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Introduction to Copulas B. Wade Brorsen Oklahoma State University
Problem Multivariate pdf or cdf when marginal distributions are not normally distributed and not independent.
Where Used? Risk and Simulation Value at Risk (VaR) Valuing Derivatives Insurance
Extreme Value Theory Tail Dependence –Housing bubble –Collateralized Debt Obligations (CDO) –Hurricane –Crop disease –Bank failures –Long Term Capital Management
Agricultural Economics Taylor (1990) Richardson/Simetar Heuristic
Gaussian Copula Multivariate-t Copula Most Copulas are Bivariate Two Main Multivariate Copulas
A copula C(u, v) is C:[0, 1] 2 →[0, 1] Other properties
Sklar’s Theorem Any cdf H(X 1, X 2 ) with margins F(X 1 ) and G(X 2 ) can be represented as H(X 1, X 2 ) = C[F(X 1 ), G(X 2 )] Where C[ ] is a unique copula function.
Gaussian Copula H(Ψ -1 (u), Ψ -1 (v)) H is bivariate normal cdf Ψ -1 is inverse of a univariate normal cdf
Estimation Inference for margins (IFM) Maximum likelihood Simulation
SAS Program u = cdf (‘normal’, x1, 2, 2); v = cdf (‘normal’, x2, 5, 5); z1 = probit (u); z2 = probit (v); PROC CORR; /* IFM Method */ Var z1, z2;
SAS Program u = cdf (‘gamma’, x1, r1, lambda1); v = cdf (‘gamma’, x2, r2, lambda 2); z1 = probit (u); z2 = probit (v); PROC CORR; Var z1, z2;
Summary Copulas can give us a multivariate cdf for nonnormal distributions Agricultural economists should use copulas
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 1 Distributions and Copulas for Integrated Risk Management Elements.
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1 Confidential – Highly Restricted Copula Representation of Joint Risk Driver Distribution Why Copulas? Copulas provide a method of joining together individual.
Simulating Exchangeable Multivariate Archimedean Copulas and its Applications Authors: Florence Wu Emiliano A. Valdez Michael Sherris.
N 3 = 7 Input (s) Output X(s) s1s1 s2s2 s3s3 sPsP m = Number of Input Points (=3) n i = Number of Outputs at Input s i X (i) = Set of Outputs X j (i) at.
Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems B. Klein, M. Pahlow, Y. Hundecha, C. Gattke and A. Schumann.
Parameter Estimation for Dependent Risks: Experiments with Bivariate Copula Models Authors: Florence Wu Michael Sherris Date: 11 November 2005.
Enterprise Risk Management in Insurance Groups July 11, Enterprise Risk Management in Insurance Groups: Measuring Risk Concentration and Default.
Correlations and Copulas 1. Measures of Dependence 2 The risk can be split into two parts: the individual risks and the dependence structure between them.
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1 Lecture Plan Modelling Profit Distribution from Wind Production (Excel Case: Danish Wind Production and Spot Prices) Reasons for copula.
1 Risk Modeling of Multi-year, Multi-line Reinsurance Using Copulas by Ping Wang St John’s University, New York on CICIRM 2011 at Beijing, China.
Latvijas Aktuāru Asociācija AN INTRODUCTION TO COPULAS Gaida Pettere Professor, Dr. Math. Riga Technical University, Chairmen of Latvian Actuarial Association.
Descriptive statistics Experiment Data Sample Statistics Experiment Data Sample Statistics Sample mean Sample mean Sample variance Sample variance.
Copula Functions and Bivariate Distributions: Applications to Political Interdependence Alejandro Quiroz Flores, Wilf Department of Politics, NYU Motivation.
Master thesis presentation Joanna Gatz TU Delft 29 of July 2007 Properties and Applications of the T copula.
CASA June 2006 BRATISLAVA Mária Bohdalová Faculty of Management, Comenius University Bratislava Oľga Nánásiová Faculty of Civil.
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Modeling Related Failures in Finance Arkady Shemyakin MFM Orientation, 2010.
Exploring Policyholder Behavior in the Extreme Tail Yuhong (Jason) Xue, FSA MAAA.
Presented by : Mohsen Ben Hassine Definitions and Basic Properties 2. Dependence 3. Important copulas 4. Methods of Constructing Copulas 5.
Models for construction of multivariate dependence Workshop on Copulae and Multivariate Probability distributions in Finance – Theory, Applications,
Dependent Variable Discrete 2 values – binomial 3 or more discrete values – multinomial Skewed – e.g. Poisson Continuous Non-normal.
Economic Capital and the Aggregation of Risks Using Copulas Dr. Emiliano A. Valdez and Andrew Tang.
Pair-copula constructions of multiple dependence Workshop on ''Copulae: Theory and Practice'' Weierstrass Institute for Applied Analysis and.
NORMAL OR GAUSSIAN DISTRIBUTION Chapter 5. General Normal Distribution Two parameter distribution with a pdf given by:
Donald F. Behan Society of Actuaries Meeting Phoenix, AZ1 Using Copulas to Model Extreme Events by Donald F. Behan and Sam Cox Georgia State University.
BY RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES Copula Regression.
The Multivariate Normal Distribution, Part 2 BMTRY 726 1/14/2014.
University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 14: Probability Wrap-Up and Statistical.
Relative Value Trading Opportunities in Portfolios Of Credits Raghunath Ganugapati (Newt) University Of Wisconsin-Madison Doctoral Candidate in Particle.
Copulas from Fokker-Planck equation Hi Jun Choe Dept of Math Yonsei University Seoul, Korea.
Maximum Likelihood Estimation Multivariate Normal distribution.
Copula functions Advanced Methods of Risk Management Umberto Cherubini.
University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 14: Probability and Statistics.
Measuring market risk: a copula and extreme value approach Supervisor Professor Moisă Altăr Academy of Economic Studies Bucharest Doctoral School of Finance.
Gra6036- Multivartate Statistics with Econometrics (Psychometrics) Distributions Estimators Ulf H. Olsson Professor of Statistics.
Probability distribution functions Normal distribution Lognormal distribution Mean, median and mode Tails Extreme value distributions.
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 1: INTRODUCTION.
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1 Multivariate Normal Distribution Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking.
Sampling Distributions A statistic is random in value … it changes from sample to sample. The probability distribution of a statistic is called a sampling.
A. The Basic Principle We consider the multivariate extension of multiple linear regression – modeling the relationship between m responses Y 1,…,Y m and.
Bayesian inference review Objective –estimate unknown parameter based on observations y. Result is given by probability distribution. Bayesian inference.
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© Prof. Jayanth R. Varma, Indian Institute of Management, Ahmedabad Risk Management at Indian Exchanges Going Beyond SPAN and VaR.
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