Risk Reduction in Style Rotation

Transcription

1 EDHEC-Risk Institute promenade des Anglais Nice Cedex 3 Tel.: +33 (0) research@edhec-risk.com Web: Risk Reduction in Style Rotation October 2010 Rodrigo Dupleich Vice President, Equities Quantitative Analysis Europe, Citi Global Markets Daniel Giamouridis Assistant Professor, Department of Accounting and Finance, Athens University of Economics and Business, and Research Associate, EDHEC-Risk Institute Chris Montagu Director, Head of Global Quantitative Research Europe, Citi Investment Research

2 Abstract We investigate the potential improvement in the implementation of style rotation strategies by techniques addressing estimation errors. We select two approaches that have recently stood out in the statistics and econometric literature and have been applied to portfolio construction literature. One builds on regularization methods which address estimation error by focusing on the weights of the constructed portfolios. And a second method that uses pooled forecasts obtained across different observation windows. Thus it focuses on minimizing estimation error in the moments of the return distribution that may arise due to structural breaks. We conclude that overall there are benefits from departing from naïve approaches which can be as significant as an improvement in the information ratio of about 54%, i.e., from 0.65 (naïve) to about 1 (dynamic). EDHEC is one of the top five business schools in France. Its reputation is built on the high quality of its faculty and the privileged relationship with professionals that the school has cultivated since its establishment in EDHEC Business School has decided to draw on its extensive knowledge of the professional environment and has therefore focused its research on themes that satisfy the needs of professionals. 2 EDHEC pursues an active research policy in the field of finance. EDHEC-Risk Institute carries out numerous research programmes in the areas of asset allocation and risk management in both the traditional and alternative investment universes. Copyright 2010 EDHEC

3 Introduction The past two years have put a lot of quantitative or systematic investment strategies to the test. The growth of quantitative investing through the early 2000s has meant that many strategies were optimally developed for a period in markets where style volatility and correlations were low and auto-correlations high. These benign market conditions meant that many strategies were almost set and forget in terms of factor/style weightings. However, August 2007 changed all of this. The product of cross-asset de-levering/sell-off, overcrowded investment strategies and a difficult macroeconomic environment has resulted in abnormal factor/style correlations and volatility, and sudden factor/style reversals. This environment has made it crucial for quantitative investors to take a more dynamic approach to their style/factor selection and or weighting. However, dynamic style selection comes with a separate set of problems. Firstly, in periods of high volatility, style rotation strategies are at the mercy of frequent tuning points in style performance. More recently, we have witnessed increased style volatility and a breakdown in typical correlation structures. In these conditions a static approach to style weighting would potentially be suboptimal, depending on how dynamic or reactive the rotation strategy is, missing turning points can severely impact portfolio performance. Secondly, and related, is that the very dynamic nature of the strategy increases portfolio turnover and therefore transaction costs. Too frequent style re-weighting will erode portfolio performance and a high noise-to-signal ratio will generate unnecessary style rotation. Thus, ideally portfolio managers should seek style rotation strategies that could be dynamic but are least vulnerable to these risks. Determining the optimal mix of assets in the context of a portfolio construction involves smart forecasts of stock returns as well as good estimates of the stock return variances and covariances. Typically, sample moments are used as best estimates of the population moments. For example, it is common practice to estimate a covariance matrix (sample covariance matrix) using historical stock returns. Several researchers have pointed out, however, that this generally acceptable practice introduces uncertainty, which can degrade the desirable properties of the constructed portfolio (see for Best and Grauer [1991], Chopra and Ziemba [1993]). The problem is termed estimation risk and is commonly addressed by employing advanced statistical techniques. 1 Style rotation is not in principle different. Smart forecasts of factor returns are necessary as well as good estimates of the factor return variances and covariances. Although researchers have been concerned with the impact of estimation risk in portfolio construction, none to our knowledge has so far been concerned with its impact on decisions involving style portfolio allocations. The vast majority of papers we have come across are concerned mainly with the development of successful factor return prediction models. Integrating remedies to reduce estimation risk is, in our view, becoming extremely important in the current market conditions. As we discussed above, style returns have been showing unprecedented levels of volatilities as well as covariances/correlations, rendering standard portfolio construction approaches subject to extreme position rebalancing. This article aims to investigate if techniques that address estimation risk, optimal prediction, and variable selection improve equity style rotation strategies. The general style rotation framework that we use is similar to that of Qian et al. [2007], in that we also focus on style portfolios with the maximum information ratio (IR hereafter). However, we integrate two recently developed/ advocated approaches that have been shown to work well in the presence of estimation risk and structural breaks in data. The first is motivated by recent research in the portfolio construction literature (see Brodie et al. [2007], De Miguel et al. [2009], Lobo et al. [2007], Welsch and Zhou [2007], Giamouridis and 1 - Giamouridis (2010) discusses this problem and provides a comprehensive review of this literature. 3

