The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk

Transcription

1 An EDHEC-Risk Institute Publication The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk August 2014 Institute

2 2 Printed in France, August Copyright EDHEC The opinions expressed in this study are those of the authors and do not necessarily reflect those of EDHEC Business School.

3 Table of Contents Executive Summary Introduction Risk Control Schemes A Conditional EVT Model Empirical Analysis...25 Conclusion...35 References...37 About EDHEC-Risk Institute...41 EDHEC-Risk Institute Publications and Position Papers ( )...45 An EDHEC-Risk Institute Publication 3

4 About the Authors Lixia Loh is a senior research engineer at EDHEC-Risk Institute Asia. Prior to joining EDHEC Business School, she was a Research Fellow at the Centre for Global Finance at Bristol Business School (University of the West of England). Her research interests include empirical finance, financial markets risk, and monetary economics. She has published in several academic journals, including the Asia-Pacific Development Journal and Macroeconomic Dynamics, and is the author of a book, Sovereign Wealth Funds: States Buying the World (Global Professional Publishing, 2010). She holds an M.Sc. in international economics, banking and finance from Cardiff University and a Ph.D. in finance from the University of Nottingham. Stoyan Stoyanov is professor of finance at EDHEC Business School and head of research at EDHEC Risk Institute Asia. He has ten years of experience in the field of risk and investment management. Prior to joining EDHEC Business School, he worked for over six years as head of quantitative research for FinAnalytica. He has designed and implemented investment and risk management models for financial institutions, co-developed a patented system for portfolio optimisation in the presence of non-normality, and led a team of engineers designing and planning the implementation of advanced models for major financial institutions. His research focuses on probability theory, extreme risk modelling, and optimal portfolio theory. He has published over thirty articles in leading academic and practitioner-oriented scientific journals such as Annals of Operations Research, Journal of Banking and Finance, and the Journal of Portfolio Management, contributed to many professional handbooks and co-authored three books on probability and stochastics, financial risk assessment and portfolio optimisation. He holds a master in science in applied probability and statistics from Sofia University and a PhD in finance from the University of Karlsruhe. 4 An EDHEC-Risk Institute Publication

5 Executive Summary An EDHEC-Risk Institute Publication 5

6 Executive Summary Empirical studies have demonstrated that cap-weighted indices do not represent efficient portfolios. The main reasons are that cap-weighting results in significant concentration, ignores correlations between stocks and, finally, cap-weighted indices do not exhibit an efficient exposure to rewarded risk factors Amenc et al. (2014b). In recent years, the industry has tried to respond by developing the so-called smart beta framework, which attempts to address the three drawbacks. To construct smart beta indices, traditional index providers usually employ a set of methodology choices for stock selection and weighting, packaged together in a single index without allowing the investor the flexibility of making separate choices. Amenc and Goltz (2013) suggest the ERI Smart Beta 2.0 approach, which represents a substantial improvement over the traditional methods. It separates the two main steps in the index construction process, allowing the investor to make an informed decision both about the factor tilt and the diversification method. The diversification method deals with the over-concentration issue and the factor tilt method leads to an exposure to better rewarded factors. In contrast to the very often opaque description of industry methodologies, the ERI Smart Beta 2.0 approach is fully transparent. Although deviating away from the cap-weighted index leads to significant risk-adjusted performance benefits, it also exposes investors to additional risks. Firstly, the alternative weighting scheme may lead to an over-weighting or under-weighting of certain sectors or countries relative to the corresponding cap-weighted index which may result in temporary underperformance. It has been shown empirically that sector and country risks are not priced in and it would therefore make sense for investors to try to avoid them. For an empirical analysis, see for example Cappiello et al. (2008). Secondly, and more generally, improved relative performance necessarily comes at the cost of tracking error (TE) risk. This risk is of course related to relative country/sector risk and arises in recognising that cap-weighted indices, although representing inefficient portfolios, will continue to be used as a reference point. Therefore, from an investor perspective, tracking error risk needs to be managed. For additional discussions, see for example Rudd (1980), Roll (1992) and Chan et al. (1999). Finally, deviating from a cap-weighted index implies also departing from the objective of representing the market. Smart beta indices require setting a specific objective which often takes the form of a goal in an optimisation problem. Solving the problem requires provision of parameters that need to be estimated from data. This exposes the optimal solution to the noise in the observed stock returns which is also known as sample risk and it can be relatively bigger or smaller depending on how many parameters need to be estimated and of what type. From an investor perspective, however, it makes sense to try to diversify away this risk as much as possible by combining different smart beta strategies into one multi-strategy index. The academic literature confirms this intuition in the face of parameter uncertainty, Kan and Zhou (2007) argue that an investor should hold a combination of the global minimum variance portfolio, the maximum Sharpe ratio portfolio, and a risk-free asset. 6 An EDHEC-Risk Institute Publication

