Maximising an Equity Portfolio Excess Growth Rate: A New Form of Smart Beta Strategy?

Transcription

1 An EDHEC-Risk Institute Publication Maximising an Equity Portfolio Excess Growth Rate: A New Form of Smart Beta Strategy? November 2017 with the support of Institute

2 Table of Contents 1. Introduction Theoretical Analysis of the Max Excess Growth Rate Portfolio Empirical Analysis on Individual Stocks Empirical Analysis on Portfolios Conclusions and Extensions...55 References...57 About EDHEC-Risk Institute...59 EDHEC-Risk Institute Publications and Position Papers ( ) This project is sponsored by BdF Gestion in the context of the Maximising and Harvesting the Rebalancing Premium in Equity Markets research chair at EDHEC-Risk Institute. Printed in France, November Copyright EDHEC The opinions expressed in this study are those of the authors and do not necessarily reflect those of EDHEC Business School.

3 Abstract It has been argued that the simple act of resetting portfolio weights back to the original weights can be a source of additional performance. This additional performance is known as the rebalancing premium and a detailed analysis (see for example Fernholz (2002)) suggests that the portfolio excess growth rate, defined as the difference between the portfolio expected growth rate and the weighted-average expected growth rate of the assets in the portfolio, is an important component of the rebalancing premium. In this context, one might wonder whether maximising a portfolio excess growth rate in the presence or in the absence of a risk-free asset would lead to an improvement in the portfolio performance or risk-adjusted performance. This paper provides a thorough empirical analysis of the maximisation of an equity portfolio excess growth rate in a portfolio construction context both for individual stocks and factor-tilted equity portfolios. In out-of-sample empirical tests conducted on individual stocks from 4 different regions (US, UK, Eurozone and Japan), we find that portfolios that maximise the excess growth rate are characterised by a strong negative exposure to the low volatility factor and a higher than 1 exposure to the market factor, implying that such portfolios are attractive alternatives to competing smart portfolios in markets where the low volatility anomaly does not hold (e.g., in the UK, or in rising interest rate scenarios) or in bull market environments. In parallel, our empirical analysis of US factor-tilted universes shows an outperformance in terms of mean return, growth rate and Sharpe ratio for portfolios that maximise the excess growth rate with respect to equally-weighted portfolios for 4 stock selections, namely past winners, past losers, high vol and high investment stocks. These results suggest in particular that maximising a portfolio excess growth rate is a welfare-improving approach for momentum portfolios. An EDHEC-Risk Institute Publication 3

4 About the Authors Jean-Michel Maeso is a senior quantitative researcher at EDHEC-Risk Institute. Previously, he spent 5 years in the financial industry specialising in research, development and implementation of investment solutions (structured products and systematic strategies) for institutional investors. He holds an engineering degree from the Ecole Centrale de Lyon with a specialisation in applied mathematics. Lionel Martellini is Professor of Finance at EDHEC Business School and Director of EDHEC-Risk Institute. He has graduate degrees in economics, statistics, and mathematics, as well as a PhD in finance from the University of California at Berkeley. Lionel is a member of the editorial board of the Journal of Portfolio Management and the Journal of Alternative Investments. An expert in quantitative asset management and derivatives valuation, his work has been widely published in academic and practitioner journals and he has co-authored textbooks on alternative investment strategies and fixed-income securities. 4 An EDHEC-Risk Institute Publication

5 1. Introduction An EDHEC-Risk Institute Publication 5

6 1. Introduction It has been argued that the simple act of resetting portfolio weights back to the original weights can be a source of additional performance. This additional performance is known as the rebalancing premium, also sometimes referred to as the volatility pumping effect or diversification bonus because volatility and diversification turn out to be key components of the rebalancing premium. A detailed analysis suggests that the portfolio excess expected growth rate (or excess growth rate in short), denoted by and defined as the difference between the portfolio expected growth rate and the weighted-average expected growth rate of the assets in the portfolio, is an important component of the rebalancing premium. In this context, one might wonder whether maximising a portfolio's excess growth rate would allow one to substantially increase its risk-adjusted performance. Besides, if all assets have the same expected growth rate, then maximising the portfolio excess growth rate is equivalent to maximising the portfolio expected growth rate, which in turn is well-known to be equivalent to maximising expected utility for an investor endowed with logarithmic preferences. Interestingly, the excess growth rate maximising portfolio, just like the minimum variance, equal risk contribution or maximum diversification portfolios, only requires estimates for the covariance matrix, and not for the vector of expected returns. The main purpose of this paper is to provide a thorough empirical analysis of the benefits of maximising an equity portfolio excess growth rate in a context where the constituents are either individual stocks or portfolios sorted according to factor/sector exposure. We formally define the Max portfolio strategy as the weighting scheme that maximises the ex-ante excess growth rate of the portfolio in the absence of a risk-free asset, and the MVR strategy (MVR stands for Maximum Volatility Return portfolio, as defined by Mantilla-Garcia (2016)) as the risky portfolio maximising the ex-ante excess growth rate in the presence of a risk-free asset. We also compare the Max and MVR portfolios to benchmark portfolios such as cap-weighted portfolios, equally-weighted portfolios, minimum variance portfolios, equal risk contribution portfolios, and maximum diversification portfolios. Our analysis relies upon an improved estimate of the covariance matrix based on an implicit factor models, with factors extracted from a principal component analysis and random matrix theory used in the robust selection of statistically relevant factors. An important contribution of the paper is to provide a detailed analysis of whether the benefits of excess growth rate maximisation can be enhanced through the selection of stocks with homogeneous characteristics. We also test whether the strategy performs better in an application to portfolios of stocks, as opposed to individual stocks. In this context, the analysis first focuses on detailed empirical tests for stocks sorted according to sector classifications or to standard characteristics such as market capitalisation, book-to-market ratio, past performance, past volatility, etc. In out-of-sample empirical tests conducted on individual stocks from 4 different regions (US, UK, Eurozone and Japan), we find that portfolios that maximise the excess growth rate are characterised by a strong negative exposure to the low volatility factor and a higher than 1 exposure to the market factor, implying that such portfolios are attractive alternatives to competing smart portfolios in markets where the low volatility anomaly does not hold (e.g., in the UK, or in rising interest rate scenarios) or in bull market environments. In parallel, our empirical 6 An EDHEC-Risk Institute Publication

