using copulas

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Using Copulas. Multi-variate Distributions. Usually the distribution of a sum of random variables is needed When the distributions are correlated, getting the distribution of the sum requires calculation of the entire joint probability distribution - PowerPoint PPT Presentation

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  • Using Copulas

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    Multi-variate DistributionsUsually the distribution of a sum of random variables is neededWhen the distributions are correlated, getting the distribution of the sum requires calculation of the entire joint probability distributionF(x, y, z) = Probability (X < x and Y < y and Z < z)Copulas provide a convenient way to do this calculationFrom that you can get the distribution of the sum

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    Correlation IssuesIn many cases, correlation is stronger for large events Can model this by copula methodsQuantifying correlationDegree of correlationPart of distribution correlatedCan also do by conditional distributionsSay X and Y are ParetoCould specify Y|X like F(y|x) = 1 [y/(1+x/40)]-2Conditional specification gives a copula and copula gives a conditional, so they are equivalentBut the conditional distributions that relate to common copulas would be hard to dream up

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    Modeling via CopulasCorrelate on probabilitiesInverse map probabilities to correlate lossesCan specify where correlation takes place in the probability rangeConditional distribution easily expressedSimulation readily available

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    Formal Rules Bivariate CaseF(x,y) = C(FX(x),FY(y))Joint distribution is copula evaluated at the marginal distributionsExpresses joint distribution as inter-dependency applied to the individual distributionsC(u,v) = F(FX-1(u),FY-1(v))u and v are unit uniforms, F maps R2 to [0,1]Shows that any bivariate distribution can be expressed via a copulaFY|X(y|x) = C1(FX(x),FY(y)) Derivative of the copula gives the conditional distributionE.g., C(u,v) = uv, C1(u,v) = v = Pr(V1 is R, and lim L(z), z->0 is L

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    LR Function(L below , R above)

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    Auto and Fire Claims in French Windstorms

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    MLE Estimates of Copulas

    Gumbel

    Normale

    HRT

    Frank

    Clayton

    Paramtre

    1,323

    0,378

    1,445

    2,318

    3,378

    Log Vraisemblance

    77,223

    55,428

    84,070

    50,330

    16,447

    ( de Kendall

    0,244

    0,247

    0,257

    0,245

    0,129

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    Modified Tail Concentration FunctionsModified function is R(z)/(1 z)Both MLE and R function show that HRT fits best

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    The t- copula adds tail correlation to the normal copula but maintains the same overall correlation, essentially by adding some negative correlation in the middle of the distributionStrong in the tailsSome negative correlation

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    Easy to simulate t-copulaGenerate a multi-variate normal vector with the same correlation matrix using Cholesky, etc.Divide vector by (y/n)0.5 where y is a number simulated from a chi-squared distribution with n degrees of freedomThis gives a t-distributed vectorThe t-distribution Fn with n degrees of freedom can then be applied to each element to get the probability vectorThose probabilities are simulations of the copulaApply, for example, inverse lognormal distributions to these probabilities to get a vector of lognormal samples correlated via this copulaCommon shock copula dividing by the same chi-squared is a common shock

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    Other descriptive functionstau is defined as 1+40101 C(u,v)c(u,v)dvdu. cumulative tau: J(z) = 1+40z0z C(u,v)c(u,v)dvdu/C(z,z)2. expected value of V given U

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