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An EDHEC-Risk Institute Publication The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk August 2014 Institute 2 Printed in France, August 2014. Copyright EDHEC 2014. The opinions expressed in this study are those of the authors and do not necessarily reflect those of EDHEC Business School. The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 Table of Contents Executive Summary.................................................................................................. 5 1. Introduction.............................................................................................................9 2. Risk Control Schemes...................................................................................... 13 3. A Conditional EVT Model..................................................................................17 4. Empirical Analysis..............................................................................................25 Conclusion................................................................................................................35 References................................................................................................................37 About EDHEC-Risk Institute.................................................................................41 EDHEC-Risk Institute Publications and Position Papers (2011-2014).........45 An EDHEC-Risk Institute Publication 3 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 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 Executive Summary An EDHEC-Risk Institute Publication 5 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 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. 6 An EDHEC-Risk Institute Publication 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. The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 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 http://www.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 follows the one developed by Loh and Stoyanov (2014b). We measure tail risk in terms of conditional Value-atRisk (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 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 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. 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 of the different smart beta strategies and different risk control schemes considered in the paper. 1. Introduction An EDHEC-Risk Institute Publication 9 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 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 10 An EDHEC-Risk Institute Publication 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 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 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 http://www. 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 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 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 2. Risk Control Schemes An EDHEC-Risk Institute Publication 13 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 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 http://www. scientificbeta.com. 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. 2.2. Sector Risk 2.1. 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 14 An EDHEC-Risk Institute Publication 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. The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 2. Risk Control Schemes 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. 2.3. Tracking Error 2 - For additional details, see http://www.scientificbeta.com 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. 2.4. 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 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 2. Risk Control Schemes 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. 3 - Additional details are available at http://www. scientificbeta.com. 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 16 An EDHEC-Risk Institute Publication 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 MultiStrategy 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. 3. A Conditional EVT Model An EDHEC-Risk Institute Publication 17 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 3. A Conditional EVT Model distribution denoted by xF, the conditional tail converges to the tail of the GPD which is defined by, 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). where 1 + ξx > 0 and β > 0 is a scale parameter. The limit results is (Embrechts et al., 1997, Chapter 3) 3.1. The Peak-over-Threshold Method 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 a constant and the tail (y) for y > u determined entirely by the GPD tail (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 18 An EDHEC-Risk Institute Publication (3.4) where β(u) is a scaling depending on the selected threshold u. 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 Fu(x) which is expressed through the unconditional distribution in the following way, (3.3) . is is . 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. The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 3. A Conditional EVT Model 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). 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 (x) and the tail empirical of the GPD with parameters estimated through the maximum likelihood method. The suggested distance is the Kolmogorov-Smirnov statistic. 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 xF. 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 X1, X2, . . . ,Xn 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 Xs,n is the s-th observation in the sample sorted in An EDHEC-Risk Institute Publication 19 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 3. A Conditional EVT Model increasing order and and of ξ and β, respectively. are estimates 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. 3.2. A GARCH-EVT Model for Tail Risk Estimation 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). 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 Xt. The GARCH(1,1) model is given by: (3.8) where , the innovations Zt are i.i.d. random variables with zero mean, unit variance and marginal distribution function FZ(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 20 An EDHEC-Risk Institute Publication residual is a sample from the distribution FZ. 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, VaRp(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 4. The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 3. A Conditional EVT Model 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. FX(x) = P(X ≤ x) which is also known as the quantile function of X. 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 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 = 0.001 which is the VaR corresponding to the 99.9% quantile. Then, X0.9n,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. 