7 1. Introduction analysis on US factor-tilted universes shows an outperformance in terms of mean return, growth rate and Sharpe ratio for portfolios that maximise the excess growth rate with respect to equally-weighted portfolios for 4 stock selections, namely past winners, past losers, high vol and high investment stocks. These results suggest in particular that maximising a portfolio excess growth rate is a welfare-improving approach for momentum portfolios. The rest of the paper is organised as follows. In Section 2, we provide a detailed theoretical analysis of the properties of the excess growth rate maximising the Max portfolio, its link with the MVR portfolio as well as its relationship with respect to other popular weighting schemes. In Section 3, we perform a series of thorough empirical tests on the basis of individual stocks sorted according to various characteristics. In Section 4, we measure the benefits of the process in an application to equity portfolios, as opposed to individual stocks. Finally our conclusions and suggestions for further research can be found in Section 5. An EDHEC-Risk Institute Publication 7

8 1. Introduction 8 An EDHEC-Risk Institute Publication

9 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio An EDHEC-Risk Institute Publication 9

10 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio We first introduce a general continuous-time framework, before presenting a detailed theoretical analysis of the portfolio excess growth rate. 2.1 General Theoretical Framework In a general setting involving n t risky assets with correlation matrix, volatility vector, covariance matrix, and weight vector, the excess growth rate can be written as half of the difference between the weighted-average variance of the assets in the portfolio and the portfolio variance: Deriving the Excess Growth Rate from the Portfolio Wealth Fernholz (2002) defines a standard Ito-process model for a financial market composed of n risky stocks and considers stock prices S i (t) as processes verifying the following stochastic differential equations: (2.1) where (W 1 (.),...,W m (.)) is a vector of m-independent Brownian motions (m n), μ(.) = (μ 1 (.),..., μ n (.)) is the vector of stocks expected return and is the (n m) matrix of volatilities representing the sensitivity of each stock to the m sources of uncertainty. The multi-dimensional Ito lemma applied to stock prices shows that (2.1) is equivalent to: (2.2) where are the individual expected growth rates of the stocks and is the covariance matrix of the stocks. The covariance matrix σ (.) is linked to the β(.) matrix as follows: (2.3) In this setup, Fernholz (2002) defines a strategy at time t where the investor put a fraction w i (t) of his wealth in asset i. The value of the investor portfolio at time t is with. Fernholz (2002) establishes that the value of the 10 An EDHEC-Risk Institute Publication

11 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio portfolio P(t) follows the process: (2.4) with the expected return of the portfolio, and the volatility of the portfolio. The multi-dimensional Ito lemma applied to the portfolio value shows that (2.1) is equivalent to: (2.5) with the expected growth rate of the portfolio, and the excess growth rate of the portfolio. Booth and Fama (1992) attribute this incremental return to diversification. They showed that the expected growth rate of a constituent and of the portfolio can be respectively approximated as: and. Then we have: can be rewritten as: and the excess growth rate Relation Between the Excess Growth Rate and the Rebalancing Premium We apply Ito s lemma to the log price of asset i ln S i (t): (2.6) The wealth P reb (t) of a continuously rebalanced portfolio to constant weights (w 1,...,w n ) verifies: (2.7) An EDHEC-Risk Institute Publication 11

12 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio where process. is a normalised Wiener Ito s lemma applied to equation (2.7) implies: (2.8) Then equations (2.6) and (2.8) imply: (2.9) and: (2.10) 1 - We assume that. (2.11) Finally: (2.12) Formally, we can write the rebalancing premium as: (2.13) where is defined as the dispersion term. Let us consider a given trajectory for the asset prices. We note the asset prices at time t: (S 1 (t),..., S n (t)). By definition of the (realised) growth rate, we have for each asset i: and An EDHEC-Risk Institute Publication

13 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio We have: and: We can then rewrite the building block of the dispersion term (i.e., without taking its expected value as: We set assets. the weighted average realised growth rate of the individual The building block of the dispersion term then becomes: An EDHEC-Risk Institute Publication 13

14 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio Finally, a second order Taylor expansion valid for gives: where is the weighted variance of the growth rates across the assets. In a general setting with n risky assets, assuming that the risky asset prices follow geometric Brownian motions, we assess that is a key component of the rebalancing premium. To see this, consider again the following equation: (2.14) Similarly, if we denote g i (t) the expected growth rates of the individual assets of the portfolio at time t, then the expected growth rate of the portfolio g(t) at time t satisfies: (2.15) 2.2 Focus on the Excess Growth Rate Term We give a proof below of the positivity of the excess growth rate for a long-only portfolio. In a general setting involving n risky assets with correlation matrix, volatility vector, covariance matrix and weights the excess growth rate verifies: 14 An EDHEC-Risk Institute Publication

