Risk Parity-based Smart Beta ETFs and Estimation Risk

Transcription

1 Risk Parity-based Smart Beta ETFs and Estimation Risk Olessia Caillé, Christophe Hurlin and Daria Onori This version: March Preliminary version. Please do not cite. Abstract The aim of this paper is to study the influence of the estimation risk on the risk parity strategy for the case of smart beta ETFs. Taking into account the estimation risk implies to test whether the risk contributions of each asset in the index are significantly different or not from fixed risk budgets. For that, we propose a formal statistical testing framework. This inference procedure allows to determine a set of index weights that satisfies the risk parity principle. Contrary to the case where the estimation risk is neglected, the risk parity allocation is not uniquely defined. Having a set of allocations that satisfies the risk parity, instead of a unique weighted index, gives the investor the opportunity to choose among them the one which minimizes the total risk, maximizes the diversification or any other secondary objective. Thus, one can define mixed strategies for smart beta ETFs. For instance, a mixed risk-parity / minimum global variance strategy consists in determining the weights of the global minimum variance index among all the weighted indices that satisfy the risk parity strategy. We show that mixed strategies based on the estimation risk improve the performance of smart beta ETFs. Keywords: Asset allocations; Risk parity; Estimation risk; Hypothesis Testing; Mixed Smart Beta strategies. JEL classification: G11; C12; C60. University of Orléans (LEO, UMRS CNRS 7332). University of Orléans (LEO, UMRS CNRS 7332). University of Orléans (LEO, UMRS CNRS 7332). 1

2 1 Introduction With their low fees and their relative performance compared to standard market capitalization-weighted indices (Maillard, Roncalli and Teiletche (2010)), smart beta exchange-traded funds (ETFs) have become popular investment vehicles among individual and institutional investors alike. Out of the approximately $1.7 trillion of assets in all exchange-traded products, there were 342 smart (or strategic) beta ETFs, with collective assets under management of about $291 billion, or 18% of that total, according to Morningstar (2014). These products are particularly popular in Europe. The survey of Russel Investments (April 2014) indicates that 40% of asset owners in Europe have smart beta allocations. Among these holders, about 50% are satisfied and plan to increase their investments. There are many definitions of smart beta ETFs. Millet (2015) defines smart beta as rules-based investment strategies that do not rely on market capitalization. More broadly, ETFs that follow this type of investment strategy generally use a method that tracks a market index which is not based on the traditional market capitalization weights. The risk-adjusted performance of the smart beta ETFs comes from the different ways they configure the underlying index. Roncalli (2014) identifies two groups of smart beta strategies. The first one is based on fundamental or factors indexation (factor-based index, equally weighted index, fundamentally weighted index, etc.). The second group corresponds to risk-based strategies which include, among others, maximum diversification, minimum global variance and risk parity strategies. Such a risk-weighted approach to smart beta necessarily involves estimating some risk measures. It is especially the case for the risk parity strategy which imply to evaluate the risk contribution of all the assets. Whatever the risk measure considered (volatility, Value-at-Risk, expected shortfall, etc.) and the model used, this induces an estimation risk. The estimation risk depends on the estimation method considered, on the sample size used for the estimation, etc., but it cannot be neglected. The aim of this paper is to study the influence of the estimation risk on the risk parity strategy for the case of smart beta ETFs. Taking into account the estimation risk implies to test whether the risk contributions of each asset in the index are significantly different (or not) from some fixed risk budgets. For that, we propose a formal statistical testing framework. This inference procedure allows to determine a set of index weights that satisfies the risk parity principle. Contrary to the case where the estimation risk is neglected, the risk parity allocation is not uniquely defined. Having a set of allocations that satisfies the risk parity, instead of a unique weighted index, gives the investor the opportunity to choose among them the one which minimizes the total risk, maximizes the diversification or any other secondary objective. Thus, one can define mixed strategies for smart beta ETFs. For instance, a mixed risk-parity / minimum global variance strategy consists in determining the 2