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 X1,n ≤ X1,n ≤ . . . ≤ Xn,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 Xs,n, which is such that p < m/n, plus a correction term obtained through the (3.12) where It 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, CVaRp(X), equals the average loss provided that the loss exceeds VaRp(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 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 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 22 An EDHEC-Risk Institute Publication (3.16) where is given in (3.15) and is estimated from the sample of the standardised residuals. 3.3. 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 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 3. A Conditional EVT Model 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 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. 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 + σ0 × where σ0 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 σ0 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). 3.4. 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 CVaRp(r) denotes the portfolio total CVaR at tail probability p, CVaRp(ri) denotes the stand-alone total CVaR of the portfolio constituents (the sub-indices) and wi 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 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 3. A Conditional EVT Model (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 10 - This is a consequence of the positive homogeneity property of CVaR, CV aRp(aX) = aCV aRp(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. 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, 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 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 r1 and r2, if both of them belong to the MDA of the Fréchet distribution with tail indices ξ1 > ξ2 > 0 respectively, then the sum r1 + r2 belongs to the same MDA with a tail index equal to ξ1. 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. 4. Empirical Analysis An EDHEC-Risk Institute Publication 25 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 4. Empirical Analysis In this section, we look at differences in tail risk across various portfolios which have adopted different risk-controlled schemes for both absolute and relative returns. The relative returns are computed as the difference between the returns of the index and the cap-weighted index corresponding to the same geographical region. 11 - Details on the index construction methodologies are available at http://www. scientificbeta.com. To compare the tail risk across different riskcontrolled schemes, we calculate annualised averages of several statistics. We provide annualised averages of volatility, constant scale tail risk (CVaR with a constant volatility of 17% for absolute returns and a constant tracking error of 3% for relative returns) and the total tail risk computed through the GARCH-based model (Total CVaR) for the diversified portfolios. This decomposition provides insight into what underlies the differences in total CVaR across portfolios with different risk-controlled schemes whether it is the average volatility (or tracking error) or whether it is the residual tail risk having explained away the clustering of volatility effect. CVaR is computed at 1% tail probability and is interpreted as the average loss provided that the loss exceeds VaR at 1% tail probability. In the sections below, we first describe the data and then we proceed to the empirical results. 4.1. Data To provide an analysis of downside risk for different types of portfolios, we use data from the Scientific Beta platform which provides indices constructed from stocks from different geographical regions using different strategies and stock-selection criteria. The daily sample covers the period 26 An EDHEC-Risk Institute Publication from June 2003 to December 2013. To carry out the tests, we express all data in returns. We consider both the absolute and relative returns, where relative return is defined as the portfolio excess return over the corresponding cap-weighted market index return. The geographies include the United States, Eurozone, the United Kingdom, Japan, Developed Asia-Pacific ex Japan and World Developed. The strategies are Efficient Minimum Volatility (MVol), Efficient Maximum Sharpe Ratio (MSR), Maximum Deconcentration (MDecon), Maximum Decorrelation (MDecor)and Diversified Risk Weighted (DRW). In addition to the these five strategies, we include Diversified MultiStrategy which combines the five in equal proportions.11 4.2. Impact of Country and Sector Risk Control on Tail Risk To examine the difference in tail risk for country risk-controlled, sector risk-controlled and non risk-controlled portfolios, we considered the four strategies for which these controls are available on the Scientific Beta platform. The strategies are Maximum Deconcentration, Maximum Decorrelation, Efficient Minimum Volatility and Efficient Maximum Sharpe Ratio. The universes studied are Eurozone, Asia-Pacific ex-Japan, and World Developed. The first two universes were selected because they cover different broad geographic regions and the last one was selected because it includes all countries. We looked at both absolute and relative returns where relative return is defined as the portfolio excess return over the corresponding cap-weighted market index return. The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 4. Empirical Analysis Table 1 shows the impact of country neutrality on tail risk of absolute and relative returns across different strategies and regions. The average volatility and constant volatility CVaR for both the country neutral and the non-neutral indices are very similar. Likewise, in the case of relative returns, the average TE and the constant TE CVaR for the country neutral and the non-neutral indices are also very similar. As a consequence, the differences in Total CVaR between country neutral indices and non country-neutral indices are insignificant. Generally, controlling country risk does not have a material effect on the tail risk for both absolute and relative returns. Table 1: Annualised risk and return statistics of country neutral indices and indices without the country neutrality constraint for different strategies and regions. The time period is from June 2003 to December 2013. Country Neutral NonNeutrality Country Neutral 0.1769 0.1782 0.5957 0.5953 0.6199 0.6242 0.0667 0.1564 0.1571 0.5992 0.6007 0.5513 0.5552 Efficient Minimum Volatility 0.0773 0.0776 0.1385 0.1413 0.6109 0.6116 0.4978 0.5083 Efficient Maximum Sharpe Ratio 0.0744 0.0672 0.1532 0.1528 0.6048 0.6058 0.5450 0.5446 Maximum Deconcentration 0.1495 0.1450 0.1926 0.1944 0.5897 0.5866 0.6682 0.6709 Maximum Decorrelation 0.1645 0.1548 0.1813 0.1855 0.5932 0.5905 0.6325 0.6442 Efficient Minimum Volatility 0.1736 0.1718 0.1551 0.1692 0.6029 0.5891 0.5502 0.5862 Efficient Maximum Sharpe Ratio 0.1622 0.1708 0.1723 0.1810 0.5963 0.5899 0.6043 0.6280 Maximum Deconcentration 0.1028 0.1027 0.1524 0.1524 