7 Executive Summary Apart from the sample risk diversification, combining smart beta indices into a multistrategy index could be appealing because of other benefits including smoothing conditional performance, see Amenc et al. (2014a,c). The goal of this paper is to check empirically if controlling the exposure to some risks such as country, sector, tracking error, or sample risk does not increase the exposure to other types of risk, such as tail risk, that may remain unaccounted for by the index construction process. The analysis in this paper complements Loh and Stoyanov (2014b) where we study the tail risk of smart beta strategies without imposing any risk controls. For the purposes of this paper, we use the data for the risk-controlled strategies available on the Scientific Beta platform at which provides a consistent index construction framework that can combine different risk control methods with popular weighting schemes subject to practical constraints guaranteeing investable indices. The methodology follows the one developed by Loh and Stoyanov (2014b). We measure tail risk in terms of conditional Value-at- Risk (CVaR) through a time series model based on a GARCH filter and Extreme Value Theory (EVT) as a probabilistic model for the tail. The model allows CVaR to be decomposed into a volatility component and a residual tail risk component. From a risk management perspective, it is important to segregate the two components because the dynamics of volatility contributes to the unconditional tail thickness phenomenon. Generally, the GARCH part is responsible for capturing the dynamics of volatility while EVT provides a model for the behaviour of the extreme tail of the distribution. To carry out the analysis on tail risk of risk-controlled portfolios, we calculate annualised averages of several statistics. We provide annualised averages of volatility, constant scale tail risk (CVaR with constant volatility of 17% for absolute returns and constant tracking error of 3% for relative returns), and the total tail risk computed through the GARCH-based model (total CVaR) for the risk-controlled portfolios. The decomposition provides insight into what underlies the differences in total CVaR across portfolios constructed using different strategies and risk-controlled schemes: whether it is the average volatility (or TE) or whether it is the residual tail risk having explained away the clustering of volatility effect. First, we examine the effect of adding a country or sector neutrality constraint. The following weighting schemes are considered: Maximum Deconcentration, Maximum Decorrelation, Efficient Minimum Volatility, and Efficient Maximum Sharpe Ratio. We look at both absolute and relative returns where relative return is defined as the portfolio excess return over the corresponding cap-weighted market index return. As a next step, we consider the effect of TE control on tail risk. Because of the use of the core-satellite technique to achieve TE control, we would expect the risk profile of the TE-controlled index to increasingly resemble that of the cap-weighted index as the TE target is reduced. Finally, we compare diversification benefits of multi-strategy indices with respect to total CVaR. An EDHEC-Risk Institute Publication 7

8 Executive Summary Our main findings in the paper can be summarised as follows. Firstly, we found no evidence that controlling for country or sector risk increases the tail risk of absolute or relative returns of smart beta strategies. Furthermore, as expected, tracking error controls reduce the total tail risk of relative returns but this is mainly through the reduction of tracking error itself with no additional benefits. For example, in the case of the Developed World universe imposing a TE of 2%, the minimum volatility portfolio results in the average TE falling from 3.56% to 1.02% and the total CVaR falling from 10.79% to 3.02%. The constant TE CVaR, however, hardly changes from 9.08% to 8.89%, which is statistically insignificant. Finally, building a multi-strategy portfolio diversifies the total tail risk of relative returns; but again, the most significant factor is the diversification of the tracking error. For the Developed World universe for instance, the constant TE CVaR ranges from 8.71% to 9.71% for the five different weighting schemes while the constant TE CVaR for the multi-strategy equals 9.60% which indicates practically no diversification benefits. On the other hand, the average TE for the five different weighting schemes ranges from 1.92% to 3.56% while the average TE for the multi-strategy equals 2.15% which is close to the lowest TE of the multi-strategy constituents. The results show that the main source of tail risk diversification is indeed the tracking error rather than the residual tail risk. of the different smart beta strategies and different risk control schemes considered in the paper. Generally, our results show that adopting risk control schemes in portfolio optimisation does not deteriorate tail risk. From a practical perspective, managing volatility and tracking error is sufficient for managing total tail risk in the context 8 An EDHEC-Risk Institute Publication