15 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio Let ƒ : X > be a differentiable convex real-valued function of a real variable such that its domain X is a subset of. More generally, given a vector of weights such that and (σ 1 (t),..., σ nt (t)) X n t and given a real convex function ƒ the Jensen s inequality can be stated as: (2.16) We apply the equation (2.16) to the convex function ƒ(x) = x 2 and obtain: (2.17) Hence the positivity of the excess growth rate. In the particular case of a portfolio made of 2 risky assets, for each time t the portfolio with the highest excess growth rate is the equally-weighted portfolio, which also is the portfolio with the highest effective number of constituents (ENC, see Section 2.3 for the definition). Now, if we come back to our more general setting with n t risky assets and assume that all the volatilities at time t are identical (equal to σ t ), the excess growth rate simplifies as: (2.18) In this last case, the excess growth rate increases with the volatility σ t, increases with the ENC (w t ) and decreases with the pairwise correlations ρ i,j (t), suggesting that the portfolio with the highest excess growth rate is a well diversified (at least with a relatively high effective number of constituents) portfolio with highly volatile and weakly correlated assets. Under the specific assumption of identical volatility for all the assets, the Max portfolio is equivalent to other well-known weighting schemes such as the Global An EDHEC-Risk Institute Publication 15

16 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio Minimum Variance and the Maximum Diversification portfolios. We notice that if we also assume that all the assets have the same pairwise correlation at time t, then the portfolio with the highest excess growth rate is the equally-weighted portfolio, which is also the portfolio with the highest effective number of constituents. In a general setting without risk-free asset involving n t risky assets with correlation matrix, volatility vector, covariance matrix, and weights, the excess growth rate can be written as: and the optimisation problem of the excess growth rate maximisation of a portfolio can be formulated as: (2.19) The Lagrange function of the optimisation problem is: (2.20) The solution satisfies the following first-order conditions: (2.21) From the first equation of (2.21), we have: From the second equation of (2.21), we have: (2.22) We retrieve from above: (2.23) 16 An EDHEC-Risk Institute Publication

17 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio We re-inject the value of λ in the first equation of (2.21): (2.24) Finally, in the stylised setting with n t risky assets, no risk-free asset and no assumptions on the covariance matrix structure, the weight vector that maximises the excess growth rate is: (2.25) where t is the (n t n t ) covariance matrix, V t the (n t 1) volatility vector, vector of ones and the Hadamard product. 2 This analytical solution can lead to possibly negative weights, so in our long-only framework we have to compute the solution numerically. If we now assume that every asset has the same volatility σ t and the same pairwise correlation ρ t, we have with I nt the (n t n t ) identity matrix and. If ρ t is different from 1, the inverse of C t is: 2 - The Hadamard product is an operation that takes as the operands two matrices of the same dimensions, and produces another matrix where each element is the product of elements of the original two matrices. (2.26) From Equation (2.25), after having noticed that and, we have: We use Equation (2.26) to simplify the following quantities: (2.27) An EDHEC-Risk Institute Publication 17

18 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio We finally obtain: In the stylised setting made of n t risky assets with the same volatility σ t and pairwise correlation ρ t, the portfolio displaying the highest excess growth rate is the equallyweighted portfolio. If we now introduce a risk-free asset in the general setting with n t risky assets as in Mantilla-Garcia (2016) then we can show that the portfolio that maximises the excess growth rate is a mixed allocation between the risk-free asset and a risky asset portfolio. This risky asset portfolio is defined by Mantilla-Garcia (2016) as the Maximum Volatility Return (MVR) portfolio. The weights of the MVR portfolio verify: (2.28) To demonstrate this result, we complete our general framework involving n t risky assets with a risk free asset, an augmented correlation matrix, an augmented volatility vector, an augmented covariance matrix and an augmented weights vector with the weight of the risk-free asset. Then the optimisation problem of the excess growth rate maximisation of the portfolio can be formulated as: and rewritten as: The Lagrange function of the optimisation problem above is: (2.29) 18 An EDHEC-Risk Institute Publication

19 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio The solution verifies the following first-order conditions: (2.30) From the first and second equations of (2.30), we have: (2.31) From the third equation of (2.30), we have: (2.32) Finally, the solution is a linear combination of the risk-free asset and a risky portfolio that is given by:. The wealth proportion invested in the risky portfolio is: is defined by Mantilla-Garcia (2016) as the (MVR) strategy.. The risky portfolio If we only assume a constant and equal pairwise correlation amongst all the risky assets but possibly different volatilities, then the MVR portfolio is a linear combination of an Equally- Weighted portfolio and an Equal Risk Contribution portfolio (see Mantilla-Garcia (2016) for further details). From a standard mean-variance perspective with a risk-free asset, Mantilla- Garcia (2016) put forward that the MVR portfolio is optimal and coincides with the tangency portfolio for a risk aversion parameter equals to 2 and if expected returns are proportional to return variances. It is widely known that the analytical formula of the portfolio that maximises the Sharpe ratio (i.e., the ratio expected excess return of the portfolio μ ρ (t) r ƒ (t) over the volatility of the portfolio σ ρ (t)) is given by where is the vector of expected excess returns. By analogy we deduce the corresponding optimisation problem for the MVR strategy: the MVR portfolio, considering a universe of n t risky assets, is the portfolio that maximises the ratio of the weighted average variance of the assets over the portfolio volatility. We note that the MVR and the Max portfolio differ. 2.3 Overview of Other Weighting Schemes, Optimisation Constraints and Reporting Indicators Other Weighting Schemes In addition to the Max and MVR portfolios, we also present results for the following weighting schemes used for benchmarking purposes: An EDHEC-Risk Institute Publication 19

