How often is the financial market going to collapse?

Gabriel Frahm; Gabriel Frahm

doi:10.3934/QFE.2018.3.590

Quantitative Finance and Economics

2018, Volume 2, Issue 3: 590-614. doi: 10.3934/QFE.2018.3.590

Previous Article Next Article

Research article Special Issues

How often is the financial market going to collapse?

Gabriel Frahm ^,

Helmut Schmidt University, Department of Mathematics and Statistics, Holstenhofweg 85, D–22043 Hamburg, Germany

Received: 26 March 2018 Accepted: 24 May 2018 Published: 01 August 2018
JEL Codes: G01, G15

Copula theory is used to investigate the phenomenon of extremal dependence. An analytical expression for the extremal-dependence coe cient (EDC) of regularly varying elliptically distributed random vectors is derived. The EDC represents a natural measure of systemic risk. Extreme value theory is applied in order to estimate the systemic risk of the G–7 countries. The given results are quite sensitive to the tail index of asset returns and thus a scenario analysis is conducted. In the worst case, the probability that the entire market crashes during 10 years exceeds 50%. Hence, we must not neglect the risk of a financial collapse during a relatively short period of time.
- copula theory,
- extremal dependence,
- extreme value theory,
- ruin,
- tail dependence
Citation: Gabriel Frahm. How often is the financial market going to collapse?[J]. Quantitative Finance and Economics, 2018, 2(3): 590-614. doi: 10.3934/QFE.2018.3.590

Related Papers:

Abstract

Copula theory is used to investigate the phenomenon of extremal dependence. An analytical expression for the extremal-dependence coe cient (EDC) of regularly varying elliptically distributed random vectors is derived. The EDC represents a natural measure of systemic risk. Extreme value theory is applied in order to estimate the systemic risk of the G–7 countries. The given results are quite sensitive to the tail index of asset returns and thus a scenario analysis is conducted. In the worst case, the probability that the entire market crashes during 10 years exceeds 50%. Hence, we must not neglect the risk of a financial collapse during a relatively short period of time.

