Brazil Risk and Alpha Factor Handbook

Transcription

1 Brazil Risk and Alpha Factor Handbook In this report we discuss some of the basic theory and statistical techniques involved in a quantitative approach to alpha generation and risk management. Focusing on the Ibovepsa and a custom universe of liquid stocks called BRA+, we show how a quantitative factor s ability to generate alpha can vary significantly according to the market capitalization of the universe on which it is tested. We examine theoretical returns to pure style factors, and observe that momentum, value, and growth appear to offer low-volatility sources of uncorrelated alpha. We show how factor returns and factor return correlations can vary significantly depending on the sectors in which the factors are tested. Focusing on a broad sector classification of defensives vs. cyclicals, we observe that earnings predictability and growth perform better among cyclicals, while value and volatility perform better among defensives. Momentum has performed well among both groups, though it has had higher risk-adjusted returns among defensives. This handbook is meant to serve as a reference document for our quantitative approach to equity investing, including techniques for managing risk and generating alpha. We will accomplish this by: 1. Providing a brief introduction to the notion of Quantitative Investing and one of its fundamental tools, a Multi- Factor Model. 2. Listing and categorizing some common quantitative factors and discussing the notion of systematic risk. 3. Examining the market decomposition of both the Ibovespa Index and a proprietary index of liquid Brazilian equities (BRA+) into various subsets that focus on size, style and sectors. 4. Describing some basic tools and statistics for analyzing quantitative factors such as quantile portfolio analysis and information coefficients. 5. Introducing a multi-factor risk model that we will use to analyze the Brazilian equity market in subsequent reports. Ilan Gleiser Chief Risk Officer ilan.gleiser@brasilplural.com Peter Goldstein Risk Manager peter.goldstein@brasilplural.com

2 Introduction: What is Quantitative Investing? A quantitative approach to asset management aims to take investment insights from a statistical analysis of the equity markets and use this information to create optimal portfolios that capture the maximum expected return (or, alpha), given one s level of tolerance for various types of risk. There is a long history of academic and professional research documenting accepted approaches and best practices, and we draw on this extensively. 1 While the factors and financial ratios that ultimately guide the investment decisions of quantitative managers are often the same as those used by fundamental managers or technical analysts such as the P/E ratio, or Relative Strength Indicator there are many advantages to a systematic and scientific approach to analyzing these factors: Discovering and targeting returns that are both independent of the direction of the general market or specific sectors or indices, and uncorrelated to equity returns generated by other portfolio managers. Creating optimal portfolios that use information more efficiently, resulting in higher risk-adjusted returns. In general, by incorporating fundamental and statistical insights about market structure in our analysis and portfolio construction process, including correlations between stocks, sectors, and styles, we are better able to: Repeatedly and systematically capture our investment insights. Tightly control risk. Eliminate any human bias or unintended bets that can appear in a portfolio when these relationships are not taken into account. In this Handbook, we hope to make the basic details of this approach as clear and simple as possible. 1 Please see Bilbiography for a list of references. 2

3 1. Brief Overview of a Multi-Factor Model A Risk Model is a mathematical model that is primarily used for: Analyzing risks inherent in an existing portfolio. Conducting performance analysis and attribution to understand the risk / return profiles of historical portfolios. Constructing optimal portfolios that maximize alpha, given the level of tolerance for various types of risk. In general, risk is defined as the uncertainty or volatility of investment returns. In a mathematical context, risk is defined using the variance or standard deviation of returns for an individual asset and covariance of returns between two or more assets. Mathematical details aside, however, it is important to emphasize that analyzing portfolio risk requires understanding the co-movements of securities in a portfolio, so that we can both (1) better assess the drivers of volatility in current and historical portfolios and (2) systematically minimize the volatility of portfolio returns, while still capturing as much alpha in the portfolio as possible. To accomplish these tasks, a multi-factor risk model decomposes the volatility and comovement of securities into a portion that is attributable to either common investment themes such as value, growth, momentum, size or sector and industry membership, and a portion that is specific to each individual security. These common investment themes are referred to as style risk factors, and they are related to intuitive dimensions of the market that help explain the co-movement of security returns. As Grinold and Kahn, pioneers in the field of multi-factor risk models, explain: A momentum factor, [for example,] taken to be a measure of the price performance of the stock for the past 12 months, is not intended as a forecast of continued success or mean reversion. It is merely a recognition that stocks that have been relatively successful (unsuccessful) during the past will quite frequently behave in a common fashion. Sometimes the momentum will be reinforced, in other times it will be reversed, and in yet other times it will be irrelevant. We are accounting for the fact that in five or six months of the year, controlling for other attributes, previously successful stocks behave in a much different manner than previously unsuccessful stocks. [Grinold and Kahn, 1994] Thus, a multi-factor risk model provides easily understandable and recognizable dimensions along which to analyze risk, allowing for better understanding and control of the risk and return characteristics of portfolios. Multi-factor models allow portfolio managers to better manage risk not simply by minimizing it though this is sometimes the goal but also by ensuring they are actively taking risks only along dimensions where they believe they can outperform. For example, if one thought that stocks with a high dividend yield were likely to outperform but had no particular opinion about the relative performance of the industries in which high yield stocks tend to be concentrated, one could use the risk model to design a low-volatility portfolio with a large exposure to stocks with higher than average dividend yield but that is neutral to all the other sector and style risk factors. Note here that we use the term exposure : the risk factors in the model are standardized so that we can more easily quantify and compare them despite the fact that the original factors for example Size and Growth are measured in different units. The risk factors in the model are specifically transformed into z-scores, so that the average exposure to a given risk factor is 0, and 95% of exposures lie between -2 and 2. 2 Thus, for example, a portfolio with an exposure of 1 to the Size risk factor would be considered to have an above average exposure to large cap stocks and thus would move in tandem with returns to that segment of the market and a portfolio with an exposure of to the Growth risk factor would be considered to have a very negative exposure to stocks with high earnings growth and total asset growth and 2 We describe the process of creating z-scores for factor exposures in detail in Section 4. 3