0.5640 0.5649 0.5058 0.5065 Maximum Decorrelation 0.1039 0.1024 0.1424 0.1428 0.5677 0.5679 0.4754 0.4770 Country Neutral NonNeutrality 0.0685 0.0724 Efficient Minimum Volatility 0.1097 0.1094 0.1229 0.1241 0.5732 0.5734 0.4143 0.4188 Efficient Maximum Sharpe Ratio 0.1054 0.1053 0.1365 0.1371 0.5716 0.5715 0.4589 0.4610 Country Neutral 0.0127 0.0149 0.0424 0.0402 0.0903 0.0916 0.1276 0.1228 0.0186 0.0132 0.0560 0.0536 0.0914 0.0931 0.1704 0.1663 Efficient Minimum Volatility 0.0233 0.0236 0.0698 0.0662 0.0890 0.0923 0.2071 0.2037 Efficient Maximum Sharpe Ratio 0.0205 0.0137 0.0583 0.0567 0.0905 0.0932 0.1760 0.1762 Country Neutral NonNeutrality Total CVaR Maximum Deconcentration Country Neutral NonNeutrality CVaR constant TE at 3% Maximum Decorrelation Country Neutral NonNeutrality Average TE NonNeutrality SciBeta Eurozone SciBeta Developed AsiaPacific ex-Japan SciBeta Developed NonNeutrality Country Neutral 0.0662 Relative returns SciBeta Eurozone Total CVaR Maximum Decorrelation Realised Return SciBeta Developed AsiaPacific ex-Japan CVaR constant vol at 17% Maximum Deconcentration Absolute returns SciBeta Developed Average Volatility NonNeutrality Realised Return Maximum Deconcentration 0.0280 0.0240 0.0483 0.0434 0.0913 0.0945 0.1470 0.1368 Maximum Decorrelation 0.0414 0.0327 0.0583 0.0558 0.0878 0.0909 0.1706 0.1689 Efficient Minimum Volatility 0.0495 0.0480 0.0688 0.0588 0.0817 0.0861 0.1873 0.1688 Efficient Maximum Sharpe Ratio 0.0394 0.0470 0.0604 0.0530 0.0844 0.0869 0.1697 0.1534 Maximum Deconcentration 0.0171 0.0169 0.0204 0.0201 0.0871 0.0884 0.0591 0.0592 Maximum Decorrelation 0.0181 0.0167 0.0231 0.0228 0.0939 0.0935 0.0723 0.0711 Efficient Minimum Volatility 0.0234 0.0232 0.0356 0.0346 0.0908 0.0920 0.1079 0.1061 Efficient Maximum Sharpe Ratio 0.0195 0.0194 0.0243 0.0239 0.0971 0.0974 0.0787 0.0776 An EDHEC-Risk Institute Publication 27 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 4. Empirical Analysis Table 2 shows the impact of sector neutrality on tail risk of absolute and relative returns across different strategies and region. The conclusions are similar: the differences in Total CVaR between sector neutral indices and the non-neutral indices are insignificant. Our finding is that controlling sector risk does not have a material effect on tail risk for both absolute and relative returns. Table 2: Annualised risk and return statistics of sector neutral indices and indices without the sector neutrality constraint for different strategies and regions. The time period is from June 2003 to December 2013. 0.1739 0.1051 0.1049 0.6199 0.6080 0.1564 0.1599 0.1057 0.1053 0.5513 0.5615 Efficient Minimum Volatility 0.0773 0.0722 0.1385 0.1458 0.1078 0.1070 0.4978 0.5199 Efficient Maximum Sharpe Ratio 0.0744 0.0708 0.1532 0.1568 0.1067 0.1054 0.5450 0.5508 Maximum Deconcentration 0.1495 0.1464 0.1926 0.1940 0.1041 0.1039 0.6682 0.6717 Maximum Decorrelation 0.1645 0.1548 0.1813 0.1863 0.1047 0.1047 0.6325 0.6504 Sector Neutral NonNeutrality Sector Neutral 0.1769 0.0697 Sector Neutral NonNeutrality 0.0643 0.0724 Efficient Minimum Volatility 0.1736 0.1574 0.1551 0.1653 0.1064 0.1052 0.5502 0.5795 Efficient Maximum Sharpe Ratio 0.1622 0.1583 0.1723 0.1776 0.1052 0.1049 0.6043 0.6211 Maximum Deconcentration 0.1028 0.1026 0.1524 0.1512 0.0995 0.0995 0.5058 0.5018 Maximum Decorrelation 0.1039 0.1021 0.1424 0.1451 0.1002 0.1002 0.4754 0.4843 Efficient Minimum Volatility 0.1097 0.1028 0.1229 0.1298 0.1011 0.1001 0.4143 0.4332 Efficient Maximum Sharpe Ratio 0.1054 0.1033 0.1365 0.1398 0.1009 0.1005 0.4589 0.4685 NonNeutrality 0.0127 0.0109 0.0424 0.0411 0.0903 0.0905 0.1276 0.1238 0.0186 0.0161 0.0560 0.0545 0.0914 0.0903 0.1704 0.1639 Efficient Minimum Volatility 0.0233 0.0184 0.0698 0.0627 0.0890 0.0886 0.2071 0.1853 Efficient Maximum Sharpe Ratio 0.0205 0.0171 0.0583 0.0559 0.0905 0.0915 0.1760 0.1704 Sector Sector Neutral Total CVaR Maximum Decorrelation Sector Neutral NonNeutrality CVaR constant TE at 3% Maximum Deconcentration Sector Neutral NonNeutrality Average TE NonNeutrality SciBeta Eurozone SciBeta Developed AsiaPacific ex-Japan SciBeta Developed SciBeta Eurozone SciBeta Developed AsiaPacific ex-Japan NonNeutrality Sector Neutral 0.0662 Relative returns SciBeta Developed Total CVaR Maximum Decorrelation Realised Return An EDHEC-Risk Institute Publication CVaR constant vol at 17% Maximum Deconcentration Absolute returns 28 Average Volatility NonNeutrality Realised Return Maximum Deconcentration 0.0280 0.0252 0.0483 0.0463 0.0913 0.0905 0.1470 0.1397 Maximum Decorrelation 0.0414 0.0328 0.0583 0.0565 0.0878 0.0903 0.1706 0.1700 Efficient Minimum Volatility 0.0495 0.0351 0.0688 0.0590 0.0817 0.0826 0.1873 0.1623 Efficient Maximum Sharpe Ratio 0.0394 0.0359 0.0604 0.0562 0.0844 0.0843 0.1697 0.1579 Maximum Deconcentration 0.0171 0.0169 0.0204 0.0190 0.0871 0.0876 0.0591 0.0555 Maximum Decorrelation 0.0181 0.0165 0.0231 0.0222 0.0939 0.0879 0.0723 0.0651 Efficient Minimum Volatility 0.0234 0.0170 0.0356 0.0278 0.0908 0.0921 0.1079 0.0853 Efficient Maximum Sharpe Ratio 0.0195 0.0176 0.0243 0.0226 0.0971 0.0944 0.0787 0.0711 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 4. Empirical Analysis 4.3. Impact of Tracking Error Control on Tail Risk With the use of the core-satellite technique to achieve TE control, we would expect the risk profile of the TE-controlled index to resemble more and more that of the cap-weighted index as the TE target is reduced from 5% to 2%. To examine the impact of controlling tracking error on tail risk, we consider the relative returns of the four strategies across the six universes: the United States, Eurozone, the United Kingdom, Japan, Developed Asia-Pacific ex Japan, and World Developed. The TE targets are as follows: uncontrolled, 5%, 3%, and 2%. The results are provided in Table 3. As expected, the total CVaR decreases with the decrease of the TE target. However, there are no significant differences in the constant TE CVaR. For example, in the case of the Developed World universe, imposing a TE of 2% in 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. As a result, it follows that the decrease in total CVaR is driven by the decrease in the average TE. In other words, reducing the TE target leads to lower TE which in turn leads to a lower total CVaR. Controlling for TE risk does not impact the tail thickness of the relative returns. The case of absolute returns is very similar in that the Total CVaR of the TE-controlled strategy converges to the Total CVaR of the cap-weighted index which is driven by the convergence of the corresponding volatility. The same effects are present in the other single-market or broader regional indices; the conclusion is not specific to a given universe. Overall, controlling for TE risk does not impact the tail thickness of the absolute returns. These conclusions are supported by the analysis by Loh and Stoyanov (2014b) that empirically the weighting scheme does not change the tail thickness. The parameter which is most affected is the volatility. 