Statistical Methods of Valuation and Risk Assessment: Empirical Analysis of Equity Markets and Hedge Fund Strategies

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Statistical Methods of Valuation and Risk Assessment: Empirical Analysis of Equity Markets and Hedge Fund Strategies

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Riskometer backtests itself daily and updates a violation count table that allows a user to see the historical performance of the method since start of 2001.

Swiss Federal Institute of Technology, Zürich University of Zürich, Swiss Banking Institute Master of Advanced Studies in Finance Master Thesis Statistical Methods of Valuation and Risk Assessment: Empirical Analysis of Equity Markets and Hedge Fund Strategies Adam Czub ***** January 2004 Acknowledgements A master thesis is usually thought as entirely individualistic work This is hardly ever the case Constant support, understanding and enlightenment are required from different people during the process I will be forever grateful for the emotional support received from my parents, Elzbieta and Wojciech, and the encouragement from my brother Tomasz Many thanks to my supervisor, Prof Alexander McNeil, for his guidance and help on my master thesis I would like to thank also my co-supervisors and colleagues Valerie ChavezDemoulin, Bernhard Brabec and Michael Heintze for their explanations and comments Finally, thanks to all of those who in one way or another helped me make this master thesis become a reality -1- Abstract The purpose of this paper is first to describe a web-based tool called Riskometer We designed and implemented its second version which has a statistical methodology implemented in S-Plus This tool, called Riskometer, returns the different Value-at-Risk and related measures of risk (Expected Shortfall, volatility) for major equity market indices using standard methods as well as the most recent state-of-the-art methods This internet tool continually backtests its own performance against the latest data We analyse the risk measures calculated by Riskometer on September 24, 2003 and January 9, 2004 In the second part of the paper, we analyse hedge fund strategies over a six years sample period using the database of indices compiled by Morgan Stanley Capital International For a better understanding about dependence structures in hedge fund strategies we focus on analysing their bivariate distributions using Archimedean copulas To identify style exposures to relevant risk factors we conduct a return-based style analysis of hedge fund strategies by relaxing the constraints of the Sharpe’s style analysis, and examine the significance of the style weights Finally, we compare these results with those obtained by applying the Kalman filter and smoother technique -2- Contents INTRODUCTION QUANTIFICATION OF EQUITY MARKET RISK: RISKOMETER 2.1 INTRODUCTION 2.2 DATA 2.3 METHODS 2.4 RISK MEASURES 2.4.1 Volatility 2.5 BACKTESTING EMPIRICAL CHARACTERISTICS OF HEDGE FUND STRATEGIES 10 3.1 INTRODUCTION 10 3.2 DATA 10 3.3 RISK-RETURN CHARACTERISTICS 15 3.4 DEPENDENCE STRUCTURE ANALYSIS 21 3.4.1 Linear Correlation as Dependence Measure 22 3.4.2 Alternative Correlation Measures 22 3.4.3 Archimedean Copulas 24 3.4.4 Statistical Significance of the Copula Parameter 27 3.4.5 Tail Dependences 28 3.5 GENERALISED STYLE ANALYSIS 30 3.5.1 Return-Based Style Analysis Model 30 3.5.2 Statistical Significance of Style Weights 32 3.5.3 Analysis of Style Weights 32 3.6 TIME-VARYING EXPOSURES ANALYSIS 34 3.6.1 Kalman Filter and Smoother Algorithm 35 3.6.2 Graphical Analysis of Time-Varying Exposures 37 CONCLUDING REMARKS 39 REFERENCES 41 GLOSSARY 43 FITTING COPULAS 46 DIGITAL FILTERING 49 -3- Introduction Much of the financial decision making by financial institutions is focuses on risk management Measuring risk and analysing ways of controlling and allocating it require a wide range of sophisticated mathematical and computational tools Indeed, mathematical models of modern finance practice contain some of the most complex applications of probability, optimisation, and estimation theories Mathematical models of valuation and risk assessment are at the core of modern risk management systems Every major financial institution in the world depends on these models and none could function without them Although indispensable, these models are by necessity abstractions of the complex real world Although there is continuing