4 thus would move in the opposite direction of returns to the average or aggregate movement of growth stocks in the market. In plain terms, we can think of the exposure to a factor as a bet on that factor, so that a portfolio with a large exposure to a risk factor would be making a large bet on that risk factor. Similarly, if a portfolio manager is unaware that his portfolio contains a certain large style or sector exposure, that manager is making an unintended bet. 2. Quantitative Factors and Systematic Risk Drawing on common usage, one can think of a factor as anything that contributes to, or has an influence on, the outcome of a particular event. We might say, The positive earnings report was the biggest factor in today s price move. More specifically for our purposes, we define a factor as any measurable, quantitative piece of information about a company or stock that can be used to explain, or forecast, equity returns. In this section, we briefly list a set of factors in our proprietary risk model and discuss the notion of systematic risk. Table 1: Some Quantitative Factors Grouped According to Style Value Growth Momentum Long Term Debt to Equity 1 Year Change in Earnings to Price 1 Month Price Momentum Price to Book 1 Year Change in Earnings Per Share 9 Month Price Momentum Capital Acquisition Ratio Asset Adjusted Change in 1 Industry Relative 1 Month Yr Free Cash Flow Price Momentum Cash Flow Return on Invested Capital Asset Adjusted Change in 1 Yr Operating Cash Flow Industry Relative 5 Day Price Momentum EBITDA to Enterprise Value Change in 1 Year Asset 12 Month minus 1 Month Turnover Lagged Price Momentum Earnings to Price Price Adjusted Change in 1 Year Sales Share Turnover Free Cash Flow to Enterprise Value Sustainable Growth Rate Relative Strength Index Free Cash Flow to Price Industry Relative Operating Cash Flow to Price Interest Coverage Ratio Earnings Predictability Volatility Size Cash Conversion Cycle 1 Month High Minus Low Logarithm of Market Cap Change in Depreciation to Logarithm of Trailing 12 2 Year Weekly Beta Capital Expenditure Month Sales Inventory to Total Assets 5 Year Monthly Beta Working Capital to Sales Annual Volatility Working Capital Accruals Earnings per Share Surprise Net Profit Margin Table 1 presents a number of common quantitative factors, grouped according to style, or investment theme. These style factors can be thought of as systematic risks inherent to entire segments of the equity market. For example, just as stocks in a specific sector are often exposed to similar risks such as regulations that affect profitability or changes in the business cycle, returns to stocks with similar style characteristics can also be highly correlated. If we read, The index suffered some profit taking and fell -1%, it may be that stocks with large positive returns over the past year as captured by a Momentum factor accounted for a large portion of the negative return. In contrast to idiosyncratic, or stock-specific risks to which only specific companies or assets are exposed, such as the success or failure of the release 4

5 of a new product, systematic risks, such as sector membership or co-movement with the market (beta), cannot be eliminated through diversification because the exposures of stocks to the systematic risk factor do not offset. 3 Rather, reducing or eliminating systematic risks often requires explicitly hedging these risks by creating a portfolio that has offsetting long and short positions in stocks that are exposed to common factors. Managing systematic risks, however, certainly does not require eliminating them in all cases or reducing one s exposure to them to zero. Clearly, a value investor will want to maintain a large positive exposure to value stocks. Sector rotation strategies and market timing will also require maintaining positive exposures to certain systematic risks. Thus, whether one wishes to target a specific systematic risk exposure, eliminate it entirely through hedging, or even simply be aware of the systematic risks inherent in one s portfolio, it is clear that identifying, monitoring, and managing exposures to systematic risk factors are essential components of a portfolio risk management process. 3. Market Decomposition: Ibovespa and BRA+ The Bovespa Index (Ibovespa) is the main indicator of the Brazilian stock market s average performance. The stocks that compose the index represent more than 8 of the number of trades and the financial value registered on BM&FBOVESPA s cash market. The index is updated every four months, and the participation of each stock in the theoretical index portfolio is directly related to its trading activity in the cash market relative to other securities. 4 Although the Ibovespa constituents represent a large portion of the liquidity available in the Brazilian equity market, in order to best manage and optimize one s risk exposures it is important to have as large of an investible universe of diverse candidate stocks as possible. For example, at the time of writing there is only one Health Care stock DASA3 in the Ibovespa. And thus if one were restricted to only holding stocks that are in the Bovespa Index, it would not be possible to create a portfolio that was long DASA3 but also neutral to general market movements in the Health Care sector. If one were considering a larger investible universe that contained more Health Care stocks, however, it would be possible to create such a portfolio. Thus to increase the investible universe beyond the Ibovespa constituents, we create a custom universe of liquid stocks that we call BRA+. First we briefly describe the screening process used to create BRA+, and then we analyze the composition and historical performance attribution of the Ibovespa and BRA+. Our database contains a total of 882 stocks that have historically ever traded on the BM&FBOVESPA. Of these 882 stocks, roughly 515 are considered to be current, or active, securities. To find an investible universe among these names that is larger than the basket of Bovespa index constituents, we filter these 515 securities for those that have an average daily trading volume over the last month of at least R$ as well as an (unadjusted) 5 closing price of at least R$1,00. This screening process is repeated monthly, and the results of this screen are combined with the Bovespa index constituents to form what we will call BRA+, our custom equity index for the Brazilian market. Finally, for purposes of performance attribution and analysis, we create a market-capitalization weighted index of BRA+ constituents. Here we briefly take a look at the composition and historical performance of the Ibovespa and BRA+. We employ a multi-factor risk model to achieve two of the objectives discussed in Section 1: (1) to analyze the sector, size, and style factor risk exposures of the indices and (2) to carry out a performance attribution of their historical returns. 3 Other systematic risks that affect all securities might include interest rates, inflation, economic growth, market sentiment. 4 Please refer to Appendix 1 for a list of index constituents as of the May 2012 rebalancing, as well as their weights in the index, sector membership, and market cap. 5 To avoid a future-snooping data bias, we must use unadjusted closing prices which were known at the time in the past when we would have run the filter to update our investible universe. 5

