? Stress tests analyze performance under extreme loss scenarios Heuristic portfolio analysis Steps in carrying out a stress test 1. Determine appropriate scenarios 2. Calculate shocks to risk factors in each scenario 3. Value the portfolio in each scenario Objectives of stress testing Address tail risk Reduce model risk by reducing reliance on models “Know the book”: stress tests can reveal vulnerabilities in specfic positions or groups of positions Criteria for appropriate stress scenarios Should be tailored to firm’s specific key vulnerabilities And avoid assumptions that favor the firm, e.g. competitive advantages in a crisis Should be extreme but not implausible 4/24 Incorporating extreme events into risk measurement Stress testing and scenario analysis Stress testing and scenario analysis Approaches to formulating stress scenarios Historical -
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. -
University of Regina Lecture Notes Michael Kozdron
University of Regina Statistics 441 – Stochastic Calculus with Applications to Finance Lecture Notes Winter 2009 Michael Kozdron [email protected] http://stat.math.uregina.ca/∼kozdron List of Lectures and Handouts Lecture #1: Introduction to Financial Derivatives Lecture #2: Financial Option Valuation Preliminaries Lecture #3: Introduction to MATLAB and Computer Simulation Lecture #4: Normal and Lognormal Random Variables Lecture #5: Discrete-Time Martingales Lecture #6: Continuous-Time Martingales Lecture #7: Brownian Motion as a Model of a Fair Game Lecture #8: Riemann Integration Lecture #9: The Riemann Integral of Brownian Motion Lecture #10: Wiener Integration Lecture #11: Calculating Wiener Integrals Lecture #12: Further Properties of the Wiener Integral Lecture #13: ItˆoIntegration (Part I) Lecture #14: ItˆoIntegration (Part II) Lecture #15: Itˆo’s Formula (Part I) Lecture #16: Itˆo’s Formula (Part II) Lecture #17: Deriving the Black–Scholes Partial Differential Equation Lecture #18: Solving the Black–Scholes Partial Differential Equation Lecture #19: The Greeks Lecture #20: Implied Volatility Lecture #21: The Ornstein-Uhlenbeck Process as a Model of Volatility Lecture #22: The Characteristic Function for a Diffusion Lecture #23: The Characteristic Function for Heston’s Model Lecture #24: Review Lecture #25: Review Lecture #26: Review Lecture #27: Risk Neutrality Lecture #28: A Numerical Approach to Option Pricing Using Characteristic Functions Lecture #29: An Introduction to Functional Analysis for Financial Applications Lecture #30: A Linear Space of Random Variables Lecture #31: Value at Risk Lecture #32: Monetary Risk Measures Lecture #33: Risk Measures and their Acceptance Sets Lecture #34: A Representation of Coherent Risk Measures Lecture #35: Further Remarks on Value at Risk Lecture #36: Midterm Review Statistics 441 (Winter 2009) January 5, 2009 Prof. -
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. -
Modeling Vehicle Insurance Loss Data Using a New Member of T-X Family of Distributions
Journal of Statistical Theory and Applications Vol. 19(2), June 2020, pp. 133–147 DOI: https://doi.org/10.2991/jsta.d.200421.001; ISSN 1538-7887 https://www.atlantis-press.com/journals/jsta Modeling Vehicle Insurance Loss Data Using a New Member of T-X Family of Distributions Zubair Ahmad1, Eisa Mahmoudi1,*, Sanku Dey2, Saima K. Khosa3 1Department of Statistics, Yazd University, Yazd, Iran 2Department of Statistics, St. Anthonys College, Shillong, India 3Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan ARTICLEINFO A BSTRACT Article History In actuarial literature, we come across a diverse range of probability distributions for fitting insurance loss data. Popular Received 16 Oct 2019 distributions are lognormal, log-t, various versions of Pareto, log-logistic, Weibull, gamma and its variants and a generalized Accepted 14 Jan 2020 beta of the second kind, among others. In this paper, we try to supplement the distribution theory literature by incorporat- ing the heavy tailed model, called weighted T-X Weibull distribution. The proposed distribution exhibits desirable properties Keywords relevant to the actuarial science and inference. Shapes of the density function and key distributional properties of the weighted Heavy-tailed distributions T-X Weibull distribution are presented. Some actuarial measures such as value at risk, tail value at risk, tail variance and tail Weibull distribution variance premium are calculated. A simulation study based on the actuarial measures is provided. Finally, the proposed method Insurance losses is illustrated via analyzing vehicle insurance loss data. Actuarial measures Monte Carlo simulation Estimation 2000 Mathematics Subject Classification: 60E05, 62F10. © 2020 The Authors. Published by Atlantis Press SARL. -
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. -
A New Heavy Tailed Class of Distributions Which Includes the Pareto