4.4. The Impact of Strategy Specific Risk Diversification on Tail Risk To see the effect of combining the five strategies in multi-strategy index, we calculate the average volatility, the constant volatility CVaR, and the total CVaR for the five strategies and the multi-strategy. We also calculate the TDR and the VDR ratios for the multi-strategy portfolio defined in (3.17) and (3.18), respectively. Table 4 provides the averaged relative return risk statistics and the two ratios computed for the six geographical regions. Regardless of region, the TDR is below 1 which indicates that the total CVaR of the Diversified Multi-Strategy is always smaller than the average of the standalone Total CVaRs; i.e. there are diversification benefits for tail risk. All TDRs are, however, very similar to the corresponding VDRs which, together with the fact that the constant scale CVaRs of the Multi-Strategy and the constituents are very similar, indicates that the main source of the diversification is in fact the tracking error. Table 5 provides the same information for the case of absolute returns which turns out to be very similar — the main source of diversification is the volatility. An EDHEC-Risk Institute Publication 29 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 4. Empirical Analysis Table 3: Annualised risk and return statistics of indices with tracking error control for different strategies and regions. The tracking error target is provided in brackets and equals: no target, 5%, 3%, and 2%. The time period is from June 2003 to December 2013. 0.0780 MDecon (5% TE) 0.0163 0.0364 0.1011 0.1227 0.0919 0.0481 MDecon (3% TE) 0.0102 0.0224 0.1009 0 . 07 5 2 0.0108 0.0924 0.0332 MDecon (2% TE) 0.0071 0.0152 0.1013 0 . 0 51 3 0.0160 0.0309 0.0968 0.0996 MDecor 0.0171 0.0506 0.0894 0 . 1 510 0.0173 0.0280 0.0958 0.0894 MDecor (5% TE) 0.0094 0.0411 0.0928 0.1272 MDecor (3% TE) 0.0105 0.0175 0.0960 0.0559 MDecor (3% TE) 0.0065 0.0255 0.0959 0 . 0 81 7 MDecor (2% TE) 0.0070 0.0120 0.0962 0.0384 MDecor (2% TE) 0.0046 0.0175 0.0975 0.0569 MVol 0.0209 0.0388 0.0917 0.1184 MVol 0.0222 0.0738 0.0872 0 . 21 4 5 MVol (5% TE) 0.0127 0.0278 0.0917 0.0850 MVol (5% TE) 0.0138 0.0419 0.0877 0.1226 MVol (3% TE) 0.0081 0.0178 0.0936 0.0555 MVol (3% TE) 0.0093 0.0265 0.0901 0 . 07 9 5 MVol (2% TE) 0.0057 0.0123 0.0955 0.0391 MVol (2% TE) 0.0068 0.0184 0.0911 0.0560 MSR 0.0166 0.0290 0.0963 0.0930 MSR 0.0197 0.0539 0.0887 0.1595 MSR (5% TE) 0.0130 0.0254 0.0907 0.0768 MSR (5% TE) 0.0079 0.0404 0.0906 0 . 1 21 9 MSR (3% TE) 0.0075 0.0157 0.0896 0.0468 MSR (3% TE) 0.0049 0.0251 0.0939 0 . 07 8 6 MSR (2% TE) 0.0050 0.0108 0.0899 0.0324 MSR (2% TE) 0.0033 0.0172 0.0956 0.0549 MDecon 0.0127 0.0424 0.0903 0.1276 MDecon 0.0280 0.0483 0.0913 0 . 1 4 70 MDecon (5% TE) 0.0091 0.0361 0.0951 0.1145 MDecon (5% TE) 0.0253 0.0429 0.0906 0.1295 MDecon (3% TE) 0.0051 0.0223 0.0964 0.0717 MDecon (3% TE) 0.0159 0.0266 0.0895 0 . 07 9 4 MDecon (2% TE) 0.0029 0.0153 0.0973 0.0497 MDecon (2% TE) 0.0114 0.0188 0.0896 0.0560 MDecor 0.0186 0.0560 0.0914 0.1704 MDecor 0.0414 0.0583 0.0878 0 . 1 70 6 MDecor (5% TE) 0.0089 0.0408 0.0925 0.1258 MDecor (5% TE) 0.0262 0.0474 0.0877 0.1385 MDecor (3% TE) 0.0042 0.0255 0.0918 0.0779 MDecor (3% TE) 0.0170 0.0293 0.0885 0.0865 MDecor (2% TE) 0.0126 0.0208 0.0889 0 . 0 61 7 MVol 0.0495 0.0688 0.0817 0.1873 MVol (5% TE) 0.0155 0.0371 0.0800 0 . 0 9 91 MVol (3% TE) 0.0100 0.0240 0.0831 0.0664 0.0257 0.0088 0.0157 MDecon (2% TE) 0.0061 MDecor MDecor (5% TE) SciBeta Japan 0.0145 MDecon (3% TE) SciBeta Asia-Pacific ex Japan MDecon (5% TE) Total CVaR 0.0912 Average TE SciBeta United States SciBeta Eurozone An EDHEC-Risk Institute Publication CVaR Constant TE at 3% 0.0196 0.0409 0.0968 0 . 1 31 8 Average TE MDecon 0.0299 MDecon Strategy Realised Returns Total CVaR 0.0897 Realised Returns 0.0901 0.0170 Strategy 30 Relative Returns CVaR Constant TE at 3% Relative Returns MDecor (2% TE) 0.0021 0.0176 0.0909 0.0532 MVol 0.0233 0.0698 0.0890 0.2071 MVol (5% TE) 0.0090 0.0353 0.0868 0.1021 MVol (3% TE) 0.0057 0.0225 0.0879 0.0658 MVol (2% TE) 0.0034 0.0154 0.0895 0.0459 MVol (2% TE) 0.0075 0.0170 0.0871 0.0495 MSR 0.0205 0.0583 0.0905 0.1760 MSR 0.0394 0.0604 0.0844 0.1697 MSR (5% TE) -0.0024 0.0402 0.0981 0.1314 MSR (5% TE) 0.0301 0.0450 0.0903 0.1354 MSR (3% TE) -0.0031 0.0253 0.0978 0.0824 MSR (3% TE) 0.0190 0.0279 0.0914 0 . 0 8 51 MSR (2% TE) -0.0033 0.0174 0.0978 0.0568 MSR (2% TE) 0.0139 0.0199 0.0927 0 . 0 61 5 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 4. Empirical Analysis Table 3: Annualised risk and return statistics of indices with tracking error control for different strategies and regions. The tracking error target is provided in brackets and equals: no target, 5%, 3%, and 2%. The time period is from June 2003 to December 2013 (Continued). CVaR Constant TE at 3% MDecon 0.0171 0.0204 0.0871 0 . 0 5 91 0.1306 MDecon (5% TE) 0.0140 0.0172 0.0929 0.0534 MDecon (3% TE) 0.0127 0.0258 0.0924 0.0794 MDecon (3% TE) 0.0086 0.0106 0.0933 0.0330 MDecon (2% TE) 0.0087 0.0177 0.0927 0.0546 MDecon (2% TE) 0.0059 0.0074 0.0933 0 . 02 31 MDecor 0.0179 0.0483 0.0912 0.1469 MDecor 0.0181 0.0231 0.0939 0 . 07 2 3 MDecor (5% TE) 0.0205 0.0368 0.0933 0.1143 MDecor (5% TE) 0.0149 0.0193 0.0966 0.0622 MDecor (3% TE) 0.0127 0.0227 0.0931 0.0706 MDecor (3% TE) 0.0089 0.0120 0.0965 0.0387 MDecor (2% TE) 0.0085 0.0156 0.0932 0.0484 MDecor (2% TE) 0.0060 0.0084 0.0956 0.0268 MVol 0.0273 0.0621 0.0892 0.1848 MVol 0.0234 0.0356 0.0908 0 . 107 9 MVol (5% TE) 0.0141 0.0377 0.0923 0.1160 MVol (5% TE) 0.0122 0.0232 0.0890 0.0689 MVol (3% TE) 0.0085 0.0236 0.0942 0.0741 MVol (3% TE) 0.0077 0.0147 0.0898 0.0440 MVol (2% TE) 0.0057 0.0163 0.0971 0.0528 MVol (2% TE) 0.0054 0.0102 0.0889 0.0302 MSR 0.0275 0.0481 0.0907 0.1454 MSR 0.0195 0.0243 0.0971 0 . 07 8 7 MSR (5% TE) 0.0104 0.0340 0.0887 0.1006 MSR (5% TE) 0.0102 0.0178 0.0917 0.0545 MSR (3% TE) 0.0062 0.0213 0.0922 0.0656 MSR (3% TE) 0.0057 0.0111 0.0905 0.0333 MSR (2% TE) 0.0044 0.0148 0.0934 0.0461 MSR (2% TE) 0.0038 0.0077 0.0894 0 . 