6 Chart 1: Number of Constituents in BRA+ and Ibovespa BRA+ # Constituents Ibovespa Index # Constituents /3/2000 1/3/2002 1/3/2004 1/3/2006 1/3/2008 1/3/2010 1/3/2012 Source: BM&FBOVESPA website, ClariFI Xpressfeed As seen in Chart 1, the Ibovespa has historically had between 50 and 70 securities in the index. As of August 1 st, 2012, there were 67 stocks in the Ibovespa index and 182 stocks in BRA+. 6 Table 2 provides a breakdown of the Ibovespa and BRA+ by market capitalization. Chart 2 provides a breakdown of the Ibovespa and BRA+ by sector, and Chart 3 provides the same sector decomposition historically. 7 Table 2: Decomposition of Ibovespa and BRA+ by Market Capitalization Ibovespa # of Companies % of Companies Mega-Cap: Over R$100 billion % Large-Cap: R$10 billion - R$100 billion % Mid-Cap: R$1 billion - R$10 billion % Small-Cap: Less than $1 billion % All-Cap: Over R$1 million BRA+ Mega-Cap: Over R$100 billion % Large-Cap: R$10 billion - R$100 billion % Mid-Cap: R$1 billion - R$10 billion % Small-Cap: Less than $1 billion % All-Cap: Over R$1 million The most recent Ibovespa rebalancing before this period occurred on May 1 st For Sector membership we follow the Global Industry Classification Standard (GICS) of MSCI: 6

7 Chart 2: Decomposition of Ibovespa and BRA+ by Sector Ibovespa Telecomm Services, 3.4 Utilities, 5.47% Consumer Discretionary, 12.16% Information Technology, 3.29% Materials, 23.09% Industrials, 4.8 Health Care, 0.63% Financials, 22.04% Energy, 14.82% Consumer Staples, 10.14% BRA+ Telecommunicati on Services 4.44% Utilities 8.68% Consumer Discretionary 4.98% Consumer Staples 18.23% Information Technology 3.07% Industrials 5.84% Health Care 1.05% Materials 14.44% Financials 23.72% Energy 15.53% Source: ClariFI Xpressfeed 7

8 Chart 3: Historical Decomposition of Ibovespa and BRA+ by Sector Ibovespa Consumer Discretionary Consumer Staples Energy Financials Health Care Industrials Information Technology Materials Telecommunication Services Utilities BRA Consumer Discretionary Consumer Staples Energy Financials Health Care Industrials Information Technology Materials Telecommunication Services Utilities Source: ClariFI Xpressfeed Table 2 and Chart 2 show that as of August 1 st 2012, the Ibovespa was primarily concentrated in mid-cap and large-cap stocks, as well as in stocks in the Materials, Financials, and Energy sectors. It is also clear that Health Care and Information Technology were the least represented, or smallest, sectors in the index. Chart 3 shows that this has also been the case historically. Finally, we can see from Chart 4 that while many sectors have had a fairly stable representation in the Ibovespa, the percentage of companies in the Ibovespa from the Telecommunication Services sector has decreased significantly over time, since the privatization of the Telebras system in the end of the 90s. In contrast to the Ibovespa, we can see from these charts that BRA+ has a larger concentration in small-cap and mid-cap stocks, smaller concentrations in the Materials and Consumer Discretionary sectors, and larger concentrations in Consumer Staples and Utilities relative to the Ibovespa. The historical sector breakdown for BRA+ has remained fairly consistent over time, though similarly to the Ibovespa, Information Technology and Health Care have tended to have a very small representation in BRA+. 8