9 1. Introduction An EDHEC-Risk Institute Publication 9

10 1. Introduction 1 - See Amenc et al. (2012a) and the references therein. Empirical studies have demonstrated that cap-weighted indices do not represent efficient portfolios. Three main reasons for this have been identified: (i) cap-weighting results in significant concentration; (ii) from a portfolio construction perspective, cap-weighting ignores correlations between stocks; and (iii) cap-weighted indices are not exposed to well-rewarded factors. In the recent years, the industry has tried to respond by developing the so-called smart beta framework that attempts to resolve the three drawbacks. Smart beta portfolios employ weighting schemes that deviate from cap-weighting addressing issues (i) and (ii) and they have been demonstrated to lead to superior risk-adjusted returns, addressing possibly to some extent issue (iii). 1 Traditional index providers usually employ a set of methodology choices for stock selection and weighting packaged together in a single index without allowing the investor the flexibility of making separate choices. Amenc and Goltz (2013) suggest the ERI Smart Beta 2.0 approach which represents a substantial improvement over the traditional methods. It separates the two main steps in the index construction process, allowing the investor to make an informed decision both about the factor tilt and the diversification method. The diversification method deals with the over-concentration issue and the factor tilt method leads to an exposure to better rewarded factors. Although deviating away from the capweighted index leads to significant risk-adjusted performance benefits, it also exposes investors to additional risks. Firstly, the alternative weighting scheme may lead to an over-weighting or under-weighting of certain sectors or countries relative to the corresponding cap-weighted index, which may result in temporary underperformance. It has been shown empirically that sector and country risks are not priced in and it would therefore make sense for investors to try to avoid them. For an empirical analysis, see for example Cappiello et al. (2008). Secondly, and more generally, improved relative performance necessarily comes at the cost of tracking error risk. This risk is of course related to relative country/sector risk and arises in recognising that capweighted indices, although representing inefficient portfolios, will continue to be used as a reference point. Therefore, from an investor perspective, tracking error risk needs to be managed. For additional discussions, see for example Rudd (1980), Roll (1992) and Chan et al. (1999). Finally, deviating from a cap-weighted index implies also departing from the objective of representing the market. Smart beta indices require setting a specific objective which often takes the form of a goal in an optimisation problem. Solving the problem requires provision of parameters that need to be estimated from data. This exposes the optimal solution to the noise in the observed stock returns which is also known as sample risk and it can be relatively bigger or smaller depending on how many parameters need to be estimated and of what type. From an investor perspective, however, it makes sense to try to diversify away this risk as much as possible by combining different smart beta strategies into one multistrategy index. The academic literature 10 An EDHEC-Risk Institute Publication

11 1. Introduction confirms this intuition in the face of parameter uncertainty, Kan and Zhou (2007) argue that an investor should hold a combination of the global minimum variance portfolio, the maximum Sharpe ratio portfolio, and a risk-free asset. Apart from the sample risk diversification, combining smart beta indices into a multistrategy index could be appealing because of other benefits including smoothing conditional performance, see Amenc et al. (2014a,c). The goal of this paper is to check empirically if controlling the exposure to some risks such as country, sector, tracking error, or sample risk does not increase the exposure to other types of risk, such as tail risk, that may remain unaccounted for by the index construction process. The analysis in this paper complements Loh and Stoyanov (2014b) where we study the tail risk of smart beta strategies without imposing any risk controls. For the purposes of this paper, we use the data for the riskcontrolled strategies available on the Scientific Beta platform at scientificbeta.com which provides a consistent index construction framework that can combine different risk control methods with popular weighting schemes subject to practical constraints guaranteeing investable indices. The methodology in this paper follows the one in Loh and Stoyanov (2014b). We measure tail risk in terms of conditional Value-at-Risk (CVaR) through a time series model based on a GARCH filter and Extreme Value Theory (EVT) as a probabilistic model for the tail. The model allows CVaR to be decomposed into a volatility component and a residual tail risk component. From a risk management perspective, it is important to segregate the two components because the dynamics of volatility contributes to the unconditional tail thickness phenomenon. Generally, the GARCH part is responsible for capturing the dynamics of volatility while EVT provides a model for the behaviour of the extreme tail of the distribution. To carry out the analysis on tail risk of risk-controlled portfolios, we calculate annualised averages of several statistics. We provide annualised averages of volatility, constant scale tail risk (CVaR with constant volatility of 17% for absolute returns and constant tracking error of 3% for relative returns), and the total tail risk computed through the GARCHbased model (total CVaR) for the riskcontrolled portfolios. The decomposition provides insight into what underlies the differences in total CVaR across portfolios constructed using different strategies and risk-controlled schemes: whether it is the average volatility (or TE) or whether it is the residual tail risk having explained away the clustering of volatility effect. First, we examine the effect of adding a country or sector neutrality constraint. The following weighting schemes are considered: Maximum Deconcentration, Maximum Decorrelation, Efficient Minimum Volatility, and Efficient Maximum Sharpe Ratio. We look at both absolute and relative returns where relative return is defined as the portfolio excess return over the corresponding capweighted market index return. We found that country and sector neutrality do not have a material effect on tail risk for both absolute and relative returns. An EDHEC-Risk Institute Publication 11

12 1. Introduction As a next step, we consider the effect of TE control on tail risk. Because of the use of the core-satellite technique to achieve TE control, we would expect the risk profile of the TE-controlled index to increasingly resemble that of the cap-weighted index as the TE target is reduced. Finally, we compare diversification benefits of multistrategy indices with respect to total CVaR. Our main findings in the paper are that there is no evidence that controlling for country or sector risk increases tail risk both in terms of absolute and relative returns. Furthermore, as expected, tracking error controls reduce the total tail risk of relative returns but this is mainly through the reduction of tracking error itself with no additional benefits. Finally, building a multi-strategy portfolio diversifies the total tail risk of relative returns; but again, the most significant factor is the diversification of the tracking error. Our results show that adopting risk control schemes in portfolio optimisation does not deteriorate tail risk. From a practical perspective, managing volatility and tracking error is sufficient for managing total tail risk in the context of the different smart beta strategies and different risk control schemes considered in the paper. The paper is organised in the following way. Section 2 briefly explains the different types of risk-controlled schemes used in the construction of the smart beta strategies. Section 3 discusses extreme value theory and its application for tail risk measurement. Section 4 briefly discusses the data and provides an analysis of the results and, finally, Section 5 concludes. 12 An EDHEC-Risk Institute Publication

