Value-at-Risk Under Different Statistical Distribution

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• Prabhat K

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The concept discussed in my presentation is value-at-risk, a risk measure mostly studied for portfolio optimization in mathematical distribution. In this, we have discussed how it depends on the different statistical distribution and has compared the results.

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1/27 Value-at-Risk Under Different Statistical Distribution Prabhat Kumar Pankaj MS 13102 Under the guidance of Dr. M. P. Rajan Indian Institute of Science Education and Research Thiruvananthpuram April 16, 2018 IISER Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Overview of The Talk O Overview of first phase work. O Introduction. O Empirical Study O VaR Estimate and Backtesting O Discussions 2/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Overview of the first phase work O We described VaR as the quantile of the projected distribution of losses and gains of an investment over a target horizon. O Algebraic properties of VaR and its shortcomings were discussed. Also, how Conditional Value-at-Risk addresses the defects of VaR. O An approach to optimizing a portfolio which focuses on minimizing conditional value-at-risk (CVaR) rather than minimizing value-at-risk (VaR) 3/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Introduction Value-at-Risk (VaR) is defined as the worst expected loss over a given period at a specified confidence level. Based on how the financial return distribution is modelled, methodology of estimating VaR can be classified into two groups: o the parametric VaR o non-parametric VaR. The statistical distribution that is commonly assumed in parametric VaR is the Normal distribution. A leptokurtic distribution has a higher peak and heavier tails than the Normal distribution. The quality of VaR is relying on how well the statistical distribution captures the leptokurtic behaviour of the financial 4/27 returns. Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Empirical Study 0 The empirical study is done using four stocks, FTSE/JSE TOP40 (J200) index and the S & P 500 index. The four shares are: O Standard Bank (SBK) O African Bank (ABL) O Merafe Resource (MRF) O and Anglo American (AGL) and are listed on the Johannesburg Stock Exchange. o Daily closing prices for the equity stocks and indexes and their daily log returns were obtained. 5/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Empirical Study o The sample period has been split into three sub-samples, that is: O Pre-crisis (from 1991 January-December 2007), O Crisis period (from January 2008-December 2009), O Post-crisis (from January 2010-July 2014). Statistical Data of the Empirical Distribution over the Period Jan- uary 1991-July 2014. Variance Skewness Excess Kurtosis S P 500 FTSE/JSE TOP40 Standard Bank African Bank Anglo American Merafe Resource Mean 0:00029 0.00048 0.00048 0.00003 0.00038 0.00054 0.00014 0.00019 0.00047 0.00082 0.00064 0.00135 0.73247 0.40611 0.20369 0.78087 0.07927 0.06429 15.95379 6.27868 4.86980 10.8220 3.93211 2.52105 No. Obs 6170 4770 4090 3944 4162 2806 Figure: Statistical data for each stock and indices for the entire period 6/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Empirical Study The excess kurtosis for each stock and the index is greater than zero. This indicates a higher peak and heavier tail then the Normal 7/27 distribution. Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Fitting the Distribution Normal Inverse Gaussian Distribution Definition [4] The random variable X follows a NIG distribution with parameters a, p, g, and 5, if its probability density function, defined for all real x G R, is given by: öa exp(ö a2 — ß2 +ß(x— g)) 52 + (x— g 1 — R exp ( — äx(T + T is a modified where the function (x) Bessel function of third order and index 1. In addition the parameters must satisfy 0 lßl a, G IR, 0 < a and 0 < 5. If a random variable X follows a NIG distribution it can be denoted in short as 8/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Fitting the Distribution The Students t—Distribution The t—distribution is characterized by the parameter known as the degree of freedom k > 0. The density function is given by: 2) j) k-kl x k7tr(f) where k is the degree of freedom and r is the gamma function. 