4 Paterlini [2010]) and involves the use of regularization procedures such as Ridge regression (Hoerl and Kennard [1970]) or Lasso regression (Tibshirani [1996]). The evidence so far suggests that such methods allow complex models to be trained on data sets of limited size without severe overfitting; essentially these methods tend to penalize complexity and aim to reduce the variance of parameter estimates. In the portfolio construction literature, regularization methods reduce the sensitivity of the optimization to possible collinearities between assets, control transaction costs by promoting stability/sparsity, and improve the out-of-sample performance relative to the classical Markowitz mean-variance approach. Furthermore, regularization methods have shown to possess interesting Bayesian and moment-shrinkage interpretations (Hastie et al. [2001], Tibshirani [1996], De Miguel et al. [2009]). This article is the first to our knowledge that uses regularization techniques in the context of style rotation. The second approach, developed by Pesaran and Timmermann [2007], involves pooling forecasts that have been obtained with the same model but across different observation windows. This approach aims to minimize the impact of structural breaks on moment estimates and can be extended to deal with different types of model uncertainty. The Pesaran and Timmermann [2007] approach does not aim to forecast breaks in the data. It does aim though to control for the possible presence of structural breaks out-of-sample. In addition, this approach resembles many of the characteristics of Bayesian model averaging. While this approach has been used in the context of predictive regressions, we, for the first time to our knowledge show how to modify it and use in the context of style rotation or more generally speaking in the context of portfolio construction. In order to control the estimation error and the speed of adjustment we evaluate performance and risk profiles of these two approaches in a standard European style universe. We also build a comparative analysis with well-known approaches suggested by Qian et al. [2007]. The results from our analysis indicate that performance/risk gains can be obtained by dynamically adjusting exposures to style factors. Departing from naïve approaches creates benefits which can be as significant as an improvement in the IR of about 54%, i.e., from 0.65 (naïve) to about 1 (dynamic). However, we show that the speed of adjustment should be limited in order to improve equally weighted schemes (i.e., naïve diversification). A fast reaction model will tend to generate unnecessary turnover and the model factor re-weighting can be triggered by factor noise. Methodology Optimal Style Portfolios Style rotation portfolios are typically based on relative scores of stocks with respect to certain factors and equal weighting schemes for aggregating the score information. Our analysis aims to suggest an alternative weighting scheme that is based on the optimization of the factor portfolio IR. In particular the optimal weightings for the factor portfolio are obtained through the following generic function: where the weights over the K factors are defined as, is the expected return of the factor portfolio and σ factors is the volatility of the factor portfolio. We define as where is the Kx1 vector of the expected returns of each factor, i.e.,, and σ factors as where Cov(R i, R j ) is the return covariance between styles i and j. The key inputs for this approach as with the meanvariance approach are the means and the covariances. Using sample moments, as best estimates of the population moments, involves, as we have already highlighted, estimation risk. (1) 4