13 2. Risk Control Schemes An EDHEC-Risk Institute Publication 13

14 2. Risk Control Schemes This section briefly discusses four types of risk control schemes which are quite common in the industry. They include country risk control, sector risk control, tracking error control, and diversifying strategy specific risk through multi-strategy indices. The risk-controlled strategies used in the study are essentially smart beta portfolios following a particular weighting scheme with additional constrains implementing the corresponding risk control. All strategies studied in the paper are implemented with quarterly rebalancing subject to a threshold constraint which aims at minimising turnover. Further details are available at scientificbeta.com Country Risk Country neutral investment strategies have grown in importance along with the increasingly international scope of investments and with the expanding use of alternative weighting schemes which can produce significant deviations from the reference index if country-neutrality is not imposed. The imposition of country neutrality is in effect region-based tracking error control, which aims at managing relative risk with respect to countries. Typically, country neutrality is achieved by imposing constraints which match the country weights of the strategy index to the reference index, while constituents within each country may be re-weighted. Upon re-balancing, weights are restored to their country-based targets. If one assumes that the cap-weighted reference index is an accurate reflection of the market, the alignment of country weights eliminates the risk of making implicit country bets; that is, a strategy can be employed while maintaining country-level economic representation. As implementing alternative weighting schemes may result in different levels of country allocation relative to a reference index, country neutral versions allow pursuit of the strategy while suppressing any deviations from the reference index's country exposure. Country risk has long been recognised as a prominent risk factor impacting equity returns (Erb et al., 1995). Country neutral weighting allows for a customised pursuit of a strategy while refraining from making any implicit country bets Sector Risk Sector neutral investment strategies have been popular among active managers who attempt to employ their stock picking skills within sectors. Sector neutrality is a risk control scheme that attempts to maintain a sector exposure which is neutral to that of the corresponding benchmark while pursuing a non-cap weighted strategy. For example, in recent years some index providers have begun to offer indices which pursue a growth or value strategy, while maintaining sector neutrality. Like country neutrality, the imposition of sector neutrality is in effect sector-based tracking error control, which aims to manage relative risk with respect to sectors. If one assumes that the cap-weighted reference index is an accurate reflection of the market, aligning sector weights of a strategy to the reference index eliminates the risk of making implicit sector bets; that is, a strategy can be employed while maintaining sector-level economic representation. 14 An EDHEC-Risk Institute Publication

15 2. Risk Control Schemes 2 - For additional details, see As implementing alternative weighting schemes may result in drastically different sector exposures from a reference index, sector neutral versions of the index allow the pursuit of the strategy while suppressing any relative sector tilts. Typically, constraints are imposed which match the sector weights of the strategy index to the reference index, while constituents within each sector may be re-weighted according to the strategy Tracking Error Relative risk controls are methods used to limit the deviation of a strategy index relative to its cap-weighted reference index. Relative risk controls are thus essentially strategies that attempt to respect an explicit tracking error constraint (e.g. tracking error limits of 3%, 4%, or 5%, etc.). Effective tracking error methods draw on hedging approaches, such as combining the strategy index (satellite) with a cap-weighted core and aligning the factor exposures within the satellite portfolio to be close to those of the reference index. A variety of methods to achieve such goals have been developed in recent years, ranging from the use of simple weight constraints on segments or stocks to the use of implicit factor models to impose factor exposure constraints on the optimised portfolio (Amenc et al., 2012b). Amenc et al. (2012b) introduce a method for relative risk control which recognises that only hedging that aligns the factor exposure of the performance-seeking (strategy) portfolio with that of the cap-weighted benchmark, can enable proper management of extreme relative risk. The hedging approach combines an optimised portfolio (strategy index) with a suitably-designed, time-varying, quantity of the cap-weighted reference portfolio so as to ensure that relative risk is kept within budgeted limits ("core-satellite approach"). Since the optimised portfolio is originally endowed with an ill-behaved tracking error process, (i.e. a tracking error that may ex-post deviate substantially from the average tracking error level), the approach also makes sure that ex-ante, the optimised portfolio risk exposures are sufficiently well-aligned with the cap-weighted reference index risk exposures, through the use of explicit tracking error constraints in the optimisation procedure as well as constraints on factor exposures relative to the cap-weighted reference index. The relative risk control methods used for the construction of Scientific Beta indices employ an explicit tracking error target which is set at 5%. To achieve this, the exposure of the strategy to implicit risk factors is aligned with the reference portfolio. This leads to a satellite portfolio with reliable target tracking error level at 5%. Tracking error is further reduced to 2% or 3% by using a core satellite approach Diversifying Strategy-Specific Risk Ever since cap-weighting has been proved to be mean-variance inefficient, several alternative weighting schemes have been proposed. These strategies differ from each other in the assumptions they make and the objectives they aim to achieve. In this paper, we use the following weighting schemes: Efficient Minimum Volatility, Efficient Maximum Sharpe Ratio, Maximum Deconcentration, Maximum Decorrelation, and Diversified Risk Weighted. 2 An EDHEC-Risk Institute Publication 15