02 31 It is curious that the constant scale CVaR does not seem to change significantly in the Multi-Strategy index. This result is consistent with the finding reported by Loh and Stoyanov (2014b) that within a given universe, the constant scale CVaR is not sensitive to the weighting scheme. Thus, diversifying risk by building a multistrategy index results in a reduction in the total CVaR but the main reason is the diversification of either TE or the volatility of the portfolios depending on whether we consider absolute or relative returns. In view of the theoretical discussion in Section 3.4 and the empirical analysis by Loh and Stoyanov (2014a,b), the multistrategy index can be viewed as another portfolio composed of the same stocks as the underlying sub-indices. Different SciBeta Developed Total CVaR 0.1491 0.0928 Average TE 0.0945 0.0422 Strategy Realised Returns 0.0473 0.0194 Total CVaR CVaR Constant TE at 3% 0.0196 MDecon (5% TE) Realised Returns MDecon Strategy SciBeta United Kingdom Relative Returns Average TE Relative Returns weights are applied to the same stocks, which has an impact on the volatility of the portfolio but has no effect on the tail index. The tail thickness of the sub-indices should be similar to the tail thickness of the multi-strategy and also to that of the cap-weighted index. The dispersion of extremes on the other hand may be affected by the weighting scheme but in this case it seems that the constant volatility CVaRs of the sub-indices are quite similar to the constant volatility CVaR of the multi-strategy, implying that the effect is minimal. An EDHEC-Risk Institute Publication 31 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 4. Empirical Analysis Table 4: Annualised risk and return statistics of the relative returns of multi-strategy indices across different regions and the corresponding sub-indices. TDR and VDR stand for tail diversification ratio and volatility diversification ratio and are defined in (3.17) and (3.18), respectively. The time period is from June 2003 to December 2013. Relative Returns SciBeta Developed SciBeta Developed AsiaPacific ex-Japan SciBeta Japan SciBeta United Kingdom SciBeta Eurozone SciBeta United States Strategy 32 An EDHEC-Risk Institute Publication Realised Returns Average TE CVaR Constant TE at 3% Total CVaR Maximum Deconcentration 0.0170 0.0299 0.0901 0.0897 Maximum Decorrelation 0.0160 0.0309 0.0968 0.0996 Efficient Minimum Volatility 0.0209 0.0388 0.0917 0.1184 Efficient Maximum Sharpe Ratio 0.0166 0.0290 0.0963 0.0930 Diversified Risk Weighted 0.0177 0.0258 0.0881 0.0759 Diversified Multi-Strategy 0.0178 0.0262 0.0931 0.0813 Maximum Deconcentration 0.0127 0.0424 0.0903 0.1276 Maximum Decorrelation 0.0186 0.0560 0.0914 0.1704 Efficient Minimum Volatility 0.0233 0.0698 0.0890 0.2071 Efficient Maximum Sharpe Ratio 0.0205 0.0583 0.0905 0.1760 Diversified Risk Weighted 0.0165 0.0443 0.0910 0.1343 Diversified Multi-Strategy 0.0186 0.0518 0.0901 0.1555 Maximum Deconcentration 0.0196 0.0473 0.0945 0.1491 Maximum Decorrelation 0.0179 0.0483 0.0912 0.1469 Efficient Minimum Volatility 0.0273 0.0621 0.0892 0.1848 Efficient Maximum Sharpe Ratio 0.0275 0.0481 0.0907 0.1454 Diversified Risk Weighted 0.0200 0.0465 0.0942 0.1459 Diversified Multi-Strategy 0.0227 0.0463 0.0925 0.1427 Maximum Deconcentration 0.0196 0.0409 0.0968 0.1318 Maximum Decorrelation 0.0171 0.0506 0.0894 0.1510 Efficient Minimum Volatility 0.0222 0.0738 0.0872 0.2145 Efficient Maximum Sharpe Ratio 0.0197 0.0539 0.0887 0.1595 Diversified Risk Weighted 0.0216 0.0446 0.0920 0.1370 Diversified Multi-Strategy 0.0203 0.0504 0.0881 0.1480 Maximum Deconcentration 0.0280 0.0483 0.0913 0.1470 Maximum Decorrelation 0.0414 0.0583 0.0878 0.1706 Efficient Minimum Volatility 0.0495 0.0688 0.0817 0.1873 Efficient Maximum Sharpe Ratio 0.0394 0.0604 0.0844 0.1697 Diversified Risk Weighted 0.0311 0.0478 0.0860 0.1370 Diversified Multi-Strategy 0.0381 0.0536 0.0847 0.1513 Maximum Deconcentration 0.0171 0.0204 0.0871 0.0591 Maximum Decorrelation 0.0181 0.0231 0.0939 0.0723 Efficient Minimum Volatility 0.0234 0.0356 0.0908 0.1079 Efficient Maximum Sharpe Ratio 0.0195 0.0243 0.0971 0.0787 Diversified Risk Weighted 0.0185 0.0192 0.0895 0.0572 Diversified Multi-Strategy 0.0195 0.0215 0.0960 0.0688 TDR VDR 0.8531 0.8495 0.9535 0.9565 0.9239 0.9165 0.9322 0.9547 0.9320 0.9446 0.9164 0.8767 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 4. Empirical Analysis Table 5: Annualised risk and return statistics of the absolute returns of multi-strategy indices across different regions and the corresponding sub-indices. TDR and VDR stand for tail diversification ratio and volatility diversification ratio and are defined in (3.17) and (3.18), respectively. The time period is from June 2003 to December 2013. Absolute Returns SciBeta Developed SciBeta Developed AsiaPacific ex-Japan SciBeta Japan SciBeta United Kingdom SciBeta Eurozone SciBeta United States Strategy Realised Returns Average Volatility CVaR Constant Volatility at 17% Total CVaR Maximum Deconcentration 0.0987 0.1845 0.5669 0.6152 Maximum Decorrelation 0.0976 0.1748 0.5640 0.5798 Efficient Minimum Volatility 0.1029 0.1503 0.5697 0.5038 Efficient Maximum Sharpe Ratio 0.0983 0.1662 0.5675 0.5549 Diversified Risk Weighted 0.0995 0.1741 0.5694 0.5832 Diversified Multi-Strategy 0.0996 0.1696 0.5676 0.5662 Maximum Deconcentration 0.0662 0.1769 0.5957 0.6199 Maximum Decorrelation 0.0724 0.1564 0.5992 0.5513 Efficient Minimum Volatility 0.0773 0.1385 0.6109 0.4978 Efficient Maximum Sharpe Ratio 0.0744 0.1532 0.6048 0.5450 Diversified Risk Weighted 0.0701 0.1686 0.5972 0.5922 Diversified Multi-Strategy 0.0724 0.1583 0.6024 0.5609 Maximum Deconcentration 0.0979 0.1768 0.5528 0.5748 Maximum Decorrelation 0.0961 0.1668 0.5568 0.5462 Efficient Minimum Volatility 0.1062 0.1426 0.5606 0.4703 Efficient Maximum Sharpe Ratio 0.1065 0.1625 0.5611 0.5364 Diversified Risk Weighted 0.0984 0.1669 0.5569 0.5469 Diversified Multi-Strategy 0.1013 0.1622 0.5583 0.5326 Maximum Deconcentration 0.0620 0.1970 0.5971 0.6920 Maximum Decorrelation 0.0594 0.1809 0.6067 0.6456 Efficient Minimum Volatility 0.0648 0.1577 0.6215 0.5766 Efficient Maximum Sharpe Ratio 0.0621 0.1741 0.6112 0.6257 Diversified Risk Weighted 0.0641 0.1880 0.6040 0.6679 Diversified Multi-Strategy 0.0628 0.1789 0.6094 0.6413 Maximum Deconcentration 0.1495 0.1926 0.5897 0.6682 Maximum Decorrelation 0.1645 0.1813 0.5932 0.6325 Efficient Minimum Volatility 0.1736 0.1551 0.6029 0.5502 Efficient Maximum Sharpe Ratio 0.1622 0.1723 0.5963 0.6043 Diversified Risk Weighted 0.1530 0.1811 0.5933 0.6321 Diversified Multi-Strategy 0.1608 0.1758 0.5956 0.6158 Maximum Deconcentration 0.1028 0.1524 0.5640 0.5058 Maximum Decorrelation 0.1039 0.1424 0.5677 0.4754 Efficient Minimum Volatility 0.1097 0.1229 0.5732 0.4143 Efficient Maximum Sharpe Ratio 0.1054 0.1365 0.5716 0.4589 Diversified Risk Weighted 0.1044 0.1444 0.5660 0.4807 Diversified Multi-Strategy 0.1054 0.1394 0.5685 0.4661 TDR VDR 0.9979 0.9976 0.9995 0.9974 0.9956 0.9942 0.9995 0.9964 0.9973 0.9959 0.9979 0.9975 An EDHEC-Risk Institute Publication 33 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 4. Empirical Analysis 34 An EDHEC-Risk Institute Publication Conclusion An EDHEC-Risk Institute Publication 35 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 Conclusion To overcome the deficiencies of cap-weighted indices, smart beta strategies have been proposed. They employ weighting schemes that deviate from cap-weighting, deal with the problem of concentration and allow for a flexible index construction process in which the index can be tilted to better rewarded factors. Along with the better risk-adjusted performance, however, investors in smart beta strategies are exposed to additional risks. The weighting scheme may lead to a temporary over-weighting or underweighting of a given sector or country relative to the corresponding cap-weighted benchmark which may lead to periodic underperformance. Also, the better risk-adjusted performance necessarily comes at the cost of some tracking error to the respective cap-weighted benchmark. Both aspects recognise that cap-weighted indices, albeit inefficient portfolios, will continue to be a reference point and therefore those relative risks need to be managed. Finally, any departure from cap-weighting is a departure from the goal of representing the market and should therefore be based on some other objective, which often takes the form of a goal in an optimisation problem. The various parameter inputs required to solve the problem expose the weighting scheme to sample risk which differs from one strategy to another. It makes sense to consider combining various strategies into a multi-strategy index in order to diversify away sample risk, but also to enjoy better risk-adjusted returns that come from smoothing the conditional performance. 