Tail Risk of Smart Beta Portfolios: An Extreme Value ... · Tail Risk of Smart Beta Portfolios: An Extreme Value Theory Approach — July 2014 About the Authors Lixia Loh is a senior

Tail Risk of Smart Beta Portfolios: An Extreme Value

Theory ApproachJuly 2014

An EDHEC-Risk Institute Publication

Institute

2 Printed in France, July 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.

3An EDHEC-Risk Institute Publication

Tail Risk of Smart Beta Portfolios: An Extreme Value Theory Approach — July 2014

Table of Contents

Executive Summary ................................................................................................. 5

1. Introduction ............................................................................................................9

2. Smart Beta Indices .......................................................................................... 15

3. A Conditional EVT Model .................................................................................21

4. Empirical Analysis ............................................................................................31

Conclusion ...............................................................................................................43

Appendix ..................................................................................................................47

References ...............................................................................................................57

About EDHEC-Risk Institute ................................................................................61

EDHEC-Risk Institute Publications and Position Papers (2011-2014) ........65

4 An EDHEC-Risk Institute Publication


About the Authors

Lixia Loh is a senior research engineer at EDHEC-Risk Institute–Asia. Prior tojoining EDHEC Business School, she was a Research Fellow at the Centre forGlobal Finance at Bristol Business School (University of the West of England).Her research interests include empirical finance, financial markets risk, andmonetary economics. She has published in several academic journals, includingthe Asia-Pacific Development Journal and Macroeconomic Dynamics, and isthe author of a book, Sovereign Wealth Funds: States Buying the World (GlobalProfessional Publishing, 2010). She holds an M.Sc. in international economics,banking and finance from Cardiff University and a Ph.D. in finance from theUniversity 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.


Executive Summary



Cap-weighted indices, although widely used as passive investment vehicles, have two important drawbacks with far-reaching consequences for investors — they represent concentrated portfolios and they are also exposed to risk factors that are not well rewarded. Both drawbacks indicate significant inefficiencies for long-term investors.

Index providers offer factor indices that aim at tilting the portfolio towards better rewarded factors. Although clearly an improvement over cap-weighting, the industry index solutions are often based on ad-hoc stock-selection and weight allocation criteria prone to data-mining risks. Because of the dangers of data-mining, investors are advised to stick to simple factor definitions rather than rely on proprietary and complex factors (see Gelderen and Huij (2013)).

Empirical research1 has demonstrated that smart beta indices offer improved performance and also sometimes lower volatility than the cap-weighted benchmarks. It is, thus, of practical and also of theoretical interest to check if smart beta indices exhibit higher extreme risk or similar extreme risk as that of the cap-weighted indices. The importance of this question stems from the fact that the superior risk-adjusted performance of smart beta indices is usually demonstrated by comparing their Sharpe ratios to that of the corresponding cap-weighted index. If,however, it turns out that smart beta returns have a substantially heavier left tail unaccounted for by volatility, then Sharpe ratios may be misleading when comparing risk-adjusted performance because a dimension of risk would be lost in

the comparison. Under this hypothesis, theimproved performance may be at the cost of an increase in tail thickness.

To study the tail risk systematically across different weighting schemes and stock selection criteria, we need a solid indexing methodology that can produce diversified factor-tilted indices consistently across different geographical universes. The Smart Beta 2.0. methodology — as put forward by EDHEC Risk Institute and applied for the production of the ERI Scientific Beta indices — allows the challenges affecting the smart beta indices provided by traditional industry vendors to be addressed.2 It separates two main steps in the index construction process and offers investors the chance to make an informed decision about the factor tilt and the diversification method. The diversification method deals with the over-concentration issue and the factor tilt method leads to exposure to better rewarded factors. Thus, the two main components in the construction process of factor indices are: (i) achieving a factor tilt through stock selection and (ii) efficiently extracting the risk premia through improved diversification, via the application of a smart weighting scheme. The two components are distinct; investors can explicitly choose which factor to tilt towards, while the diversification method reduces the impact of specific or unrewarded risks. The stock-selection criteria considered are size, liquidity, momentum, volatility, value, and dividend yield and the weighting schemes are Efficient Minimum Volatility, Efficient Maximum Sharpe Ratio, Maximum Deconcentration, Maximum Decorrelation and Diversified Risk Weighted. We examine the tail risk of these strategies with and without a factor tilt for the following

Executive Summary

1 - See Amenc et al. (2012) and the references therein.2 - See Amenc and Goltz (2013) for further details. An implementation of the methodology with complete and transparent documentation is available at http://www.scientificbeta.com.



Scientific Beta universes: USA (500 stocks), Eurozone (300 stocks), UK (100 stocks), Japan (500 stocks), Developed Asia-Pacific ex-Japan (400 stocks), and World Developed (2,000 stocks). The data used cover the full sample period from June 2003 to December 2013 and also two sub-sample periods, the pre-crisis period from June 2003 to June 2007 and the turbulent period from July 2007 to December 2013.

To compare extreme risk across different smart beta indices, we use a statistical methodology based on extreme value theory (EVT) and conditional value-at-risk (CVaR) at 1% tail probability as a downside risk measure. EVT has been used for a long time in areas other than finance to study the probabilities of extreme events, and in the area of risk measurement it has been used to describe the probabilistic behaviour of tail losses. On the other hand, both themore common value-at-risk (VaR) and CVaR are risk measures used to estimate the tail risk, or downside risk, of portfolio losses. They are designed to exhibit a degree of sensitivity to large portfolio losses; in practice, VaR provides a loss threshold exceeded with some small predefined probability such as 1% or 5%, while CVaR measures the average loss higher than VaR and is, therefore, more informative about extreme losses. In our risk model, we choose to work with CVaR because of its higher sensitivity to the extreme tail.

The comparison across different smart beta strategies is performed by decomposing their tail risk into a volatility component and a residual component through a two-step process. First, the clustering of volatility is explained away by applying the standard econometric framework of the Generalised

Autoregressive Conditional Heteroskedastic (GARCH) model and, second, the remaining tail risk is estimated from the residual process using EVT. From a risk managementperspective, it is important to segregate the two components because the dynamics ofvolatility 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.

We carry out the comparison by, first, looking at the differences in tail risk of absolute and relative returns by varying the weighting scheme using all stocks in the corresponding universe.3 Our main finding is that the CVaR across strategies is primarily driven by the average volatility or the average tracking error for the case of absolute and relative returns, respectively. The results show that adopting a different weighting scheme allows an investor to achieve superior performance compared to that of the corresponding cap-weighted index without any deterioration in the behaviour of the left tail of the smart beta return distribution. The additional performance does not come at the cost of an increase in tail thickness. As a consequence, from a long-term investor perspective, focusing on volatility or tracking error management on a strategy level appears to be of first-order importance for CVaR management. Across geographies, all strategies in Asia tend to have relatively higher total absolute returnCVaR than those in Europe and the US, a finding which extends earlier empirical results for cap-weighted indices (see Loh and Stoyanov (2013)). Also, in a broader context, our results indicate comparing risk-adjusted performance of smart

Executive Summary

3 - Relative returns are defined as the difference between the returns of the strategy and the returns of the corresponding cap-weighted benchmark.



betas through the Sharpe ratio (or the information ratio respectively) would not mislead investors.

Second, we look at the differences in tail risk of different stock-selection criteria using one and the same weighting scheme — the Maximum Deconcentration (or Equally-Weighted) scheme — in order to avoid introducing bias among stocks. In contrast to the first set of examples, our results indicate that the stock-selection criteria can make a statistically significant difference to the residual tail risk of relative returns. The impact varies across geographies, the most affected universe being Asia-Pacific ex Japan. The impact also varies across different market conditions and it is difficult to isolate the single stock-selection criterion with the biggest impact. The results show that an investor can use a factor-tilted portfolio to manage extreme risk exposure during different market conditions. With a factor-tilted portfolio, the investor can achieve superior performance on the investment and manage the tail risk exposure. For most of these criteria, the differences in the residual CVaR are amplified further by the average tracking error.

As far as absolute returns are concerned, we find no evidence of statistically significantdifferences in the tail risk. Investing in a non-cap-weighted portfolio would result in higher return and possibly lower volatility without changing the tail risk. This is to say investors can have a higher Sharpe ratio than the cap-weighted portfolio while maintaining a tail risk similar to that of the cap-weighted portfolio. We attempt to explain the lack of difference in the residual tail risk in the case of absolute returns through a CAPM-type one-factor model

for the factor-tilted portfolios and also the residuals from the GARCH model. The strong significance of the factor models and the near linear behaviour of the extreme losses suggests that a possible explanation is the relatively limited impact of the regression residual on the response variable. This confirms the previous conclusion that managing volatility or the tracking error is of first-order importance.

Executive Summary


1. Introduction



There are two important drawbacks of cap-weighted indices with far-reaching consequences for investors — they are concentrated portfolios and they are exposed to risk factors that are not well rewarded. In an effort to try to resolve the problems of cap-weighted indices, industry vendors have started offering smart beta indices using a framework known as smart factor investing. From an index construction viewpoint, industry providers have adopted three main methods which are, essentially, combinations of different stock selection methods and weighting schemes — (i)stock selection based on some stock characteristics and cap-weighting (i.e. different universe but the weighting scheme of the cap-weighted index), (ii) universe as in the cap-weighted index but weighting based on a stock characteristic (i.e. same universe, different weighting scheme), and (iii) both stock selection and weighting done according to stock characteristics (different universe, different weighting scheme).

The approach in (i) attempts to tilt the portfolio towards a rewarded factor but maintaining cap-weighting does not resolve the general concentration problem. The approach in (ii) relies entirely on the weighting scheme to achieve the desired tilt (e.g. weight by volatility to achieve a low volatility exposure) without any stock selection. The stock characteristics employed for stock selection or weighting are often based on accounting information (e.g. sales, cash flow, book value, dividend), but they can also be based on stock price information included in a risk metric (e.g. volatility, market beta).

Although an improvement over the traditional cap-weighted approach, big disadvantages of these three standard methods include unclear justification of stock selection and weight allocation criteria which are possibly based on ad-hoc decisions made in the presence of data mining risks. Also, there is no specification of a clear investment objective. Further to that, from a portfolio construction perspective none of these approaches include information related to joint behaviour of stock prices which is critical for prudent portfolio construction. As faras stock selection goes, because of the dangers of data-mining, investors are generally advised to stick to simple factor definitions rather than rely on proprietary and complex factors (see Gelderen and Huij (2013)).