3 weights of the global minimum variance index among all the weighted indices that satisfy the risk parity strategy. One objective of our research proposal consists in determining if these mixed strategies based on the estimation risk improve the performance of smart beta ETFs. One advantage of the risk-based smart betas is that they imply a lower parameter estimation risk than the alternative strategies based on the expected return estimates. Nevertheless they still use a statistical estimation of the covariance and thus have some estimation risk. This problem is particularly relevant when correlations are high, exhibiting weight instability especially when the volatilities are similar. A solution consists in using techniques like shrinkage of the covariance matrix or thresholding methods. However, even in these cases, the estimation risk cannot be overlooked. One could also argue that the estimation risk can be neglected since it is generally possible to collect a large sample (in the time dimension) of returns to precisely estimate the risk measures. But there is always a trade-off between accuracy and data issues, since a large sample size may induce nonstationarity, missing values (apparition or vanishing of firms, merges, etc.), outliers, breaks, etc. Most of times, this leads to consider relatively small samples, with less than 5 years of data, for which the estimation risk cannot be considered as negligible. Our study is related to the current literature devoted to the estimation risk in the risk measures. This risk is generally assessed through asymptotic confidence intervals. For instance, Chan et al. (2007) and Francq and Zakoïan (2015) derive the asymptotic confidence intervals for the conditional VaR estimator in the specific context of heavy-tailed GARCH models. Gouriéroux and Zakoïan (2013) consider a different approach based on an Estimation adjusted VaR (EVaR). Alternatively, several papers propose resampling methods to carry out inference on risk measures. Danielson and Zhou (2016) in their paper untitled Why risk is so hard to measure, assess the estimation error of VaR and ES forecasts. Here, our approach is different since we do not measure the estimation risk, but we propose a test of equality of risk measures that takes into account the estimation risk. The logic is similar to Hurlin et al. (2016) who propose a bootstrap-based test of the null hypothesis of equality of two firms conditional Risk Measures (RMs) at a single point in time. The rest of the research proposal is organized as follows. In Section 2 we define the risk parity strategy and the corresponding estimation risk. Section 3 introduces the test for risk parity. An illustration is given in Section 4. Section 5 introduces the mixed strategies for risk parity smart beta ETFs and the last section concludes. 2 Smart Beta ETFs and Risk Parity Strategy The risk parity (RP thereafter) strategy consists in choosing optimal weights in order to equalize risk contributions of assets composing the portfolio to a set risk 3

4 budgets expressed in nominal terms or as a percentage of the total portfolio risk. These contributions can be computed with different measures of risk such as the volatility, the Value-at-Risk or the expected shortfall, as long as the chosen measure respects the coherency and convexity properties and is homogenous of degree 1 (Roncalli, 2014). 1 The RP strategy is then designed to reduce the lack of diversification which occurs with cap-weighted indices. Figure 1 illustrates the risk contributions obtained in a traditional 60% Equities 40% Bonds portfolio (panel A) where equities represent 90% of the total risk (panel B). On the contrary, a particular RP-based portfolio, called the Equal Risk Contribution (ERC) portfolio, implies to determine the weights (panel C) so as to obtain the same risk contribution for each asset (panel D). By definition, this RP portfolio is more diversified in terms of risk. Furthermore, the RP appears to be a good alternative to usual diversification strategies such as minimum global variance or equal weights for instance. Some empirical studies show that a RP portfolio has a better mean return and Sharpe ratio, and also a weak turnover (Maillard, Roncalli and Teiletche, 2010; Ferrarese, Khan and Buckle 2015). Figure 1: Illustration of the Risk Parity Strategy Formally, the optimal weights associated with a RP strategy (or a Risk Budgeting strategy) can be obtained as the solution of a simple optimization program. Let us consider a universe of n risky assets indexed by i and denote by r t = (r 1t,..., r nt ) the vector of assets returns at time t. Let σ 2 i be the variance of asset i, σ ij the covariance 1 Notice that the variance cannot be a potential risk measure to apply because it is homogeneous of degree two with respect to the weights. 4