20 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio Capitalisation Weighted (CW), which weights stocks by their capitalisation. Equally-Weighted (EW), which attributes the same weight to each constituent in the investment universe. Global Minimum Variance (GMV), which aims to minimise the ex-ante volatility by using the covariance matrix structure. Equal Risk Contribution (ERC), which equalises the contribution of each constituent to the portfolio volatility (see Maillard et al. (2010) for more details). Maximum Diversification (MD), which aims at maximising the diversification ratio defined as the weighted average of the volatilities of the constituents over the portfolio volatility (see Choueifaty and Coignard (2008) for more details). 3 - Cauchy-Schwarz inequality states that for any two vectors (x 1,..., x nt ) and (y 1,..., y nt ), we have Equality is achieved if, and only if, one of the two vectors is zero or the two vectors are parallel. Here, we take y i = 1.. Each portfolio is rebalanced at quarterly intervals in March, June, September and December. The rebalancing dates coincide with the third Friday of the corresponding month. The CW portfolio is computed by extracting the stocks capitalisation. The EW portfolio computation is straightforward. On the other hand, the covariance matrix needs to be estimated for the ERC, GMV, MD, MVR, and Max portfolios. The main challenge in covariance matrix estimation is the curse of dimensionality: because the number of stocks in the investment universe can be large (up to 500 stocks) and exceeds the number of observations, the historically-estimated sample covariance matrix will be non-invertible; this is particularly disturbing since weighting schemes such as the minimum variance GMV or Max investor s unconstrained portfolios are functions of the inverse of the covariance matrix. We will therefore estimate the covariance matrix at each step by using a statistical factor model where each factor is modelled as a linear combination of returns of the stocks. We use the principal component analysis to extract the factors and consider the past 2-year weekly returns (104 data points): we thus extract the factors from the data rather than use explicit factors. This choice is prompted by the desire to avoid imposing any view on which factors matter most in explaining the variability of stock returns. We then limit the number of statistical factors using a criterion from Random Matrix Theory in order to achieve parsimony and robustness (see Plerou et al. (1999) and Amenc et al. (2011) for more details) Optimisation Constraints We constrain the ERC portfolio weights to be positive. We also constrain the GMV, MD, MVR and Max portfolio weights (1) to be positive and (2) to verify that their effective number of constituents (ENC in short) is greater than or equal to, where (2.33) and where denotes the Euclidian norm. An application of Cauchy-Schwarz inequality shows that ENC (w t ) n t, and that equality holds if, and only if, all weights are equal. 3 This second constraint is meant to ensure a minimum level of naive diversification in the portfolios. The benefits of weight constraints have been widely documented in the litterature (see Jagannathan and Ma (2003) for hard minimum and maximum weight constraints and DeMiguel et al. (2009) for flexible constraints applying to the norm of the weight vector). 20 An EDHEC-Risk Institute Publication

21 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio Reporting Indicators In our analysis, we will compare these different portfolios by reporting commonly used performance and risk measures such as volatility (a measure of average risk) or Valueat-Risk (a measure of extreme risk). These measures, however, are typically backwardlooking risk indicators computed over one historical scenario. As a result, they provide very little information, if any, regarding the possible causes of the portfolio riskiness, the probability of a severe outcome in the future, or the reward that an investor can expect in exchange for bearing those risks. In this context, we propose to also report forward-looking risk indicators for the tested portfolios. To try and identify a meaningful measure of the number of bets to which investors dollars are allocated, one can first define the contribution of each constituent to the overall volatility of the portfolio σ ρ (t) (see Roncalli (2013) for further details) as: (2.34) 4 - An application of Cauchy- Schwarz inequality shows that ENCB n t, and that equality holds if, and only if, all risk contributions are equal. This leads to the following scaled contributions: (2.35) where To account for the presence of cross-sectional dispersion in the correlation matrix, one can apply the naive measure of concentration ENC introduced above to the scaled contributions to portfolio risk. This allows us to define the effective number of correlated bets in a portfolio as the dispersion of the volatility contributions of its constituents: (2.36) where denotes again the Euclidian norm. 4 This measure takes into account not only the number of available assets but also the correlation properties between them. More specifically, a constituent that is highly positively correlated with all the other constituents will tend to have a higher contribution to the volatility (considering a long-only portfolio for simplicity), leading to a lower effective number of correlated bets since most of the portfolio risk is concentrated in that constituent. A shortcoming of the ENCB is that it can give a misleading picture for highly correlated assets. For instance, an equally-weighted portfolio of two highly correlated bonds with similar volatilities is well diversified in terms of dollars and volatility contributions, but portfolio risk is extremely concentrated in a single interest rate risk factor exposure. To better assess the contributions of underlying risk factors, Meucci An EDHEC-Risk Institute Publication 21

22 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio (2009) and Deguest et al. (2013) propose to decompose the portfolio returns (which can be seen as combinations of correlated asset returns or correlated factor returns) as a combination of the contributions of n t uncorrelated implicit factors. In other words, baskets should be interpreted as uncorrelated risk factors, as opposed to correlated asset classes, and it is only if the distribution of the contributions of various factors to the risk of the portfolio is well-balanced that the investor s portfolio can truly be regarded as well-diversified. Putting all these elements together, we propose using the effective number of uncorrelated bets (ENUB) in our empirical analysis, which would serve as a meaningful measure of diversification for investors portfolios (see Meucci (2009) and Deguest et al. (2013) for more details). First we decompose the portfolio return rρ as the weighted sum of n t correlated asset returns r 1 (t),..., r n (t) and also the weighted sum of t n t uncorrelated factor returns r (t),..., r F (t) (see also Deguest et al. (2013) or Meucci 1 Fnt et al. (2015)): where r t = (r 1 (t),..., r n (t)) denotes the vector t of the original constituents returns, r = (r F t F1 (t),..., r (t)) the vector of uncorrelated Fnt factors returns, w t = (w 1 (t),...,w n (t)) the weight vector of correlated components and t w = (w F t F1 (t),...,w (t)) the weight vector of uncorrelated factors. Note that to simplify Fnt we assess that the number of original correlated assets and the number of uncorrelated factors is the same. The main challenge with this approach is to turn correlated asset returns into uncorrelated factor returns. The volatility of the portfolio is: where is the covariance matrix of the n t uncorrelated factors. The factor returns can typically be expressed as a linear transformation of the original returns: for some well chosen transformation A t guaranteeing that the covariance matrix of the factors is a diagonal matrix. A t is a n t n t transition matrix from the assets to the factors and it is therefore critical that it be invertible since. Then, we define the contribution of each factor to the overall variance of the portfolio as: (2.37) 22 An EDHEC-Risk Institute Publication

