Factor copulas through a vine structure


Abstract


Copula functions have been widely used in actuarial science, nance andeconometrics. Though multivariate copulas allow for a flexible specication of the dependence structure of economic variables, they are not particularly tempting in high dimensional contexts. A factor model which involves copula functions has already proved to be a powerful tool in credit risk applications.We exploit a recent approach to obtain a factor copula model based on a vine structure, which enables to model the dependence and conditional dependence of variables through a representation of a cascade of arbitrary bivariate copulas. The contribution of this paper consists into applying the vine copula model in order to derive a non linear three factor model. In particular, we draw the three factor model of Fama and French (1992). According to the Inference for Margins (IFM) method, we have computed, separately, the margins and the copula parameters via maximum likelihood estimation. Finally, tail dependence measures are given for the implied estimated copula.

DOI Code: 10.1285/i20705948v8n2p246

Keywords: Factor copula model; Vines; Tail dependence; Tail density functions.

References


Aas, K., Berg, D., 2009. Models for construction of multivariate dependence: A comparison study. European Journal of Finance 15, 639-659.

DOI:10.1080/13518470802588767

Aas, K., Czado, C., Frigessi, A., Bakken, H., 2006. Pair-copula constructions of multiple dependence. Insurance: Mathematics & Economics 44, 182-198.

Andersen, L., Sidenius, J., 2004. Extensions of the Gaussian Copula Model, Journal of Credit Risk 1, 29{70.

Banz, W., 1981. The relationship between return and market value of common stocks, Journal of Financial Economics 9, 3{18.

Bedford, T., Cooke, R. M., 2001. Probability density decomposition for conditionally

dependent random variables modeled by vines. Annals of Mathematics and Articial

Intelligence 32, 245-268. DOI:10.1023/A:1016725902970

Bedford, T., Cooke, R. M., 2002. Vines - a new graphical model for dependent

random variables. Annals of Statistics 30, 1031-1068. DOI:10.1214/aos/1031689016

Brechmann, E.C., Czado, C., 2011. Risk management with high-dimensional vine

copulas: An analysis of the Euro Stoxx 50, Working paper, Technische Universitat

Munchen.

Brechmann, C., Czado, C., Aas, K., 2012. Truncated Regular Vines in High Dimensions

with Applications to Financial Data, Canadian Journal of Statistics 40, 68{85.

Campbell, R., Koedijk, K., Kofman, P., 2002. Increased correlation in bear markets.

Financial Analysts Journal 58, January-February, 87{94. ISSN: 0015198X

Carhart, M.M., 1997. On persistence in mutual fund performance, Journal of Finance

, 57{82.

Chan, L.K., Hamao, Y., Lakonishok, J., 1991. Fundamentals and stock returns in

Japan, Journal of Finance 46, 1739{1789.

Cherubini, U., Luciano, E., Vecchiato, W., 2004. Copula Methods in Finance. John

Wiley & Sons.

Cherubini, U., Gobbi, F., Mulinacci, S., Romagnoli, S., 2011. Dynamic Copula

Methods in Finance, John Wiley & Sons.

Chollete, L., Ning, C., 2010. Asymmetric Dependence in US Financial Risk Factors?,

US Working Papers in Economics and Finance 2/11, University of Stavanger.

Dias, A., Embrechts, P., 2010. Modeling exchange rate dependence dynamics at

Rivieccio

dierent time horizons. Journal of International Money and Finance 29, 1687{1705.

DOI: 10.1016/j.jimonn.2010.06.004

Engle, R F., 2002. Dynamic conditional correlation: a simple class of multivariate

generalized autoregressive conditional heteroscedasticity models. Journal of Business

and Economic Statistics 20, 339{350. DOI:10.1198/073500102288618487

Fama, E.F., French, K.R., 1992. The cross-section of expected stock returns, Journal

of Finance 47, 427-465.

Genest, G., Favre, A., 2007. Everything You Always Wanted to Know about Copula

Modeling but Were Afraid to Ask, J. Hydrol. Eng. 12, Special issue: Copulas in

Hydrology, 347-368. DOI: 10.1061/(ASCE)1084-0699(2007)12:4(347)

Granger, C.W.J., Terasvirta, T., Patton, A., 2006. Common factors in conditional

distributions for bivariate time series, Journal of Econometrics 132, 43-57.