5 between assets i and j and Σ be the covariance matrix. Denote ω = (ω 1,..., ω n ) the portfolio weights with ω i 0 (no short-selling), n i=1 ω i = 1 and σ (ω) = (ω Σω) 1/2 the volatility of the portfolio returns. The risk contribution of the asset i is defined by the product of the weight ω i by the marginal risk contribution σ (x) / ω i. Define S i (ω, Σ) the risk contribution standardized by the portfolio risk as S i (ω, Σ) ω i σ (x) σ (x) ω i = ω i (Σω) i ω Σω (1) where (Σω) i the i th row of the n 1 vector Σω. The Euler s theorem implies that n i=1 S i (ω, Σ) = 1. (2) Denote by S (ω, Σ) = (S 1 (ω, Σ),..., S n (ω, Σ)) the vector of risk contributions of the n assets. The RP strategy consists in determining an index such that the risk contributions S (ω, Σ) are equal to a target of risk contributions, denoted b = (b 1,..., b n ) with n i=1 b i = 1 and b i > 0, fixed by the ETF issuer and the investor (Roncalli, 2014). 2 The simplest RP strategy is the ERC which consists in finding a portfolio such that the risk contribution is the same for all assets of the index (cf. Figure 1). It can be viewed as a special case with b i = 1/n for i = 1,..., n. The RP strategy-based portfolio, denoted ω, is defined as follows ω = {ω ]0, 1] n : n i=1 ω i = 1, S i (ω, Σ) = b i i = 1,..., n}. (3) Notice that, under mild assumptions on the covariance matrix Σ, the risk budgeting weight vector ω exists and is uniquely defined. For n = 2, there is a closed form solution for ω. For the general case with n > 2, there is no analytical solution. However, the risk budgeting index can be obtained by numerical optimization (see Roncalli 2014, for more details). In practice, the risk contributions S (ω, Σ) are unobservable and have to be estimated from a sample {r t } T t=1. If we denote Σ a consistent estimator of Σ, then the feasible risk budgeting strategy-based index becomes ω = {ω ]0, 1] n : n i=1 ω i = 1, S i (ω, Σ) = b i i = 1,..., n}. (4) This empirical approach based on the point estimate Σ leads to neglect the estimation risk due to the distance between Σ and Σ. However, it is obvious that some portfolios ω that are close to ω, also satisfy definition 3 of RP given the estimation risk in Σ. The first issue consists in determining this set of elligible portfolios with a formal inference procedure. 2 We exclude the particular case b i = 0 since it necessarily implies a null weight for the asset i. 5

6 3 Risk Parity and Estimation Risk Taking into account the estimation risk within the RP strategy leads to define a formal test procedure that can be defined as H 0 : S (ω, Σ) = b. (5) The need for statistical inference comes from the fact that the risk contributions S i (ω, Σ) are not observed, since Σ is unknown and is replaced by a consistent estimator Σ. So, the null hypothesis is based on the true risk contribution implied by Σ, while the estimated value S(ω, Σ) is affected by the estimation risk on Σ. Our test boils down to the question of whether Ŝ (ω) b is large enough relative to the parameter estimation error coming from Σ to reject the null. In order to test for H 0, we can define a simple Wald test statistic given by W (ω, Σ) = (S(ω, Σ) b) V asy (S(ω, Σ)) 1 (S(ω, Σ) b) (6) where V asy (S(ω, Σ)) denotes the asymptotic covariance matrix of the estimator S(ω, Σ). Under some regularity conditions, W (ω) converges in distribution to a chi-squared distribution with n 1 degrees of freedom, since n i=1 b i = 1 and n i=1 S i (ω, Σ) = 1 for any matrix Σ that belongs to the set of positive definite matrices. For a risk level α [0, 1], the set of portfolios that are significantly not different from a RP strategy is then defined as Ω (α) = {ω ]0, 1] n : n i=1 ω i = 1, W (ω, Σ) < χ 2 1 α (n 1)} (7) where χ 2 1 α (n 1) denotes the 1 α quantile of the chi-squared distribution with n 1 degrees of freedom. The main difference with the standard RP strategy based on the point estimate Σ, is that the optimal portfolio ω is not uniquely defined. Taking into account the estimation risk in the risk contribution S(ω, Σ) leads to define a set Ω (α) of optimal portfolios for which we cannot reject the null of RP. This is the main advantage of our approach. This set of RP portfolios Ω (α) has some desirable properties. First, when the sample size T tends to infinity, the continuous mapping theorem implies that S(ω, Σ) p S(ω, Σ 0 ), where Σ 0 denotes the true value of the covariance matrix. Thus, it can be shown that Ω (α) p ω. (8) When the sample size T tends to infinity, i.e. when there is no estimation risk, the set of optimal portfolios shrinks to a single portfolio. This portfolio corresponds to the standard RP portfolio ω, based on the point estimate Σ. Second, we can show that for a given finite sample size T, the size of the set Ω (α) decreases with the risk level α, i.e. card (Ω (α)) / α < 0. For a fixed sample size T, when α tends to 1, the set of optimal portfolios shrinks to the standard RP portfolio ω. 6