9 Chart 4: Ibovespa and BRA+ Exposure to Style Factors of the Multi-Factor Risk Model as of 08/01/ Foreign Exposure Growth Interest Rate Exposure Leverage Liquidity Momentum New York Listing Size Value Volatility BRA+ Exposure Ibovespa Exposure Source: ClariFI Xpressfeed, BARRA Chart 4 shows the exposures of the Ibovespa to various style factors in a multi-factor risk model. Recall from Section 1 that the exposure is a standardized measure that relates a stock or portfolio to a specific dimension of the market or common investment theme that helps explain the co-movement of security returns. From this perspective, focusing on the three largest exposures of Liquidity, New York Listing, and Foreign Exposure, one can see that the Ibovespa is concentrated in liquid securities, many of which are traded on the New York Stock Exchange (e.g., possibly through an ADR) and whose prices exhibit a negative correlation to changes in the exchange rate between the Brazilian Real and the US Dollar (though, as seen through Interest Rate Exposure, not to changes in domestic interest rates). BRA+, on the other hand, generally has smaller absolute exposures, though it also contains many securities that also trade on the NYSE. Finally, as we saw from Table 2, we can also see here from the Size factor exposure being close to 0 for both the Ibovespa and BRA+ that neither index is overly concentrated in Small- or Mega-Cap stocks. Charts 5-7 display the historical exposures of the Ibovespa and BRA+ to various style factors in the multi-factor risk model. We have separated Liquidity and New York Listing from the remaining risk factors simply to improve the appearance of the charts. Chart 5: Historical Ibovespa Exposure to Liquidity and New York Listing Factors Liquidity New York Listing Source: ClariFI Xpressfeed, BARRA 9

10 Chart 6: Historical Ibovespa Exposure to Style Factors other than Liquidity and New York Listing Foreign Exposure Growth Interest Rate Expsoure Leverage Momentum Size Value Volatility Source: ClariFI Xpressfeed, BARRA Chart 7: BRA+ Exposure to Style Factors of the Multi-Factor Risk model Foreign Exposure Growth Interest Rate Exposure Leverage Liquidity Momentum New York Listing Size Value Volatility Source: BARRA, ClariFI Xpressfeed As we might expect from the fact that the stocks that compose the Ibovespa represent a significant portion of the value traded on the BM&FBOVESPA, we can see from Chart 5 that the Ibovespa has historically maintained a consistent, large positive exposure to the Liquidity factor of the risk model. It also appears from Charts 5 and 7 that many of the companies that have issues listed in the US have historically been represented in both the Ibovespa and BRA+. From Chart 6 we can see that the Ibovespa s exposure to the remaining risk factors can vary significantly over time. From Chart 7 we can see that BRA+ has historically had fairly muted risk exposures compared to the Ibovespa, though they too are dynamic and vary over time. Having analyzed the composition of the Ibovespa and BRA+ by company size, sector, and risk factor exposures, we will now briefly present and discuss a performance attribution of their historical returns. 10

11 Chart 8: Ibovespa Daily Index Value Index /1/2000 1/1/2001 1/1/2002 1/1/2003 1/1/2004 1/1/2005 1/1/2006 1/1/2007 1/1/2008 1/1/2009 1/1/2010 1/1/2011 1/1/2012 Source: ClariFI Xpressfeed Chart 9: Ibovespa and BRA+ Cumulative Returns /1/2000 1/1/2001 1/1/2002 1/1/2003 1/1/2004 1/1/2005 1/1/2006 1/1/2007 1/1/2008 1/1/2009 1/1/2010 1/1/2011 1/1/2012, ClariFI Xpressfeed Chart 8 shows the historical index value of the Ibovespa, and Chart 9 shows the cumulative returns to the Ibovespa and BRA+ since January 1 st, Looking back to the historical risk factor exposures in Charts 5-7, as well as the historical sector breakdown in Chart 3, we can ask, What portion of the risk and return of the Ibovespa and BRA+ is attributable to, or explained by, these sub-components? And what are the biggest drivers of risk and return for the indices? Chart 10: Decomposition of Monthly Ibovespa Returns into Risk Factor Returns and Stock Specific Returns 25% 2 15% 1 5% -5% -1-15% -2-25% -3 BRA+ Cumulative Returns Ibovespa Cumulative Returns Source: ClariFI Xpressfeed, BARRA Bovespa Risk Factor Return Stock Specific Return 11

12 Chart 11: Decomposition of Monthly BRA+ Returns into Risk Factor Returns and Stock Specific Returns 2 15% 1 5% -5% -1-15% -2-25% -3 Source: BARRA, ClariFI Xpressfeed BRA+ Risk Factor Return Stock Specific Return Chart 10 shows the decomposition of monthly Ibovespa returns into a portion of returns attributable to factors in the risk model, which are called risk factor return, and a portion of returns that is not explained by the common risk factors, which are called stock specific return. 8 Chart 11 shows the same data for BRA+. Note that in each period the index return is equal to the risk factor return plus the stock specific return. Furthermore, the risk factor return in a given month is calculated from the index s exposure to the risk factors in that month (as shown in Charts 6-7) and the returns to the common investment themes (e.g., Value or Growth) represented in the risk model. 9 Two things are clear from the chart: (1) the risk factor returns tend to be more volatile than the stock specific returns, and (2) the stock specific returns are fairly flat throughout the entire history. These same facts are apparent in Charts 12-13, where we plot the cumulative versions of the single-period returns in Charts 10-11, respectively. Chart 12: Ibovespa Cumulative Risk Factor and Stock Specific Returns Risk Factor Return Stock Specific Return Source: ClariFI Xpressfeed, BARRA 8 Note that in Charts we are analyzing returns in excess of the CDI or risk-free rate. 9 While an explanation of the exact methodology for computing risk factor and stock specific returns is beyond the scope of this paper, we can briefly say that these returns are derived from cross-sectional regressions in which we regress realized single-period stock returns against beginning of period factor exposures; the coefficients of the regression represent the factor returns, and the residuals of the regression are the individual stock specific returns, which are not explained by the risk model. Further details can easily be provided upon request. 12