36 An EDHEC-Risk Institute Publication Country or sector risk can be avoided through standard techniques. Tracking error risk can be managed efficiently through the classical core-satellite method and sample risk can be diversified away through a multi-strategy index. From a practical perspective, however, it is important to verify whether by diversifying some aspects of risk we are not magnifying others, such as tail risk. 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. In summary, 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. References An EDHEC-Risk Institute Publication 37 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 References • Amenc, N. and F. Goltz. 2013. Smart beta 2.0. Journal of Index Investing 4(3): 15-23. • Amenc, N., F. Goltz and A. Lodh. 2012a. Choose your betas: Benchmarking Alternative Equity Strategies. Journal of Portfolio Management 39(1): 88-111. • Amenc, N., F. Goltz, A. Lodh and L. Martellini. 2012b. Diversifying the Diversifiers and Tracking the Tracking Error: Outperforming Cap-Weighted Indices with Limited Risk of Underperformance. Journal of Portfolio Management 38(3): 72-88. • Amenc, N., F. Goltz, A. Lodh and L. Martellini. 2014a. Scientific Beta Multi-Strategy Factor Indices: Combining Factor Tilts and Improved Diversification. ERI Scientific Beta Publication (May). • Amenc, N., F. Goltz, A. Lodh and L. Martellini. 2014b. Towards Smart Equity Factor Indices: Harvesting Risk Premia without Taking Unrewarded Risks. Forthcoming in Journal of Portfolio Management. • Amenc, N., F. Goltz and A. Thabault. 2014c. Scientific Beta Multi-Beta Multi-Strategy Indices: Implementing Multi-Factor Equity Portfolios with Smart Factor Indices. ERI Scientific Beta Publication (May). • Badaoui, S. and A. Lodh. 2014. Scientific Beta Diversified Multi-Strategy Index. ERI Scientific Beta Publication (March). • Cappiello, L., M. L. Duca and A. Maddaloni. 2008. Country and Industry Equity Risk Premia in the Euro Area - An Intertemporal Approach. European Central Bank Working Paper Series No. 913/ June 2008. • Chan, L., J. Karceski and J. Lakonishok. 1999. On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model. Review of Financial Studies 12: 937-974. • Chavez-Demoulin, V. and P. Embrechts. 2004. Smooth Extremal Models in Finance. Journal of Risk and Insurance 71(2): 183-199. • Embrechts, P., C. Klüppelberg and C. Mikosch. 1997. Modelling Extremal Events for Insurance and Finance. Springer-Verlag. • Erb, C., C. Harvey and T. Viskanta. 1995. Country Risk and Global Equity Selection. Journal of Portfolio Management 21(2): 74-83. • Gonzalo, J. and J. Olmo. 2004. Which Extreme Values are Really Extreme? Journal of Financial Econometrics 2(3): 349-369. • Kan, R. and G. Zhou. 2007. Optimal Portfolio Choice with Parameter Uncertainty. Journal of Financial and Quantitative Analysis 42(3): 621-656. • Loh, L. and S. Stoyanov. 2014a. Tail Risk of Equity Market Indices: An Extreme Value Theory Approach. EDHEC-Risk Institute Publication (February). • Loh, L. and S. Stoyanov. 2014b. Tail Risk of Smart Beta Portfolios: An Extreme Value Theory Approach. EDHEC-Risk Institute Publication (July). 38 An EDHEC-Risk Institute Publication The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 References • Maddipatla, S., R. Sreenivasan and V. Rasbagh. 2011. On Sums of Independent Random Variables whose Distributions belong to the Max Domain of Attraction of Max Stable Laws. Extremes 14: 267-283. • McNeil, A. and R. Frey. 2000. Estimation of Tail-Related Risk Measures for Heteroskedastic Financial Time Series: An Extreme Value Approach. Journal of Empirical Finance 7(3-4): 271-300. • McNeil, A., R. Frey and P. Embrechts. 2005. Quantitative Risk Management. Princeton. • Pflug, G. and W. Römisch. 2007. Modeling, measuring and managing risk. World Scientific. • Roll, R. 1992. A Mean / Variance Analysis of Tracking Error. Journal of Portfolio Management 18: 13-22. • Rudd, A. 1980. Optimal Selection of Passive Portfolios. Financial Management 9(1): 57-66. An EDHEC-Risk Institute Publication 39 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 References 40 An EDHEC-Risk Institute Publication About EDHEC-Risk Institute An EDHEC-Risk Institute Publication 41 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 About EDHEC-Risk Institute Founded in 1906, EDHEC is one of the foremost international business schools. Accredited by the three main international academic organisations, EQUIS, AACSB, and Association of MBAs, EDHEC has for a number of years been pursuing a strategy of international excellence that led it to set up EDHEC-Risk Institute in 2001. This institute now boasts a team of over 95 permanent professors, engineers and support staff, as well as 48 research associates from the financial industry and affiliate professors.. The Choice of Asset Allocation and Risk Management EDHEC-Risk structures all of its research work around asset allocation and risk management. This strategic choice is applied to all of the Institute's research programmes, whether they involve proposing new methods of strategic allocation, which integrate the alternative class; taking extreme risks into account in portfolio construction; studying the usefulness of derivatives in implementing asset-liability management approaches; or orienting the concept of dynamic “core-satellite” investment management in the framework of absolute return or target-date funds. Academic Excellence and Industry Relevance In an attempt to ensure that the research it carries out is truly applicable, EDHEC has implemented a dual validation system for the work of EDHEC-Risk. All research work must be part of a research programme, the relevance and goals of which have been validated from both an academic and a business viewpoint by the Institute's advisory board. This board is made up of internationally recognised researchers, the Institute's business partners, and representatives of major international institutional investors. Management of the research programmes respects a rigorous validation process, which guarantees the scientific quality and the operational usefulness of the programmes. 