7 The computation of the Wald statistics W (ω, Σ) requires to derive the asymptotic covariance matrix of S(ω, Σ). This derivation can be obtained under mild assumptions. Assume that Σ is a consistent and asymptotically normally distributed estimator with T (vec( Σ) vec (Σ0 )) d N (0, Δ) (9) where Δ is a n 2 n 2 matrix and vec (.) the vectorialization operator. Then, the delta method implies that ( ( T S(ω, Σ) d S(ω, Σ 0 )) N 0, S(ω, Σ 0) vec (Σ) Δ S(ω, Σ ) 0) vec (Σ). (10) Denoting by Δ a consistent estimator of Δ, a consistent estimator of the the covariance matrix of S(ω, Σ) is given by 4 Illustration V asy (S(ω, Σ)) = 1 T S(ω, Σ) vec (Σ) Δ S(ω, Σ) vec (Σ) (11) In order to illustrate the influence of the estimation risk on the Equal Risk Contributions (ERC) strategy, we consider a simple framework with three assets (n = 3) whose returns are normally distributed with μ = r t N (μ, Σ) (12) Σ = (13) The covariance matrix Σ is estimated by maximum likelihood. Figure 2 displays the mean and the variance of all the feasible portfolios. Those for which the ERC hypothesis (equation 5 with b i = 1/3) is rejected for a significant level α are represented by blue points, whereas those for which the ERC cannot be rejected are colored in green. The point that corresponds to the standard ERC portfolio based on the point estimate Σ is colored in red. For a sample size T = 50 and a significant level α = 1% (panel A), the set Ω (α) of ERC portfolios is relatively large. When the sample size increases to T = 2500 (about ten years of daily observations), Ω (α) approximately corresponds to the standard ERC portfolio (panel B). A similar result (not reported) can be obtained for a smaller sample size as soon as the risk level α is sufficiently high. 7

8 Figure 2: ERC strategy and Estimation Risk Panel A: T = 50, α = 1% Panel B: T = 2500, α = 1% Mean 0.06 Mean Portfolios RB Portfolios with estimation risk RB portfolio Variance Portfolios RB Portfolios with estimation risk RB portfolio Variance 5 Mixed Strategies for Smart Beta ETFs Given the set of eligible portfolios for a RP strategy, it is possible to build a mixed strategy for funds portfolios or smart beta ETFs. Indeed, one can combine the RP strategy with a secondary investment objective. This objective maybe risk-based. For instance, one can choose the minimum global variance portfolio among all the risk parity portfolios. Alternatively, it is possible to use a fundamental/factor based indexation approach or a performance criteria. To elaborate further this point, consider the example of a mixed strategy RP - minimum variance. The optimal portfolio ω is then defined as the solution of the following optimization problem ω = arg min ω Σω (14) ω Ω (α) u.c. ω 0 1 ω = 1 When T or α 1 the optimal portfolio ω corresponds to the standard RP portfolio ω, since the set Ω (α) is reduced to one point. On the contrary, when the estimation risk is high or when the investor accepts a low probability of type I error (reject the RP null hypothesis when it is true), both portfolios may be largely different. For instance, in the limit case where α tends to 0, the optimal portfolio ω corresponds to the global minimum variance portfolio. One objective of this research consists in comparing the relative performances of RP smart beta ETFs to smart beta based on mixed strategies that take into account the estimation risk. Such a comparison requires to adapt our methodology to realistic configurations (i) with a large number of assets, (ii) a realistic time 8

9 dimension for the estimation sample and (iii) a reasonable risk level. This implies for instance to consider either non-parametric approaches for the estimation of the risk contributions, shrinkage estimators (Candelon, Hurlin and Tokpavi, 2012) or Bayesian approaches. Besides, we will use the fast algorithms for computing highdimensional RP portfolios proposed by Griveau-Billion, Richard and Roncalli (2013). This comparison will be based on the performance criteria proposed for the ETFs by Hassine and Roncalli (2013). 6 Conclusions In this paper we have studied the influence of the estimation risk on the risk parity strategy for the case of Smart Beta ETFs. To do so, we have proposed a statistical testing framework which aims to test if the risk contributions of each assets in the index are significantly equal to some fixed risk budgets, chosen by the fund or the investor. The main advantage of this approach is that it allows to determine a set of index weights that satisfies the risk parity strategy, contrary to optimization techniques which neglect the estimation risk. Having a set of portfolios gives the investor the opportunity to combine the risk parity strategy with other criteria commonly used in Smart Beta strategies. These mixed strategies include for example the selection of the portfolio which minimizes global variance among those which satisfy the risk parity principle. 9

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