13 Chart 13: BRA+ Cumulative Risk Factor and Stock Specific Returns Risk Factor Return Stock Specific Return Source: BARRA, ClariFI Xpressfeed We can also examine the cumulative returns to individual sectors and style factors in the risk model to get a better sense of what is driving risk and return for the indices. Charts 14 and 15 show the decomposition of the aggregate risk factor returns in Charts 12 and 13 into returns attributable to individual style factors in the risk model. Charts 16 and 17 show a decomposition of the cumulative index returns of Chart 9 into those attributable to each sector represented in the index. Chart 14: Ibovespa Decomposition of Cumulative Risk Factor Returns by Factor Foreign Exposure Growth Interest Rate Exposure Leverage Liquidity Momentum New York Listing Size Value Volatility Source: ClariFI Xpressfeed, BARRA Chart 15: BRA+ Decomposition of Cumulative Risk Factor Returns by Factor Foreign Expsoure Growth Interest Rate Exposure Leverage Liquidity Momentum New York Listing Size Value Volatility Source: BARRA, ClariFI Xpressfeed 13

14 Chart 16: Ibovespa Decomposition of Cumulative Returns by Sector 105% 85% 65% 45% 25% 5% -15% -35% Consumer Discretionary Consumer Staples Energy Financials Health Care Industrials Information Technology Materials Telecommunication Services Utilities Source: ClariFI Xpressfeed, BARRA Chart 17: BRA+ Decomposition of Cumulative Returns by Sector Consumer Discretionary Consumer Staples Energy Financials Health Care Industrials Information Technology Materials Telecommunication Services Source: BARRA, ClariFI Xpressfeed Utilities Finally, Charts 18 and 19 show both the realized volatility and the volatility as predicted by the risk model, for the Ibovespa and BRA+, respectively. 10 Overall the risk model does a fairly accurate job of forecasting volatility (i.e., risk), though it is conservative in so far as it does tend to slightly over-estimate volatility on average. It is clear from both Charts that the risk model significantly over-estimated volatility following the 2008 market crash A multi-factor model facilitates volatility forecasting by allowing one to express a large portion a portfolio s expected volatility in terms of its exposure to the factors and the covariance matrix of the factors, rather than requiring one to estimate the full covariance matrix of the individual securities. 11 The Ibovespa bottomed at on 10/27/

15 Volatility (Standard Deviation %) Volatility (Standard Deviation %) Brazil Risk and Alpha Factor Handbook Chart 18: Risk-Model Forecast and Realized Volatility of the Ibovespa 6 5 Risk Model Forecast of Ibovespa Volatility Realized Ibovespa Volatility Source: ClariFI Xpressfeed, BARRA Chart 19: Risk-Model Forecast and Realized Volatility of BRA+ 6 5 Risk Model Forecast of BRA+ Volatility Realized BRA+ Volatility Source: BARRA, ClariFI Xpressfeed 4. Factor Analysis Methodology Having introduced some common quantitative factors in Section 2 and analyzed the composition and return history of the Ibovespa and BRA+ according to similar themes in Section 3, let us now review some of the basic methods for measuring whether a quantitative factor has any predictive power, using Return on Equity (ROE) as an example. In this section we first cover ranks, z-scores, quantile portfolios, and information coefficients. We then address the issue of having unintended systematic risk exposures in quantile portfolios, and we discuss two simple approaches for addressing this issue. Finally we run a performance attribution on the ROE quantile portfolio and discuss the results. 15

16 Brazil Risk and Alpha Factor Handbook 4.1 Ranks and Z-Scores To begin our analysis, we source from our database the ROE values for every stock in the investment universe. 12 Then we simply rank the stocks according to their ROE value, where the stock with the lowest worst ROE value is ranked 1, and the stock with the highest best ROE value is assigned a number that depends on the number of stocks in the investment universe for which we have ROE values, as well as the number of tied values. 13 Finally, we compute a z-score by subtracting the average rank from each individual rank and then dividing the resulting value by the standard deviation of ranks for our universe. Clearly, the order of the stock ranking is preserved after performing this transformation; instead of having integer ranks that start with 1, however, we now have a distribution of scores with mean 0 and standard deviation 1, as can be seen in Table 3. We can refer to these scores as exposures. Note that while computing the z-score here is not essential for some of the following applications, it often has the benefit of mathematical simplicity, allowing us to more easily compare the magnitudes of different stocks exposures to different factors. 14 Chart 20: Distribution of ROE values for the BRA+ Universe 6 5 ROE Table 3: Raw ROE values, Ranks, and Z-Scores ROE Rank Z-Score Coverage 98.4% 98.4% 98.4% Min Max Median Mean E-17 Std Dev Source: ClariFI Expressfeed, ROE values as of 08/10/2012 Having processed the raw fundamental factor value into an exposure in this way, we now want to determine if the factor has any predictive power in explaining, or forecasting, future stock returns. While there are a number of techniques of varying levels of mathematical sophistication that can be used including ordinary linear regression, multiple or generalized linear regression, or panel data analysis, to name a few we here describe the commonly used practices of analyzing quantile portfolio back-tests and examining information coefficients. 4.2 Quantile Portfolios Quantiles are simply values taken at regular intervals in an ordered set of data. These values are essentially cut-off points that divide the data into nearly equally sized subsets. For instance, percentiles divide the data into 100 roughly equally spaced baskets, so that 5% of the data has values above the 95 th percentile. To create quantile portfolios we simply choose the number of quantiles to use, and then take our ranks or equivalently, exposures and create a portfolio that is long the stocks in top quantile on an equal-weighted basis and short the stocks in the bottom quantile on an equal-weighted basis, thus resulting in a roughly dollar-neutral long-short portfolio. The number of quantiles will depend on the total number of data points or stocks in our universe, since we want the quantile portfolios to be diversified and not contain 12 Return on Equity is equal to Net Income divided by Shareholder s Equity. 13 In this example there were 189 stocks in the investment universe, and our database had ROE values for 186 (or 98.4%). As you can see from the maximum rank being equal to 170, there were 16 ties. This occurred because there are 16 companies for which 2 classes of shares appear in the investment universe, and both share classes are assigned the same raw value for ROE. 14 Also note that ranking factor values prior to taking z-scores helps mitigate issues with outliers in the data. An alternative method to handle outliers would be to winsorize the raw data, possibly using some multiple of the median absolute deviation (MAD) as a cutoff. 16