42 An EDHEC-Risk Institute Publication Six research programmes have been conducted by the centre to date: • Asset allocation and alternative diversification • Style and performance analysis • Indices and benchmarking • Operational risks and performance • Asset allocation and derivative instruments • ALM and asset management These programmes receive the support of a large number of financial companies. 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EDHEC-Risk Institute: Key Figures, 2011-2012 Nbr of permanent staff 90 Nbr of research associates 20 Nbr of affiliate professors 28 Overall budget €13,000,000 External financing €5,250,000 Nbr of conference delegates 1,860 Nbr of participants at research seminars 640 Nbr of participants at EDHEC-Risk Institute Executive Education seminars 182 An EDHEC-Risk Institute Publication 43 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 About EDHEC-Risk Institute The EDHEC-Risk Institute PhD in Finance The EDHEC-Risk Institute PhD in Finance is designed for professionals who aspire to higher intellectual levels and aim to redefine the investment banking and asset management industries. It is offered in two tracks: a residential track for high-potential graduate students, who hold part-time positions at EDHEC, and an executive track for practitioners who keep their full-time jobs. Drawing its faculty from the world’s best universities, such as Princeton, Wharton, Oxford, Chicago and CalTech, and enjoying the support of the research centre with the greatest impact on the financial industry, the EDHEC-Risk Institute PhD in Finance creates an extraordinary platform for professional development and industry innovation. Research for Business The Institute’s activities have also given rise to executive education and research service offshoots. EDHEC-Risk's executive education programmes help investment professionals to upgrade their skills with advanced risk and asset management training across traditional and alternative classes. In partnership with CFA Institute, it has developed advanced seminars based on its research which are available to CFA charterholders and have been taking place since 2008 in New York, Singapore and London. In 2012, EDHEC-Risk Institute signed two strategic partnership agreements with the Operations Research and Financial Engineering department of Princeton University to set up a joint research programme in the area of risk and investment management, and with Yale 44 An EDHEC-Risk Institute Publication School of Management to set up joint certified executive training courses in North America and Europe in the area of investment management. As part of its policy of transferring knowhow to the industry, EDHEC-Risk Institute has also set up ERI Scientific Beta. ERI Scientific Beta is an original initiative which aims to favour the adoption of the latest advances in smart beta design and implementation by the whole investment industry. Its academic origin provides the foundation for its strategy: offer, in the best economic conditions possible, the smart beta solutions that are most proven scientifically with full transparency in both the methods and the associated risks. EDHEC-Risk Institute Publications and Position Papers (2011-2014) An EDHEC-Risk Institute Publication 45 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 EDHEC-Risk Institute Publications (2011-2014) 2014 • Loh, L., and S. Stoyanov. Tail Risk of Smart Beta Portfolios: An Extreme Value Theory Approach (July). • Foulquier, P. M. Arouri and A. Le Maistre. P. A Proposal for an Interest Rate Dampener for Solvency II to Manage Pro-Cyclical Effects and Improve Asset-Liability Management (June). • Amenc, N., R. Deguest, F. Goltz, A. Lodh, L. Martellini and E.Schirbini. Risk Allocation, Factor Investing and Smart Beta: Reconciling Innovations in Equity Portfolio Construction (June). • Martellini, L., V. Milhau and A. Tarelli. Towards Conditional Risk Parity — Improving Risk Budgeting Techniques in Changing Economic Environments (April). • Amenc, N., and F. Ducoulombier. Index Transparency – A Survey of European Investors Perceptions, Needs and Expectations (March). • Ducoulombier, F., F. Goltz, V. Le Sourd, and A. Lodh. The EDHEC European ETF Survey 2013 (March). • Badaoui, S., Deguest, R., L. Martellini and V. Milhau. Dynamic Liability-Driven Investing Strategies: The Emergence of a New Investment Paradigm for Pension Funds? (February). • Deguest, R., and L. Martellini. Improved Risk Reporting with Factor-Based Diversification Measures (February). • Loh, L., and S. Stoyanov. Tail Risk of Equity Market Indices: An Extreme Value Theory Approach (February). 2013 • Lixia, L., and S. Stoyanov. Tail Risk of Asian Markets: An Extreme Value Theory Approach (August). • Goltz, F., L. Martellini, and S. Stoyanov. Analysing statistical robustness of crosssectional volatility. (August). • Lixia, L., L. Martellini, and S. Stoyanov. The local volatility factor for asian stock markets. (August). • Martellini, L., and V. Milhau. Analysing and decomposing the sources of added-value of corporate bonds within institutional investors’ portfolios (August). • Deguest, R., L. Martellini, and A. Meucci. Risk parity and beyond - From asset allocation to risk allocation decisions (June). • Blanc-Brude, F., Cocquemas, F., Georgieva, A. Investment Solutions for East Asia's Pension Savings - Financing lifecycle deficits today and tomorrow (May) • Blanc-Brude, F. and O.R.H. Ismail. Who is afraid of construction risk? (March) • Lixia, L., L. Martellini, and S. Stoyanov. The relevance of country- and sector-specific model-free volatility indicators (March). 46 An EDHEC-Risk Institute Publication The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 EDHEC-Risk Institute Publications (2011-2014) • Calamia, A., L. Deville, and F. Riva. Liquidity in european equity ETFs: What really matters? (March). • Deguest, R., L. Martellini, and V. Milhau. The benefits of sovereign, municipal and corporate inflation-linked bonds in long-term investment decisions (February). • Deguest, R., L. Martellini, and V. Milhau. Hedging versus insurance: Long-horizon investing with short-term constraints (February). • Amenc, N., F. Goltz, N. Gonzalez, N. Shah, E. Shirbini and N. Tessaromatis. The EDHEC european ETF survey 2012 (February). • Padmanaban, N., M. Mukai, L . Tang, and V. Le Sourd. Assessing the quality of asian stock market indices (February). • Goltz, F., V. Le Sourd, M. Mukai, and F. Rachidy. Reactions to “A review of corporate bond indices: Construction principles, return heterogeneity, and fluctuations in risk exposures” (January). • Joenväärä, J., and R. Kosowski. An analysis of the convergence between mainstream and alternative asset management (January). • Cocquemas, F. Towar¬ds better consideration of pension liabilities in european union countries (January). • Blanc-Brude, F. Towards efficient benchmarks for infrastructure equity investments (January). 