23 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio which leads to the following percentage contributions for each factor: where The effective number of uncorrelated bets (ENUB) can be written as follows: 5, where (2.38) 5 - The ENCB and ENUB indicators, just like the ENC indicator, are not only meaningful diversification statistics, they can also be used as constraints to construct well-diversified portfolios. 6 - A matrix Q t is said to be orthogonal if it satisfies. The ENUB reaches a minimum equal to 1 if the portfolio is loaded in a single risk factor, and a maximum equal to n t, the nominal number of constituents, if the risk is evenly spread amongst factors. The portfolios named factor risk parity portfolios in Deguest et al. (2013) are built such that the contribution ρ i (t) of each factor to the variance are all equal. This methodology relies on uncorrelated implicit factors which are not uniquely defined. In fact, it is easy to see that if A is a change of basis matrix from constituents to factors and Q t is any orthogonal matrix, then the matrix is also a change of basis matrix such that is diagonal. 6 Thus, one has to specify an orthogonalisation procedure. An option is to perform principal component analysis on the covariance matrix, so as to sequentially extract uncorrelated factors that have the maximum marginal explanatory power with respect to asset returns. This decomposition, however, has some undesirable properties: Carli et al. (2014) note that the principal factors lack interpretability and Meucci et al. (2015) show that if all assets have the same volatility and the same pairwise correlation, then an equally-weighted portfolio of the assets is fully invested in the first principal factor (i.e., the one with the largest variance) and that the other factors have zero weight. As a result, the portfolio risk is entirely explained by the first factor, regardless of the value of the common correlation. This property is counterintuitive, as one would expect uncorrelated factors to have more balanced contributions when the correlation across assets shrinks to zero. To overcome these problems, Meucci et al. (2015) introduce an alternative method for extracting uncorrelated factors, known as Minimum Linear Torsion. The MLT algorithm seeks to extract the matrix A t such that the factor covariance matrix is diagonal while keeping the distance between the factors and the asset returns as small as possible, thus involving the smallest deformation of the original components. In detail, A t is solution to the following program: subject to and (2.39) where diag( t ) denotes a diagonal matrix with diagonal elements equal to those of t. Note that the second constraint implies two properties: first, the factor covariance matrix is diagonal, and second, the factor variances are equal to the asset variances, which is a natural requirement since the factors should resemble asset returns as much as possible. In the previous example (uniform volatilities and correlations and equally-weighted portfolio), it can be shown that all uncorrelated factors have the same weight, which is more in line with the intuition. Fortunately, the solution to the optimisation problem is known in closed form, up to the singular value decompositions of and an auxiliary matrix: An EDHEC-Risk Institute Publication 23

24 2. Theoretical Analysis of the Max Excess Growth Rate Portfolio expressions for the change of basis matrix A t can be found in Meucci et al. (2015) and Carli et al. (2014). Since efficient numerical algorithms exist for computing singular value decompositions, the solution is straightforward to obtain numerically. We use in our studies this least intrusive orthogonalisation procedure to extract uncorrelated factors starting from correlated factor indices. We will also compute the annualised out-of-sample excess growth rate between two quarterly rebalancing dates of each strategy for the different universes as follows: (2.40) g strat,t >t+1 denotes the realised growth rate of the strategy between two rebalancing dates, n t the number of stocks in the portfolio at time t, w i (t) the weight of stock i in the portfolio at time t and g i,t >t+1 the growth rate of asset i between two rebalancing dates. We finally report the average out-of-sample excess growth rate of every strategy across all the periods. This indicator will allow us (1) to assess if the Max strategy, which by construction is the strategy with the highest ex-ante excess growth rate, has the highest average ex-post excess growth rate and (2) to compare the amplitude of the out-ofsample excess growth rate of the MVR strategy with the out-of-sample excess growth rates of the Max and the other strategies. 24 An EDHEC-Risk Institute Publication

25 3. Empirical Analysis on Individual Stocks An EDHEC-Risk Institute Publication 25