17 just a few securities, though using 5 or 10 quantiles is fairly common. 15 Using the data in Table 3 for illustration, if we were to use the median to create just 2 quantiles, we would create a portfolio that is long all stocks in the universe with an ROE above 2.88% corresponding to a rank above 83.5 and an exposure above and short all stocks with an ROE value below 2.88%. We then calculate the returns to this portfolio over some holding period, and repeat the procedure at the end of the holding period when we rebalance the portfolio and recalculate the ranks. We refer to this process as a quantile portfolio back-test, and we refer to the returns from the backtest as quantile portfolio returns. Charts show cumulative, yearly, and monthly returns to a quantile portfolio backtest for the ROE factor that uses 5 quintiles from January 2000 to July Chart 21: Quantile Portfolio Backtest for ROE Factor Quantile Portfolio Quantile 1 (Long Portfolio) Quantile 5 (Short Portfolio) Chart 22: Yearly Quantile Portfolio Returns for ROE Factor Yearly Returns 15.63% 32.27% 50.79% -4.29% 39.22% % 16.95% 23.63% -3.95% 7.45% 48.99% 29.56% 2012 (Jan- July) 15 Kenneth French uses both quintile (5) and decile (10) portfolios in examining returns to the Fama/French research factors: 17

18 Chart 23: Monthly Quantile Portfolio Returns for ROE Factor The factor has performed fairly well over the period. The average monthly return is 1.57%, with a monthly volatility of 5.53%, and the quantile portfolio returns were positive in 63% of the months in the period. The t-statistic of monthly quantile portfolio returns is 3.47, which suggests that these returns are statistically significant at the 95% confidence level. It is interesting to note that the cumulative returns to the sub-portfolio that is just long the top quantile is significantly higher than the aggregate quantile portfolio that is both long the top quantile and short the bottom quantile. The average monthly return to the long-only top quantile portfolio is 2.53%; however the monthly volatility of that portfolio is 10.42%, thus resulting in lower risk-adjusted returns over the period. 4.3 Information Coefficient By comparing returns to stocks in the extremes of factor score distribution, the longshort quantile portfolio gives us some idea of whether the factor ranking is successful in distinguishing winners from losers. Further evidence of this is available if we compare average monthly returns of the stocks in each quantile of our factor score distribution, as opposed to just examining the top-bottom spread. Annualized average monthly returns for the stocks in quantiles 1-5 are presented in Chart 24. Chart 24: Annualized Average Monthly Quantile Returns for ROE Factor Quantile 1 Quantile 2 Quantile 3 Quantile 4 Quantile 5 Here we see that average monthly returns to the stocks in each quantile decrease in a fairly monotonic way, with quantile 1 outperforming quantile 2, quantile 2 outperforming quantile 3, and so on, on average, over the entire period. This suggests that the factor may be doing an accurate job of ranking stocks across the entire universe, and not just getting it right in the extremes. Taking this one step further, we 18