improvement in those models, their accuracy as useful approximations varies significantly across time and situation Quantification of Equity Market Risk: Riskometer 2.1 Introduction The ETHZ Riskometer consists of a web-based tool which was designed and implemented using a statistical methodology implemented in S-Plus It returns the different Value-at-Risk and related measures of risk (expected shortfall, volatility) using standard methods and the most recent state-of-the-art methods This educational tool continually backtests its own performance against the latest data In the present version Riskometer focuses principally on three major stock indices: the Dow Jones Industrial Average (DJIA), the Standard&Poors 500 (S&P500), and the Deutsche Aktienindex (DAX) The data are collected daily and added to the historical daily time series dataset, providing risk measures that may be interpreted as prognoses for a one-day time horizon The underlying methods included in the Riskometer may be applied to any stock price, exchange rate, commodity price or portfolio comprising combinations of these underlying risk factors, whether linear or non-linear They may also be applied to daily data or to both higher or lower frequency time series data In the market risk area, Value-at-Risk estimation involves portfolios of more than one asset The Riskometer can be extended to such multivariate series 2.2 Data The Riskometer works with return data since the beginning of 1998 what represents somewhat less than 1500 days of historical returns We believe that our time window is long enough for not losing statistical accuracy in measuring risk and includes relevant data representing a period characterised by important market up and down moves additionally to high and low volatility times Furthermore, we make the assumption that our amount of data may be considered to be a realisation of a stationary time series model 2.3 Methods The underlying methods implemented in the Riskometer provide either unconditional quantile estimation or conditional quantile estimation Unconditional methods for calculation of market risk measures are -4- Plain Vanilla: Variance – Covariance Historical Simulation And the conditional methods are EWMA (Exponentially Weighted Moving Average) GARCH modelling GARCH-style time series modelling with extreme value theory Point process approach Standard methods (variance-covariance, historical simulation, EWMA, and GARCH modelling) are described in the technical document of Riskmetrics Group, Inc (2001) available on-line Concerning more sophisticated methodologies, GARCH-style time series modelling with extreme value theory, which still needs to be implemented in the present version of Riskometer, is detailed in the McNeil and Frey (2000) and Point process approach is detailed in the Chavez-Demoulin, Davison and McNeil (2003) 2.4 Risk Measures We describe the risk measures calculated by the different methods of the Riskometer on September 24, 2003 and January 9, 2004 On September 24, 2003 Riskometer gave the results shown in Table For each index and each method five numbers have been calculated, excepting for Historical Simulation, GARCH modelling and Point process approach methods which not provide volatility figures The first four are estimates of Value-at-Risk (VaR) and Expected Shortfall (ES) at probability levels of 95% and 99%, respectively Each of these numbers may be interpreted as potential daily percentage losses for September 24, being based on closing data up to September 23 Table 1: Risk Measures on September 24, 2003 Method DJIA DAX S&P500 VaR 95% ES 95% VaR 99% ES 99% Volatility 6 2.11 1.97 2.04 1.50 1.88 3.06 2.90 3.04 2.67 3.96 2.19 2.15 2.14 1.63 1.99 2.64 2.76 2.56 2.29 2.71 3.84 3.92 3.81 3.94 5.59 2.75 2.93 2.68 2.34 2.81 2.98 3.16 2.89 2.25 3.22 4.34 4.86 4.30 3.87 6.64 3.10 3.62 3.02 2.45 3.30 3.41 4.13 3.31 3.04 4.03 4.97 5.82 4.93 4.97 7.94 3.55 4.21 3.47 3.14 4.12 20.17 NA 19.63 NA NA 29.67 NA 29.22 NA NA 21.06 NA 20.56 NA NA -5- If we concentrate on the DJIA, for the Exponentially Weighted Moving Average method (method no 3), a 95% VaR number of 2.04% indicates that the estimated 5th percentile of the predictive return distribution form day was –2.04%; we estimate that there is one chance in 20 that the return is a loss of magnitude greater than 2.04% An ES number of 2.56% indicates that, in the event that such a 20-day loss occurs, this will be its expected