26 3. Empirical Analysis on Individual Stocks We consider, in this section, several universes of individual stocks from four developed regions: United States, Eurozone, UK and Japan. The individual stock returns data used in our study emanates from the Scientific Beta database Long Term US Equity Universe We consider the ERI Scientific Beta Long-Term Track Record (LTTR in short) US Universe as our base case investment universe over the period June December This universe is composed every quarter of the 500 largest US capitalisations. The strategies studied are rebalanced quarterly and a calibration period of two years of weekly returns is used for the covariance matrix estimation when needed. 7 - See com for more details. Table 1 reports risk, performance and diversification indicators relative to the CW, EW, GMV, ERC, MD, MVR and the Max weighting schemes applied to this universe. The Max portfolio has the worst risk-adjusted performance of all the portfolios with a Sharpe ratio of 0.29 (versus 0.39 for the CW portfolio and 0.54 for the ERC portfolio) and displays the highest extreme risk of all the portfolios with a maximum drawdown of 77.7 %. The MVR portfolio exhibits a risk-adjusted performance (Sharpe ratio of 0.45) close to that of the EW portfolio (Sharpe ratio of 0.46). The growth rates of the MVR and Max portfolios are lower than that of the EW portfolio (11.0% for MVR and 9.0% for the Max versus 11.4% for the EW strategy). We acknowledge the domination of the MVR and Max portfolios by the EW portfolio in terms of growth rate and risk-adjusted return: the heterogeneity of the universe or a high negative exposure to one or many rewarded risk factors could be possible explanations of these relative underperformances. We notice that the Max allocation is the one with the highest out-of-sample average annualised excess growth rate (7.8%), followed by the MVR strategy (6.0%), which is with their corresponding ex-ante optimisation functions. The CW and EW strategies have out-of-sample average annualised growth rates that are significantly lower at respectively 3.1% and 4.2%. Table 1: Risk, Performance and Diversification Statistics on the LTTR US Universe Universe: US LTTR (500 stocks) Time period: 19/6/70-31/12/15 CW EW GMV ERC MD MVR Max Ann. mean return 11.5% 12.7% 12.3% 12.7% 12.7% 12.4% 11.5% Ann. volatility 16.8% 16.7% 11.9% 14.4% 13.6% 16.5% 22.3% Ann. growth rate 10.1% 11.4% 11.6% 11.7% 11.8% 11.0% 9.0% Sharpe ratio Average ENC 23% 100% 33% 65% 33% 33% 33% Average ENCB 23% 84% 40% 100% 33% 16% 18% Average ENUB 40% 38% 51% 44% 43% 23% 20% Max drawdown 54.3% 58.3% 43.9% -53.0% 50.1% 59.6% 77.7% Daily VaR 1% -2.8% -2.8% -2.0% -2.5% -2.4% -2.8% -3.9% Ann.excess growth rate 3.1% 4.2% 2.7% 3.5% 4.2% 6.0% 7.8% This table reports the risk, performance and diversification statistics for the CW, EW, GMV, ERC, MD, MVR and the Max portfolios built on the ERI Scientific Beta Long Term US Equity universe over the period June 1970-December The yield on Secondary US Treasury Bills is used as a proxy for the risk-free rate. We use daily total return series in US dollars. All statistics are annualised. 26 An EDHEC-Risk Institute Publication

27 3. Empirical Analysis on Individual Stocks We now investigate the factor exposures of the different portfolios. We focus on the wellacademically documented following factors: market, size, momentum, value and low volatility. To observe the factor exposures of the portfolios, we first perform a five-factor OLS linear regression of the portfolios excess returns. We use the yield on Secondary Market US Treasury Bills (3M) as a proxy for the risk-free rate. The market factor is proxied by the CW strategy excess returns time series. The size, value, momentum and low volatility factors are proxied by returns time series of long/short equity portfolios. We borrow the series from the long-term ERI Scientific Beta database. Each long/short factor has a long leg, which is formed with the top or bottom 30% in terms of factor exposure, and a short leg, which is the other bottom or top 30% group. The choice of the criterion and the definition of the long leg is the following: Size: stocks are sorted on their market capitalisation and the long leg is the 30% lowest market cap stocks group. Value: stocks are sorted on their book-to-market (BE/ME) ratio and the long leg is the 30% highest BE/ME group. Momentum: stocks are sorted on their past 52-week returns, skipping the last four weeks and the long leg is the 30% highest past winners group. Low volatility: stocks are sorted on their past 104-week volatility and the long leg is the 30% low volatility group. The factor exposures of the different weighting schemes are reported in Table 2. The Max portfolio has a null alpha and negative exposures to the value, momentum and low volatility factors, which are positively rewarded in the long run (see Table 3 for the positivity and the amplitude of the risk premia). The MVR portfolio displays a positive annualised alpha of 0.76% but lower than that of the EW portfolio (0.97%). As expected, a common pattern of the MVR and Max portfolios is their negative exposures to the low volatility risk factor. Given that the low volatility risk factor is positively rewarded with a risk premia of 0.5%, the high negative exposure of the Max strategy to this risk factor ( ) besides its negative exposure to the positively rewarded value risk factor ( ) mostly explain the low risk-adjusted performance of this strategy (Sharpe ratio of 0.29). Table 2: Factor Exposures on the US LTTR Universe CW EW GMV ERC MD MVR Max Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Alpha 0.0% 0.00% % % % % % 0.00 Market Value Size Momentum Low Vol Adj- R % 98.1% 90.7% 95.7% 91.5% 91.2% 94.4% Ann. alpha 0.00% 0.97% 1.54% 1.17% 1.55% 0.76% 0.00% This table reports the results of the regression of the daily excess returns time series of the CW, EW, GMV, ERC, MD, MVR and Max weighting schemes over the market, value, size, momentum and low volatility factors. The universe considered is the ERI Scientific Beta Long Term US Equity Universe over the period June 1970-December An EDHEC-Risk Institute Publication 27