16 2. Risk Control Schemes 3 - Additional details are available at scientificbeta.com. Their objectives are as follows: the Efficient Minimum Volatility portfolio has minimal volatility, the objective of the Maximum Sharpe Ratio is to maximise the Sharpe ratio of the portfolio, Maximum Deconcentration is an implementation of the equally weighted scheme, Maximum Decorrelation has the objective to build a portfolio of the least correlated stocks, and finally the constituents of the Diversified Risk Weighted portfolio have equal risk contributions under the assumption of equal correlations among stocks. Depending on which parameters need to be estimated to construct the corresponding portfolio, different features of the input sample would be critical for the weighting scheme. For example, Efficient Minimum Volatility relies on the covariance matrix only while the Efficient Maximum Sharpe Ratio relies on both the covariance matrix and the vector of expected returns. One aspect of strategy-specific risk is exactly the sample risk to which a given strategy is exposed. The combination of these different strategies allows the diversification of the risks that are specific to each strategy by exploiting the imperfect correlation between the different strategies' parameter estimation errors and the differences in their underlying optimality assumptions. Kan and Zhou (2007) argue that in the presence of parameter uncertainty, an investor should hold a combination of the Minimum Variance and Maximum Sharpe Ratio strategies, along with a risk-free asset. The idea of diversifying across the two different strategies stems from the fact that the parameter estimation errors are not perfectly correlated and can hence be diversified away. Moreover, as the single strategy's performance will show different profiles of dependence on market conditions, a multi-strategy approach can help investors smooth the overall performance across market conditions. Scientific Beta's Diversified Multi-Strategy is a combination five different weighting strategies Maximum Deconcentration, Maximum Decorrelation, Efficient Minimum Volatility, Efficient Maximum Sharpe Ratio and the Diversified Risk Weighted strategy. All the portfolio construction steps are applied separately to each of those five strategies, which results in five sets of weights. Only then, the Diversified Multi- Strategy is made of the equal-weighted combination of the resulting sets. In the Diversified Multi-Strategy weighting scheme, five Scientific Beta strategies are combined in order to diversify away individual strategies' specific risks and to mix strategies with different sensitivities to market conditions. 3 Badaoui and Lodh (2014) demonstrate the potential for diversification of the Scientific Beta Diversified Multi-Strategy index by showing that it presents a good trade-off between return and relative risk as the strategy has a return that corresponds to the average return of its five components and a tracking error level that is lower than the average tracking error of its constituents. In this paper, our focus is on tail risk. 16 An EDHEC-Risk Institute Publication

17 3. A Conditional EVT Model An EDHEC-Risk Institute Publication 17

18 3. A Conditional EVT Model In finance, EVT has been traditionally applied to estimate probabilities of extreme losses or loss thresholds such that losses beyond it occur with a predefined small probability, which are also known as high quantiles of the portfolio loss distribution. In fact, EVT provides a model for the extreme tail of the distribution which turns out to have a relatively simple structure described through the corresponding limit distributions such as the Generalised Extreme Value (GEV) distribution or the Generalised Pareto Distribution (GPD) The Peak-over-Threshold Method The approach in this paper is based on the peak-over-threshold method (POT), see Loh and Stoyanov (2014a) and the references therein. Suppose that we have selected a high loss threshold u and we are interested in the conditional probability distribution of the excess losses beyond u. We denote this distribution by F u (x) which is expressed through the unconditional distribution in the following way, (3.1) where x > 0. Because we are interested in the extreme losses, we need to gain insight into the probability that the excesses beyond u, X u, can exceed a certain loss level. Thus, (3.1) is re-stated in terms of the tail (3.2) There is a celebrated limit result in EVT which states that as u increases towards the right endpoint of the support of the loss distribution denoted by x F, the conditional tail converges to the tail of the GPD which is defined by, (3.3) where 1 + ξx > 0 and β > 0 is a scale parameter. The limit results is (Embrechts et al., 1997, Chapter 3) (3.4) where β(u) is a scaling depending on the selected threshold u. The limit result in (3.4) can be used to construct an approximation for the tail of the losses exceeding a high threshold u. If we denote by y = u + x and express x in terms of y in (3.2), we obtain (3.5) after substituting the limit law for. For a fixed threshold u, note that is a constant and the tail (y) for y > u is determined entirely by the GPD tail. It is possible to define sets of portfolio loss distributions also known as maximum domains of attraction (MDA) such that the limit relation in (3.4) leads to a GPD with one and the same tail parameter ξ. Since EVT is used to study rare events, characteristic of the tail behaviour of the portfolio loss distribution turns out to be the important feature; other features of F are not relevant. We distinguish between three different classes of portfolio loss distributions. 18 An EDHEC-Risk Institute Publication