Amenc and Goltz (2013) suggest the ERI Smart Beta 2.0 approach4, which is a substantial improvement over the traditional methods. It separates the two main steps in the index construction process offering investors the chance 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. Thus, the two main components in the construction process of factor indices are: (i) achieving a factor tilt through stock selection and (ii) efficiently extracting the risk premia through improved diversification, via the application of a smart weighting scheme. The two components are distinct; investors can explicitly choose which factor to tilt towards, while the diversification method reduces the impact of specific or unrewarded risks.

1. Introduction

4 - An implementation of the methodology with complete and transparent documentation is available at http://www.scientificbeta.com.



In this research paper, we address the following important question for smart beta investing: Is smart beta, which produces better performance and sometimes lower volatility, nonetheless not more exposed to extreme risk? The importance of this question stems from the fact that the superior risk-adjusted performance of smart beta indices is usually demonstrated by comparing their Sharpe ratios to that of the corresponding cap-weighted index. If, however, it turns out that smart beta returns have a substantially heavier left tail unaccounted for by volatility, then Sharpe ratios may be misleading when comparing risk-adjusted performance because a dimension of risk would be lost in the comparison. To find an answer to this question, we follow the method outlined in Loh and Stoyanov (2014) adapted to an in-sample analysis to compare the tail risk of different smart beta strategies. The methodology relies on a choice of risk measure — Value-at-risk (VaR) and Conditional Value-at-risk (CVaR) — and a model for the tail behaviour based on Extreme Value Theory (EVT).

VaR and CVaR are risk measures widely used to estimate the tail risk, or downside risk, of portfolio losses. These measures are designed to exhibit a degree of sensitivity to large portfolio losses whose frequency of occurrence is described by what is known as the tail of the distribution. In practice, VaR provides a loss threshold exceeded with some small predefined probability such as 1% or 5%, while CVaR measures the average loss higher than VaR and is, therefore, more informative about extreme losses.

Recent studies on predictive performance of various VaR methods have found EVT-based method to be particularly accurate (Danielsson and Vries, 1997; J.Pownall and Koedij, 1999; McNeil and Frey, 2000; Bekiros and Georgoutsos, 2005; Fernandez, 2005; Tolikas et al., 2007; Lohand Stoyanov, 2013, 2014). These studies on tail risk, however, focus primarily on cap-weighted stock indices and there is only limited empirical research on tail risk of diversified portfolios. There are a few papers in this area (Susmel, 2001; Butler and Joaquin, 2002; Chollete et al., 2012) but they emphasise geographical diversification rather than if and how different weighting schemes and stock-selection criteria within one geography influence tail risk.

In this paper, we apply the EVT-based approach to compare the tail risk of different smart beta strategies. For risk measurement purposes, we select CVaR over VaR because of its higher sensitivity to tail losses; we use CVaR at 1% tail probability. We look at differences in tail risk across various strategies within the same geography for both absolute and relative returns and also differences in tail risk of factor-tilted portfolios. Relative returns are defined as the difference between the returns of the strategy and the returns of the corresponding cap-weighted benchmark. The strategies considered are Efficient Minimum Volatility, Efficient Maximum Sharpe Ratio, Maximum Deconcentration, Maximum Decorrelation and Diversified Risk Weighted. The stock-selection criteria considered are size, liquidity, momentum, volatility, value, and dividend yield. We examine the tail risk of these

1. Introduction



strategies for the following Scientific Beta universes: USA (500 stocks), Eurozone (300 stocks), UK (100 stocks), Japan (500 stocks), Developed Asia-Pacific ex-Japan (400 stocks), and World Developed (2,000 stocks). The data cover the entire sample period June 2003 to December 2013 and we also consider two sub-sample periods, the pre-crisis period from June 2003 to June 2007 and the turbulent period from July 2007 to December 2013.

The comparison is performed by decomposing tail risk into a volatility component and a residual component through a two-step process. First, the clustering of volatility is explained away applying the standard econometric framework of the Generalised Autoregressive Conditional Heteroskedastic (GARCH) model and, second, the remaining tail risk is estimated from the residual process using extreme value theory (EVT). 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 model part is responsible for capturing the dynamics of volatility while EVT provides a model for the behaviour of the extreme tail of the distribution.

We carry out the comparison by, first, looking at the differences in tail risk of absolute and relative returns by varying the weighting scheme using all stocks in the corresponding universe. Our main finding is that the CVaR across strategies is primarily driven by the average volatilityor the average tracking error (TE) for the case of absolute and relative returns,

respectively. Adopting a different weighting scheme leads to a higher return and lower volatility compared to those of the corresponding cap-weighted portfolio but at the same time does not deteriorate the thickness of the left tail of the smart beta return distribution. As a consequence, from a long-term investor perspective, focusing on volatility or tracking error management on a strategylevel appears to be of first-order importance for CVaR management. Across geographies, all strategies in Asia tend to have relatively higher total absolute return CVaR than those in Europe and the US which extends earlier empirical results for cap-weighted indices (see Loh and Stoyanov (2013)).

Second, we look at the differences in tail risk of different stock-selection criteria using one and the same weighting scheme — the Maximum Deconcentration (or Equally-weighted) scheme — in order to avoid introducing bias among stocks. In contrast to the first set of examples, our results indicate that the stock-selection criteria can make a statistically significant difference to the residual tail risk of relative returns. The impact varies across geographies, the most affected universe being Asia-Pacific ex Japan. The impact also varies across different market condition and it is difficult to isolate the single stock-selection criterion with the biggest impact. For most of these criteria, the differences in the residual CVaR are amplified further by the average tracking error.

As far as absolute returns are concerned, we find no evidence of statistically significant differences in the tail risk.

1. Introduction



Investing in a non-cap-weighted portfolio would result in higher return and lower volatility without changing the tail risk. This is to say the investors can have a higher Sharpe ratio than the cap-weighted portfolio but at the same time have a similar tail risk to the cap-weighted portfolio. We attempt to explain the lack of difference in the residual tail risk in the case of absolute returns through a CAPM-type one-factor model for the factor tiltedportfolios and also the residuals from the GARCH model. The strong significance of the factor models and the near linear behaviour of the extreme losses suggests that a possible explanation is the relatively limited impact of the regression residual on the response variable. This confirms the previous conclusion that managing volatility or the tracking error is of first-order importance.

In contrast, the results from relative returns of factor-tilted portfolios show a different story. The results show that beside excess return over cap-weighted indices and lower TE, tail risk exposure can be reduced. While the sub-period analyses further show that the investor can use a factor-tilted portfolio to manage tail risk exposure during different market conditions.

Overall this research provides evidence that adopting a smart beta strategy can result in superior performance in terms of returns and volatility as compared to a cap-weighted index, while maintaining a tail risk exposure similar to that of the cap-weighted index. On the other hand, an investor who adopts a smart beta strategy by sub-selecting stocks based on a ranking criterion would be able to

achieve superior performance compared to the a cap-weighted indices, while possibly slightly decreasing the tail risk of the relative return.

The paper is organised in the following way: Section 2 briefly explains the different types of strategies; Section 3 discusses the conditional EVT risk model; Section 4 analyses the empirical results; and Section 5 concludes.

1. Introduction



1. Introduction


2. Smart Beta Indices



This section begins with a brief discussion of the different weighting schemes used to construct diversified portfolios, which are Efficient Minimum Volatility, Efficient Maximum Sharpe Ratio, Maximum Deconcentration, Maximum Decorrelation and Diversified Risk Weighted. Having described the weighting schemes, we brieflydiscuss the criteria used to construct factor-tilted portfolios.

All methodologies described below are implemented with quarterly rebalancing subject to a threshold constraint which aims at minimising turnover. Detailed explanations of the strategies and their implementation are available at http://www.scientificbeta.com.

2.1. Efficient Minimum VolatilityThe theoretical basis for minimum volatility portfolios lies in the seminal work of Markowitz (1952) where the minimum volatility portfolio is a mean-variance efficient portfolio. Efficient Minimum Volatility is a weighting scheme that attempts to minimise the overall portfolio volatility based on correlations and volatilities of stocks in the universe. The true minimum volatility portfolio lies on the efficient frontier, and is the result of optimisation without any expected return estimation and thus the only necessary inputs are estimates of volatilities and correlations of constituent stocks. A common problem cited for the Minimum Volatility strategy is that of concentration in low risk (low volatility or low beta) stocks, which in turn leads to pronounced sector biases towards defensive sectors such as utilities (see Chan et al. (1999)). A possible remedy to this problem of concentration in

low volatility stocks is to introduce weightconstraints. Jagannathan and Ma (2003) show that weight constraints not only control the concentration but also improve the performance of Minimum Volatility portfolios. DeMiguel et al. (2009) go beyond considering rigid constraints at the individual stock level and introduce flexible constraints on overall portfolio concentration (so-called "norm constraints"). They show that using such flexible constraints leads to better out-of-sample risk and return properties ofMinimum Volatility portfolios.

To construct the portfolio, we use an optimisation technique that aims at minimising the portfolio volatility. The norm-constrained optimisation problem can be stated as follows:

where the set

defines the set of feasible portfolios. In this optimisation problem, Σ is a n × n covariance matrix and is a vector of ones. The constraint 'e = 1 ensures that the weights sum up to 1 and the constraint

' ≤ δ is a quadratic norm constraint designed to limit the overall concentration of the portfolio. The quadratic norm constraint is related to the inverse of the Herfindahl index (I = ' ) which is a measure of concentration. In the implementation, we use δ = 3/n which means the concentration of the optimal portfolio does not exceed three times the concentration of the equally weighted portfolio. Theoptimal weights are additionally adjusted to satisfy the box constraints

.5


5 - For additional details on the estimation of the covariance matrix, universe and index construction rules, and also liquidity adjustments, see the available documentation at http://www.scientificbeta.com



2.2. Efficient Maximum Sharpe RatioEfficient Maximum Sharpe Ratio is a weighting scheme that weights stocks in order to achieve the maximum possible risk-adjusted portfolio performance within a given stock universe. In line with modern portfolio theory, the Maximum Sharpe Ratio strategy is an implementable proxy for the tangency portfolio. As it relies on mean-variance optimisation, and in contrast to minimum volatility strategies which only estimate risk parameters (volatilities and correlations), the Maximum Sharpe Ratio strategy attempts to estimate both risk parameters and expected returns. This tangency portfolio can be used by investors who may differ with respect to their target volatility levels. Investors can combine the tangency portfolio with an investment in a risk-free asset to obtain their desired level of volatility. The combination of the tangency portfolio and the risk-free asset will provide the maximum reward for the given level of volatility. The strategy requires estimated risk and expected return, however, as discussed in Merton (1980), it is difficult to estimate expected returns in a reliable manner. To solve the problem, Amenc et al. (2011) propose a novel way to obtain a proxy for the tangency portfolio, where they estimate expected returns indirectly by assuming a relation between risk and return. They group all stocks in the universe by semi-deviation into deciles, then assign the decile's median to each stock within each decile. The expected return of a stock in a decile is then assumed to be equal to the median semi-deviation of that decile.