28 3. Empirical Analysis on Individual Stocks Table 3: Risk Premia on the US LTTR Universe Market Size Value Momentum Low Vol Ann. Mean Excess Return 6.5% 1.2% 1.2% 4.7% 0.5% This table reports the risk premia on the US LTTR Universe over the period June 1970-December All statistics are annualised. If we keep the same investment universe but change the investment period to June 2002-December 2015 we notice that the low volatility risk factor is negatively rewarded (-3.2%, see Table 4). The annualised mean return, the annualised volatility, the annualised growth rate and the Sharpe ratio statistics relative to the EW, GMV, ERC, MD, MVR and the Max weighting schemes for the US LTTR universe over this shorter investment period are given in Table 5. We assess that the Max and MVR strategies display better growth rates (respectively 8.9% and 10.2% versus 7.5%) and Sharpe ratio (respectively 0.42 and 0.53 versus 0.41) than the CW portfolio, suggesting that the performances of these weighting schemes are enhanced when the low volatility risk premium is negative. Table 4: Risk Premia on the US LTTR Universe Market Size Value Momentum Low Vol Ann. Mean Excess Return 6.5% 1.2% 1.2% 4.7% 0.5% This table reports the risk premia on the US LTTR Universe over the period June 2002-December All statistics are annualised. Table 5: Risk, Performance and Diversification Statistics on the LTTR US Universe Universe: US LTTR (500 stocks) Time period: 19/6/02-31/12/15 CW EW GMV ERC MD MVR Max Ann. mean return 9.5% 11.3% 11.0% 11.2% 12.2% 12.4% 12.3% Ann. volatility 19.7% 21.3% 15.3% 18.7% 17.4% 21.1% 26.1% Ann. growth rate 7.5% 9.1% 9.9% 9.5% 10.7% 10.2% 8.9% Sharpe ratio This table reports the annualised mean return, the annualised volatility, the annualised growth rate and the Sharpe ratio for the CW, EW, GMV, ERC, MD, MVR and the Max portfolios built on the ERI Scientific Beta Long Term US Equity universe over the period June 2002-December The yield on Secondary US Treasury Bills is used as a proxy for the risk-free rate. We use daily total return series in US dollars. All statistics are annualised. We will thereafter compare these results to that obtained in other regions where the low volatility risk factor can be negatively rewarded. 3.2 Empirical Tests on Regional Universes We apply the same experimental protocol described above to the following universes for the period June 2002-February 2017: Eurozone stocks (300 stocks). UK stocks (100 stocks). Japanese stocks (500 stocks from June 2002 to June 2016 and 300 stocks from June 2016). Table 6 shows the factor exposures of the different weighting schemes for the Eurozone universe. The Max and MVR portfolios, with respective Sharpe ratios of 0.20 and 28 An EDHEC-Risk Institute Publication

29 3. Empirical Analysis on Individual Stocks 0.15, lower growth rates of respectively 3.7% and 2.6%, and higher respective maximum drawdowns of 60.0% and 68.1%, both display lower risk-adjusted performances than the CW portfolio (Sharpe ratio of 0.29, growth rate of 5.5% and Maximum Drawdown 58.3%). The growth rates of the MVR and Max portfolios are also lower than that of the EW portfolio (6.8%). In the previous US LTTR universe, the risk-adjusted performance and the growth rate of the Max strategy was sharply below those of the EW and even the CW strategies; while the MVR strategy displays risk and performance statistics close to those of the EW strategy. On that universe, the Sharpe ratios and the growth rates of the MVR and Max portfolios are clearly below those of EW and CW portfolios. Table 6: Risk, Performance and Diversification Statistics on the Eurozone Universe Universe: Eurozone (300 stocks) Time period: 21/6/02-28/2/17 CW EW GMV ERC MD MVR Max Ann. mean return 7.7% 8.7% 9.2% 8.6% 7.5% 5.2% 5.2% Ann. Volatility 21.1% 19.5% 12.9% 16.5% 14.3% 17.6% 22.7% Ann. growth rate 5.5% 6.8% 8.4% 7.2% 6.5% 3.7% 2.6% Sharpe ratio Average ENC 31% 100% 33% 63% 33% 33% 33% Average ENCB 28% 83% 41% 100% 30% 12% 13% Average ENUB 41% 37% 55% 44% 39% 16% 15% Max drawdown 58.3% 61.9% 50.1% 59.0% 53.2% 60.0% 68.1% Daily VaR 1% -3.7% -3.5% -2.4% -3.0% -2.7% -3.2% -4.0% Ann.excess growth rate 3.0% 4.2% 3.2% 3.8% 5.2% 7.0% 8.1% This table compares the risk, performance and diversification statistics for the CW, EW, GMV, ERC, MD, MVR and the Max portfolios built on the ERI Scientific Beta Eurozone Equity universe over the period June 2002-February The Euribor 3 Months is used as a proxy for the risk-free rate. We use daily total return series in Euros. All statistics are annualised. Table 7 displays the factor exposures of the different weighting schemes for the Eurozone universe. The MVR and Max portfolios have strong negative annualised alphas (respectively -3.35% and -3.79%) and negative exposures to the low volatility factor (respectively and ). The positivity of the low volatility risk premium (1.0%, see Table 8) is a source of explanation of the underperformance of the MVR and Max portfolios relative to the other weighting schemes. Table 7: Factor Exposures for the Eurozone Universe CW EW GMV ERC MD MVR Max Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Alpha 0.0% 0.00% % % 0.00% -0.01% % Market Value Size Momentum Low Vol Adj- R % 98.3% 92.5% 96.4% 90.8% 89.5% 93.6% Ann. alpha 0.00% 0.12% 1.68% 0.20% -0.44% -3.35% -3.79% This table reports the results of the regression of the daily excess returns time series of the CW, EW, GMV, ERC, MD, MVR and Max weighting schemes over the market, value, size, momentum and low volatility factors. The universe considered is the ERI Scientific Beta Eurozone Equity Universe over the period June 2002-February An EDHEC-Risk Institute Publication 29