19 3. A Conditional EVT Model 4 - An approach based on adaptive calibration of the threshold is adopted by some authors. Gonzalo and Olmo (2004) describe a method based on minimising the distance between the empirical (x) and the tail of the GPD with parameters estimated through the maximum likelihood method. The suggested distance is the Kolmogorov-Smirnov statistic. The Fréchet MDA, ξ > 0 A loss distribution belongs to this domain of attraction if and only if X has a tail decay dominated by a power function in the following sense, The link between α and ξ is ξ = 1/α. It is possible to demonstrate that this MDA consists of fat-tailed distributions F that have unbounded moments of order higher than α, i.e. E X k < if k < α. For applications in finance, it is safe to assume that volatility is finite which implies α > 2 and ξ < 1/2, respectively. For further detail, see (Embrechts et al., 1997, Section 3.3.1). The Gumbel MDA, ξ = 0 This MDA is much more diverse. A portfolio loss distribution belongs to the MDA of the Gumbel law if and only if in which β(u) is a scaling function and can be chosen to be equal to the average excess loss provided that the loss exceeds the threshold x, (3.6) This choice of β(u) is also known as the mean excess function. This MDA is characterised in terms of excess losses that exhibit an asymptotic exponential decay and consists of distributions with a diverse tail behaviour: from moderately heavytailed such as the log-normal to light-tailed distributions such as the Gaussian or even distributions with bounded support having an exponential behaviour near the upper end of the support x F. For further detail, see (Embrechts et al., 1997, Section 3.3.3). The Weibull MDA, ξ < 0 This MDA consists entirely of distributions with bounded support and is, therefore, not interesting for modelling the behaviour of risk drivers. Distributions that belong to this MDA include for example the uniform and the beta distribution. For further detail, see (Embrechts et al., 1997, Section 3.3.2). Finally, we should note that one distribution can be in only one MDA. There are examples of distributions that are not in any of the three MDAs but they are, however, rather artificial. To apply (3.5) in practice, we need to choose a high threshold u and also to estimate the probability F(u). In addition, we also need estimates of ξ and β(u). Regarding the choice of u, different strategies have been adopted in the academic literature. One general recommendation is to set it so that a given percentage of the sample are excesses. Chavez-Demoulin and Embrechts (2004) report that a 10% threshold provides a good trade-off between the bias and variability of the estimator of the important shape parameter ξ when the sample size is of about 1,000 observations. A similar guideline is provided by McNeil and Frey (2000). 4 If the threshold is allowed to vary, then the probability can be estimated through the empirical c.d.f. as suggested for example in McNeil and Frey (2000). For instance, suppose that X 1, X 2,...,X n is a sample of i.i.d. portfolio losses. If u is chosen such that exactly m observations are excesses, then the approximation in (3.5) becomes, (3.7) where s = 1 m/n and X s,n is the s-th observation in the sample sorted in An EDHEC-Risk Institute Publication 19

20 3. A Conditional EVT Model 5 - See the related comments in Loh and Stoyanov (2014a). 6 - The GARCH(1,1) model turns out to be quite robust in cases of model mis-specification, see the related comments and additional references in Loh and Stoyanov (2014a). increasing order and and are estimates of ξ and β, respectively. Regarding estimation, a variety of estimators can be employed to estimate ξ and β. We use the maximum likelihood estimator (MLE) which is rationalised by the uniform convergence in (3.4). For additional details, see Loh and Stoyanov (2014a) and the references therein A GARCH-EVT Model for Tail Risk Estimation Instead of applying the POT method to the time series directly, we prefer to build a model for the time-varying characteristics and apply EVT to the residuals of the model having explained away, at least partly, the temporal structure of the time series. 5 In line with McNeil and Frey (2000) we estimate a GARCH model to explain away the time structure of volatility. To make things simple, we fit a GARCH(1,1) model to the portfolio return time series as a general GARCH filter. 6 Denote the time series of portfolio losses by X t. The GARCH(1,1) model is given by: (3.8) where, the innovations Z t are i.i.d. random variables with zero mean, unit variance and marginal distribution function F Z (x) and K, a, and b are the positive parameters with a+b < 1. The model in (3.8) is fitted to the data and then the standardised residual is derived. If we assume that the data is generated by the model in (3.8), then the standardised residual is a sample from the distribution F Z. EVT is applied by fitting the GPD to the residual using an approximate MLE. Apart from the probabilistic model, the other key component of a risk model is the measure of risk. We use two measures of risk: VaR and CVaR at the tail probability of 1%. In this section, we provide definitions and explicitly state the risk forecasts built through the probabilistic model. The discussion below assumes that the random variable X describes portfolio losses and VaR and CVaR are defined for the right tail of the loss distribution which translates into the left tail of the portfolio return distribution. The same quantities for the right tail of the return distribution (left tail of the loss distribution) are obtained from the definitions below by considering X instead of X; that is, the downside of a short position is the upside of the corresponding long position. The risk functionals are, however, multiplied by 1 to preserve the interpretation that negative risk means a potential for profit. Value-at-Risk The VaR of a random variable X describing portfolio losses at a tail probability p, VaR p (X), is implicitly defined as a loss threshold such that over a given time horizon losses higher than it occur with a probability p. By construction, VaR is the negative of the p-th quantile of the portfolio return distribution or the (1 p)-th quantile of the portfolio loss distribution. In the industry, VaR is often defined in terms of a confidence level but we prefer to reserve the term confidence level for the context of statistical testing which we need in Section An EDHEC-Risk Institute Publication