DOI:10.1016/j.jeconom.2005.01.022

Gregory, J., Laurent, J.P., 2005. Basket Default Swaps, CDOs and Factor Copulas,

Journal of Risk 7, 8-23.

Heinen, A., Valdesogo, A., 2009. Asymmetric CAPM dependence for large dimensions:

the Canonical Vine Autoregressive Model, CORE Discussion Papers 2009069,

Universit catholique de Louvain, Center for Operations Research and Econometrics

(CORE). Available at http://dx.doi.org/10.2139/ssrn.1297506

Heinen, A., Valdegoso, A., 2011. Dynamic D-vine Model. In: Kurowicka, D., Joe,

H. (Eds.). Dependence Modeling: Vine Copula Handbook. World Scientic, 329{353.

Hull, J., White, A., 2004. Valuation of a CDO and nth to default CDO without

Monte Carlo Simulation, Journal of Derivatives 12, 10-23.

Hull, J., White, A., 2006. Valuing Credit Derivatives Using an Implied Copula

Approach, Journal of Derivatives 14 2, 8-28.

Hull, J., White, A., 2010. An improved implied copula model and its application to

the valuation of bespoke CDO tranches, Journal of investement management.

Joe, H., 1996. Families of m-variate distributions with given margins and m(m-

/2 bivariate dependence parameters. In: Ruschendorf, L., Schweizer, B., Taylor,

M.D.(Eds.). Distributions with Fixed Marginals and Related Topics.

Joe, H., 1997. Multivariate Models and Dependence Concepts. Chapman &

Hall/CRC, New York.

Joe, H., Li, H., Nikoloulopoulos, A. K., 2010. Tail dependence functions

and vine copulas. Journal of Multivariate Analysis 101, 252{270. Available at

http://dx.doi.org/10.1016/j.jmva.2009.08.002

Joe, H., Li, H., Nikoloulopoulos, A.K., 2010. Vine copulas with asymmetric tail

dependence and applications to nancial return data, Computational Statistics and

Data Analysis.

Kurowicka, D., Cooke, R. M., 2004. Distribution-free continuous bayesian belief

nets. In Fourth International Conference on Mathematical Methods in Reliability

Methodology and Practice, Santa Fe, New Mexico.

Electronic Journal of Applied Statistical Analysis 15

Kurowicka, D., Cooke, R. M., 2006. Uncertainty Analysis with High Dimensional

Dependence Modelling. New York: Wiley.

Li, D.X., 2000. On Default Correlation: A Copula Approach, Journal of Fixed

Income 9, 43{54.

Li, D.X., Wu, P., 2011. Extremal Dependence of Copulas: A Tail Density Approach,

Working Paper.

Lintner, J., 1965. The valuation of risk assets and the selection of risky investments

in stock portfolios and capital budgets, Review of Economics and Statistics 47, 13{37.

Longin, F., Solnik, B., 2001. Extreme correlation of international equity markets.

The Journal of Finance 56, 649{676. DOI: 10.1111/0022-1082.00340.

McNeil, A. J., Frey, R., Embrechts, P., 2005. Quantitative Risk Management,

Princeton University Press, New Jersey.

Nelsen, R. B., 2006. An Introduction to Copulas. Springer-Verlag, New York.

Oh, D.H., Patton, A., 2011. Modelling Dependence in High Dimensions with Factor

Copulas, Working Paper. Revised April 2012.

Patton, A., 2006. Modelling asymmetric exchange rate dependence, International

Economic Review 47, 527{556. DOI:10.1111/j.1468-2354.2006.00387.x

Richardson, M., Smith, T., 1993. A test for multivariate normality in stock returns,

Journal of Business 66, 2, 295{321.

Sharpe, W.F., 1964. Capital asset prices: a theory of market equilibrium under

conditions of risk, Journal of Finance 19, 425{442.

Sklar, A., 1959. Fonctions de repartition a n dimensions et leurs marges. Publ. Inst.

Statist. Univ. Paris 8, 229{231.

Vasicek, O., 1987. The Loan Loss Distribution, Working Paper, KMV.

van der Voort, M., 2005. Factor copulas: totally external defaults, Working Paper,

Erasmus University Rotterdam.


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