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Optimal portfolio selection under Expected Shortfall optimisation with Random Matrix Theory denoising
Název práce v češtině: Optimal portfolio selection under Expected Shortfall optimisation with Random Matrix Theory denoising
Název v anglickém jazyce: Optimal portfolio selection under Expected Shortfall optimisation with Random Matrix Theory denoising
Klíčová slova: Teorie portfolia, Random Matrix Theory, Stabilní rozdělení, Expected Shortfall
Klíčová slova anglicky: Portfolio management, Random Matrix Theory, Lévy stable distributions, Expected Shortfall
Akademický rok vypsání: 2016/2017
Typ práce: diplomová práce
Jazyk práce: angličtina
Ústav: Institut ekonomických studií (23-IES)
Vedoucí / školitel: PhDr. Boril Šopov, M.Sc., LL.M.
Řešitel: skrytý - zadáno vedoucím/školitelem
Datum přihlášení: 21.01.2017
Datum zadání: 21.02.2017
Datum a čas obhajoby: 31.01.2018 08:30
Místo konání obhajoby: Opletalova - Opletalova 26, O105, Opletalova - místn. č. 105
Datum odevzdání elektronické podoby:05.01.2018
Datum proběhlé obhajoby: 31.01.2018
Oponenti: doc. PhDr. Jozef Baruník, Ph.D.
 
 
 
Kontrola URKUND:
Zásady pro vypracování
Much of the intellectual energy in finance of the 20th century was spent on the problem of optimal portfolio selection. Historically, the cornerstone of the solution, presented by Markowitz in 1952, has been the variance-covariance matrix – a measure of co-movement of financial time series. However, there are inherent weaknesses in this approach that has been challenged since the introduction of methods originating in other fields – physics and signal processing. Random Matrix Theory is one of the most often scrutinised in its ability to offer better estimates of the ‘true’ covariances. This work shall implement RMT denoising method, in order to improve the solution of the portfolio problem, particularly with respect to the risk.
Furthermore, the original problem setup minimises the risk expressed as the standard deviation of the returns of the portfolio, given a level of return. This thesis looks into the problem of minimising the Expected shortfall of the portfolio instead.

With the advent of Basel III accord, the mainstream measure of risk shifts from Value at Risk to a more sophisticated Expected shortfall. Also, with soaring power of today’s standard hardware, we are able to employ more computationally exhaustive methods in portfolio optimization and risk management as such.

Instead of assuming that financial data follow Gaussian laws, semi-closed form of alpha-Stable Lévy distributions can be taken into account. Unlike in the world of Normal distributions, Stable distributions have infinite variance and are a generalization of several important distributions. Hence modelling the conditional Value at Risk, we can account for the fat tails more precisely than in the usual framework.

This work will test the optimization method on both simulated with known covariance structure and real data to empirically test the hypotheses described below.

The method of Random Matrix Theory has not been inspected in quantitative finance journals over the past decade very frequently. It has found its core authors and there have been some interesting empirical results.

The methodology is not however very clear on the method of clipping the noisy eigenvalus of the covariance matrix. Hence it will be inspected and implemented from scratch to avoid any possible flaws in reusing external code.

The semi-closed solution to the ES minimisation will be presented with adequate mathematical background and implemented in respective software as well.

The stable distributions will be studied and for there is quite a lot of confusion in the literature, and particularly implementation, the work will follow Nolan (2003, 2012). A Maximum Likelihood Estimate of the stable distribution fits to the time series will be inspected, despite its computational exhaustiveness.

Lastly, the thesis shall follow and verify the Stoyanov’s method of calculating the expected shortfall for Stable distributions.

The author is concerned with its own implementation of the methods in R programming language.

Seznam odborné literatury
Bouchaud, J.-P. & M. Potters (2009): “Financial applications of random matrix theory: a short review.” arXiv preprint arXiv:0910.1205 .
Edelman, A. & N. R. Rao (2005): “Random matrix theory.” Acta Numerica 14: pp. 233–297.

Gnedenko, B. V. & A. N. Kolmogorov (1954): Limit distributions for sums of independent random variables. Addison-Wesley Mathematics Series. Cambridge, MA: Addison-Wesley. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. MR:0062975. Zbl:0056.36001.
Laloux, L., P. Cizeau, M. Potters, & J.-P. Bouchaud (2000): “Random matrix theory and financial correlations.” International Journal of Theoret- ical and Applied Finance 3(03): pp. 391–397.
Marčenko, V. A. & L. A. Pastur (1967): “Distribution of eigenvalues for some sets of random matrices.” Mathematics of the USSR-Sbornik 1(4): p. 457.
Nolan, J. P. (2003): “Modeling financial data with stable distributions.” Handbook of Heavy Tailed Distributions in Finance, Handbooks in Finance: Book 1: pp. 105–130.
Nolan, J. P. (2012): Stable distributions, volume 1177108605. ISBN.
Stoyanov, S. V., G. Samorodnitsky, S. Rachev, & S. Ortobelli Lozza (2006): “Computing the portfolio conditional value-at-risk in the alpha- stable case.” Probability and Mathematical Statistics 26(1): pp. 1–22.

Zolotarev, V. M. (1986): One-dimensional stable distributions, volume 65. American M
athematical Soc.
Předběžná náplň práce
Chapter 1 - Introduction
Chapter 2 – Literature Review – almost finished
Chapter 3 – Methodology – mostly done
Chapter 4 – Data and results
Chapter 5 - Conclusion


1. Hypothesis #1: Random Matrix Theory denoising of the covariance matrix will result in portfolios with lower Expected shortfall
2. Hypothesis #2: RMT will be tested as a better approximation of the data-generating covariance structure that Pearsonian covariance matrix.
3. Hypothesis #3: The ratio of number of observations and number of stocks influences the estimate of covariance matrix


The work will summarise the research in Random Matrix Theory, Stable distributions and modern Portfolio Theory. It attempts to firstly verify the relevance of RMT with simulating multivariate finance time series. Secondly will inspect, if relevant, the effect of using this method on creating a portfolios with respect to minimal Expected Shortfall of the resulting portfolio. This will be compared to portfolios calculated as Global Minimum Variance from the usual Markowitz’s framework.
Moreover, the thesis will implement an MLE estimate engine for fitting stable distributions and to evalute Expected Shortfalls for various portfolios.
Předběžná náplň práce v anglickém jazyce
Chapter 1 - Introduction
Chapter 2 – Literature Review – almost finished
Chapter 3 – Methodology – mostly done
Chapter 4 – Data and results
Chapter 5 - Conclusion


1. Hypothesis #1: Random Matrix Theory denoising of the covariance matrix will result in portfolios with lower Expected shortfall
2. Hypothesis #2: RMT will be tested as a better approximation of the data-generating covariance structure that Pearsonian covariance matrix.
3. Hypothesis #3: The ratio of number of observations and number of stocks influences the estimate of covariance matrix


The work will summarise the research in Random Matrix Theory, Stable distributions and modern Portfolio Theory. It attempts to firstly verify the relevance of RMT with simulating multivariate finance time series. Secondly will inspect, if relevant, the effect of using this method on creating a portfolios with respect to minimal Expected Shortfall of the resulting portfolio. This will be compared to portfolios calculated as Global Minimum Variance from the usual Markowitz’s framework.
Moreover, the thesis will implement an MLE estimate engine for fitting stable distributions and to evalute Expected Shortfalls for various portfolios.
 
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