21 3. A Conditional EVT Model 7 - The correction term is obtained from the GPD and could make sense for very small values of p as well; values that may extend beyond the available observations in the sample. For example, suppose that the sample contains 100 portfolio losses, n = 100, and set p = which is the VaR corresponding to the 99.9% quantile. Then, X 0.9n,n is the 90th observation in the sorted sample and the empirical approximation to would be the largest observation in the sample. As a consequence, the correction term in (3.11) allows us to go beyond the available data points in the sample which emphasises a key advantage of EVT to the historical method. Thus, to map the terms properly, in the industry we talk about VaR at 95% and 99% confidence level which corresponds to VaR at 5% and 1% tail probability. Formally, if we suppose that X describes portfolio losses, then VaR at tail probability p is defined as (3.9) where F -1 denotes the inverse of the c.d.f. F X (x) = P(X x) which is also known as the quantile function of X. As explained earlier, we employ EVT to estimate high quantiles of the loss distribution. To this end, we adopt the approximation of the tail in (3.5). Solving for the value of y yielding a tail probability of p, we get (3.10) The estimator is derived from (3.7) in the same way. Suppose that X 1,n X 1,n... X n,n denote the order statistics, then following (3.7) we get (3.11) where s = 1 m/n and m denotes the number of observations that are considered excesses. The approximation in (3.11) is usually interpreted in the following way: the estimate of VaR equals the empirical quantile X s,n, which is such that p < m/n, plus a correction term obtained through the GPD. In the implementation, we set m/n = 0.1 and, thus, in terms of quantiles the 99% quantile ( ) equals the 90% quantile (X (0,9xn) ) plus the corresponding correction term. 7 As mentioned before, we assume that the portfolio loss distribution is dynamic and follows the GARCH(1,1) process. Under this assumption, the conditional VaR model is given by (3.12) where I t denotes the information available at time t, is given in (3.11) and is calculated from the sample of the standardised residuals. Conditional Value-at-Risk An important criticism of VaR in the academic literature is that it is uninformative about the extreme losses beyond it. Indeed, the only information provided is the probability of losing more than VaR which is equal to the tail probability level p but should any such loss occur, there is no information about its possible magnitude. Conditional value-at-risk is constructed to overcome this deficiency: CVaR at tail probability p, CVaR p (X), equals the average loss provided that the loss exceeds VaR p (X). CVaR is formally defined as an average of VaRs, (3.13) and if we assume that the portfolio loss distribution has a continuous c.d.f. then CVaR can be expressed as a conditional An EDHEC-Risk Institute Publication 21

22 3. A Conditional EVT Model expectation, (3.14) In the academic literature, CVaR is also known as average value-at-risk or expected shortfall. Average value-at-risk corresponds directly to the quantity in (3.13) while expected shortfall is the quantity in (3.14). Although (3.13) is more general and average value-at-risk seems to be a better name for the quantity, we stick to the widely accepted CVaR; see for example Pflug and Römisch (2007) for further discussion. Since CVaR integrates the entire tail, an asymptotic model for the tail in areas where no data points are available is even more important than for VaR. Assuming that ξ < 1, the expectation in (3.14) can be calculated explicitly through the GPD, where Plugging in from (3.11) and the corresponding estimates, we get. (3.15) For derivations and further detail, see (McNeil et al., 2005, Section 7.2.3). Under the assumption of a GARCH(1,1) process for the portfolio loss distribution, the counterpart of (3.12) for CVaR equals (3.16) where is given in (3.15) and is estimated from the sample of the standardised residuals Comparing the Tail Risk of Different Strategies Equations (3.12) and (3.16) indicate that regardless of the adopted risk measure, the conditional tail risk depends linearly on the conditional volatility. Therefore, the objective to compare the tail risk of different strategies makes sense only for a given point in time t and a given risk horizon (e.g. one time step ahead, t + 1). Then, the problem reduces to comparing the two forecasts produced by equation (3.12) or (3.16). If it turns out that the tail risk of strategy X is bigger than that of strategy Y, then this may be because (i) X is more volatile and they have equal residual tail risk; (ii) X has a higher residual tail risk and their volatilities are equal; or (iii) a combined effect which cannot be decomposed into a volatility and a residual tail effect. If the comparison involves a time period, then we face a bigger problem because we need to compare a sequence of risk forecasts. To resolve this issue, we adopt the following approach. Instead of looking at an out-of-sample comparison which would involve calibration and forecasting in a rolling time-window, we employ an in-sample approach. That is, we fit the GARCH model to the selected time period, extract the residual, and apply the described methodology to it. Tail risk is calculated 22 An EDHEC-Risk Institute Publication