The optimal weights are obtained by solving the following optimisation problem:

where the set defines the set of feasible portfolios, μ denotes the expected returns as proxied by the median semi-deviations, and denotes the n × n covariance matrix. Like the Efficient Minimum Volatility portfolio, the optimal weights are additionally adjusted to satisfy the same box constraints. Liquidity and turnover adjustments are also applied.

2.3. Maximum DeconcentrationOriginating from the equally-weighted portfolio, Maximum Deconcentration is a weighting scheme that maximises the effective number of stocks — which is equivalent to minimising the concentration as measured by the Herfindahl index — of an equity portfolio.

The optimal weights are obtained by solving the following optimisation problem:

where D describes the set of feasible portfolios. Equal weighting is a simple way of deconcentrating a portfolio and allows investors to benefit from systematic rebalancing back to fixed weights. Depending on the universe and on whether additional implementation rules are used,the rebalancing feature of equal weighting can be associated with relatively high turnover and liquidity problems. Maximum Deconcentration owes its popularity mainly to its robustness and it has been shown to deliver attractive performance despite highly unrealistic conditions of optimality,even when compared to sophisticated portfolio optimisation strategies (DeMiguel et al., 2009). In the absence of any other constraints, this index coincides with the




equal-weighted portfolio. In the presence of tracking error, country neutrality or sector neutrality constraints with respect to the cap-weighted reference index, the optimisation results in weights different from those of an equally-weighted portfolio. Lastly, turnover and liquidity rules are imposed using the cap-weighted reference index weights.

2.4. Maximum DecorrelationMaximum Decorrelation is a weighting scheme which weights stocks so as to exploit the risk reduction effect resulting from low correlations between stocks. The Maximum Decorrelation strategy aims at minimising the volatility of a portfolio of stocks under the assumption that individual volatilities are identical, thus only exploiting the correlation structure. Conventional minimum volatility weighting schemes use estimates of volatilities and correlations, and refrain from estimating expected returns. The result of the unconstrained application of this optimisation is typically a portfolio that is highly concentrated in low volatility stocks. The Maximum Decorrelation approach, in contrast, assumes equal volatilities, and attempts to achieve reduced portfolio volatility by exploiting the correlation properties of constituent stocks. The approach was introduced to measure the diversification potential within a given asset universe (Christoffersen et al., 2012). Just as Maximum Deconcentration reduces concentration in a nominal sense, Maximum Decorrelation reduces the correlation-adjusted concentration.

More formally, the optimal weights are obtained by solving the following

optimisation problem:

where C denotes the correlation matrix and D denotes the set of feasible portfolios. Like the other strategies, the optimal weights are additionally adjusted to satisfy the same box constraints. Liquidity and turnover adjustments are also applied.

2.5. Diversified Risk WeightedDiversified Risk Weighted (DRW) is a weighting scheme which aims to achieve diversification by balancing the constituents' contributions to the total portfolio volatility. Each constituent is weighted according to its contribution to the overall portfolio risk, so that each stock contributes an equal amount of estimated risk. The underlying theory is laid out in Maillard et al. (2010). This weighting scheme attempts to equalise the individual stock contributions to the total risk of the index, assuming uniform correlations across stocks. It is a specific case of the DRW approach where one makes the explicit assumption that all pairwise correlation coefficients are identical. In this case, and without any constraint or adjustment, the portfolio weights are proportional to the inverse of the stock individual volatilities. The Diversified Risk Weighted approach can be seen as a special case of Sharpe Ratio maximisation where one assumes that Sharpe Ratios of stocks are identical, and correlations across stocks are identical for all pairs of stocks.

2.6. Factor-tilted Smart Beta IndicesApart from studying the tail risk resulting from different weighting schemes, we also explore how tilting the portfolio




towards a rewarded factor changes tail risk. To construct factor-tilted indices, we follow the two-step approach of Smart Beta 2.0. First, the stocks in the universe are ranked according to a criterion and then a half-universe is selected. We use market cap, a liquidity score, a momentum score, volatility, the book-to-market ratio, and dividend yield. The second step consists of applying one of the weighting schemes described in this section to the half-universe.

There are two reasons why these factors are considered important and are commonly used. First, these factors are standard in the academic literature and have been extensively tested. Traditional models in the academic literature include the three-factor Fama-French or the four-factor Carhart models. Even as the literature grows and more and more factors are tested, the statistical significance of value, momentum, size and volatility remains substantially high even after adjusting the criterion for the total number of tested factors. This result implies that those factors are expected to be very robust and reliable out-of-sample (see Harvey et al. (2014)).

Second, regardless of empirical research, investors should always question the persistence of factors and consider if there is economic rationale behind them. These are also the questions of robustness: (i) Would the premium disappear if an increasing number of investors were trying to capture it? (ii) Was the discovery of the premium a result of data mining? As regards the former aspect, investors should require a sound economic rationale. Regarding the latter aspect, investors should rely on simple definitions that

have been widely studied in the academic literature, rather than resort to complex and proprietary definitions.

In particular, in case there is solid economic rationale, then this would decrease the likelihood that these factors are significant in-sample by pure chance. Furthermore, a solid economic rationale would guarantee that if all investors know about a given factor, its effect will not disappear. From the standpoint of the theory of finance, a factor has a high expected reward if it provides pay-offs in those states of the world in which the marginal utility of consumption is high, i.e. in "bad times" when the investor wealth is low. Factors that do not have this property are not rewarded; that is, an asset exposed to them may still be risky but investors would notbe willing to pay a premium to hold it.

The following economic rationales have been suggested in the academic literature:• Value: an investor would demand a premium to hold value stocks because their price is driven by assets that are hard to reduce in bad times, which is also known as costly reversibility (see Zhang (2005)). Because of this, the prices of value stocks tend to decline more in bad times and the value premium can thus be interpreted as compensation for risk in bad times.• Momentum: the premium can be viewed as reward for taking macro-economic risks — past winners appear to have higher loadings on the growth rate of industrial production and are more sensitive to changes in the expected growth rate (see Liu and Zhang (2008)).• Size: an investor would require compensation because small stocks tend to have lower profitability and greater





uncertainty of earnings and are, thus, more sensitive to recessions.• Low risk: there have been several attempts to explain the puzzle. One of them is related to preferences of investors to lottery-like pay-offs (see for example Baker et al. (2011)). Presence of preferences for lottery-type investments pushes the price of high-volatility stocks higher and, as a consequence, generates lower expected return. A similar argument can be developed in the presence of limited leverage: investors willing to implement aggressive strategies cannot leverage the investment in the max Sharpe ratio portfolio (which would be optimal); rather, they overweight high volatility stocks which increases demand andpushes the price up.


3. A Conditional EVT Model



In finance, EVT has been traditionally applied to estimate probabilities of extreme losses or loss thresholds such that losses beyond it occur with a predefined small probability, which are also known as high quantiles of the portfolio loss distribution. In fact, EVT provides a model for the extreme tail of the distribution which turns out to have a relatively simple structure described through the corresponding limit distributions such as the Generalised Extreme Value (GEV) distribution or the Generalised Pareto Distribution (GPD).

3.1. The Peak-over-Threshold MethodThe approach in this paper is based on the peak-over-threshold method (POT), see Loh and Stoyanov (2014) 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,

(3.1)

where x > 0. Because we are interested in the extreme losses, we need to gain insight into the probability that the excesses beyond u, X − u, can exceed a certain loss level. Thus, (3.1) is re-stated in terms of the tail

,

(3.2)

There is a celebrated limit result in EVT which states that as u increases towards the right endpoint of the support of the loss distribution denoted by xF, the conditional

tail converges to the tail of the GPD which is defined by,

(3.3)

where 1 + ξx > 0 and β > 0 is a scale parameter. The limit results is (Embrechts et al., 1997, Chapter 3)

(3.4)

where β(u) is a scaling depending on the selected threshold u.

The limit result in (3.4) can be used to construct an approximation for the tail of the losses exceeding a high threshold u. If we denote by y = u + x and express x in terms of y in (3.2), we obtain (3.5)

after substituting the limit law for .For a fixed threshold u, note that is a constant and the tail for y > u is determined entirely by the GPD tail .

It is possible to define sets of portfolio loss distributions also known as maximum domains of attraction (MDA) such that the limit relation in (3.4) leads to a GPD with one and the same tail parameter ξ. Since EVT is used to study rare events, characteristic of the tail behaviour of the portfolio loss distribution turns out to be the important feature; other features of F are not relevant. We distinguish between three different classes of portfolio loss distributions.




The Frechet MDA, ξ > 0 A loss distribution belongs to this domain of attraction if and only if X has a tail decay dominated by a power function in the following sense,

The link between α and ξ is ξ = 1/α. It is possible to demonstrate that this MDA consists of fat-tailed distributions F that have unbounded moments of order higher than α, i.e. E|X|k < ∞ if k < α. For applications in finance, it is safe to assume that volatility is finite which implies α > 2 and ξ < 1/2, respectively. For further detail, see (Embrechts et al., 1997, Section 3.3.1).

The Gumbel MDA, ξ = 0 This MDA is much more diverse. A portfolio loss distributionbelongs 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 heavy-tailed 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 examplesof 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 theprobability . 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).6

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)


6 - An approach based on adaptive calibration of the threshold is adopted by some authors. Gonzalo andOlmo (2004) describe a method based on minimising the distance between the empirical and the tail of the GPD with parameters estimated through the maximum likelihood method. The suggested distance is theKolmogorov-Smirnov statistic.



where s = 1 − m/n and Xs,n is the s-th observation in the sample sorted in increasing order and and are estimates of ξ and β, respectively.