size The 99% VaR and ES estimates are 2.89% and 3.31%, respectively The VaR and ES estimates are all driven by the annualised volatility estimate It is obtained by taking the standard deviation of the one-day distribution and multiplying it by the square root of 260 representing the approximate number of trading days in a year This number is best interpreted in relation to other annualised volatility numbers and not necessarily as an absolute measurement From Table we see that annualised volatility of the DAX on September 24 is considerably larger than that of the two American indices, showing that the German market was more turbulent on this date Table shows the daily summary for January 9, 2004, by which time American and European markets had calmed down Table 2: Risk Measures on January 9, 2004 Method DJIA DAX S&P500 VaR 95% ES 95% VaR 99% ES 99% Volatility 6 2.08 1.96 2.01 1.29 1.71 3.03 2.84 2.99 1.52 2.82 2.16 2.12 2.10 1.30 1.77 2.60 2.73 2.53 2.06 2.42 3.81 3.89 3.74 2.71 3.82 2.71 2.89 2.63 2.06 2.31 2.93 3.16 2.85 1.94 2.85 4.30 4.74 4.22 2.20 4.46 3.06 3.48 2.97 1.95 2.64 3.36 4.13 3.26 2.65 3.57 4.92 5.82 4.84 3.36 5.28 3.50 4.17 3.40 2.67 3.17 19.81 NA 19.36 NA NA 29.31 NA 28.70 NA NA 20.70 NA 20.16 NA NA All three annualised volatilities have fallen For instance, the EWMA method indicates that the Dow Jones annualised volatility has fallen from 19.63% to 19.36%, the DAX volatility has fallen from 29.22% to 28.70%, and the S&P500 volatility has fallen from 20.56% to 20.16% Correspondingly, all risk measures are now lower The 95% VaR for the DJIA is now 2.01% and the ES 2.53% The 99% numbers for VaR and ES are 2.85% and 3.26% 2.4.1 Volatility Riskometer allows us to explore the recent historical development of the annualised volatility Daily annualised volatilities estimates of our equity market indices since the start of 1998 are graphically represented online and in Figure -6- Figure 1: Annualised Volatility Figures of Equity Market Indices The graph ends with the volatility estimates of January 8, 2004 The forecasts for January are not shown, but the decrease of volatilities for the three indices after an extremely volatile period is obvious The peak volatility for the year 2003 for both American indices occurs on March 24 The peak volatility for the DAX occurs on April 7, so it is clear that the two American indices follow each other closely, but the DAX has a somewhat different behaviour Furthermore, we clearly observe that the DAX has been the most volatile index during 2003 Concentrating on the American indices, we see that the relatively calm present period comes after a long period of extreme volatility Indeed, it is the first low volatility period since autumn of 2002 Throughout the second part of 2002 and the first three quarters of 2003, volatilities attained spectacular levels The highest peaks on both the DJIA and S&P500 indices occurs on July 29, 2002 and August 8, 2002 respectively On those days, volatility figures reached around 40% Then, during the fourth quarter of 2002 we peaks of 15 October for S&P500 and October 17 for DJIA Once again, after a slight decrease, volatility figures reached levels of 40% Concerning DAX, in the second part of 2002, its volatility peaked on August 8, as S&P500 and on October 17, as DJIA with values over 60% The last high volatility period of March – April 2003 of American indices has been followed by the DAX around two weeks later Indeed, on April 7, the German index annualised volatility attained once again impressing figures over 50% Since autumn 2003 Equity markets indices volatilities decayed to more modest levels and settle again below 20% -7- 2.5 Backtesting Riskometer backtests itself daily and updates a violation count table that allows a user to see the historical performance of the method since start of 2001 To appreciate what happens we look again at January 9, 2004 By the end of that day it was possible to evaluate what had happened on the markets and to compare risk measures with reality In fact, all of the indices had decreased in value, by 1.27%, 0.73% and 0.89%, respectively, for the DJIA, DAX, and S&P500 indices Thus, for all indices, we observe that the actual losses did not exceed the VaR estimates As an example for DJIA, the violations of method are shown graphically in Figure We observe that the last violation of the 95% VaR was on May 19, 2003 and this of the 99% VaR on March 24, 2003 Note that in this