19 can look at the time-series of cross-sectional correlations between our beginning of period factor ranks and the ranks of subsequent realized one-period returns. 16 That is, in each period, we rank the stocks according to their one-period forward return (giving the stock with the lowest, or largest negative return a value of 1), and we calculate the correlation coefficient between our factor ranking (as depicted in Table 3) and the ranking of forward returns. The value of this correlation coefficient is often referred to as the information coefficient, or IC. 17 Chart 25 presents the monthly ICs for the ROE factor. Chart 25: Monthly IC for ROE Factor from 01/ / The IC is often interpreted as the forecasting skill of a factor, or of a quantitative portfolio manager. The average monthly IC of the ROE factor is 0.069, and the IC is positive in 67% of months in the period. Typically, our separate assessments of factor performance based on quantile portfolio returns and IC will agree, as they are both estimates of the general relationship between our factor ranking and forward returns, over the analysis period. But these two estimates can of course disagree. 18 In any given period, the correlation between our factor ranks and the ranks of subsequent returns might be high, while a stock in the top (bottom) quantile portfolio might have a large negative (positive) return, causing the quantile portfolio returns to be negative. Conversely, we can easily construct instances where the quantile portfolio returns would be consistently positive, even though the IC is not. On average, however, one would expect these metrics to agree, as the IC gives an indication of how well the factor ranking maps to the distribution of forward returns, and the quantile portfolio returns represent returns to holding a long-short portfolio based on the extremes of the factor ranks Systematic Risks in Quantile Portfolios Having focused much of our discussion in Section 3 on systematic risks, it would be remiss to fail to acknowledge that quantile portfolios are likely to have significant exposures to systematic risks. For example, being equal-weighted rather than weighted by market-cap, the quantile portfolio could have a larger than average 16 This is referred to as a cross-sectional correlation, as opposed to a time-series correlation, because it is a correlation between factor values and forward returns at a given point in time, rather than, for example, the time-series correlation between an individual stocks factor score and that same stocks monthly forward returns. As we are taking the cross-sectional correlation at each point in time in the analysis period, the result is a time-series of cross-sectional correlation coefficients. 17 There are a number of ways one can perform this calculation. For example, we may want to look at the time-series of correlations between our factor score and total raw returns, excess returns to a benchmark, or residual returns to a CAPM model, and so forth. What is most important is that we understand the assumptions of our methods so that we can draw valid conclusions. For example, a useful benefit of using the non-parametric rank correlation is that it is better suited than Pearson correlation measures for capturing non-linear monotonic relationships. 18 For instance, we can see that the IC was positive 67% of the months in the analysis period, while the quantile portfolio returns were positive 63% of the months in the analysis period, thus indicating that there were some months were the IC was positive but the quantile portfolio returns were not. By comparing Charts 28 and 30 we could find other instances where the signs of the IC and the quantile portfolio return did not agree. 19 There is much more to be said about the relationship between the information coefficient and the alpha of a factor. Grinold s (1994) Forecasting Rule of Thumb states that the alpha of an individual security is equal to volatility times IC times score, where the score is the factor z-score or exposure to the factor, the IC is the IC of a quantitative factor, and, depending on certain mathematical assumptions, volatility is either stock specific volatility or the cross-sectional dispersion (variance) of returns at a given point in time. We can use Grinold s to examine alphas of aggregate portfolios rather than individual securities, though deriving and further analyzing these results are beyond the scope of this paper. 19

20 exposure to small-cap stocks. Additionally, even though the two equal weighted long and short portfolios that comprise the quantile portfolio are roughly dollar neutral, we made no attempt when designing these portfolios to ensure that the betas of these portfolios would be the same, and so it is evidently possible that the aggregate quantile portfolio carries market risk. Similarly, it is clearly possible that all of the stocks in one extreme of the distribution of factor scores are all in the same sector or industry, and so these portfolios might carry significant sector or industry risk. Whether these risks are beneficial or detrimental to quantile portfolio returns, or simply add noise to our analysis, they are clearly something we want to keep in mind and be prepared to address when analyzing a quantitative factor. There are three simple and related ways to analyze the bias and interaction between the factor we are studying and other systematic risks that we will now describe: double-sorting based on multiple factors, creating separate quantile portfolios in different subsets of our investible universe, and finally running a performance attribution on the quantile portfolio. We examine these in turn Double-Sorting One simple approach to managing these exposures in quantile portfolios is to use a double-sort. 20 For example, to create a ROE quantile portfolio that is beta- as well as dollar-neutral, we could first sort the universe of stocks according their betas, then separate the stocks with betas above the 50 th percentile (median) from those with values below, and finally rank the high beta stocks and low beta stocks separately according to their ROE factor score. The result would be 4 portfolios: (a) high-beta, high ROE, (b) high-beta, low ROE, (c) low-beta, high ROE, and (d) low-beta low ROE. The modified quantile portfolio that went long portfolios (a) and (c) and short portfolios (b) and (d) on an equal weighted basis would represent a roughly dollarneutral and beta-neutral portfolio that had a high exposure to the ROE factor. Clearly a similar process could be carried out for any factor, or number of factors, including industry or size. Removing bias, or systematic risk exposures, from a quantile portfolio in this way would help us to more clearly capture risk or return characteristics that are attributable exclusively to the factor under study, net of other effects. For this purpose double-sorting can be a simple and convenient approach Separate Sub-Universes Another approach to examining systematic risk exposures in quantile portfolios involves analyzing the interaction between two factors, rather than explicitly removing systematic bias. For this we would simply create sub-sets of the investible universe according to their exposure to one factor such as beta, industry, or size and then run separate backtests for each sub-universe, so that we can examine quantile portfolio returns and information coefficients from the separate backtests. As an example, to assess if the ROE factor s effectiveness in ranking stocks is driven by its performance among small-cap or large- cap stocks, we separate the universe into two subsets determined by median market cap, and rerun our backtest (Charts 26 and 27 and Table 4). Table 4: Summary Statistics of ROE Quantile Portfolio Returns Average Monthly Average Monthly Average IC Return Volatility Sharpe Ratio Large Caps % Small Caps % 7.43% As an example this procedure is used by Kenneth French to create value portfolios without a size-bias, and size portfolios without a value bias: 20

