QRM - Copulas and Dependence

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VaR of Portfolio Z at probability level ？
VaRα ( Z ) ≡ FZ−1 (α ) = inf{z | z ∈ R : FZ ( z ) ≥ α } VaRα ( X ) ≡ FZ−1 (α ) = inf{x | x ∈ R n : Fx ( x ) ≥ α }
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Gap between Risk Model and the World
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Traditional Gaussian Hypothesis on the distribution of logarithm returns or the complete market Typical Model –Markowitz Mean-variace portfolio, optimized, the variance corresponds to the risk measure but it implies the world is Gaussian Two difficulties •Gaussian Assumption •Joint Distribution Modelling One of the main issues of risk management is the aggregation of individual risks.
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the dependence structure
•Linear correlation •Rank Dependence •Tail Dependence •… … … … ..
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Chapter 2 Copulas
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Def. Let S1,S2,… ,Sn be nonempty subsets of R, and let H be an n-place real function such that Dom H=S1×S2×… ×Sn. Let B=[a,b] be an n-box all of whose vertices are in Dom H. Then the H-volume of B is given by where the sum is taken over all vertices c of B, and sgn is given by Equivalently, the H-Volume of an n-box B=[a,b] is the n’ th order difference of H on B
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Copulas and VaR
Let X be random multivariate with distribution function Fi and Ui be standard-uniformly distributed at [0,1] Probability-integral
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See www.math.ethz.ch/~embrechts
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Modelling by use of Copulas
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Modelling by use of Copulas
•Identification of the marginal distributions •Selection/Definition/Construction of appropriate copula to
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Potential Copula Applications
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Insurance • Life (multi-life products) • Non-life (multi-line covers) • Integrated risk management (Solvency 2/偿付能力) • Dynamic Financial analysis (assets and liabilities Model, ALM) Finance • Stress testing (Credit) • Risk aggregation • Pricing/Hedging basket derivatives • Risk measure estimation under incomplete information Other Fields • Reliability, Survival analysis • Environmental science, Genetics
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Functions of Copulas
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Relationship between a multi-dimensional probability distribution function and its lower dimensional margins Functions of Copula Link multivariate distributions to their on-dimensional margins D.X. Li (1998, 2000) On default correlation: A copula function approach. Working paper 99-07, RiskMetrics Group. /picsresources/dlc.pdf P.Embrechts, A.Mcneil and D.Straumann, Correlation and dependence in risk management: properties and pitfalls. In: M.A.H.Dempster(Ed.),Value at risk and beyond, Cambridge University Press, Cambridge,1999,pp.176-223.
Fi ( xi ) ~ U [0,1]; Fi −1 (U i ) ~ Fi n
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Quantile Function
Fi −1 (α ) ≡ inf{xi | Fi ( x ) ≥ α } F −1 (α ) ≡ inf{x | F ( x ) ≥ α }
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Joint Distribution
F ( x1 , x2 , L , xn ) = ψ ( F1 ( x1 ), F2 ( x 2 ), L , Fn ( x n ))
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Joint Probability Distribution Function is of dependence structure of multivariate To Analyze the dependence structure of multivariate distributions without studying marginal distributions
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Motivations
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Consider a portfolio of financial risks {X1,..,Xn} •a portfolio of traded financial assets e.g. equities; •a credit portfolio of loans to various counter-parties; •a portfolio of potential insurance losses in various lines of business or geographical areas. Sampling –i.i.d. Suppose that the distribution of pay-off n-dimensional f(X1, . ,Xn) representing the risk of the portfolio or some contract written on the portfolio. It’ s VERY difficulty to estimate joint distribution f(… )!! How incorporate the marginal distribution of {Xi} to describe the dependence structure of multivariate {X1,..,Xn}?
f ( x, y ) =
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Nearly Half-century of Copulas
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1959, M.Fré chet and A.Sklar introduced the concepts of copulas Fré chet,M.,1951, Sur Les tableaux de corré lation dont les marges sont donné es,Ann.Univ. Lyon 9,Sec.A,53-57 Fré chet,M.,1957,Les tableaux de corré lation et les programmes lineaires, Revue Inst. Int. Statist. 25,23-40 Sklar,A.,1959, Fontions de repartition a n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8, 229-231 Sklar,A.,1973, Random variables,joint distribution functions and copulas, Kybernetika 9, 449-460