picture negative returns are shown as positive values, and positive returns as negative values Figure 2: EWMA 95% and 99% VaR estimates If VaR is being estimated successfully violations of the 95% VaR should occur once every 20 days on average, and violations of the 99% VaR once every 100 days Whether this is approximately true is more easily judged in Table -8- Table 3: Backtesting since start of 2001 Method DJIA DAX S&P500 95% observed expected P-value 99% observed expected P-value 6 44 46 38 32 38 74 62 69 56 59 44 43 40 36 42 36 36 36 36 36 37 37 37 37 37 36 36 36 36 36 0.18 0.10 0.74 0.50 0.74 0 0 0.18 0.24 0.50 0.31 14 6 38 14 26 16 11 5 7 7 7 7 7 7 7 0.02 0.71 0.71 0.71 0.02 0.72 0.15 0.46 0.46 0.71 0.46 It shows the results of a backtest from the start of 2001 It covers a period in which we might have expected 36 violations of the 95% VaR for the American indices and 37 violations of the 95% VaR for the German index and violations of the 99% VAR for all indices In the 95% observed column we see the actual numbers of violations incurred by Riskometer at the 95% level and in 99% observed column we see the actual numbers of violations incurred by Riskometer at the 99% level In the case of DJIA for method it is 38 at the 95% level, and at the 99% level Although the 95% number is slightly higher than expected, we observe that VaR is being estimated accurately A binomial test has been carried out and expressed in the table as a p-value A p-value less than or equal to 0.05 would be interpreted as evidence against the null hypothesis of reliable VaR estimation For our example this is not the case In the case of DJIA for method we observe that, for VaR 99%, the binomial test p-value indicates evidence against the null hypothesis For S&P500, we notice that all methods succeed the binomial test This confirms a general good performance of the Riskometer methods at the 95% level and at the 99% level Finally, for DAX, we observe that only the GARCH modelling method succeed the binomial test and only at the 99% level Concerning the remaining VaR estimates we observe that Riskometer methods perform very badly at the 95% level and at the 99% level presenting a systematic underestimation of the VaR and too many violations -9- data or fundamental economic data Fund manger intervention is limited to selecting trades and applying risk management disciplines Convertible Arbitrage A conservative, market-neutral approach that aims to profit from pricing differences or inefficiencies between the values of convertible bonds and common stock issued by the same company Managers of such funds generally purchase undervalued convertible bonds and short-sell the same issuers' stock The approach typically involves a medium-term holding period and results in low volatility Fixed Income Arbitrage This approach employ a variety of fixed income related strategies ranging from relative value based trades to directional bets on interest rate shifts Style also includes credit-related arbitrage, which typically involves the purchasing or selling of corporate issues and the simultaneous selling or purchasing of government issues Merger Arbitrage A hedge fund investment approach is considered Merger Arbitrage if at least 80% of positions not based upon cash deals are entered into in a hedged manner, replicating the contemplated transaction It exploits merger activity to capture the spread between current market values of securities and their values after successful of a merger, restructuring or similar corporate transaction Long Bias Long Bias portfolios have net long exposure to the underlying market in all conditions These funds not have zero betas to the overall market, but they are generally aiming to provide a higher beta in rising markets and a lower beta in falling markets The fund’s portfolio must be at least partially hedged or use short sales to be considered Long Bias Short Bias Short Bias portfolios maintain a significant net short market exposure Short Bias managers specialize in short selling opportunities, and are not necessarily bearish about the market Few will use leverage and most have large cash positions from the proceeds of short selling This designation also includes short-only funds, where the portfolio is not required to contain any long positions Variable Bias Variable Bias is the designation for Security Selection funds that not conform to a constant specific market exposure but are