9/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Fitting the Distribution The Skew t—Distribution The density function of Skew t is given by st(x : k,ß) = ßx where tk(.) is the density function of the t—distribution with degrees of freedom k and is the skewness parameter. When = 0, the above equation reduces to the t-distribution. 10/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Maximum Likelihood Estimation We need to fit the NIG, Skew t, Normal and t—distribution model to the empirical data. Maximum likelihood estimation is used to find the parameters of the distribution. Let (E, be a statistical model associated with a sample of i.i.d. r.v. Xl , ...,Xn. Assume that there exists 9* G @ such that Xl rx-I : 9* is the true parameter. Statisticians goal: given Xl , ...,Xn, find an estimator 9 such that IPO is close to P for the true parameter 9 11/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Maximum Likelihood Estimation o To find the parameters close to true parameter, we maximize the likelihood function. 0 11} | pe (Xi) is the likelihood function. This is Obtained by minimizing the distance between the two distribution. o This is the maximum likelihood principle. 12/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Maximum Likelihood Estimation Paramotor Estimates ovor 'tho Period January 1991-JuIy 2014 S P 500 FTSE/JSE TOP 40 Standard Bank African Bank Anglo American Merafe Resources NİG 58.3541 67.8875 44.7345 32,372 43.1323 41.3981 NİG 5.0934 5.109 0.3524 0.6613 0.9557 4.5965 NİG 0.0075 0.0123 o. 0208 0.025 0.027 o. 0527 NİG 0.0009 0.0014 0.0003 0.0005 0.0002 0.0054 t Dist- 2.7962 4.0000 4.oooo 3.8414 4.6143 6.5225 Skew t 2.8038 3.8419 4.0006 4,0000 4.6127 6.2649 Skew t 0.9519 0.9533 1.0145 ı.oooo 1.0208 1.1460 Figure: Maximum likelihood parameter estimates for the entire period 13/27 Prabhat Kumar Pankaj IMS13102 VaR Under DifTerent Statistical Distribution
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Parametric hypothesis testing Consider a sample Xl , ...,Xn of i.i.d. random variables and a statistical model (E, and Let 00 and be disjoint subsets of O. Consider the two hypotheses: Ho is the null hypothesis, HI is the alternative hypothesis. If we believe that the true 9 is either in 00 or in 01, we may want to 14/27 test Ho against HI. Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Parametric hypothesis testing A test is a statistic G {0, 1} such that: o If = 0, Ho is not rejected; o If V = 1, Ho is rejected. Rejection region of a test RIP = : v(x) 1}. Type 1 error of a test (rejecting Ho when it is actually true): 15/27 Prabhat Kumar Pankaj 1MS13102 pe(v— 1). VaR Under Different Statistical Distribution
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Parametric hypothesis testing A test has level a if av(e) a, YO G 00 and in general, a test has the form = 1 (Tn > c) , for some statistic Tn and threshold c G IR. Definition The (asymptotic) p—value of a test is the smallest (asymptotic) level a at which rejects Ho. It is random, it depends on the sample. 16/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Kolmogorov-Smirnov test statistic values Parameter Estimates for the Period January 1991—July 2014 Kolmogorov-Smirnov Critical Value S & P 500 FTSFJJSE TOP40 Standard Bank African Bank Anglo American Merafe Resources NIG 0.0058 0.0081 0.0201 0.0173 0.0114 0.0809 t-Dist. 0.0135 0.0082 0.0190 0.0150 0.0101 0.0862 Skew t 0.0113 0.0046 0.0184 0.0149 0.0088 0.0828 Normal 0.0875 0.0580 0.0560 0.0657 0.0443 0.0765 Co.9 0.0156 0.0177 0.0191 0.0195 0.0190 0.0232 Co.95 0.0173 0.0197 0.0212 0.0216 0.0211 0.0257 Co.975 0.0188 0.0214 00231 0.0236 0.0229 0.0280 co.99 0.0207 0.0236 0.0255 0.0259 0.0252 0.0308 Figure: Kolmogorov-Smirnov test statistics and estimated critical values for the fitted distributions. 17/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Kolmogorov-Smirnov test statistic values O The results of the table shows that we do not reject the null hypothesis for the NIG, Skew t and t-distribution. O However, the null hypothesis is rejected for the Normal distribution for all the shares with exception of Merafe Resources and in some cases the other three distributions are rejected for Merafe Resources as well. 18/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Empirical data with the fitted distribution NIG Dist. . • t—dist. NIG Dist. 15 —Ono (b) S & P 500 0.10 (a) Standard Bank Figure: The log-density of the empirical data with the fitted NIG, Normal, Skew t and t—distribution for one of the stocks and index. 19/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Empirical data with the fitted distribution NIG Dist. S I—dist. I—dist_ —0. IS (a) Standard Bank Skew 0.05 —0.20 0.10 oas -0.05 coo (b) S P 500 Figure: Comparison of histograms with the fitted NIG, Skew t, t-distribution and Normal distribution for one stock and index. 