What Drives the Performance of Efficient Indices? The Role of Diversification Effects, Sector Allocations, Market Conditions, and Factor Tilts

Transcription

1 An EDHEC-Risk Indices & Benchmarks Publication What Drives the Performance of Efficient Indices? The Role of Diversification Effects, Sector Allocations, Market Conditions, and Factor Tilts April 2011

2 We thank Eric Shirbini and Vijay Vaidyanathan for their useful comments. We also thank John St. Hill for the insightful discussions on this topic. Printed in France, April Copyright EDHEC The opinions expressed in this survey are those of the authors and do not necessarily reflect these of EDHEC Business School.

3 Table of Contents Abstract...5 Introduction How Diversification Generates Performance Sector Exposures and Sector Attribution Adjusting Performance for Market Conditions and Factor Exposures When Do Efficient Indices Underperform? A Focus on the Run-up to the Tech Bubble...35 Conclusion...41 References...47 About EDHEC-Risk Indices & Benchmarks...49 EDHEC-Risk Indices & Benchmarks Publications An EDHEC-Risk Indices & Benchmarks Publication 3

4 About the Authors Felix Goltz is head of applied research at EDHEC-Risk Institute and director of research and development at EDHEC-Risk Indices & Benchmarks. He does research in empirical finance and asset allocation, with a focus on alternative investments and indexing strategies. His work has appeared in various international academic and practitioner journals and handbooks. He obtained a PhD in finance from the University of Nice Sophia-Antipolis after studying economics and business administration at the University of Bayreuth and EDHEC Business School. Dev Sahoo is a quantitative analyst at EDHEC-Risk Institute. He does research in empirical finance, focusing equity indexing strategies. He has a master's in risk and asset management from EDHEC Business School. Dev worked as a software developer for several years after obtaining a master's degree from Florida International University. He also has has a bachelor's degree in engineering, with first-class honors, from Indian Institute of Technology. 4 An EDHEC-Risk Indices & Benchmarks Publication

5 Abstract An EDHEC-Risk An Indices EDHEC-Risk & Benchmarks Institute Publication 5

6 Abstract Capitalisation-weighted indices are known to suffer from problems associated with high concentration; they fail to take full advantage of the diversification opportunities offered by equity markets. Efficient indices draw on portfolio construction techniques to provide risk/ return tradeoffs better than those of their cap-weighted counterparts. Efficient indices provide significant improvements of risk/return properties across different regions and various economic and market conditions. This paper, then, provides a detailed comparison of the performance of efficient indices and that of the respective cap-weighted indices to account for the sources of this outperformance. The efficient index is built on the concept of finding a proxy for the tangency portfolio by maximising the Sharpe ratio, and better diversification is a major contributor to the performance of efficient indexation. Efficient indices are at their best when market conditions reward optimal diversification strategies, that is, when cap-weighted indices tend to be most heavily concentrated and when correlations are stable. In addition, we look at the way sector performance contributes to the overall performance of both efficient and cap-weighted indices. Unlike that of the cap-weighted index, the performance of efficient indices is driven more by better diversification across stocks than by shifts in sector weights. Finally, an analysis of the equity risk factor exposures for both efficient and cap-weighted indices confirms that though exposures of efficient indices are obviously different from those of cap-weighted indices the performance of efficient indices cannot be explained entirely by simple factor tilts. On the whole, our results suggest that improved diversification is a key source of the outperformance of efficient indexing. 6 An EDHEC-Risk Indices & Benchmarks Publication

7 xxxxxxxxxxxxxxxxxxxxxx Introduction An EDHEC-Risk Indices & Benchmarks Publication 7

8 Introduction 1 - Taken from Amenc et al. (2010) Amenc et al. (2010) provide a method for the construction of efficient equity indices. These indices weight stocks by their risk-and-return properties to obtain an improved risk/return tradeoff. The aim of these efficient indices is to extract the risk premium of investing in equity markets without suffering from the now widely recognised inefficiencies of cap-weighted indices (see Amenc, Goltz, and Le Sourd 2009 and Amenc et al and the references therein for an overview of both theoretical and empirical findings on the inefficiencies of cap-weighted indices). Rather than weight stocks by the economic footprint of companies in the broad economy (as the Research Affiliates Fundamental Index does), efficient indices weight them by their risk/return footprint in an investor s portfolio. Amenc et al. (2010) show that efficient indexation has, over the last fifty years, generated out-of-sample Sharpe ratios significantly higher than those of the corresponding cap-weighted indices. These results are reproduced in table 1. They also show that higher risk/return efficiency is achieved consistently within subsamples and across varying market conditions and geographic regions. In terms of extreme risk, efficient indices have results similar to those of cap-weighted indices, so the increase in average risk/ return efficiency does not come at the cost of increases in extreme risks. In another paper, Amenc, Goltz, and Martellini (2011) compare efficient indices and other non-cap-weighted indices such as equal-weighted, minimum-variance, and fundamentally weighted indices. The results show that efficient indices not only obtain the highest Sharpe ratios but also appear to have the most diversified factor exposures. Their volatility is lower than that of equalweighted and fundamentally weighted indices, and since they do not have the same tilt towards low-beta stocks as that of minimum-volatility indices, their average returns are higher. The question we try to address in this paper is that of the sources of the superior performance of efficient indices. Since the aim of efficient indices is risk/return efficiency, this paper will often consider risk-adjusted outperformance rather than standalone returns. In a nutshell, our results suggest that the superior performance of efficient indices stems mainly from this indexing method s ability Table 1: US long-term results: performance of efficient index versus that of cap-weighted index 1 The table shows risk and return statistics for efficient portfolios constructed with the same set of constituents as the capweighted S&P 500 index from 01/1959 to 12/2008 Index Ann. average return (compounded) Ann. standard deviation Sharpe ratio (compounded) Information ratio Tracking error Efficient index 11.63% 14.65% % Cap-weighted 9.23% 15.20% % Difference 2.40% -0.55% (efficient minus cap-weighted) p-value for difference 0.14% 6.04% 0.04% - - The results are based on weekly return data. P-values for differences are computed using the paired t-test for the average, the F-test for volatility, and a Jobson-Korkie test for the Sharpe ratio. 8 An EDHEC-Risk Indices & Benchmarks Publication