23 3. A Conditional EVT Model 8 - The GLR measure is defined as the ratio of the portfolio variance to the weighted variance of its constituents. 9 - We use VDR instead of the standard GLR measure because the scale of CVaR is defined in terms of volatility rather than variance. through (3.12) or (3.16) but instead of using forecasted volatility, we use the estimated through the GARCH model. For CVaR, for example, the corresponding formula is To compare tail risk in-sample, we consider the following three aggregated quantities: (a) total CVaR over the period, which is the average of over the sample period; (b) the average estimated volatility, i.e. the average of ; and (c) constant volatility CVaR, which equals + σ 0 where σ 0 is one and the same number across all strategies. The rationale is as follows. Since for daily returns is very close to zero, a comparison of the constant volatility CVaR across strategies is essentially a comparison of the residual tail risks. The term σ 0 is supposed to scale the quantity into a meaningful risk number. Furthermore, combining (a) with (b) and (c) we are able to tell if the differences in total CVaR are primarily caused by differences in the average volatility or the residual tail risk. Loh and Stoyanov (2014b) test the validity of the in-sample approach for this data set using an out-of-sample back-testing for VaR and CVaR at 1% tail probability and conclude that the risk model is realistic. A more detailed back-testing only for cap-weighted indices supporting the same conclusion is available in Loh and Stoyanov (2014a) Measuring the Effects of Diversification The different weighting schemes considered in the paper are exposed to different sample risk depending on the input parameters they rely on. Combining the five strategies into one multi-strategy index is supposed to diversify away some of the sample risk because the estimators of the different parameters are not perfectly correlated. From a tail risk perspective, sample risk in this context can materialise either as higher volatility, a fatter tail, or as both together. Although it would be interesting to develop a measure of sample risk and check the diversification benefits of the multistrategy index, this is not simple because sample risk would need to be isolated. In this paper, our objective is to measure the diversification of the total tail risk only one component of which is sample risk. CVaR, like any other risk measure, is supposed to be able to identify diversification opportunities if they exist; that is, the CVaR of any portfolio must not exceed the weighted average of the stand-alone CVaRs of the constituents. Thus, to explore the effect of the Diversified Multi-Strategy Index on tail risk, we calculate a Tail-risk Diversification Ratio (TDR) similar to the GLR measure for variance: 8 (3.17) in which CVaR p (r) denotes the portfolio total CVaR at tail probability p, CVaR p (r i ) denotes the stand-alone total CVaR of the portfolio constituents (the sub-indices) and w i denote the weights of the constituents,. In this case, the portfolio is the Diversified Multi-Strategy Index, the constituents are the five sub-indices, and the weights are equal by construction. Since volatility is a component in tail risk, we compare TDR to the Volatility Diversification Ratio (VDR) 9 : An EDHEC-Risk Institute Publication 23

24 3. A Conditional EVT Model 10 - This is a consequence of the positive homogeneity property of CVaR, CV ar p (ax) = acv ar p (X), a > 0 where X denotes a random variable and a is a positive multiplier. The interpretation of this property is that if we double the portfolio positions, then the risk should also double. (3.18) where σ denotes volatility. By the sub-additivity property of CVaR and volatility, it follows that both ratios cannot exceed 1 and the lower they are, the bigger the diversification effect. Comparing the two to one another is useful because if the main driver of diversification is the volatility component of CVaR, then the two measures take similar values. By the sub-additivity property of CVaR, the following inequality holds for any portfolio The inequality can be rested equivalently in terms of the constant volatility CVaRs, 10 If the constant volatility CVaRs are roughly the same, then those terms cancel and we get the same inequality expressed in terms of volatility only, of the sum of the risks is dominated by the heaviest tail of the stand-alone risks. We reproduce the result for the Fréchet MDA only; results for the other MDAs and further examples are provided by Maddipatla et al. (2011). Theorem 3.1. For independent random variables r 1 and r 2, if both of them belong to the MDA of the Fréchet distribution with tail indices ξ 1 > ξ 2 > 0 respectively, then the sum r 1 + r 2 belongs to the same MDA with a tail index equal to ξ 1. As a consequence, combining different indices in a multi-index diversifies away volatility but may or may not change the constant volatility CVaR depending on the interplay between the parameters ξ and β. As far as tail thickness alone goes, in theory the tail index of the multi-index equals the tail index of the heaviest tail of the sub-indices even assuming that the sub-indices are independent which, although highly unrealistic, represents a condition in which diversification is supposed to work best. In other words, if TDR is similar to VDR then the main effect of diversification is in the reduction of volatility. In theory, apart from the volatility parameter GPD implies that there are two other parameters that determine tail risk: the tail index (ξ) and the dispersion of extremes (β). Results in probability theory suggest that diversification may influence the β parameter but has no influence on ξ. In the case of independent risks, Maddipatla et al. (2011) prove that the tail behaviour 24 An EDHEC-Risk Institute Publication