20/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Value-at-Risk Estimates VaR Estimates over the Period January 1991-July 2014 Historical NIG S P 500 FTSE/JSE TOP 40 Standard Bank African Bank Anglo American Merafe Resources 3.15% 3.79% 5.81% 7.35% 6.60% 9.02% 3.75% 5.81% 7.53% 6.38% 8.98% t—Dist. 3.04% 3.56% 5.65% 7.73% 6.73% 9.36% Skew t 3.45% 3.84% 5.86% 7.54% 6.46% 8.13% Normal 2.68% 3.11% 4.87% 6.37% 5.97% 8.36% Figure: Comparison of the Value-at-Risk estimates, the non-parametric estimates are calculated using the Historical Simulation approach. The VaR estimates were calculated using the Monte Carlo simulation. 21/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Backtesting the Model o To verify the correctness of the VaR models we perform backtesting method. o The number of daily losses exceeding VaR estimate are referred to as violations. o The regulatory backtesting procedure is performed over the last 250 trading days with the 99% oneday VaR compared to the observed daily profits and losses over the period. 22/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Backtesting the Model Number of Violations for 99% Historical Daily-van. NIG t S & P 500 FTSE/JSE TOP 40 Standard Bank African Bank Anglo American Merafe Resources 63 49 41 40 42 28 54 51 41 37 48 30 —Dist. 72 56 46 37 38 25 Skew t 51 43 39 37 45 36 Normal 102 98 72 60 63 34 Figure: Number of violations for each VaR model and the expected violations at the 99% confidence level 23/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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Discussions o In this study we modelled selected equity stocks, the FTSE/JSE TOP40 and S & P 500 indices using the NIG, t-distribution, Skew t and Normal distribution. Each of these statistical distributions captures different features of the financial returns. o The regulatory backtesting procedure is performed over the last 250 trading days with the 99% oneday VaR compared to the observed daily profits and losses over the period. o The NIG, Skew t and t-distribution fitted the financial returns better both in the center and tails as compared to the classic Normal distribution, with the Skew t and t-distribution showing heavier tails then the NIG semi-heavy tails. 0 we calculated VaR under the NIG, Normal, Skew t and t-distribution assumptions. The results obtained showed that the VaR calculated under the three distributions outperformed those 24/27 under the Normal distribution. Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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References Uryasev, S.; Rockafellar, R.T. Optimisation of Conditional Value-at-Risk. J. Risk 2000, 2, 21-41. Alexander, C. Market Risk Analysis Value-at-Risk Models; Volume IV; John Wiley Sons Ltd, England.: Hoboken, NJ, USA, 2008 15. https://ocw.mit.edu/courses/mathematics/18-650-statistics- for-applications-fall-2016/1ecture- slides/MIT18.650F16NaximumXE.pdf Barndorff-Nielsen, O.E. Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling. Board Found. Scand. J. Stat. 25/27 1997, 24, 113. Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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ACKNOWLEDGEMENT First and foremost, I want to express my unending gratitude to the Almighty. With people having to face so much anguish in this World, I am blessed to be endowed with an enriching environment where I can pursue my interests. Next, I am indebted to Dr. M. P. Rajan for providing me an opportunity to work under his guidance. His valuable guidance whenever I needed it is what helped me to work at my own pace and his support during the project was second to none. I sincerely acknowledge his trust in me and patience with me. I also wish to mention the support from my friends: Prakhar Swarup (School of Chemistry), Ravi Kant (School of Physics) and Anubhab Dey (School of Physics). Last but not the least, I am obliged towards School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram (IISER- TVM) for all the resources 1 used and all the knowledge I gained. My upcoming career would definitely benefit lea s and bounds from this ro•ect. 26/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution
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THANK YOU 27/27 Prabhat Kumar Pankaj 1MS13102 VaR Under Different Statistical Distribution

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