Estimating the Value at Risk: A Dynamic Extreme Value Theory Approach Author: Supervisor: Lauren McGeever Dr. S. U. Can 10464832 Second Reader: [email protected] Prof. R. J. A. Laeven Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam Faculty of Economics and Business Amsterdam School of Economics December 17, 2015 iii Abstract Since first being introduced globally in 1996 by the Basel committee, the value at risk has become “the first line of defence against financial risk” (Jorion 1996). Although the Value-at-Risk is undeniably a useful tool for estimating risk, it still has several shortcomings. This paper will examine these criticisms and proposes extreme value theory as a more sophisticated approach to calculating the Value- at-Risk. We will also examine the role of stochastic volatility models in scaling estimates by the volatility, in order to improve the prediction of the models. The data to be examined belongs to three of the most well-known stock indices in the world. We will be using: FTSE 100, S&P 500 and Nikkei 225. We will be using two periods, an in-sample period of September 4th, 1984 to September 3rd, 2010 in which to calculate the VaR estimates and an out-of-sample period from 4th September 2010 to the 3rd September 2015 which will be used to back test our estimates. From this investigation we can conclude that use of Extreme Value The- ory provides better estimates for the Value-at-Risk, but only for higher confidence levels. We also conclude that models fitted with a stochastic volatility model offer better predictions. iv Contents Abstract iii 1 Introduction1 2 The Value at Risk3 2.1 Introducing the Value at Risk..................3 2.2 Traditional VaR Methods.....................3 2.2.1 The Normal based VaR approach............3 2.2.2 Historical simulation approach.............4 2.3 A Coherent Risk Measure....................4 3 An Introduction to Extreme Value Theory6 3.1 Introduction to Extreme Value Theory.............6 3.1.1 Extremal Types Theorem................7 3.2 Classical Extreme Value theory (The block maxima approach)8 3.3 The Peak-over-the-Threshold approach............9 3.3.1 The Generalized Pareto distribution..........9 3.3.2 Threshold selection and Parameter estimation for the GPD............................9 3.3.3 The Return level..................... 10 3.4 Selecting an EVT method..................... 11 4 Data and Methodology 12 4.1 The data.............................. 12 4.1.1 Using R Studio for analysis............... 13 4.2 Data transformation....................... 13 4.3 Fitting a stochastic volatility model............... 14 4.4 Fitting a generalized Patero model............... 15 4.4.1 Checking the goodness of fit for parameter estimates 16 5 Results and Back-testing 19 5.1 Examining Return levels..................... 19 5.1.1 Return level estimates for static and dynamic models 21 5.2 VaR estimates from the in sample period............ 21 5.3 Back-Testing............................ 23 5.3.1 The Portion of Failures Unconditional Coverage test 23 5.3.2 The Duration Based Conditional Coverage test.... 24 6 Conclusion 26 7 References 28 A Tables of results 30 B R code 34 v C Plots 37 vi List of Figures 4.1 Plot of the NIKKEI 225 log returns against observation days 14 4.2 Mean residual life plot for the NIKKEI 225, and 95% confi- dence interval........................... 15 4.4 Diagnostic plots for static EVT model of the NIKKEI 225.. 17 4.5 Diagnostic plots for dynamic EVT model of the NIKKEI 225 18 5.1 Return level plot the FTSE 100 with static EVT........ 19 5.2 Return level plot the FTSE 100 with dynamic EVT...... 20 C.3 Fitting the GPD over a range of thresholds for the S&P 500. 37 C.4 Fitting the GPD over a range of thresholds for the FTSE 100. 38 C.5 Fitting the GPD over a range of thresholds for the NIKKEI 225 38 C.6 Diagnostic plots for S&P 500................... 39 C.7 Diagnostic plots for FTSE 100.................. 39 C.8 Return levels plot for the S&P 500 and NIKKEI 225...... 40 vii List of Tables 4.1 Threshold choices and number of exceedances for: FTSE 100, S& P 500, NIKKEI 225...................... 16 4.2 Maximum likelihood estimates of the shape and scale pa- rameters.............................. 16 5.1 Return level results for Static EVT models........... 21 5.2 Return level results for dynamic EVT models......... 21 5.3 S&P 500 VaR estimates...................... 22 5.4 S& P 500 VaR Violations..................... 23 5.5 S&P 500 Conditional Coverage Back-test............ 25 A.1 NIKKEI 225............................ 30 A.2 FTSE 100.............................. 30 A.3 VaR Violations FTSE 100..................... 31 A.4 VaR Violations for NIKKEI 225................. 31 A.5 FTSE 100.............................. 32 A.6 NIKKEI 225............................ 