Regarding estimation, a variety of estimators can be employed to estimate ξ and β. We use the maximum likelihood estimator (MLE) which is rationalised by the uniform convergence in (3.4). Under the assumption that data are distributed exactly according to the GPD, then given a sample of i.i.d. observations Y = (Y1, . . . ,Yn1) the log-likelihood function equals

and can be maximised numerically. The MLE , where

D = (−1/2,∞) × (0,∞), satisfies the following asymptotic property

where

(3.8)

and (0, Σ) denotes a bivariate normal distribution. For additional details, see (Embrechts et al., 1997, Section 6.5). Since data do not exactly follow the GPD law but are in its MDA we use the GPD log-likelihood and the result in (3.8) only as an approximation. In practice, the GPD is estimated from the sample Yi = Xs+i,n − Xs,n, where i = 1, . . . , n1 = n−s and s is defined as s = 1−m/n. Information about other estimators, such as the Hill and the Pickandsestimator, and further detail on the

relevance of the MLE are available in de Haan and Ferreira (2006).

3.2. A GARCH-EVT Model for Tail Risk EstimationInstead 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.7 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 generalGARCH filter.8

Denote the time series of portfolio losses by Xt. The GARCH(1,1) model is given by:

(3.9)

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 (3.9) is fitted to the data and then the standardised residual is derived. If we assume that the data is generated by the model in (3.9), then the standardised residual is a sample from the distribution FZ. EVT is applied by fitting the GPD to the residual using 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


7 - See the related comments in Loh and Stoyanov (2014).8 - 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 (2014).



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. Thus, to map the terms properly, in the industry we talk about VaR at the 95% and 99% confidence levels, which respectively correspond 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

(3.10)

where F−1 denotes the inverse of the c.d.f. FX(x) = P(X ≤ x) which is also known as thequantile function of X.

As explained earlier, we employ EVT to estimate high quantiles of the loss distribution. To this end, we adopt the approximation of the tail in (3.5). Solving for the value of y yielding a tail probability of p, we get

(3.11)

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

(3.12)

where s = 1 − m/n and m denotes the number of observations that are considered excesses.

The approximation in (3.12) 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 GPD. In the implementation, we set m/n = 0.1 and, thus, in terms of quantiles the 99% quantile equals the 90% quantile (X(0.9 × n)) plus the corresponding correction term.9

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


9 - 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, X0.9 x n,n is the 90-th observation in the sorted sample and the empirical approximation to

wouldbe the largest observation in the sample. As a consequence, the correction term in (3.12) allows us to go beyond the available data points in the sample which emphasises a key advantage of EVT to the historical method.



given by

(3.13)

where It denotes the information available at time t, is given in (3.12) 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, CV aRp(X), equals the average loss provided that the loss exceeds V aRp(X).

CVaR is formally defined as an average of VaRs,

(3.14)

and if we assume that the portfolio loss distribution has a continuous c.d.f. then CVaR can be expressed as a conditional expectation,

(3.15)

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 (3.14) while expected shortfall is the quantity in (3.15). Although (3.14) 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 (3.15) can be calculated explicitly through the GPD,

where

Plugging in from (3.12) and the corresponding estimates, we get

(3.16)

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 (3.13) for CVaR equals

(3.17)

where is given in (3.16) and is estimated from the sample of the standardised residuals.




3.3. Comparing the Tail Risk of Different StrategiesEquations (3.13) and (3.17) 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 is reduced to comparing the two forecasts produced by equation (3.13) or (3.17). 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 forecastingin 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 through (3.13) or (3.17) but instead of using forecasted volatility, we use the estimated

through the GARCH model. For CVaR, for example, the corrsponding formula is

.

To compare tail risk in-sample, we consider the following three aggregated quantities: (a) total CVaR over the period, which is

the average of over the sample period, (b) the average estimated volatility, i.e. the average of , and (c) constant volatility CVaR, which equals

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.

In order for this approach to make sense, however, we need to demonstrate first that the methodology behind (3.13) and (3.17) is realistic which requires an out-of-sample back-testing. This is needed because although Loh and Stoyanov (2014) show that the GARCH-EVT method is reliable out-of-sample for a large set of cap-weighted market indices, in this paper we consider different types of strategies which implies the back-testing needs to be repeated to make sure that the model has an acceptable out-of-sample performance.

To validate the in-sample approach, we back-tested the methodology on smart beta indices with and without factor tilting constructed from long-term US data spanning about 40 years which is a much bigger sample than the ones used in Loh and Stoyanov (2014). The weighting schemes are Efficient Minimum Volatility (MVol), Efficient Maximum Sharpe Ratio (MSR), Maximum Deconcentration (MDecon), Maximum




Decorrelation (MDecor) and Diversified Risk Weighted (DRW).10 We adopted the same back-testing statistical tests as in Loh and Stoyanov (2014). A standard procedure is followed in which we forecast VaR and CVaR at 1% tail probability in the left and the right tail with a daily risk horizon at each day in a given backtesting window by rolling a time-window of 1,000 days for estimation purposes. Then, we compare the forecasted VaR numbers to the realised returns and compute the exceedances. We run Kupiec's test, which concerns the average number of exceedances, Christoffersen's test, which checks if the exceedances are independent, and the combined test. Apart from VaR, wealso run a test for CVaR which compares the average forecasted loss to the average realised loss beyond the 1% VaR threshold both for the left and the right tail. See (Loh and Stoyanov, 2014, Section 5) for further details on the the tests and the additional statistics.

The results for the full period from January 1973 to December 2012 are provided in Table 1, which provides p-values for the corresponding tests as well as some additional statistics such as the average estimated tail index ξ for the left and the right tail, the average forecasted CVaR (Avg CVaRf), and the average realised conditional loss (Avg C-loss). The results show that GARCH-EVT performs well across the different strategies. In the VaR-based statistical tests, there is one failure in Kupiec's test and three failures in the combined test for the left tail. For the right tail, there are two failures in the combined test. For the CVaR-based tests, there are three failures for left tail and one failure for right tail.

To examine if the methodology is robust across different sample periods, we repeated the same tests on the data decade by decade. The results from the different decades lead to similar conclusions for the methodology. Among the four decades, the method has the worst performance for the period from January 1973 to December 1982 and has the best performance from January 2003 to December 2012. Overall, the back-testing results show that the EVT-GARCH method performs well for smart beta indices. See the Appendix to this paper for the corresponding tables with detailed results decade by decade.


10 - Table 1 includes also results for the diversified multi-strategy weighting scheme which is an equally weighted combination of MVol, MSR, MDecon, MDecor, and DRW. The rationale for constructing a diversified multi-strategy index is to diversify away strategy-specific risks which include, for example, estimation risk.




Table 1: VaR- and CVaR-based statistics of the out-of-sample performance of the GARCH-EVT risk model on different smart beta strategies constructed from the long-term US universe (500 stocks) in the period 1973-2012. The VaR-based tests include the number of exceedances (Exc), the p-value of Kupiec's test (Ku-test), the p-value of Christoffersen's test (Ch-test), and the p-value of the combined test (KC-test). The CVaR-based tests include the average estimated tail index (Avg ), the average forecasted CVaR (Avg CVaRf), the average realised conditional loss (Avg C-Loss), and the p-value of the CVaR t-test. The numbers in bold indicate failures at the 95% confidence level. The 95% confidence interval for the number of exceedances is [84, 124].

VaR-based statistics (1973-2012)

Left-1% Right-1%

Strategies Exc Ku-test Ch-test KC-test Exc Ku-test Ch-test KC-test

SB US Cap-weighted 115 0.3026 0.5436 0.4888 105 0.9491 0.4086 0.7093

SB US MDecon 115 0.3026 0.0494 0.0852 117 0.2221 0.2008 0.2094

SB US MDecor 122 0.0908 0.0731 0.0480 121 0.1101 0.2353 0.1380

SB US MVol 116 0.2602 0.0523 0.0808 113 0.4010 0.1698 0.2739

SB US MSR 122 0.0908 0.0179 0.0145 16 0.2602 0.1928 0.2272

SB US DRW 121 0.1101 0.0692 0.0536 121 0.1101 0.2353 0.1380

SB US Diversified Multi-Strategy 121 0.1101 0.0692 0.0536 121 0.1101 0.0692 0.0536

SB US CW, Large-Cap 114 0.3495 0.5294 0.5298 117 0.2221 0.7738 0.4554

SB US CW, Mid-Cap 123 0.0743 0.0040 0.0032 113 0.4010 0.0092 0.0237

SB US CW, Value 111 0.5173 0.0388 0.0959 112 0.4570 0.1626 0.2860

SB US CW, Growth 105 0.9491 0.1177 0.2935 107 0.7951 0.9241 0.9625

SB US CW, High-Momentum 107 0.7951 0.1295 0.3064 112 0.4570 0.1190 0.2249

SB US CW, Low-Momentum 126 0.0391 0.7078 0.1110 119 0.1586 0.0144 0.0186

CVaR-based statistics (1973-2012)

Left-1% Right-1%

Avg Avg Avg T-test Avg Avg Avg T-test

Strategies L

CVaRf C-Loss p-value R CVaRf C-Loss p-value

SB US Cap-weighted 0.0214 0.0320 0.0296 0.1292 -0.1204 -0.0274 -0.0306 0.0968

SB US MDecon 0.0140 0.0320 0.0295 0.0376 -0.1300 -0.0262 -0.0290 0.0884

SB US MDecor 0.0198 0.0310 0.0286 0.0641 -0.1260 -0.0250 -0.0261 0.0740

SB US MVol 0.0198 0.0274 0.0260 0.0536 -0.1164 -0.0225 -0.0237 0.0747

SB US MSR 0.0250 0.0298 0.0273 0.0913 -0.1275 -0.0241 -0.0253 0.0517

SB US DRW 0.0142 0.0309 0.0285 0.0981 -0.1258 -0.0255 -0.0278 0.3290

SB US Diversified Multi-Strategy 0.0262 0.0301 0.0275 0.0799 -0.1266 -0.0245 -0.0266 0.1913

SB US CW, Large-Cap 0.0136 0.0323 0.0306 0.0868 -0.1265 -0.0283 -0.0302 0.6791

SB US CW, Mid-Cap -0.0034 0.0339 0.0299 0.0743 -0.0986 -0.0282 -0.0298 0.0192

SB US CW, Value -0.0131 0.0354 0.0328 0.0497 -0.1046 -0.0310 -0.0333 0.2945

SB US CW, Growth -0.0001 0.0306 0.0297 0.0418 -0.1255 -0.0267 -0.0286 0.3190

SB US CW, High-Momentum -0.0087 0.0350 0.0339 0.1014 -0.1101 -0.0302 -0.0320 0.6532

SB US CW, Low-Momentum -0.0076 0.0349 0.0315 0.1673 -0.0953 -0.0321 -0.0338 0.3639





4. Empirical Analysis



In this section, we look at differences in tail risk across various strategies within the same geography for both absolute and relative returns followed by the differences in tail risk of factor-tilted portfolios. 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.