still focused on individual security selection Some Variable Bias managers will alter the fund’s market exposure dramatically in response to perceive opportunities, moving from a large net long position to substantially net short within a short period of time They are generally looking to time the market and make money from both their timing ability and their security selection skill Others will allow their net market position to be dictated by the balance of security and sector ideas they have Distressed Securities Distressed Securities funds invest in the securities of firms in or near bankruptcy Bankrupt firms will typically have defaulted on their debt and may have also filed for protection from their creditors under bankruptcy laws Firms near bankruptcy may be approaching these - 44 - situations through severe operating and/or financial difficulties Investors in Distressed Securities are seeking capital appreciation rather than high yields Event Driven Event Driven encompasses a combination of investment processes targeting securities which experience a change in valuation due to corporate transactions For instance, a strategy focusing on acquisitions and bankruptcies combines elements of two investment processes: the merger arbitrage and distressed securities - 45 - Fitting Copulas Table 18: Discretionary Trading Copula No Strategy AIC ˆ θ () ˆ ˆ se B θ Systematic Trading 0.39 0.49 1.17 Convertible Arbitrage -10.21 2.24 0.41 Fixed Income Arbitrage -5.44 0.50 0.29 Merger Arbitrage -16.47 2.62 0.42 Long Bias -10.41 2.25 0.45 Short Bias -1.93 -3.48 1.31 Variable Bias -28.38 6.12 1.25 Distressed Securities -1.78 1.62 0.41 Event Driven -11.94 2.36 0.45 Table 19: Systematic Trading Strategy Copula No AIC ˆ θ () ˆ ˆ se B θ Convertible Arbitrage 1.34 1.09 0.12 Fixed Income Arbitrage 1.56 -0.09 0.15 Merger Arbitrage 1.95 -0.02 0.15 Long Bias -3.52 -0.25 0.08 Short Bias -1.89 1.20 0.17 Variable Bias 1.97 -0.02 0.13 Distressed Securities -1.56 0.30 0.22 Event Driven -3.63 -0.27 0.07 - 46 - Table 20: Convertible Arbitrage Copula No Strategy AIC ˆ θ () ˆ ˆ se B θ Fixed Income Arbitrage 10 -21.89 0.34 0.05 Merger Arbitrage 10 -7.51 0.22 0.08 Long Bias 10 -2.99 0.16 0.09 Short Bias -0.32 0.61 0.20 Variable Bias -8.42 2.76 1.19 Distressed Securities -19.24 0.03 0.03 Event Driven -11.90 2.33 0.44 Table 21: Fixed Income Arbitrage Copula No Strategy AIC ˆ θ () ˆ ˆ se B θ Merger Arbitrage -3.30 0.38 0.19 Long Bias -0.23 1.18 0.14 Short Bias 1.72 0.85 0.28 Variable Bias -5.54 1.37 0.17 10 -7.94 0.22 0.06 -4.59 1.32 0.16 Distressed Securities Event Driven Table 22: Merger Arbitrage Strategy Copula No AIC ˆ θ () ˆ ˆ se B θ Long Bias -21.49 0.06 0.04 Short Bias -6.93 -0.30 0.04 Variable Bias -13.76 1.65 0.29 Distressed Securities -42.00 0.00 0.03 Event Driven -44.73 0.10 0.04 - 47 - Table 23: Long Bias Copula No Strategy AIC ˆ θ () ˆ ˆ se B θ Short Bias -87.49 0.01 0.00 Variable Bias -89.64 2.35 0.22 Distressed Securities -42.66 1.63 0.41 Event Driven -73.70 0.24 0.07 Table 24: Short Bias Copula No Strategy AIC ˆ θ () ˆ ˆ se B θ Variable Bias -37.48 -11.55 2.34 Distressed Securities -16.78 0.10 0.15 Event Driven -22.96 -8.18 1.55 Table 25: Variable Bias Copula No Strategy AIC ˆ θ () ˆ ˆ se B θ Distressed Securities -21.85 5.03 1.74 Event Driven -44.37 8.08 1.82 Table 26: Distressed Securities Strategy Event Driven Copula No - 48 - AIC -50.02 ˆ θ 1.49 () ˆ ˆ se B θ 0.17 Digital Filtering Figure 9: Discretionary Trading - 49 - Figure 10: Systematic Trading - 50 - Figure 11: Convertible Arbitrage - 51 - Figure 12: Fixed Income Arbitrage - 52 - Figure 13: Merger Arbitrage - 53 - Figure 14: Long Bias - 54 - Figure 15: Short Bias - 55 - Figure 16: Variable Bias - 56 - Figure 17: Distressed Securities - 57 - Figure 18: Event Driven - 58 - ... insight and extend our understanding of hedge fund risks to a wide range of equity oriented hedge fund strategies They characterised the risk exposures of hedge funds using buy -and- hold and option-based... detailed analysis of hedge fund risks and returns is also important from the standpoint of asset pricing theory Understanding the hedge fund risks exposures is also a key feature to the design of optimal... Value-atRisk and related measures of risk (expected shortfall, volatility) for major equity market indices using standard methods as well as the most recent state -of- the-art methods The analysis of the risk

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