risks Article A New Heavy Tailed Class of Distributions Which Includes the Pareto Deepesh Bhati 1 , Enrique Calderín-Ojeda 2,* and Mareeswaran Meenakshi 1 1 Department of Statistics, Central University of Rajasthan, Kishangarh 305817, India; [email protected] (D.B.); [email protected] (M.M.) 2 Department of Economics, Centre for Actuarial Studies, University of Melbourne, Melbourne, VIC 3010, Australia * Correspondence: [email protected] Received: 23 June 2019; Accepted: 10 September 2019; Published: 20 September 2019 Abstract: In this paper, a new heavy-tailed distribution, the mixture Pareto-loggamma distribution, derived through an exponential transformation of the generalized Lindley distribution is introduced. The resulting model is expressed as a convex sum of the classical Pareto and a special case of the loggamma distribution. A comprehensive exploration of its statistical properties and theoretical results related to insurance are provided. Estimation is performed by using the method of log-moments and maximum likelihood. Also, as the modal value of this distribution is expressed in closed-form, composite parametric models are easily obtained by a mode matching procedure. The performance of both the mixture Pareto-loggamma distribution and composite models are tested by employing different claims datasets. Keywords: pareto distribution; excess-of-loss reinsurance; loggamma distribution; composite models 1. Introduction In general insurance, pricing is one of the most complex processes and is no easy task but a long-drawn exercise involving the crucial step of modeling past claim data. For modeling insurance claims data, finding a suitable distribution to unearth the information from the data is a crucial work to help the practitioners to calculate fair premiums. -
An Introduction to Risk Measures for Actuarial Applications
An Introduction to Risk Measures for Actuarial Applications Mary R Hardy CIBC Professor of Financial Risk Management University of Waterloo 1 Introduction In actuarial applications we often work with loss distributions for insurance products. For example, in P&C insurance, we may develop a compound Poisson model for the losses under a single policy or a whole portfolio of policies. Similarly, in life insurance, we may develop a loss distribution for a portfolio of policies, often by stochastic simulation. Profit and loss distributions are also important in banking, and most of the risk measures we discuss in this note are also useful in risk management in banking. The convention in banking is to use profit random variables, that is Y where a loss outcome would be Y<0. The convention in insurance is to use loss random variables, X = −Y . In this paper we work exclusively with loss distributions. Thus, all the definitions that we present for insurance losses need to be suitably adapted for profit random variables. Additionally, it is usually appropriate to assume in insurance contexts that the loss X is non-negative, and we have assumed this in Section 2.5 of this note. It is not essential however, and the risk measures that we describe can be applied (perhaps after some adaptation) to random variables with a sample space spanning any part of the real line. Having established a loss distribution, either parametrically, non-parametrically, analyti- cally or by Monte Carlo simulation, we need to utilize the characteristics of the distribution for pricing, reserving and risk management. -
On Approximations of Value at Risk and Expected Shortfall Involving Kurtosis
On approximations of Value at Risk and Expected Shortfall involving kurtosis Maty´ as´ Barczy∗;, Ad´ am´ Dudas´ ∗∗,Jozsef´ Gall´ ∗∗∗ * MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi v´ertan´uktere 1, H{6720 Szeged, Hungary. ** Former master student of Faculty of Science and Technology, University of Debrecen, Hun- gary. *** Faculty of Informatics, University of Debrecen, Pf. 12, H{4010 Debrecen, Hungary. e-mail: [email protected] (M. Barczy), [email protected] (A.´ Dud´as), [email protected] (J. G´all). Corresponding author. Abstract We derive new approximations for the Value at Risk and the Expected Shortfall at high levels of loss distributions with positive skewness and excess kurtosis, and we describe their precisions for notable ones such as for exponential, Pareto type I, lognormal and compound (Poisson) distributions. Our approximations are motivated by that kind of extensions of the so-called Normal Power Approximation, used for approximating the cumulative distribution function of a random variable, which incorporate not only the skewness but the kurtosis of the random variable in question as well. We show the performance of our approximations in numerical examples and we also give comparisons with some known ones in the literature. 1 Introduction Value at Risk (VaR) and Expected Shortfall (ES) are standard risk measures in financial and insurance mathematics. VaR permits to measure the maximum aggregate loss of a portfolio with a given confidence level, while ES can be defined as the conditional expectation of the loss arXiv:1811.06361v3 [q-fin.RM] 23 Dec 2020 for losses beyond the corresponding VaR-level (see Definitions 3.1 and 3.2). -
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.