References

[1]	Adrover J (1998) Minimax bias-robust estimation of the dispersion matrix of a multivariate distribution. Ann Stat 26: 2301–2320.
[2]	Aeschliman C, Park J, Kak A (2010) A novel parameter estimation algorithm for the multivariate t-distribution and its application to computer vision, in K. Daniilidis, P. Maragos, N. Paragios (editors), European Conference on Computer Vision 2010, Springer, 594–607.
[3]	Artzner P, Delbaen F, Eber JM, et al. (1999) Coherent measures of risk. Math Financ 9: 203–228. doi: 10.1111/1467-9965.00068
[4]	Barndorff-Nielsen O, Kent J, Sørensen M (1982) Normal variance-mean mixtures and ɀ distributions. Int Stat Rev 50: 145–159. doi: 10.2307/1402598
[5]	Bingham N, Kiesel R (2002) Semi-parametric modelling in finance: theoretical foundation. Quant Financ 2: 241–250. doi: 10.1088/1469-7688/2/4/201
[6]	Bouchaud JP, Cont R, Potters M (1997) Scaling in stock market data: stable laws and beyond, in B. Dubrulle, F. Graner, D. Sornette (editors), Scale Invariance and Beyond, Proceedings of the CNRS Workshop on Scale Invariance, Les Houches, March 1997, EDP-Springer.
[7]	Breymann W, Dias A, Embrechts P (2003) Dependence structures for multivariate high-frequency data in finance. Quant Financ 3: 1–14. doi: 10.1080/713666155
[8]	Cambanis S, Huang S, Simons G (1981) On the theory of elliptically contoured distributions. J Multivariate Anal 11: 368–385. doi: 10.1016/0047-259X(81)90082-8
[9]	Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quant Financ 1: 223–236. doi: 10.1080/713665670
[10]	De Luca G, Rivieccio G (2012) Multivariate tail dependence coe_cients for Archimedean copulae, in A. Di Ciaccio, M. Coli, J. Ibanez(editors), Advanced Statistical Methods for the Analysis of Large Data-Sets, Studies in Theoretical and Applied Statistics, Springer, 287–296.
[11]	Ding Z, Granger C, Engle R (1993) A long memory property of stock market returns and a new model. J Empir Financ 1: 83–106. doi: 10.1016/0927-5398(93)90006-D
[12]	Dobrić J, Frahm G, Schmid F (2013) Dependence of stock returns in bull and bear markets. Dependence Modeling 1: 94–110.
[13]	Dümbgen L, Tyler D (2005) On the breakdown properties of some multivariate M-functionals. Scand J Stat 32: 247–264. doi: 10.1111/j.1467-9469.2005.00425.x
[14]	Eberlein E, Keller U (1995) Hyperbolic distributions in finance. Bernoulli 1: 281–299. doi: 10.2307/3318481
[15]	Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling Extremal Events (for Insurance and Finance), Springer.
[16]	Engle R (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50: 987–1007.
[17]	Fama E (1965) The behavior of stock market prices. J Bus 38: 34–105. doi: 10.1086/294743
[18]	Fang K, Kotz S, Ng K (1990) Symmetric Multivariate and Related Distributions, Chapman & Hall.
[19]	Ferreira H, Ferreira M (2012) On extremal dependence of block vectors. Kybernetika 48: 988–1006.
[20]	Frahm G (2004) Generalized Elliptical Distributions: Theory and Applications, Ph.D. thesis, University of Cologne.
[21]	Frahm G (2006) On the extremal dependence coeffcient of multivariate distributions. Stat Probabil Lett 76: 1470–1481. doi: 10.1016/j.spl.2006.03.006
[22]	Frahm G (2009) Asymptotic distributions of robust shape matrices and scales. J Multivariate Anal 100: 1329–1337. doi: 10.1016/j.jmva.2008.11.007
[23]	Frahm G, Jaekel U (2010) A generalization of Tyler's M-estimators to the case of incomplete data. Comput Stat Data An 54: 374–393. doi: 10.1016/j.csda.2009.08.019
[24]	Frahm G, Jaekel U (2015) Tyler's M-estimator in high-dimensional financial-data analysis, in K. Nordhausen, S. Taskinen (editors), Modern Nonparametric, Robust and Multivariate Methods, Chapter 17, Springer, 289–305.
[25]	Frahm G, Junker M, Schmidt R (2005) Estimating the tail-dependence coeffcient: properties and pitfalls. Insur Math Econ 37: 80–100. doi: 10.1016/j.insmatheco.2005.05.008
[26]	Frahm G, Junker M, Szimayer A (2003) Elliptical copulas: applicability and limitations. Stat Probabil Lett 63: 275–286. doi: 10.1016/S0167-7152(03)00092-0
[27]	Hettmansperger T, Randles R (2002) A practical affine equivariant multivariate median Biometrika 89: 851–860.
[28]	Hult H, Lindskog F (2002) Multivariate extremes, aggregation and dependence in elliptical distributions. Adv Appl Probab 34: 587–608. doi: 10.1239/aap/1033662167
[29]	Joe H (1997) Multivariate Models and Dependence Concepts, Chapman & Hall.
[30]	Jorion P (1985) International portfolio diversification with estimation risk. J Bus 58: 259–278. doi: 10.1086/296296
[31]	Junker M, May A (2005) Measurement of aggregate risk with copulas. Economet J 8: 428–454. doi: 10.1111/j.1368-423X.2005.00173.x
[32]	Kelker D (1970) Distribution theory of spherical distributions and a location-scale parameter generalization. Sankhya A 32: 419–430.
[33]	Kent J, Tyler D (1988) Maximum likelihood estimation for the wrapped Cauchy distribution. J Appl Stat 15: 247–254. doi: 10.1080/02664768800000029
[34]	Kent J, Tyler D (1991) Redescending M-estimates of multivariate location and scatter. Ann Stat 19: 2102–2119. doi: 10.1214/aos/1176348388
[35]	Laub P, Taimre T, Pollett P (2015) Hawkes processes, Technical report, Cornell University Library, arXiv:1507.02822.
[36]	Mandelbrot B (1963) The variation of certain speculative prices. J Bus 36: 394–419. doi: 10.1086/294632
[37]	Markowitz H (1952) Portfolio selection. J Financ 7: 77–91.
[38]	Maronna R, Yohai V (1990) The maximum bias of robust covariances. Commun Stat-Theor M 19: 3925–3933. doi: 10.1080/03610929008830422
[39]	McNeil A, Frey R, Embrechts P (2005) Quantitative Risk Management, Princeton University Press.
[40]	Mikosch T (2003) Modeling dependence and tails of financial time series, in B. Finkenstaedt, H. Rootz´en (editors), Extreme Values in Finance, Telecommunications, and the Environment, Chapman & Hall.
[41]	Nelsen R (2006) An Introduction to Copulas, Springer, 2nd edition.
[42]	Paindaveine D (2008) A canonical definition of shape. Stat Probabil Lett 78: 2240–2247. doi: 10.1016/j.spl.2008.01.094
[43]	Rachev S, Mittnik S (2000) Stable Paretian Models in Finance, Wiley.
[44]	Schmidt R (2002) Tail dependence for elliptically contoured distributions. Math Method Oper Res 55: 301–327. doi: 10.1007/s001860200191
[45]	Shorrocks A, Davies J, Lluberas R (2017) Global Wealth Databook 2017, Technical report, Credit Suisse Research Institute.
[46]	Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges, Research report, University of Paris, Institute of Statistics, France.
[47]	Tyler D (1983) Robustness and efficiency properties of scatter matrices Biometrika 70: 411–420.
[48]	Tyler D (1987a) A distribution-free M-estimator of multivariate scatter Ann Stat 15: 234–251.
[49]	Tyler D (1987b) Statistical analysis for the angular central Gaussian distribution on the sphere Biometrika 74: 579–589.

Reader Comments

Your name:*

Email:*
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)