5 Our approach for determining optimal style weights and ultimately constructing the equity portfolio involves two steps. First, we employ IR optimization to determine the optimal factor weightings for a given point in time (as per the above objective function). We use historical style return data to compute the expected returns and variances/covariances for the styles. 2 We constrain the portfolio optimization problem such that the weights sum to one and the weights are positive. Second, using the optimal style weights from the above procedure, we calculate composite scores by weighting factor scores across a cross-section of stocks. With these scores we can transform into alphas such that we can measure the return spread between portfolios of high and low scores; and/or build an optimized stock portfolio. In the following sections we present the two approaches we use to address estimation risk in the context of style rotation, i.e., factor weighting. Ridge Regression In the introduction we discussed some basic empirical properties of what are termed regularization methods. Our analysis is based on Ridge regression. In the context of a regression, the Ridge approach shrinks the regression coefficients by imposing a penalty on their size. This approach effectively addresses weight volatility or weights estimation error. In particular, we consider the following modification of Equation (1):, subject to: and where λ is a non-negative constant (or complexity parameter) that controls the amount of shrinkage and is the weight for the i th factor. Brodie et al. [2007] highlight that it is likely that a standard numerical procedure for the optimization in equations in the form of Equation (1) may amplify the effects of noise anisotropically, leading to an unstable and unreliable estimate of the vector w. The fact that a large number of factors may be considered in practice by equity portfolio managers makes the optimization problem more computationally complex and increases the possibility that unstable estimates of the vector w be obtained. These issues are tackled with the introduction of the penalty in the objective function which effectively renders the optimization problem more stable (Daubechies et al. [2004]). The larger the value of λ the more intense the shrinkage and hence the more weights are shrunk towards zero. Technically, the Ridge regression aims to shrink combinations of factors that tend to explain less the variability of data (or directions in the space of factors that have the smallest variance). Therefore this technique can be also linked to principal component analysis (PCA). From the Bayesian point of view, the Ridge approach can be seen as adopting a Gaussian prior on the factor weights (see Hastie et al. [2001], Bishop [2006]) and has been shown to possess interesting moment-shrinkage interpretations (De Miguel et al. [2009]). The evidence, so far, suggests that these methods reduce the sensitivity of the optimization to possible collinearities between assets, control transaction costs by promoting stability/sparsity, and improve the out-of-sample performance relative to the classical Markowitz mean-variance approach. Multiple Estimation Windows While addressing weight volatility through regularization has benefits and some attractive interpretations, dealing explicitly with the inefficiencies of moment estimation, i.e., moment estimation errors may be more beneficial. Recently Pesaran and Timmermann [2007] and Pesaran and Pick [2008], proposed a method on constructing out-of-sample forecasts when data is subject to structural breaks. Pesaran and Timmermann [2007] argue that forecast averaging procedure can be extended to deal with different types of model uncertainty, such as the size of the 2 - Style returns are calculated using a factor mimicking portfolios (FMP); i.e., the (daily) returns between the top and low baskets based on a style factor. 5

6 estimation window. A common approach relies on estimating the parameters using data available after the most recent break (e.g., rolling window); however, Pesaran and Timmermann [2007] showed that this approach need not be optimal when the objective is to optimize forecasting performance and forecast averaging can reduce the mean square forecast error (MSFE). Their idea, also called average windows forecasts (AveW), is to pool forecasts obtained using the same model but estimated across different observation windows. This forecast combination strategy can also be seen as a risk diversification strategy in the face of uncertainty regarding possible breaks. An interesting feature of these methods is that no exact information about the structural break is necessary. As in Pesaran and Timmermann [2007], we compute average block (window) return for the i th style factor for the most recent (T-m) block data and denote by μ i (m) the vector of expected returns of each factor, that is: (2) We also denote by Σ i,j (m) the covariance of the factor returns, which is: This approach is implemented in two stages. Exhibit 1 shows that each model calculates expected returns and variance/covariances based on different estimation windows or time periods. The individual model s expected return vectors and covariance matrices are then used as inputs into individual mean-variance optimizations. The end result is N (depending on the number of models) optimal style weight vectors which are then used in the next part of the approach. (3) Exhibit 1: Pooled Forecasts Approach, Stage 1 In the second stage we perform the optimization scheme of Equation (1) for m-different rolling windows, that is:, subject to: and where is defined through Equation (2), i.e., and σ factors through Equation (3), i.e., as. This stage involves averaging across the optimal weights for different estimation windows, as follows: Exhibit 2 depicts this procedure. For each model there is an individual optimization based on different sample moments. The end results are a number of model weights vectors which we then calculate the average factor weight across models. Exhibit 2: Pooled Forecasts Approach, Stage 2 6