To compare the tail risk across strategies and factors, we calculate annualised averages ofseveral 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 strategies or geographical regions: 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. DataTo 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 June 2003 to December 2013. To carry out the tests, we express all data in returns. Weconsider 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 United States, Eurozone, United Kingdom, Japan, Developed Asia-Pacific ex Japan and World Developed. The strategies are Efficient Minimum Volatility, Efficient Maximum Sharpe Ratio, Maximum Deconcentration, Maximum Decorrelation, and Diversified Risk Weighted. For different stock-selection criteria we consider size, liquidity, momentum, volatility, value, and dividend across different geographies.11

4.2. Tail Risk of Different Weighting Schemes without Factor-TiltingWe first examine the tail risk of portfolios constructed for different markets using different strategies without a stock-selection criterion. We consider two cases: (i) absolute returns and (ii) relative returns where relative return is defined as the portfolio excess return over the cap-weighted market index return.

Table 2 provides the numerical results. In the absolute return case, there is little difference between the strategies' constant volatility CVaR within each geography indicating that the differences in the total CVaR are primarily caused by the differences in the average volatility. Comparing the different strategies and cap-weighted portfolio, we can see an increase in realised returns and a fall in volatility while maintaining a tail risk


11 - Details on the index construction methodologies are available at http://www.scientificbeta.com




Table 2: Risk and return of diversified portfolios across different strategies for different stock universes for both absolute and relative returns. Relative returns are computed by subtracting the returns of the corresponding cap-weighted benchmark from the returns of the strategy. The period of analysis is from June 2003 to December 2013.

Strategy Absolute Returns Relative Returns

RealisedReturns

AverageVolatility

CVaR Constant Volatility at 17%

TotalCVaR

RealisedReturns

AverageTE

CVaR Constant TE at 3%

TotalCVaR

SciB

eta

Unite

d St

ates

Cap-Weighted 0.0804 0.1736 0.5749 0.5872 - - - -

Maximum Deconcentration 0.0987 0.1845 0.5669 0.6152 0.0170 0.0299 0.0901 0.0897

Maximum Decorrelation 0.0976 0.1748 0.5640 0.5798 0.0160 0.0309 0.0968 0.0996

Efficient Minimum Volatility 0.1029 0.1503 0.5697 0.5038 0.0209 0.0388 0.0917 0.1184

Efficient Maximum Sharpe Ratio 0.0983 0.1662 0.5675 0.5549 0.0166 0.0290 0.0963 0.0930

Diversified Risk Weighted 0.0995 0.1741 0.5694 0.5832 0.0177 0.0258 0.0881 0.0759

SciB

eta

Euro

zone

Cap-Weighted 0.0528 0.1935 0.5765 0.6561 - - - -






SciB

eta

Unite

d Ki

ngdo

m Cap-Weighted 0.0768 0.1759 0.5420 0.5608 - - - -






SciB

eta

Japa

n

Cap-Weighted 0.0416 0.2067 0.5799 0.7053 - - - -






SciB

eta

Deve

lope

d As

ia-P

acifi

c ex

-Jap

an Cap-Weighted 0.1182 0.2038 0.5713 0.6849 - - - -






SciB

eta

Deve

lope

d

Cap-Weighted 0.0843 0.1491 0.5590 0.4903 - - - -








exposure similar to the cap-weighted indices when these strategies are adopted. In term of differences between the geographical regions, Table 2 shows that the total CVaRs of all strategies in Asia (Japan and Asia Pacific ex Japan) are consistently higher than those in Europe and the US. This is in line with the empirical results by Loh and Stoyanov (2013) which are limited to cap-weighted indices only.

Like the case of absolute returns, the differences in the total CVaR of the relative returns are driven primarily by the difference in the average tracking error (TE). The results show that investing in a portfolio using different strategy instead of the cap-weighted indices resulted in higher return and lower TE without significant increase in tail risk. In fact, the different strategies produced almost similar tail risk exposure across strategies in most regions. The only geographies for which the differences between the maximal and the minimal constant volatility CVaR are statistically significant are Japan and Asia Pacific ex Japan. For the two geographical regions, the minimal and the maximal constant volatility CVaRs are attained at the Efficient Minimum Volatility strategy and the Maximum Deconcentration strategy. Across all strategies, the only market that stands out is that of the US for which the average TE appears to be relatively smaller and leads to consistently smaller average total relative return CVaRs than those of the other geographical regions. Although empirical results are not provided in this paper, the same conclusion holds for the two sub-periods from 2003 to 2007 and from 2007 to 2013.

As a next step, we look at the tail risk of all Scientific Beta strategies within a given geography. The set of all strategies includes all combinations of weighting scheme, stock-selection method, and risk control.12

For all geographical regions, if we rank strategies by constant volatility CVaR, we find that the difference between the one on top and the one at the bottom is at the border of statistical significance at the 95% confidence level. The relationship betweenstrategies' average volatility, constant volatility CVaR, and Total CVaR is very similar across geographical regions and is illustrated in Figure 1 and 2 for all the regions.

There appears to be a slightly negative statistical relationship between average volatility and constant volatility CVaR although the difference between the minimum and the maximum on the vertical axis is in this case is not statistically significant (note the difference in scale ofthe two axes). On the other hand, the statistical relationship between the average volatility and Total CVaR is highly significant; average volatility is the main determinant of Total CVaR. The strategies that have lowest volatility are among those with a low volatility stock selection or with Efficient Minimum Volatility weighting.

4.3. Tail Risk of Equally Weighted Factor-Tilted PortfoliosThe first set of experiments indicates that the most important factor that influences total CVaR is volatility or tracking error depending on whether we consider the case of absolute or relative returns. In this section, we focus specifically on factor-tilted portfolios. Although the stock-


12 - Risk controls include sector neutrality, geographical neutrality (which makes sense for composite indices), and tracking error control at pre-defined levels of 5%, 3%, and 2%.



selection criteria have been designed with harvesting risk premia in mind, it is of practical interest to see if there are any implications for tail risk when portfolios are built from the half-universes defined by a stock-selection criterion. We consider the tail risk of portfolios constructed from the different universe using the Maximum Deconcentration strategy and all stock-selection criteria: size, liquidity, momentum, volatility, value, and dividend. Although Table 2 suggests that any weighting scheme would be just fine, Maximum Deconcentration was selected because it represents, essentially, an equally weighted

portfolio and does not discriminate between stocks; e.g. it would not underweight by chance individual stocks with significant tail risk.

The risk and return statistics for absolute returns are included in Table 3. The results shows that Table 3 is very similar to Table 2 in that the main differences in total CVaR are driven by differences in average volatility. Across factor-tilted portfolios, the average volatility appears to be the main driver of the total CVaR across the different geographies. There is little difference in the constant scale CVaR between the factor-


Figure 1: The relationship between the average volatility and Total CVaR for all Scientific Beta Developed strategies. The red triangle indicates the position of the corresponding cap-weighted portfolio. The period of analysis is from June 2003 to December 2013.




13 - The cap-weighted index used in this calculation is not factor tilted, i.e. we use the natural benchmark forthe corresponding geographical region.

Figure 2: The relationship between the average volatility and constant scale CVaR for all Scientific Beta Developed strategies. The red triangle indicates the position of the corresponding cap-weighted portfolio. The period of analysis is from June 2003 to December 2013.

tilted portfolio within each geography. In cases where there are differences in total CVaR, we can see that the differences are due to average volatility. The factor-tilted portfolios produced different realised returns and average volatility but quite similar tail risk. This allows investors to select a factor-tilted portfolio to suit their risk appetite with a similar tail risk exposure to the cap-weighted indices.

However, we see a different picture in the relative returns defined as the difference

of the index returns and the returns of the cap-weighted index for the corresponding geographical region.13 The numerical results in Table 4 indicate that factors can cause significant differences in residual tail risk for some regions such as the US, Japan, Asia-Pacific ex Japan and World Developed. The numbers in bold in the table indicate the statistically significant differences in the constant volatility CVaR. For example, in the US the size and volatility criteria affectthe residual tail risk while in Japan those criteria are the liquidity and momentum



ones. In the case of Asia-Pacific ex Japan, our results indicate that all factors have significant influence on residual tail risk. In the World Developed universe, the corresponding factors are volatility, value, and dividend yield. The results in Table 4 imply that stock-selection criteria influencethe total CVaR through the volatility and residual tail risk depending on geography. This provides evidence that factor-tilted portfolio allows the investor to manage tail risk exposure after removing the cap-weighted factor.

To examine if the influence changes under different market conditions, we divide our analysis into two sample periods: the pre-crisis period (from June 2003 to June 2007) and the turbulent period (July 2007 — December 2013). The results for the two sub-periods are available in Table 5. In the pre-crisis period, the US and UK are the only regions where the stock-selection criteria does not affect the residual tail risk. For the Eurozone, only volatility significantly affects the residual tail risk while in Japan and Asia-Pacific ex Japan, most stock-selectioncriteria significantly influence the residual tail risk. For the World Developed universe, the corresponding criteria are the momentum and the dividend yield.

For the turbulent period, the only region that is not affected by stock-selection tail risk is the Eurozone. The residual tail risk of the US is affected by size, volatility and dividend yield while in the UK, the factors are momentum and value/growth. Residual tail risk in Japan is affected by liquidity and volatility. In Asia-Pacific ex Japan universe, all the stock-selection criteria except liquidity affect the residual tail risk. In the World Developed universe, volatility,

value, and dividend yield lead to significant difference in residual tail risk. Generally the sub-period analyses show that the investor can use a factor-tilted portfolio to manage tail risk exposure during different market conditions.

Overall, the results show that different stock-selection criteria can influence the total CVaR in different ways depending on geography and market conditions. The differences in the constant volatility CVaR are not very big although they are statistically significant. In most cases, thedifferences in the constant scale CVaR are amplified further by the differences in the average tracking error.

The results in Tables 3 and 4 beg the question of why differences in constant-scale CVaRappear in the case of relative returns and not for absolute returns. In fact, if we compare the constant-scale CVaR of the regional cap-weighted indices in Table 2 to the corresponding constant-scale CVaRs of the factor-tilted indices in Table 3 it turns out the numbers are fairly similar and the differences are statistically insignificant.