33 viii List of Abbreviations ARCH Autoregressive Conditional Heteroskedasticity CDF cumulative distribution function CVaR Conditional Value at Risk EVT Extreme Value Theory iid independent and identically distributed GARCH Generalized Autoregressive Conditional Heteroskedasticity GEV Generalized Extreme Value distribution GPD Generalized Pareto distribution MRL Mean Residual Life plot POT Peak-Over-the-Threshold P-P plots Probability-Probability plots Q-Q plots Quantile-Quantile plots r.v. Random Variable VaR Value-at-Risk 1 Chapter 1 Introduction Over the last century financial crises have occurred in greater frequency and magnitude, highlighting the need for stricter risk management. Some of largest the and most devastating crashes include: the Wall Street crash of 1929; Black Monday of 1987; the Asian crisis of 1997; the credit crisis of 2007; and even more recently the Chinese stock market crash of 2015. Primarily due to advances in modern technology, financial institutions are more interconnected than ever. This often creates a domino effect in which other financial institutions are affected by the failure of another. Looking at the credit crisis of 2007 this is very apparent. The bursting of the Ameri- can housing bubble was chiefly facilitated by sub-prime mortgages and the relaxation of lending standards. After this bubble burst large financial insti- tutions such as Merrill Lynch, Citigroup and UBS incurred huge losses be- cause of their large positions in the securitization of these sub-prime mort- gages. Governments were obliged to rescue many of these financial institu- tions, however Lehman Brothers was allowed to fail. This created a ripple effect, which was felt by other institutions world wide. The formation of the Basel committee on banking supervision was the first attempt in creating an international regulatory framework; their aim was to implement risk management policy to reduce financial crises. In 1987 they proposed the Value-at-risk as a model for market risk, this was finally implemented in the 1996 amendment of the Basel accord. It required banks to hold capital for market and credit risk. VaR models are essential to risk management. Jorion, 1996 describes them as “the first line of de- fense against financial risk” and sees them as "improving transparency and stability in financial markets". This is because they have the capability to summarize the total risk in a single number. A VaR model can be inter- preted as fullfilling the requirement: At X% certainty there will be no loss of more than Y over N days. Other favorable points are that this number is easy to interpret and that standard methods of calculation are easy to use. Value-at-Risk may appear a very favorable risk measure for downside risk however it is widely accepted that VaR may regularly underestimate risk. This underestimation can have damaging effects on institutions, leav- ing them unprepared for adverse market movements. Traditional methods of calculating VaR often rely on the unrealistic assumption that the under- lying distribtuion is normal. For average returns (center of the distribution) this is a reasonable assumption. Unfortunately it has been observed by many that market returns often have heavier tails and are more leptokur- tic than the normal distribution. Research by Mandelbrot, 1963 and Fama, 1965 did much in the way of proving this. The underestimation of the tails is a major problem because “financial solubility of an investment is likely to 2 Chapter 1. Introduction be determined by extreme changes in market conditions rather than typical changes” (Coles, 2001). These returns found in the tails are often referred to as extreme values because they are very large losses and occur rarely; though not as rare as we would like. Understanding that the normal distribution is inadequate for model- ing the tails of financial returns, we can ask the question what distribution would be a better fit? One of the more popular solutions leads us to extreme value theory, “Letting the tails speak for themselves” (Coles, 2001) . This al- lows us to create an asymptotic distribution for the tails. EVT has been used in numerous fields of science for many years. These include: Ther- modynamic of earthquakes; Ocean wave modeling and even food science. Since the 1980’s EVT has also grown in popularity in the field of financial risk management, especially after the discovery of EVT’s second theorem: peak over the threshold by Pickands–Balkema–de Haan, 1974. This is be- cause “ the generalized Pareto threshold model provides a direct method for the estimation of value-at-risk.” (Coles, 2001). In this thesis we will compare the standard ways of modeling VaR with the newer peak over the threshold EVT approach with and without the help of a stochastic volatility model to deal with changes in the volatility. We are aiming to prove that EVT gives more accurate results for value at risk than the standard methods and even better estimates when combined with a GARCH(1,1) model.