7 The idea of this approach is that we remove model uncertainty by averaging all possible models, i.e., in our case different rolling windows for mean and covariance matrices. As mentioned previously, averaging across windows can be useful for forecasting when faced with structural breaks in the data. This case is particularly important in volatile market environments where sudden factor performance reversals have been common. Technically, when the data is subject to break(s), a full sample estimator, that is, an expanding window, will yield to low variance but biased forecasts. On the other hand, forecasts based on a sub-sample, that is, a recent rolling window, will tend to reduce bias but inefficiencies arise due to higher variance. Therefore, in the absence of full information regarding the point and size of break(s), an optimal approach is to use different sub-windows to forecast and then average the outcomes, either by simply using equal weights or by cross-validated weights. Using equal weights over forecasts can be seen as setting a uniform prior (uninformative prior) distribution over our mean and covariance estimates/forecasts. This approach allows us to use short-, medium- and long-term information about factor s performance and the correlation structure. We (optimally) equally weight this information given our uncertainty about the out-of-sample possible outcomes. In addition, given that factors are subject to sudden reversals (breaks), this approach allows us to control for possible reversals during the out-of-sample period. Empirical Analysis We empirically investigate whether controlling for estimation risk with either of the two approaches improves the performance of style rotation portfolios. We employ the style rotation methodology that we have described in the previous section. Data and Empirical Setup The focus of our analysis is not on the style/factors that we have chosen for our style rotation strategy these factors could be anything. However, we have selected styles that the typical systematic investment managers would possibly use in their investment process. We have also been careful to include a blend of style/factors with different alpha decays (or volatilities). Our base universe is the MSCI Europe and stocks are assigned scores with respect to thirteen standard style factors and fall within eight broad investment themes. These are GARP, Value, Growth, Low Risk, Quality, Reversals, Momentum, and Estimates. 3 Our analysis extends from While we consider all thirteen factors when constructing portfolios, optimal style weights are determined by aggregating the individual factors into their style composite groupings. The backtest methodology consists of ranking all stocks in the universe by their respective aggregate style score at the end of each month. We then form equally weighted quintile portfolios and focus on the top and bottom quintile portfolios. Subsequently we calculate the next month s total return for each portfolio and rebalance monthly. Survivorship bias has been addressed by using the index constituents at the time of rebalance. Statistical significance for the mean and median are tested with a t-test and a sign test respectively. The statistical significance of the IRs is tested on the basis of 10,000 bootstrapped samples from the strategies original return sample. Optimisation Process To solve the optimization problem, i.e., the two variants of Equation (1), we use NUOPT (SPLUS) numerical optimizer. As our objective function is non-linear we employ trust-region methods to tackle the optimisation problem. As mentioned earlier, we apply a hard constraint to the problem: minimum and maximum factor weights/exposure, i.e., w min w i w max (for i=1,...,k). For our analysis this constraint plays a significant role in performance/risk statistics in factor dynamic 3 - Further details on the construction of the style factors are available upon request. 7

8 approaches. Removing this constraint tends to generate slightly higher factor exposures (even under the Ridge approach) which tend to undermine performance and risk profile of our strategy. Although allowing factor shorting could be debatable, 4 and in some cases, it seems that it could break some intuition behind finance/quant models (e.g., the logic behind combined value and momentum signals buying expensive stocks with poor price and earnings momentum), 5 from our analysis we find that based on monthly rebalances, removing the factor short constraints does not greatly contribute to the overall performance of a given factor model and it seems that the improvements are observed only during specific time periods. Results We compare our results with two style portfolio construction strategies. The first one is an equallyweighted scheme of all factors (EW hereafter). The second one involves weighting factors with respect to their most recent IRs (NOIR hereafter). This approach resembles the characteristics of factor-level momentum strategies and it does not take into account the correlation structure across factors. The set of statistics we use for comparison involve descriptive measures of the empirical return distribution, measures of extreme risk, drawdown statistics, and stock turnover. 6 Ridge Approach The results of the backtesting analysis involving the regularization weighting scheme discussed above are tabulated in Exhibit 3. As well as applying the Ridge penalty in the objective function, as mentioned before, we also constrain the factor weightings between 0% and 15%. Note that lower bound provides a short constraint but the upper bound was selected on an ad hoc basis ad hoc, in so far as we use a rough rule of thumb being that we did not want exposure that was double the average factor weighting. This being, we consider 13 factors, 1/13 ~ 7.5%, 7.5 x 2 ~ 15%. We also recognize that to some degree this is an additional parameter that can influence the overall results of our analysis. To evaluate the results from our analysis, we report different statistics related to risk and performance profiles. The first group of statistics shows the empirical distribution of the monthly returns, i.e., the difference in returns between the top and bottom quintile baskets. It is interesting to see in Exhibit 3 that the Ridge approach tends to improve the return distribution. If we set the Ridge parameter λ to 8.0, the average monthly return equals 1.15% which is approximately 50% higher than the equally weighted strategy (0.77%) and non-optimized IR strategy (0.74%). In the second group of performance and risk statistics, we show that volatility increases to 13.8% from 13.6% and 12.8 for the NOIR and EW cases, respectively. However the improvement in IR remains economically significant, i.e., from 0.65 and 0.72 for EW and NOIR to a range of 0.87 to 0.98 for the Ridge approach depending on the value of λ. The third group shows the semi-variance and Conditional Value at Risk (CVaR) statistics; both statistics tend to show mixed results. Volatility of negative returns (semi-volatility) moves from 12.7% (NOIR) and 9.0% (EW) to 10.9%; while CVaR is -8.06% from the -9.83% (NOIR) and -7.41% (EW). Drawdown statistics are presented in the table s fourth group, it is interesting to verify that losses from the high water level reduces to 20.5%, a level lower than the 41.5% and 25.0% shown by the NOIR and EW schemes respectively. It is also important to evaluate the stock turnover. Obviously a more dynamic or flexible approach will tend to generate higher stock/style rotation. It is interesting to see that the Ridge approach shows a turnover of around 85% an amount higher than 75% (EW) but lower than 86% (NOIR) levels of the two benchmark approaches. The turnover statistic also shows the direct impact of Especially when using our Ridge approach as Jagannathan and Ma [2003] argue that with the short-sale constraint in place, the sample covariance matrix performs as well as covariance matrix estimates based on, for example, shrinkage estimators. 5 - This point is also related on how we should interpret a negative IC/IR, on one hand a negative IC/IR could be seen as a short signal if the aim of the investment process is to track factor synthetic assets (e.g., a Value ETF). From this point of view, see, for example, Section (page 205) in Qian et al. [2007] where factor shorting is allowed. On the other hand, a negative IC/IR for a signal could be interpreted as the signal having low forecasting power and therefore its exposure to this factor/signal should be decreased. 6 - We measure stock turnover as the percentage of in/out stocks for each of the top and low basket, we divide by two in order to obtain one statistic. This approach resembles.