A possible explanation is provided by the classical CAPM-type factor model in which the factor-tilted index excess return is the dependent variables and the corresponding cap-weighted index excess return is the explanatory variable, (4.1)

where r denotes the factor-tilted index return, rm denotes the cap-weighted index return for the corresponding geography, rf denotes the risk-free rate, a and b are the intercept and the slope coefficients, and is the residual.14 The model in (4.1)


14 - Although the standard notation in the one-factor model is α for the intercept and β for the slope coefficient,we use a and b instead to avoid confusion because the same notation is standard also for EVT.




Table 3: Risk and return statistics of the Maximum Deconcentration strategy with different stock selection criteria for absolute returns and different stock universes. The numbers in bold indicate statistically significant differences in the constant volatility CVaR for a given stock selection criterion at the 95% confidence level. The period of analysis is from June 2003 to December 2013.

Stock Selection Absolute Returns Stock Selection Absolute Returns

RealisedReturns

AverageVolatility


TotalCVaR

RealisedReturns

AverageVolatility


TotalCVaR

SciB

eta

Uni

ted

Stat

es

Large-Cap 0.0917 0.1791 0.5705 0.6009

SciB

eta

Japa

n

Large-Cap 0.0597 0.2059 0.5797 0.7019

Mid-Cap 0.1099 0.1896 0.5585 0.6230 Mid-Cap 0.0615 0.1915 0.6128 0.6902

High-Liquidity 0.0968 0.1978 0.5656 0.6581 High-Liquidity 0.0562 0.2214 0.5770 0.7515

Mid-Liquidity 0.1035 0.1712 0.5670 0.5712 Mid-Liquidity 0.0656 0.1788 0.6174 0.6495

High-Momentum 0.0909 0.1829 0.5596 0.6019 High-Momentum 0.0593 0.1929 0.6115 0.6938

Low-Momentum 0.1023 0.1923 0.5660 0.6400 Low-Momentum 0.0631 0.2077 0.5761 0.7040

High-Volatility 0.0985 0.2233 0.5632 0.7397 High-Volatility 0.0531 0.2389 0.5789 0.8135

Low-Volatility 0.0959 0.1541 0.5720 0.5186 Low-Volatility 0.0686 0.1651 0.6187 0.6010

Value 0.1058 0.1893 0.5717 0.6367 Value 0.0865 0.1954 0.6028 0.6928

Growth 0.0905 0.1808 0.5576 0.5931 Growth 0.0372 0.2018 0.5922 0.7030

High-Div-Yield 0.1001 0.1755 0.5759 0.5944 High-Div-Yield 0.0758 0.1864 0.6138 0.6728

Low-Div-Yield 0.0992 0.1983 0.5568 0.6493 Low-Div-Yield 0.0465 0.2092 0.5881 0.7238

SciB

eta

Euro

zone

Large-Cap 0.0596 0.1919 0.5803 0.6549

SciB

eta

Dev

elop

ed A

sia-

Paci

fic e

x-Ja

pan

Large-Cap 0.1281 0.1946 0.5844 0.6690

Mid-Cap 0.0743 0.1656 0.6038 0.5883 Mid-Cap 0.1666 0.1934 0.5936 0.6751







Value 0.0667 0.1952 0.5847 0.6712 Value 0.1800 0.2013 0.5837 0.6913

Growth 0.0663 0.1609 0.5991 0.5669 Growth 0.1271 0.1830 0.5965 0.6421



SciB

eta

Uni

ted

King

dom

Large-Cap 0.0983 0.1758 0.5411 0.5596

SciB

eta

Dev

elop

ed

Large-Cap 0.0963 0.1527 0.5569 0.5001

Mid-Cap 0.0976 0.1815 0.5570 0.5948 Mid-Cap 0.1111 0.1531 0.5674 0.5108







Value 0.0865 0.1948 0.5504 0.6308 Value 0.1098 0.1608 0.5603 0.5299

Growth 0.1077 0.1643 0.5539 0.5354 Growth 0.0941 0.1452 0.5640 0.4819






Table 4: Risk and return statistics of the Maximum Deconcentration strategy with different stock selection criteria for relative returns and different stock universes. The numbers in bold indicate statistically significant differences in the constant volatility CVaR for a given stock selection criterion at the 95% confidence level. The period of analysis is from June 2003 to December 2013.

Stock Selection Relative Returns Stock Selection Relative Returns

RealisedReturns

AverageTE

CVaRConstant TE at 3%

TotalCVaR

RealisedReturns

AverageTE

CVaR Constant TE at 3%

TotalCVaR

SciB

eta

Uni

ted

Stat

es

Large-Cap 0.0105 0.0220 0.0872 0.0640

SciB

eta

Japa

n

Large-Cap 0.0174 0.0265 0.0893 0.0787

Mid-Cap 0.0274 0.0465 0.0975 0.1511 Mid-Cap 0.0191 0.0658 0.0933 0.2045







Value 0.0235 0.0399 0.0916 0.1219 Value 0.0431 0.0532 0.0977 0.1732

Growth 0.0094 0.0373 0.0918 0.1142 Growth -0.0043 0.0449 0.0948 0.1418



SciB

eta

Euro

zone

Large-Cap 0.0065 0.0272 0.0952 0.0865

SciB

eta

Dev

elop

ed A

sia-

Paci

fic e

x-Ja

pan

Large-Cap 0.0089 0.0412 0.0845 0.1162

Mid-Cap 0.0204 0.0686 0.0907 0.2074 Mid-Cap 0.0433 0.0666 0.0952 0.2115




Low-Momentum -0.0084 0.0525 0.0940 0.1644 Low-Momentum -0.0062 0.0617 0.0857 0.1761

High-Volatility -0.0016 0.0592 0.0943 0.1860 High-Volatility 0.0222 0.0938 0.0994 0.3107


Value 0.0132 0.0509 0.0897 0.1522 Value 0.0553 0.0642 0.0896 0.1919

Growth 0.0128 0.0568 0.0917 0.1738 Growth 0.0080 0.0549 0.0933 0.1708



SciB

eta

Uni

ted

King

dom

Large-Cap 0.0200 0.0352 0.0911 0.1071

SciB

eta

Dev

elop

ed

Large-Cap 0.0111 0.0155 0.0865 0.0446

Mid-Cap 0.0193 0.0734 0.0920 0.2251 Mid-Cap 0.0248 0.0328 0.0879 0.0962







Value 0.0090 0.0641 0.0896 0.1915 Value 0.0236 0.0291 0.0868 0.0843

Growth 0.0286 0.0547 0.0930 0.1695 Growth 0.0090 0.0266 0.0971 0.0861






Table 5: Relative return tail risk of the Maximum Deconcentration weighting scheme applied on halfuniverses obtained by different stock selection criteria under different market conditions. The numbers in bold indicate statistically significant differences in the constant tracking error CVaR for a given stock selection criterion.

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considers the excess returns directly and the dependence in the extremes would be driven by the clustering of volatility, i.e. most of the extreme losses for the strategy and the cap-weighted benchmark would occur in turbulent high-volatility market conditions. To get an idea of this dependence without the clustering of volatility, we consider the same model but with the corresponding residuals from the GARCH process which for daily data is similar to using the excess returns normalised by the estimated GARCH volatilities.

Numerical results for the value and growth factor-tilted portfolios for the SB World Developed universe are provided in Figure 3. We choose the World Developed universe because it is the biggest and the most diverse universe of stocks of the ones that we consider in this paper. Figure 3 includes scatter-plots of the daily excess

returns of the factor-tilted indices and the cap-weighted benchmark, estimated regression coefficients, the regression line superimposed on the scatter-plots, and also the coefficient of determination (R2). The same information is provided for the regression based on the GARCH residuals. The analysis is based on the full sample from June 2003 to December 2013. Scatter-plots for the other stock-selection criteria are provided in Appendix 2.

The one-factor models in Figure 3 and in all the figures in Appendix 2 describe the empirical data quite well; the coefficient of determination is very high, in excess of 94%. We first look at the factor models based on the excess returns. Although the one-factor is highly significant, this does not imply that factor-tilted portfolios are equally volatile; the estimated slope coefficients and the intercept terms


Figure 3: Scatter-plots of the daily excess returns (in percentage) of the growth and value factor-tilted portfolios (vertical axis) of the SB World Developed universe and the daily excess returns (in percentage) of the cap-weighted portfolio constructed from the same universe (horizontal axis) covering the period from June 2003 to December 2013. The plots on the left are constructed from the returns and the ones on the right are constructed from the GARCH residuals. All plots include a regression line and estimates of the regression coefficients together with the coefficient of determination (R2).



(measured in daily frequency) are quite different from strategy to strategy (see the plots on the left side of Figure 3 and Figures 4 and 5 in Appendix 2). As far as the left tail goes, however, the excess returns of the factor-tilted indices are quite closeto the estimated regression line which implies that the extreme losses are dominated by the extreme losses of the cap-weighted index scaled by the corresponding slope term. Although this interpretation is simplistic15 since volatility is proxied by the slope coefficient and the clustering of volatility effect is unaccounted for, it indicates a possible reason for the similar tail risks in the case of absolute returns. In another word, this means that tail risk is mainly due to the market factor. Investors can achieve better performance in terms of returns and volatility compared tocap-weighted portfolio, but maintain a similar tail risk exposure.

The same conclusion is drawn by looking at the plots on the right side in Figure 3. Because they are based on the GARCH residuals, the clustering of volatility effect has already been explained away and the extremes are not necessarily occurring in a high-volatility regime. Nevertheless, the high significance of the factor model indicates that the regression residual hasa limited impact on the response variable.

On the other hand, the normalised relative returns used to calculate the constant-scaleCVaR would be represented in this description by the error term in the factor model using the GARCH residuals. Table 4 indicates that the tail behaviour of the error terms can be different across opposite factor-tilted strategies (e.g. high- vs low-volatility) and under the standard

one-factor model assumptions, the results in Maddipatla et al. (2011) imply the heaviest tail should dominate in the response variable. This is not observed in Table 3 because of the high significance of the factor model, i.e. impact of the residual on the response variable is limited and cannot be properly estimated with the available sample.


15 - In fact, the extreme losses shown on the scatter-plots are most likely observed in high-volatility regimes.


Conclusion



Cap-weighted indices, although widely used as a passive investment vehicle, have two important drawbacks with far-reaching consequences — they represent concentrated portfolios and they are also exposed to risk factors that are not well rewarded. Smart beta indices have been introduced in an effort to improve on the two disadvantages; industry index providers have designed a new framework called smart factor investing that aims at tilting the portfolio towards better rewarded factors through stock selection or using different weighting schemes. Although clearly an improvement over cap-weighting, the industry index solutions are often based on ad-hoc stock-selection and weight allocation criteria prone to data-mining risks.