21 Chart 26: Cumulative Returns of ROE Factor by Market Capitalization Top Half of Universe by Market Cap Bottom Half of Universe by Market Cap Chart 27: Yearly Returns of ROE Factor by Market Capitalization Top Half of Universe by Market Cap Bottom Half of Universe by Market Cap (Jan - July) 13.69% 83.57% 32.14% -5.23% 43.46% % 17.75% % -0.12% 41.89% 26.51% 20.49% % % 55.76% % % % 5.04% 20.11% 57.16% 35.59% In doing this exercise, we note that there is a large differential in performance of the ROE factor depending on the market capitalization of the investible universe, as the ROE factor has a larger IC and larger absolute and risk adjusted returns in the top half of the universe as determined by market cap over the aggregate period. Recent returns, however, appear to have been better among smaller-cap stocks. 4.5 Performance Attribution of ROE Factor Finally, to understand the role that systematic risks play in driving the risk and return of our ROE factor, we can take the quantile portfolio resulting from our factor backtest and analyze it in terms of the performance of the long and short side of our portfolio, sector exposures, and exposures to other risk factors in a multi-factor risk model. Note that here we re analyzing the performance of the factor across our whole universe and not just in a specific sector or market cap. For ease of presentation we here analyze the performance over the last 5 years. We can see from the performance breakdown of the quantile portfolio in Chart 28 that the long (quantile 1) portfolio and the short (quantile 5) portfolio are both importantly contributing to the aggregate quantile portfolio return. We can also see from Chart 29 that returns attributable to factors in the risk model play an important role in driving aggregate quantile portfolio returns. 21

22 Chart 28: Cumulative Returns to ROE Factor by Long and Short Quantile Portfolios Quantile 1 (Long Portfolio) Quantile 5 (Short Portfolio) Quantile Portfolio Chart 29: Cumulative Returns to ROE Factor by risk factor and stock specific return Risk Factor Return Stock Specific Return Total Return 5-5 Examining sector exposures in Charts 30 and 31, it is clear that the long and short portfolios do in fact have significant sector exposures which do not offset, so the longshort aggregate quantile portfolio has large and often persistent sector concentrations. Chart 30: Sector exposures of the long ROE quintile portfolios Consumer Discretionary Consumer Staples Energy Financials Health Care Industrials Information Technology Materials Telecommunication Services Utilities 22

23 Chart 31: Sector exposures of the short ROE quintile portfolios Consumer Discretionary Consumer Staples Energy Financials Health Care Industrials Information Technology Materials Telecommunication Services Utilities As we have pointed out, these systematic risks can introduce sector-specific effects into the returns we observe for the ROE factor quantile portfolio. We can see the individual contributions that sector and various other risk factor exposures make to the aggregate quantile portfolio in Chart 32. Chart 32: Contributions of sectors and styles to ROE factor quintile portfolio 2 15% 1 5% -5% -1 Sector Growth Interest Rate Exposure Liquidity New York Listing Size Value Volatility Foreign Expsoure We are also interested in the systematic risks inherent in the ROE factor quantile portfolio due to its exposures to factors in the risk model. Again for ease of presentation, we chose Leverage and Momentum as two risk factors to which the ROE factor quantile portfolio had large, dynamic exposures, and we present those exposures in Chart 33. We present the portfolio exposures and risk-factor returns for other factors separately in Chart

24 Chart 33: ROE Factor Quantile Portfolio Exposure to Leverage and Momentum Factors % 2% -2% -4% -6% -8% -1-12% -14% Momentum Exposure Leverage Exposure Momentum Return Leverage Return We can tell from Chart 33 and the raw underlying data that the large -12% drawdown in the Momentum component occurred between 01/01/2009 and 09/01/2009, when the average exposure of the ROE quantile portfolio to Momentum was 0.9; the exposure climbed as high as 1.15 in 01/05/2009. The drawdown ended as the exposure to Momentum decreased below 0 to by 01/09/2009. And even as the exposure to this factor became negative this did not negatively contribute to the performance of the ROE quintile portfolio. Thus it is clear from this specific example, and the more general return decomposition in Chart 32, that the ROE quantile portfolio s exposure to systematic risk factors significantly affects its returns. Performance attribution is a useful tool to help us visualize and analyze these exposures, to explore potentially interesting interactions, and to begin to isolate returns attributable to the factor under study, independent of other entangled effects. 5. Plural Multi-Factor Risk Model In the preceding sections we have attempted to motivate a scientific approach to equity analysis that focuses on understanding and explaining market movements in terms of systematic factors to which many securities are exposed. We have discussed a number of such factors related to sector membership, size, and common investment themes or styles, and we have given numerous examples of how to analyze a portfolio or index along those lines. Finally, we have provided a brief introduction to some of the basic techniques one can use to analyze the risk and return characteristics of quantitative factors, including how to explore, and mitigate, the entangled effects that other systematic risks can have on the analysis. Drawing on this approach, we introduce a proprietary multi-factor risk model that we will use to analyze the BRA+ Index and the Ibovespa Index in subsequent reports. The style factors of this model are those listed in Table 1: Value, Growth, Momentum, Earnings Predictability, Volatility, and Size. These style factors are roughly comprised of equally weighted combinations of exposures (z-scores) of the underlying quantitative descriptors in each group. (Thus, for example, the Size style factor is a simple equalweighted combination of Log of Market Cap and Log of Trailing 12 Month Sales.) These style factors, together with GIC sectors, comprise all the factors in the model. Here, for the sake of completeness, we present cumulative sector and style factor returns over the last 10 years. It is important to note that, unlike the ROE quantile portfolio returns analyzed above, these factor returns are pure factor returns in the sense that the theoretical portfolios from which they are derived are completely 24