2012 • Arias, L., P. Foulquier and A. Le Maistre. Les impacts de Solvabilité II sur la gestion obligataire (December). • Arias, L., P. Foulquier and A. Le Maistre. The Impact of Solvency II on Bond Management (December). • Amenc, N., and F. Ducoulombier. Proposals for better management of non-financial risks within the european fund management industry (December). • Cocquemas, F. Improving Risk Management in DC and Hybrid Pension Plans (November). • Amenc, N., F. Cocquemas, L. Martellini, and S. Sender. Response to the european commission white paper "An agenda for adequate, safe and sustainable pensions" (October). • La gestion indicielle dans l'immobilier et l'indice EDHEC IEIF Immobilier d'Entreprise France (September). • Real estate indexing and the EDHEC IEIF commercial property (France) index (September). • Goltz, F., S. Stoyanov. The risks of volatility ETNs: A recent incident and underlying issues (September). • Almeida, C., and R. Garcia. Robust assessment of hedge fund performance through nonparametric discounting (June). An EDHEC-Risk Institute Publication 47 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 EDHEC-Risk Institute Publications (2011-2014) • Amenc, N., F. Goltz, V. Milhau, and M. Mukai. Reactions to the EDHEC study “Optimal design of corporate market debt programmes in the presence of interest-rate and inflation risks” (May). • Goltz, F., L. Martellini, and S. Stoyanov. EDHEC-Risk equity volatility index: Methodology (May). • Amenc, N., F. Goltz, M. Masayoshi, P. Narasimhan and L. Tang. EDHEC-Risk Asian index survey 2011 (May). • Guobuzaite, R., and L. Martellini. The benefits of volatility derivatives in equity portfolio management (April). • Amenc, N., F. Goltz, L. Tang, and V. Vaidyanathan. EDHEC-Risk North American index survey 2011 (March). • Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, L. Martellini, and S. Sender. Introducing the EDHEC-Risk Solvency II Benchmarks – maximising the benefits of equity investments for insurance companies facing Solvency II constraints - Summary - (March). • Schoeffler, P. Optimal market estimates of French office property performance (March). • Le Sourd, V. Performance of socially responsible investment funds against an efficient SRI Index: The impact of benchmark choice when evaluating active managers – an update (March). • Martellini, L., V. Milhau, and A.Tarelli. Dynamic investment strategies for corporate pension funds in the presence of sponsor risk (March). • Goltz, F., and L. Tang. The EDHEC European ETF survey 2011 (March). • Sender, S. Shifting towards hybrid pension systems: A European perspective (March). • Blanc-Brude, F. Pension fund investment in social infrastructure (February). • Ducoulombier, F., Lixia, L., and S. Stoyanov. What asset-liability management strategy for sovereign wealth funds? (February). • Amenc, N., Cocquemas, F., and S. Sender. Shedding light on non-financial risks – a European survey (January). • Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, Martellini, L., and S. Sender. Ground Rules for the EDHEC-Risk Solvency II Benchmarks. (January). • Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, Martellini, L., and S. Sender. Introducing the EDHEC-Risk Solvency Benchmarks – Maximising the Benefits of Equity Investments for Insurance Companies facing Solvency II Constraints - Synthesis -. (January). • Amenc, N., F. Cocquemas, R. Deguest, P. Foulquier, Martellini, L., and S. Sender. Introducing the EDHEC-Risk Solvency Benchmarks – Maximising the Benefits of Equity Investments for Insurance Companies facing Solvency II Constraints (January). • Schoeffler.P. Les estimateurs de marché optimaux de la performance de l’immobilier de bureaux en France (January). 48 An EDHEC-Risk Institute Publication The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 EDHEC-Risk Institute Publications (2011-2014) 2011 • Amenc, N., F. Goltz, Martellini, L., and D. Sahoo. A long horizon perspective on the cross-sectional risk-return relationship in equity markets (December 2011). • Amenc, N., F. Goltz, and L. Tang. EDHEC-Risk European index survey 2011 (October). • Deguest,R., Martellini, L., and V. Milhau. Life-cycle investing in private wealth management (October). • Amenc, N., F. Goltz, Martellini, L., and L. Tang. Improved beta? A comparison of indexweighting schemes (September). • Le Sourd, V. Performance of socially responsible investment funds against an Efficient SRI Index: The Impact of Benchmark Choice when Evaluating Active Managers (September). • Charbit, E., Giraud J. R., F. Goltz, and L. Tang Capturing the market, value, or momentum premium with downside Risk Control: Dynamic Allocation strategies with exchange-traded funds (July). • Scherer, B. An integrated approach to sovereign wealth risk management (June). • Campani, C. H., and F. Goltz. A review of corporate bond indices: Construction principles, return heterogeneity, and fluctuations in risk exposures (June). • Martellini, L., and V. Milhau. Capital structure choices, pension fund allocation decisions, and the rational pricing of liability streams (June). • Amenc, N., F. Goltz, and S. Stoyanov. A post-crisis perspective on diversification for risk management (May). • Amenc, N., F. Goltz, Martellini, L., and L. Tang. Improved beta? A comparison of indexweighting schemes (April). • Amenc, N., F. Goltz, Martellini, L., and D. Sahoo. Is there a risk/return tradeoff across stocks? An answer from a long-horizon perspective (April). • Sender, S. The elephant in the room: Accounting and sponsor risks in corporate pension plans (March). • Martellini, L., and V. Milhau. Optimal design of corporate market debt programmes in the presence of interest-rate and inflation risks (February). An EDHEC-Risk Institute Publication 49 The Impact of Risk Controls and Strategy-Specific Risk Diversification on Extreme Risk — August 2014 EDHEC-Risk Institute Position Papers (2011-2014) 2012 • Till, H. Who sank the boat? (June). • Uppal, R. Financial Regulation (April). • Amenc, N., F. Ducoulombier, F. Goltz, and L. Tang. What are the risks of European ETFs? (January). 2011 • Amenc, N., and S. Sender. Response to ESMA consultation paper to implementing measures for the AIFMD (September). • Uppal, R. A Short note on the Tobin Tax: The costs and benefits of a tax on financial transactions (July). • Till, H. A review of the G20 meeting on agriculture: Addressing price volatility in the food markets (July). 50 An EDHEC-Risk Institute Publication For more information, please contact: Carolyn Essid on +33 493 187 824 or by e-mail to: [email protected] EDHEC-Risk Institute 393 promenade des Anglais BP 3116 - 06202 Nice Cedex 3 France Tel: +33 (0)4 93 18 78 24 EDHEC Risk Institute—Europe 10 Fleet Place, Ludgate London EC4M 7RB United Kingdom Tel: +44 207 871 6740 EDHEC Risk Institute—Asia 1 George Street #07-02 Singapore 049145 Tel: +65 6438 0030 EDHEC Risk Institute—North America One Boston Place, 201 Washington Street Suite 2608/2640 — Boston, MA 02108 United States of America Tel: +1 857 239 8891 EDHEC Risk Institute—France 16-18 rue du 4 septembre 75002 Paris France Tel: +33 (0)1 53 32 76 30 www.edhec-risk.com