Since it has been demonstrated that smart beta indices have improved performanceand also sometimes lower volatility than the cap-weighted benchmarks, it is of practical and also theoretical importance to check if they exhibit higher extreme risk. This is an important question because the improvement in the risk-adjusted performance of smart beta strategies is usually demonstrated in terms of the Sharpe ratio. If, however, there is a substantial change in the thickness of the left tail of the smart beta return distribution which may be underestimated by volatility, then Sharpe ratios may mislead investors into thinking that smart beta strategies are superior. To study the tail risk systematically across different weighting schemes and stock selection criteria, we need a solid indexing methodology that can produce diversified factor-tilted indices consistently across different geographical universes. The Smart Beta 2.0 approach represents such a methodology and is based on two distinct

steps, stock selection for factor tilting andweight allocation for diversification, both of which use reliable criteria showing robust out-of sample performance.

To compare extreme risk across different smart beta indices, we use a statistical methodology based on Extreme Value Theory and Conditional Value-at-Risk (CVaR) at 1% tail probability as downside risk measure. We choose CVaR to Value-at-Risk (VaR) because it is more sensitive to the tail of the distribution. Furthermore, we decompose the tail risk into two components by means of a GARCH model — a conditional volatility component and a residual tail risk component. The rationale for this decomposition is that from a risk management perspective the time structure of volatility contributes to the unconditional fat tail of the return distribution and, thus, we can isolate the effect of volatility and the effect of the residual tail thickness.

In this paper, we studied the in-sample extreme risk of smart beta portfolios using the GARCH-EVT model. To validate the in-sample approach, we back-tested the methodology on smart beta indices constructed from long-term US data spanning 40 years and found that the methodology is robust and reliable. The VaR- and CVaR-based tests for the case of 1% tail probability indicated that, with a couple of exceptions, the model is statistically acceptable for all portfolios for both the left and the right tail.

Our main finding is that the total CVaR across strategies is primarily driven by the average volatility or the average tracking error for the case of absolute and relative

Conclusion



returns, respectively. As a consequence, adopting a different weighting scheme can lead to superior performance compared to that of the corresponding cap-weighted index without a deterioration of the tail thickness of the left tail of smart beta returns. In other words, the additional performance does not come at the cost of an increase in tail thickness. Therefore, from a long-term investor perspective, focusing on volatility or tracking error management on a strategy level appears to be of first-order importance for total CVaR for the respective stock universes. Across geographies, all strategies in Asia tend to have higher total absolute return CVaR than those in Europe and the US which extends earlier empirical results for cap-weighted indices. In a broader context,investors would not be misled by using the Sharpe ratio or the information ratio in comparing risk-adjusted returns.

In contrast, the stock-selection criteria can make a statistically significant difference to residual tail risk and the impact varies across geographies; the most affected universe being Asia-Pacific ex Japan. The impact also varies under different market conditions and it is difficult to isolate the single stock-selection criterion with the biggest impact. For most of the criteria, the differences in the residual CVaR are amplified further by the average tracking error. Although the differences in tail risk appear statistically significant, they are rather small in magnitude and the tracking error is still of first-order importance for risk management.

To attempt to explain the lack of difference in the constant-scale residual tail risk in thecase of absolute returns, we considered a

CAPM-type one-factor model for the factor-tilted portfolios. The strong significance of the model and the near linear behaviour of the extreme losses suggests that a possible explanation is the relatively limited impact of the residual on the response variable. This confirms the previous conclusion that managing volatility or the tracking error is of first-order priority.

Conclusion



Conclusion


Appendix



Appendix 1: Out-of-Sample Tests for the Long-Term US Data Decadeby Decade

Table 6: VaR- and CVaR-based statistics of the out-of-sample performance of the GARCH-EVT risk model on different smart beta strategies constructed from the long-term US universe (500 stocks) in the period 1973-1982. The VaR-based tests include the number of exceedances (Exc), the p-value of Kupiec's test (Ku-test), the p-value of Christoffersen's test (Ch-test), and the p-value of the combined test (KC-test). The CVaR-based tests include the average estimated tail index (Avg ), the average forecasted CVaR (Avg CVaRf), the average realised conditional loss (Avg C-Loss), and the p-value of the CVaR t-test. The numbers in bold indicate failures at the 95% confidence level. The 95% confidence interval for the number of exceedances is [16 36].


Left-1% Right-1%



SB US MDecon 34 0.1365 0.0802 0.0715 33 0.1909 0.3576 0.2784

SB US MDecor 40 0.0111 0.1520 0.0142 35 0.0952 0.0901 0.0591

SB US MVol 33 0.1909 0.0711 0.0834 34 0.1365 0.4658 0.2529

SB US MSR 38 0.0281 0.0182 0.0055 31 0.3471 0.3835 0.4396

SB US DRW 36 0.0649 0.1008 0.0474 32 0.2605 0.3725 0.3568


SB US CW, Large-Cap 29 0.5724 0.4192 0.6153 32 0.2605 0.4102 0.3783

SB US CW, Mid-Cap 41 0.0067 0.0036 0.0004 29 0.5724 0.3322 0.5329

SB US CW, value 30 0.4512 0.0481 0.1068 34 0.1365 0.0802 0.0715

SB US CW, Growth 31 0.3471 0.3877 0.4426 33 0.1909 0.4377 0.3145




Left-1% Right-1%


Strategies L


SB US Cap-weighted -0.1153 0.0239 0.0212 0.3776 -0.0418 -0.0258 -0.0288 0.3301

SB US MDecon -0.0950 0.0240 0.0227 0.1573 -0.0721 -0.0233 -0.0250 0.3317

SB US MDecor -0.1313 0.0230 0.0212 0.4231 -0.0657 -0.0221 -0.0236 0.3701

SB US MVol -0.1143 0.0199 0.0186 0.0338 -0.0843 -0.0194 -0.0205 0.3290

SB US MSR -0.0939 0.0221 0.0205 0.2314 -0.0837 -0.0211 -0.0231 0.1517

SB US DRW -0.0984 0.0235 0.0217 0.2896 -0.0850 -0.0231 -0.0251 0.4237

SB US Diversified Multi-Strategy -0.0817 0.0225 0.0209 0.2161 -0.0899 -0.0217 -0.0239 0.1834

SB US CW, Large-Cap -0.1338 0.0242 0.0214 0.1823 -0.0727 -0.0270 -0.0289 0.6569

SB US CW, Mid-Cap -0.0896 0.0262 0.0249 0.4145 0.0105 -0.0255 -0.0260 0.2654

SB US CW, Value -0.1724 0.0271 0.0242 0.0654 -0.0921 -0.0274 -0.0283 0.5574

SB US CW, Growth -0.1005 0.0236 0.0223 0.3793 -0.0561 -0.0265 -0.0273 0.6859

SB US CW, High-Momentum -0.1234 0.0268 0.0239 0.0315 -0.1235 -0.0282 -0.0277 0.7307

SB US CW, Low-Momentum -0.2211 0.0267 0.0241 0.9789 -0.0858 -0.0303 -0.0310 0.7553

Appendix





Left-1% Right-1%



SB US MDecon 25 0.8290 0.0223 0.0717 22 0.4082 0.0128 0.0321

SB US MDecor 26 0.9859 0.0263 0.0848 21 0.2999 0.1615 0.2193

SB US MVol 27 0.8587 0.0309 0.0959 22 0.4082 0.1795 0.2886

SB US MSR 27 0.8587 0.0309 0.0959 23 0.5349 0.1985 0.3609

SB US DRW 26 0.9859 0.0263 0.0848 22 0.4082 0.0128 0.0321


SB US CW, Large-Cap 22 0.4082 0.1795 0.2886 24 0.6768 0.5043 0.7336

SB US CW, Mid-Cap 29 0.5738 0.3321 0.5333 27 0.8587 0.0309 0.0959

SB US CW, Value 23 0.5349 0.0156 0.0442 20 0.2114 0.1444 0.1579

SB US CW, Growth 19 0.1426 0.1284 0.1075 19 0.1426 0.5974 0.2970




Left-1% Right-1%


Strategies L



SB US MDecon 0.1007 0.0327 0.0290 0.4691 -0.1582 -0.0239 -0.0266 0.6249

SB US MDecor 0.1070 0.0316 0.0288 0.4434 -0.1550 -0.0230 -0.0237 0.5034

SB US MVol 0.0875 0.0276 0.0251 0.5907 -0.1100 -0.0209 -0.0221 0.3700

SB US MSR 0.1022 0.0302 0.0270 0.6944 -0.1450 -0.0222 -0.0227 0.7313

SB US DRW 0.1039 0.0312 0.0278 0.7091 -0.1210 -0.0231 -0.0247 0.6414


SB US CW, Large-Cap 0.1144 0.0363 0.0307 0.8933 -0.1810 -0.0280 -0.0308 0.8187

SB US CW, Mid-Cap 0.0737 0.0329 0.0292 0.3097 -0.1134 -0.0253 -0.0280 0.0867

SB US CW, Value 0.1256 0.0409 0.0330 0.8067 -0.1073 -0.0328 -0.0375 0.7409

SB US CW, Growth 0.0737 0.0334 0.0305 0.5083 -0.1794 -0.0251 -0.0270 0.4799

SB US CW, High-Momentum 0.0869 0.0417 0.0356 0.8056 -0.2032 -0.0314 -0.0370 0.3703

SB US CW, Low-Momentum 0.0974 0.0341 0.0289 0.5081 -0.0441 -0.0287 -0.0315 0.3419