30 3. Empirical Analysis on Individual Stocks Table 8: Risk Premia on the Eurozone Universe Market Size Value Momentum Low Vol Ann. Mean Excess Return 6.0% 2.1% 2.6% 1.5% 1.0% This table reports the risk premia on the Eurozone Universe over the period June 2002-February All statistics are annualised. The risk, performance and diversification statistics relative to the CW, EW, GMV, ERC, MD, MVR and the Max weighting schemes for the UK universe are summarised in Table 9. The Max and the MVR portfolios, with respective Sharpe ratios of 0.50 and 0.41, have better risk-adjusted performances than the CW portfolio (Sharpe ratio of 0.35). They also exhibit higher growth rates (respectively 10.0% and 9.0%) than that of the EW portfolio (8.7%): these results, favourable to the MVR and the Max strategies, discernibly contrast with the results obtained in the US LTTR and Eurozone universes. Table 9: Risk, Performance and Diversification Statistics on UK Universe Universe: UK Live (100 stocks) Time period: 21/6/02-28/2/17 CW EW GMV ERC MD MVR Max Ann. mean return 9.0% 10.5% 10.4% 10.5% 10.7% 11.8% 11.8% Ann. volatility 19.1% 18.9% 14.4% 17.0% 15.6% 19.4% 23.4% Ann. growth rate 7.2% 8.7% 9.3% 9.1% 9.5% 10.0% 9.0% Sharpe ratio Average ENC 32% 100% 33% 83% 33% 33% 33% Average ENCB 30% 85% 37% 100% 33% 19% 18% Average ENUB 48% 49% 75% 61% 55% 27% 22% Max drawdown 44.4% 50.1% 33.0% 46.0% 40.9% 52.2% 59.6% Daily VaR 1% -3.4% -3.4% -2.7% -3.1% -2.8% -3.4% -4.1% Ann.excess growth rate 2.5% 3.1% 2.0% 2.7% 3.1% 4.9% 5.7% This table compares the risk, performance and diversification statistics for the CW, EW, GMV, ERC, MD, MVR and the Max portfolios built on the ERI Scientific Beta UK Equity Universe over the period June 2002-February The UK Treasury Bill Tender (3M) is used as a proxy for the risk-free rate. We use daily total return series in British Pounds. All statistics are annualised. The factor exposures of the different weighting schemes for the UK universe are listed in Table 10. The MVR and Max portfolios have higher annualised alphas (respectively 2.30% and 0.96%) than the EW portfolio (0.37%) and display negative exposures to the low volatility factor (respectively and ). Unlike the US LTTR and the Eurozone universes, the low volatility risk premium is negative (-3.2%, see Table 11). That particular configuration where high volatility stocks outperform low volatility stocks on the long run can explain the good performances of the MVR and Max portfolios relative to the other weighting schemes. 30 An EDHEC-Risk Institute Publication

31 3. Empirical Analysis on Individual Stocks Table 10: Factor Exposures for the UK Universe CW EW GMV ERC MD MVR Max Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Alpha 0.0% 0.00% % % % % % 0.55 Market Value Size Momentum Low Vol Adj- R % 96.5% 89.1% 95.7% 88.2% 89.0% 91.9% Ann. alpha 0.00% 0.37% 2.33% 1.05% 2.21% 2.30% 0.96% This table reports the results of the regression of the daily excess returns time series of the CW, EW, GMV, ERC, MD, MVR and Max weighting schemes over the market, value, size, momentum and low volatility factors. The universe considered is the ERI Scientific Beta UK Equity universe over the period June 2002-February Table 11: UK Risk Premia Market Size Value Momentum Low Vol Ann. Mean Excess Return 6.6% 3.3% -0.1% 3.0% -3.2% This table reports the risk premia on the UK universe over the period June 2002-February All statistics are annualised. Table 12 contains the risk, performance and diversification statistics relative to the CW, EW, GMV, ERC, MD, MVR and the Max weighting schemes for the Japanese universe. The Max portfolio exhibit a risk-adjusted performance (Sharpe ratio of 0.32) close to that of the CW portfolio (Sharpe ratio of 0.31). The MVR portfolio has a risk-adjusted performance (Sharpe ratio of 0.42) equal to that of the EW portfolio. The MVR and Max also have lower growth rates (respectively 6.4% and 4.8%) than that of the EW portfolio (6.8%). Table 12: Risk, Performance and Diversification Statistics on Japanese Universe Universe: Japan (500 and then 300 stocks) Time period: 21/6/02-28/2/17 CW EW GMV ERC MD MVR Max Ann. mean return 7.0% 9.0% 9.6% 9.1% 9.0% 8.3% 7.6% Ann. volatility 22.2% 21.2% 15.6% 18.8% 17.0% 19.2% 23.8% Ann. growth rate 4.6% 6.8% 8.4% 7.4% 7.5% 6.4% 4.8% Sharpe ratio Average ENC 24% 100% 33% 73% 33% 33% 33% Average ENCB 20% 87% 40% 100% 32% 16% 17% Average ENUB 47% 49% 62% 56% 48% 25% 22% Max drawdown 60.1% 57.8% 41.6% 51.8% 48.0% 58.8% 71.1% Daily VaR 1% -3.7% -3.6% -2.8% -3.3% -3.2% -3.6% -4.2% Ann.excess growth rate 2.9% 3.7% 2.7% 3.3% 4.0% 5.6% 6.5% This table compares the risk, performance and diversification statistics for the CW, EW, GMV, ERC, MD, MVR and the Max portfolios built on the ERI Scientific Beta Japanese Equity universe over the period June 2002-February The Japan Gensaki T-Bill (1M) is used as a proxy for the risk-free rate. We use daily total return series in Japanese Yen. All statistics are annualised. The factor exposures of the different weighting schemes for the Japanese universe are displayed in Table 13. The MVR and Max portfolios have lower annualised alphas (respectively 0.18% and -0.87%) than the EW portfolio (0.32%). The low volatility risk An EDHEC-Risk Institute Publication 31