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

Transcription

1 An EDHEC-Risk Institute Publication Tail Risk of Smart Beta Portfolios: An Extreme Value Theory Approach July 2014 Institute

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

3 Table of Contents Executive Summary Introduction Smart Beta Indices A Conditional EVT Model Empirical Analysis...31 Conclusion...43 Appendix...47 References...57 About EDHEC-Risk Institute...61 EDHEC-Risk Institute Publications and Position Papers ( )...65 An EDHEC-Risk Institute Publication 3

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

5 Executive Summary An EDHEC-Risk Institute Publication 5

6 Executive Summary 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 com. 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 research 1 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, the improved 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 6 An EDHEC-Risk Institute Publication

7 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. 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 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 the more 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 management perspective, it is important to segregate the two components because the dynamics of volatility contributes to the unconditional tail thickness phenomenon. Generally, the GARCH part is responsible for capturing the dynamics of volatility while EVT provides a model for the behaviour of the extreme tail of the distribution. 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 return CVaR 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 An EDHEC-Risk Institute Publication 7

8 Executive Summary 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 factortilted 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. 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. As far as absolute returns are concerned, we find no evidence of statistically significant differences 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 8 An EDHEC-Risk Institute Publication

9 1. Introduction An EDHEC-Risk Institute Publication 9

10 1. Introduction 4 - An implementation of the methodology with complete and transparent documentation is available at scientificbeta.com. 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 adhoc 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 far as 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 approach 4, 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. 10 An EDHEC-Risk Institute Publication

11 1. Introduction 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 Valueat-risk (VaR) and Conditional Value-atrisk (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 EVTbased 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; Loh and 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 stockselection 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 An EDHEC-Risk Institute Publication 11

12 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 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 volatility or 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 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 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 stockselection 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. 12 An EDHEC-Risk Institute Publication

13 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 capweighted 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 CAPMtype 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 firstorder importance. 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. 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 An EDHEC-Risk Institute Publication 13

14 1. Introduction 14 An EDHEC-Risk Institute Publication

15 2. Smart Beta Indices An EDHEC-Risk Institute Publication 15

16 2. Smart Beta Indices 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 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 briefly discuss the criteria used to construct factortilted 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 scientificbeta.com Efficient Minimum Volatility The 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 weight constraints. 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 of Minimum 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. The optimal weights are additionally adjusted to satisfy the box constraints An EDHEC-Risk Institute Publication

17 2. Smart Beta Indices 2.2. Efficient Maximum Sharpe Ratio Efficient 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 Maximum Deconcentration Originating 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 An EDHEC-Risk Institute Publication 17

18 2. Smart Beta Indices 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. 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 Maximum Decorrelation Maximum 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 correlationadjusted concentration. More formally, the optimal weights are obtained by solving the following 2.5. Diversified Risk Weighted Diversified 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 Factor-tilted Smart Beta Indices Apart from studying the tail risk resulting from different weighting schemes, we also explore how tilting the portfolio 18 An EDHEC-Risk Institute Publication

19 2. Smart Beta Indices 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 threefactor 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 not be 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 An EDHEC-Risk Institute Publication 19

20 2. Smart Beta Indices 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 and pushes the price up. 20 An EDHEC-Risk Institute Publication

21 3. A Conditional EVT Model An EDHEC-Risk Institute Publication 21

22 3. A Conditional EVT Model In finance, EVT has been traditionally applied to estimate probabilities of extreme losses or loss thresholds such that losses beyond it occur with a predefined small probability, which are also known as high quantiles of the portfolio loss distribution. In fact, EVT provides a model for the extreme tail of the distribution which turns out to have a relatively simple structure described through the corresponding limit distributions such as the Generalised Extreme Value (GEV) distribution or the Generalised Pareto Distribution (GPD) The Peak-over-Threshold Method The approach in this paper is based on the peak-over-threshold method (POT), see Loh and Stoyanov (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 F u (x) which is expressed through the unconditional distribution in the following way, (3.1) where x > 0. Because we are interested in the extreme losses, we need to gain insight into the probability that the excesses beyond u, X u, can exceed a certain loss level. Thus, (3.1) is re-stated in terms of the tail, (3.2) There is a celebrated limit result in EVT which states that as u increases towards the right endpoint of the support of the loss distribution denoted by x F, the conditional tail converges to the tail of the GPD which is defined by, (3.3) where 1 + ξx > 0 and β > 0 is a scale parameter. The limit results is (Embrechts et al., 1997, Chapter 3) (3.4) where β(u) is a scaling depending on the selected threshold u. The limit result in (3.4) can be used to construct an approximation for the tail of the losses exceeding a high threshold u. If we denote by y = u + x and express x in terms of y in (3.2), we obtain (3.5) after substituting the limit law for. For a fixed threshold u, note that is a constant and the tail 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. 22 An EDHEC-Risk Institute Publication