Appendix





Left-1% Right-1%



SB US MDecon 26 0.9874 0.4692 0.7695 35 0.0952 0.4945 0.1969

SB US MDecor 27 0.8571 0.4522 0.7417 37 0.0432 0.3020 0.0760

SB US MVol 29 0.5724 0.4192 0.6153 30 0.4512 0.3574 0.4930

SB US MSR 29 0.5724 0.4192 0.6153 36 0.0649 0.5238 0.1485

SB US DRW 30 0.4512 0.4033 0.5309 38 0.0281 0.5838 0.0772


SB US CW, Large-Cap 31 0.3471 0.3835 0.4396 33 0.1909 0.3576 0.2784

SB US CW, Mid-Cap 25 0.8305 0.2395 0.4894 33 0.1909 0.0711 0.0834

SB US CW, Value 30 0.4512 0.4033 0.5309 34 0.1365 0.3432 0.2106

SB US CW, Growth 29 0.5724 0.0418 0.1074 29 0.5724 0.4192 0.6153




Left-1% Right-1%


Strategies L



SB US MDecon 0.1007 0.0327 0.0290 0.4691 -0.1582 -0.0239 -0.0266 0.6249

SB US MDecor 0.1070 0.0316 0.0288 0.4434 -0.1550 -0.0230 -0.0237 0.5034

SB US MVol 0.0875 0.0276 0.0251 0.5907 -0.1100 -0.0209 -0.0221 0.3700

SB US MSR 0.1022 0.0302 0.0270 0.6944 -0.1450 -0.0222 -0.0227 0.7313

SB US DRW 0.1039 0.0312 0.0278 0.7091 -0.1210 -0.0231 -0.0247 0.6414


SB US CW, Large-Cap 0.1144 0.0363 0.0307 0.8933 -0.1810 -0.0280 -0.0308 0.8187

SB US CW, Mid-Cap 0.0737 0.0329 0.0292 0.3097 -0.1134 -0.0253 -0.0280 0.0867

SB US CW, Value 0.1256 0.0409 0.0330 0.8067 -0.1073 -0.0328 -0.0375 0.7409

SB US CW, Growth 0.0737 0.0334 0.0305 0.5083 -0.1794 -0.0251 -0.0270 0.4799

SB US CW, High-Momentum 0.0869 0.0417 0.0356 0.8056 -0.2032 -0.0314 -0.0370 0.3703

SB US CW, Low-Momentum 0.0974 0.0341 0.0289 0.5081 -0.0441 -0.0287 -0.0315 0.3419

Appendix





Left-1% Right-1%



SB US MDecon 30 0.4536 0.4035 0.5327 27 0.8602 0.4524 0.7424

SB US MDecor 29 0.5751 0.4194 0.6169 28 0.7118 0.4357 0.6894

SB US MVol 27 0.8602 0.4524 0.7424 27 0.8602 0.4524 0.7424

SB US MSR 28 0.7118 0.4357 0.6894 26 0.9843 0.4694 0.7696

SB US DRW 29 0.5751 0.4194 0.6169 29 0.5751 0.4194 0.6169


SB US CW, Large-Cap 32 0.2623 0.372 7 0.3585 28 0.7118 0.4357 0.6894

SB US CW, Mid-Cap 28 0.7118 0.4357 0.6894 24 0.6754 0.5044 0.7331

SB US CW, Value 28 0.7118 0.4357 0.6894 24 0.6754 0.5044 0.7331

SB US CW, Growth 26 0.9843 0.4694 0.7696 26 0.9843 0.4694 0.7696




Left-1% Right-1%


Strategies L


SB US Cap-weighted -0.0399 0.0359 0.0290 0.8913 -0.2329 -0.0288 0.0290 0.0632

SB US MDecon -0.0715 0.0387 0.0300 0.6330 -0.2159 -0.0307 0.0300 0.1392

SB US MDecor -0.0354 0.0370 0.0303 0.7722 -0.2086 -0.0288 0.0303 0.1308

SB US MVol -0.0580 0.0316 0.0271 0.7069 -0.2154 -0.0249 0.0271 0.2941

SB US MSR -0.0491 0.0351 0.0282 0.6859 -0.2182 -0.0273 0.0282 0.1454

SB US DRW -0.0696 0.0366 0.0300 0.6539 -0.2257 -0.0290 0.0300 0.2997

SB US Diversified Multi-Strategy -0.0506 0.0357 0.0283 0.6882 -0.2239 -0.0280 0.0283 0.1855

SB US CW, Large-Cap -0.0490 0.0352 0.0291 0.8879 -0.2333 -0.0285 0.0291 0.2432

SB US CW, Mid-Cap -0.0928 0.0428 0.0310 0.6264 -0.1797 -0.0349 0.0310 0.0467

SB US CW, Value -0.0740 0.0398 0.0347 0.7185 -0.1766 -0.0334 0.0347 0.9533

SB US CW, Growth -0.0867 0.0326 0.0263 0.7273 -0.1865 -0.0264 0.0263 0.3531

SB US CW, High-Momentum -0.0726 0.0358 0.0346 0.5624 -0.0531 -0.0298 0.0346 0.5658

SB US CW, Low-Momentum -0.0783 0.0453 0.0349 0.7791 -0.1410 -0.0393 0.0349 0.4870

Appendix



Appendix 2: One-Factor Models for the GARCH Residuals of Factor-Tilted Strategy Returns

Figure 4: Scatter-plots of the daily excess returns (in percentage) of different factor-tilted portfolios (vertical axis) of the SB World Developed universe and the daily excess returns (in percentage) of the cap-weighted portfolio constructed from the same universe (horizontal axis) covering the period from June 2003 to December 2013. The plots include a regression line and estimates of the regression coefficients together with the coefficient of determination (R2).

Appendix



Figure 5: Scatter-plots of the daily excess returns (in percentage) of different factor-tilted portfolios (vertical axis) of the SB World Developed universe and the daily excess returns (in percentage) of the cap-weighted portfolio constructed from the same universe (horizontal axis) covering the period from June 2003 to December 2013. The plots include a regression line and estimates of the regression coefficients together with the coefficient of determination (R2).

Appendix



Figure 6: Scatter-plots of the daily GARCH residuals (in percentage) of the daily returns of different factor-tilted portfolios (vertical axis) of the SB World Developed universe and the daily GARCH residuals (in percentage) of the daily returns of the cap-weighted portfolio constructed from the same universe (horizontal axis) covering the period from June 2003 to December 2013. The plots include a regression line and estimates of the regression coefficients together with the coefficient of determination (R2).

Appendix



Figure 7: Scatter-plots of the GARCH residuals (in percentage) of the daily returns of different factortilted portfolios (vertical axis) of the SB World Developed universe and the daily GARCH residuals (in percentage) of the daily returns of the cap-weighted portfolio constructed from the same universe (horizontal axis) covering the period from June 2003 to December 2013. The plots include a regression line and estimates of the regression coefficients together with the coefficient of determination (R2).

Appendix



Appendix


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About EDHEC-Risk Institute





The Choice of Asset Allocation and Risk ManagementEDHEC-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 RelevanceIn 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.

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. The results of the research programmes are disseminated through the EDHEC-Risklocations in Singapore, which was established at the invitation of the Monetary Authority of Singapore (MAS); the City of London in the United Kingdom; Nice and Paris in France; and New York in the United States.

EDHEC-Risk has developed a close partnership with a small number of sponsors within the framework of research chairs or major research projects:• Core-Satellite and ETF Investment, in partnership with Amundi ETF• Regulation and Institutional Investment, in partnership with AXA Investment Managers• Asset-Liability Management and Institutional Investment Management, in partnership with BNP Paribas Investment Partners• Risk and Regulation in the European Fund Management Industry, in partnership with CACEIS• Exploring the Commodity Futures Risk Premium: Implications for Asset Allocation and Regulation, in partnership with CME Group

Founded in 1906, EDHEC is one of the foremost international

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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

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• Asset-Liability Management in Private Wealth Management, in partnership with Coutts & Co.• Asset-Liability Management Techniques for Sovereign Wealth Fund Management, in partnership with Deutsche Bank• The Benefits of Volatility Derivatives in Equity Portfolio Management, in partnership with Eurex• Structured Products and Derivative Instruments, sponsored by the French Banking Federation (FBF)• Optimising Bond Portfolios, in partnership with the French Central Bank (BDF Gestion)• Asset Allocation Solutions, in partnership with Lyxor Asset Management• Infrastructure Equity Investment Management and Benchmarking, in partnership with Meridiam and Campbell Lutyens• Investment and Governance Characteristics of Infrastructure Debt Investments, in partnership with Natixis• Advanced Modelling for Alternative Investments, in partnership with Newedge Prime Brokerage• Advanced Investment Solutions for Liability Hedging for Inflation Risk, in partnership with Ontario Teachers’ Pension Plan• The Case for Inflation-Linked Corporate Bonds: Issuers’ and Investors’ Perspectives, in partnership with Rothschild & Cie• Solvency II, in partnership with Russell Investments• Structured Equity Investment Strategies for Long-Term Asian Investors, in partnership with Société Générale Corporate & Investment Banking

The philosophy of the Institute is to validate its work by publication in international academic journals, as well as to make it available to the sector through its position papers, published studies, and conferences.

Each year, EDHEC-Risk organises three conferences for professionals in order to present the results of its research, one in London (EDHEC-Risk Days Europe), one in Singapore (EDHEC-Risk Days Asia), and one in New York (EDHEC-Risk Days North America) attracting more than 2,500 professional delegates.

EDHEC also provides professionals with access to its website, www.edhec-risk.com, which is entirely devoted to international asset management research. The website, which has more than 65,000 regular visitors, is aimed at professionals who wish to benefit from EDHEC’s analysis and expertise in the area of applied portfolio management research. Its monthly newsletter is distributed to more than 1.5 million readers.

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




The EDHEC-Risk Institute PhD in FinanceThe 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 BusinessThe 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

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 know-how 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)




2014• 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 cross-sectional 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).

• Calamia, A., L. Deville, and F. Riva. Liquidity in european equity ETFs: What really matters? (March).

EDHEC-Risk Institute Publications (2011-2014)



• 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).

• 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).

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 index-

weighting 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 index-weighting 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).




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).

EDHEC-Risk Institute Position Papers (2011-2014)

For more information, please contact: Carolyn Essid on +33 493 187 824 or by e-mail to: [email protected]

EDHEC-Risk Institute393 promenade des AnglaisBP 3116 - 06202 Nice Cedex 3FranceTel: +33 (0)4 93 18 78 24

EDHEC Risk Institute—Europe 10 Fleet Place, LudgateLondon EC4M 7RBUnited KingdomTel: +44 207 871 6740

EDHEC Risk Institute—Asia1 George Street#07-02Singapore 049145Tel: +65 6438 0030

EDHEC Risk Institute—North AmericaOne Boston Place, 201 Washington StreetSuite 2608/2640 — Boston, MA 02108United States of America Tel: +1 857 239 8891

EDHEC Risk Institute—France 16-18 rue du 4 septembre75002 Paris FranceTel: +33 (0)1 53 32 76 30

www.edhec-risk.com

Tail Risk of Smart Beta Portfolios: An Extreme Value ... · Tail Risk of Smart Beta Portfolios: An Extreme Value Theory Approach — July 2014 About the Authors Lixia Loh is a senior

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