9 the Ridge hard constraint. As the Ridge parameter, λ, increases, the Ridge constraint becomes tighter and factor weights tend to shrink faster. As a result, we allow for a lower flexibility or dispersion in our factor allocation which directly translates into better performance. Exhibit 3: Performance and Risk Statistics (Ridge Approach) Note: ** and * indicate statistical significance at the 1% and 5% significance levels respectively. On the flip-side as λ approaches zero the penalty becomes less binding. Setting the λ= 0.02 can be considered as the benchmark to conducting the analysis in the absence of the constraint. In additional analysis we investigated Lasso as an alternative to Ridge. Given our constraints on the weights, i.e. 0 w i 15%, we obtained similar results in terms of factor exposures but slightly less compelling results in terms of performance/risk profiles. Finally, in Exhibit 4 we plot the cumulative performance of the strategy using λ=8 as a Ridge (complexity) parameter. It can be seen that this strategy tends to outperform the two benchmarks EW and NOIR. Exhibit 4: Wealth Curves of Ridge Approach (vs. Benchmark) Multiple Estimation Windows As before, we conduct the same analysis and show the results of our empirical analysis in Exhibit 5. For this analysis the overall estimation window is 130 days regardless of the window block size. The number of window blocks that we use determines the number of estimation windows that we consider in the context of the overall estimation window. For example, given the overall estimation window size of 130 days, a 3 Block analysis refers to average conducted across a window of 43 days, 87 days and 130 days. 9

10 Specifically, in Exhibit 5 we observe that the monthly mean (median) returns increase to 1.16% (1.17%) for 3 block case, higher than the 0.74% (1.07%) and 0.77% (0.53%) obtained through NOIR and EW. Volatility tends to increase slightly to 14.1% from the 13.6% (NOIR) and 12.8% (EW). These two components have a significant impact on the IR. In particular, there is an improvement from 0.65 and 0.72 for EW and NOIR to a range of 0.89 to 1.00 for the pooled forecasts approach depending on the number of used blocks. Furthermore, in terms of the extreme risk profile, semi-volatility measures and CVaR at 5% show mixed results around 10.2% and -8.23%, respectively; while the NOIR (EW) benchmark presents a 12.7% (9.0%) and -9.83% (-7.41%), for the semi-volatility and CVaR statistics, respectively. In addition, the duration of drawdowns tends to be shorter combined with drops from a highwater mark of -21.6%, a level lower than the -41.5% and -25.0% observed by the NOIR and EW schemes respectively. The stock turnover remains relatively stable around 0.80 from the 0.75 and 0.86 shown by the NOIR and EW benchmarks. Averaging over different rolling window blocks, i.e., 1 to 10, varies the performance and risk profile of the strategy. Our analysis suggests that averaging optimized factor weights over three different blocks, results in an optimal balance between performance and risk. Despite ten blocks giving the highest IR and average return, three blocks seems to provide slightly better drawdown and volatility statistics. Therefore, three blocks shows an IR of 0.99 combined with a drawdown of 21.6% which is an improvement to the standard case when only one rolling window is taken into account and IR and drawdown are 0.89 and -22.8% respectively. Exhibit 5 Performance and Risk Statistics (Pooled Forecasts Approach) Note: ** and * indicate statistical significance at the 1% and 5% significance levels respectively. Exhibit 6 depicts the cumulative performance of the backtesting analysis using three rolling blocks. This approach also outperforms the two benchmarks in the study period. 10