23 3. A Conditional EVT Model 6 - An approach based on adaptive calibration of the threshold is adopted by some authors. Gonzalo and Olmo (2004) describe a method based on minimising the distance between the empirical and the tail of the GPD with parameters estimated through the maximum likelihood method. The suggested distance is the Kolmogorov-Smirnov statistic. 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 distribution belongs to the MDA of the Gumbel law if and only if in which β(u) is a scaling function and can be chosen to be equal to the average excess loss provided that the loss exceeds the threshold x, (3.6) This choice of β(u) is also known as the mean excess function. This MDA is characterised in terms of excess losses that exhibit an asymptotic exponential decay and consists of distributions with a diverse tail behaviour: from moderately heavytailed such as the log-normal to light-tailed distributions such as the Gaussian or even distributions with bounded support having an exponential behaviour near the upper end of the support x F. For further detail, see (Embrechts et al., 1997, Section 3.3.3). The Weibull MDA, ξ < 0 This MDA consists entirely of distributions with bounded support and is, therefore, not interesting for modelling the behaviour of risk drivers. Distributions that belong to this MDA include for example the uniform and the beta distribution. For further detail, see (Embrechts et al., 1997, Section 3.3.2). Finally, we should note that one distribution can be in only one MDA. There are examples of distributions that are not in any of the three MDAs but they are, however, rather artificial. To apply (3.5) in practice, we need to choose a high threshold u and also to estimate the probability. 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 X 1, X 2,...,X n is a sample of i.i.d. portfolio losses. If u is chosen such that exactly m observations are excesses, then the approximation in (3.5) becomes (3.7) An EDHEC-Risk Institute Publication 23

24 3. A Conditional EVT Model 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). where s = 1 m/n and X s,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 = (Y 1,..., Y n1 ) 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 Y i = X s+i,n X s,n, where i = 1,..., n 1 = n s and s is defined as s = 1 m/n. Information about other estimators, such as the Hill and the Pickands estimator, and further detail on the relevance of the MLE are available in de Haan and Ferreira (2006) A GARCH-EVT Model for Tail Risk Estimation Instead of applying the POT method to the time series directly, we prefer to build a model for the time-varying characteristics and apply EVT to the residuals of the model having explained away, at least partly, the temporal structure of the time series. 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 general GARCH filter. 8 Denote the time series of portfolio losses by X t. The GARCH(1,1) model is given by: (3.9) where, the innovations Z t are i.i.d. random variables with zero mean, unit variance and marginal distribution function F Z (x) and K, a, and b are the positive parameters with a+b < 1. The model in (3.9) 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 F Z. 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 24 An EDHEC-Risk Institute Publication

25 3. A Conditional EVT Model 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, X 0.9 x n,n is the 90-th observation in the sorted sample and the empirical approximation to would be the largest observation in the sample. As a consequence, the correction term in (3.12) allows us to go beyond the available data points in the sample which emphasises a key advantage of EVT to the historical method. and explicitly state the risk forecasts built through the probabilistic model. The discussion below assumes that the random variable X describes portfolio losses and VaR and CVaR are defined for the right tail of the loss distribution which translates into the left tail of the portfolio return distribution. The same quantities for the right tail of the return distribution (left tail of the loss distribution) are obtained from the definitions below by considering X instead of X; that is, the downside of a short position is the upside of the corresponding long position. The risk functionals are, however, multiplied by 1 to preserve the interpretation that negative risk means a potential for profit. Value-at-Risk The VaR of a random variable X describing portfolio losses at a tail probability p, VaR p (X), is implicitly defined as a loss threshold such that over a given time horizon losses higher than it occur with a probability p. By construction, VaR is the negative of the p-th quantile of the portfolio return distribution or the (1 p)-th quantile of the portfolio loss distribution. In the industry, VaR is often defined in terms of a confidence level but we prefer to reserve the term confidence level for the context of statistical testing which we need in Section 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. F X (x) = P(X x) which is also known as the quantile function of X. As explained earlier, we employ EVT to estimate high quantiles of the loss distribution. To this end, we adopt the approximation of the tail in (3.5). Solving for the value of y yielding a tail probability of p, we get (3.11) The estimator is derived from (3.7) in the same way. Suppose that X 1,n X 1,n... X n,n denote the order statistics, then following (3.7) we get (3.12) 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 X s,n, which is such that p < m/n, plus a correction term obtained through the GPD. In the implementation, we set m/n = 0.1 and, thus, in terms of quantiles the 99% quantile equals the 90% quantile (X(0.9 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 An EDHEC-Risk Institute Publication 25