11 Exhibit 6: Wealth Curves of Pooled Forecasts Approach (vs. Benchmark) Cross-sectional Comparisons This section explores the benefits of the highlighted approaches further by comparing the performance of the best of them with two well established active style rotation approaches. Qian et al. [2007] advocate that style rotation portfolio strategies can benefit by incorporating information coefficient (IC hereafter) covariances/correlations. While the reader is referred to the original work of Qian et al. [2007] for the details of the approach, we highlight that optimal portfolios are obtained through (TS Sigma hereafter): where w* denotes the optimal weights. For a model with K factors, i.e. the average IC by a vector, Σ IC denotes the IC covariance structure of the K factors, and s is an arbitrary, generally positive constant. A second approach involves the factor correlation matrix Φ IC that makes use of the cross-sectional factor covariance (CS Omega hereafter), such that: In both cases we average ICs over recent observations (six months, as before). Moreover, given that the above equations are already solutions (closed-form) of IR optimizations, weights can be obtained directly by inverting the covariance matrix. Exhibit 7: Performance and Risk Statistics (Comparison with Other Approaches) Note: ** and * indicate statistical significance at the 1% and 5% significance levels respectively. 11

12 In Exhibit 7, we make a comparison on the analysis between our results and these benchmarks. In terms of the return profile, the PT 3 Block and Ridge (lambda=8) tend to outperform these other approaches. The IR ratios also tend to show better performance on average our approaches show an IR of close to 1 while the other approaches show on average However, our approach(s) tends to show slightly higher volatilities. Other risk measures such as semi-variance and CVaR at 5% also tend to increase to an average -8.1% for the CVaR measure from the -7.0% showed by the QHS ICs approaches. Maximum drawdown statistics are lower in our approach while QHS ICs while turnover statistics tend to increase. Note that we are not suggesting that the QHS IC optimisation approach is inferior and indeed the PT and Ridge techniques could easily be applied to the IC opt. approach. We merely highlight that this is an alternative approach to dynamic style weighting. Conclusion In this article we investigate if techniques that address estimation risk, optimal prediction, and variable selection improve equity style rotation strategies. We employ integrate two recently developed/advocated approaches that have been shown to work well in the presence of estimation risk and structural breaks in data. The first involves using of regularization procedures which in the context of portfolio construction have shown to possess very attractive properties. They reduce the sensitivity of the optimization to possible collinearities between assets, control transaction costs by promoting stability/sparsity, and improve the out-of-sample performance relative to the classical Markowitz mean-variance approach. The second approach we used involves pooling forecasts that have been obtained with the same model but across different observation windows. This approach minimizes the impact of structural breaks on moment estimates and can be extended to deal with different types of model uncertainty. The results from our analysis indicate that performance/risk gains can be obtained by dynamically adjusting exposures to style factors. Departing from naïve approaches generates benefits which can be as significant as an improvement in the IR of about 54%, i.e., from 0.65 (naïve) to about 1 (dynamic). However, we show that the speed of adjustment should be limited in order to improve equally weighted schemes, i.e., naïve diversification. A fast reaction model will tend to generate unnecessary turnover and the model factor re-weighting can be triggered by factor noise. While we believe our results are overall compelling, we are cognizant that using a dynamic approach is not a panacea. Forecasting style returns and therefore identifying style turning points is not a trivial task and being able to do this successfully is the key. It s the same as any quantitative investment strategy there is not much point having the world s best portfolio construction technique or having an accurate market impact cost forecasting model if you don t have alpha or skill. We thus believe we have a good framework for determining optimal style weights but recognize that the key input is the style return. Our future research will integrate this issue in the analysis. 12

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