9 Introduction to exploit the correlation structure of index constituents and the diversification benefits offered by the investment set. Cap-weighted indices, by contrast, are heavily concentrated in few stocks and unable to exploit diversification benefits properly. Efficient indices are less heavily concentrated and exploit diversification well. Diversification, of course, is not simply having a non-concentrated index; it is about profiting from the correlation of the constituents. Efficient indexation will avoid concentration, especially in stocks that are highly correlated with the rest of the universe. In the following sections, we provide several analyses and comparisons of efficient indices and their cap-weighed counterparts in order to examine their performance. In addition, we show where and when outperformance is most pronounced and account for the sources of this outperformance. In the subsequent sections, we attribute the performance to sector exposures and several economic factors. The motivation for sector analysis is to see whether the superior performance of efficient indexation, when compared to cap-weighting, is driven mainly by changes in sector weights or by better diversification across constituent stocks. Similarly, factor analysis and conditional performance analysis provide a better understanding of the performance of efficient indices in various economic conditions and shed light on the dependency of these indices on several risk factors. using weekly returns and quarterly rebalancing, is from January 1959 to December Data is obtained from CRSP and Kenneth French s data library. For the analysis of efficient indices across several regions we rely on short-term international data, which uses eleven efficient indices from different regions. The analysis period for this dataset runs from 2002-Q4 to 2010-Q2 as a result of limited data availability for these indices. But for the analysis of efficient indices over time we use long-term US data, which runs from January 1959 to December The same dataset is used to assess conditional performance and do factor analyses of efficient indices. Throughout the entire document, the benchmark index, unless explicitly stated otherwise, is the S&P 500, and the efficient index is constructed using S&P 500 index constituents. The time period examined, An EDHEC-Risk Indices & Benchmarks Publication 9

10 Introduction 10 An EDHEC-Risk Indices & Benchmarks Publication

11 1. How Diversification xxxxxxxxxxxxxxxxxxxxxx Generates Performance An EDHEC-Risk Indices & Benchmarks Publication 11

12 1. How Diversification Generates Performance 2 - A short squeeze occurs when short sellers generate buying pressure by covering their positions, buying back stocks to return to lenders in anticipation of a lack of outstanding stocks or increases in stock price. Inevitably, these purchases lead to steep but momentary increases in prices for no fundamental reason Cap-Weighting versus Building Well-Diversified Portfolios Historically, cap-weighted indices have tended to concentrate heavily in few stocks, leading to under-diversification and risk/reward inefficiency. Since one of the objectives of indices is to allow investors to seek broad exposure to the equity asset class, investing in highly concentrated indices runs counter to the investor s investment objective. This excessive concentration tends to become even more pronounced in specific episodes. Take, for example, the case of the DAX when Volkswagen experienced a short squeeze 2 on 28 October This squeeze was triggered by Porsche s announcement that it would increase its holdings of Volkswagen to 75%, sending the price of the Volkswagen stock up by 72% in a single day and its weight in the DAX 30 to 27%. The effect was so acute that the DAX rose by 11% even though the shares of twenty-one of its thirty constituents fell. The Deepwater Horizon oil spill had a similar effect on the FTSE 100: BP s stock price fell by 74.8% and the index fell by 6.7% from 2010-Q2 to 2010-Q3. If BP had a 1% equal weight in the FTSE 100 index during the oil spill, the index would have fallen by only 0.3% over the same period. This stark contrast between a 6.7% loss and a 0.3% loss highlights the over-concentration problem afflicting cap-weighted indices. In addition, during the tech bubble, when the price of most technology stocks rose, Nokia, at its peak, had a price-earnings ratio of (in 2000-Q1) and a weight of 10.5% in the EURO STOXX 50 index. When the bubble burst, Nokia was back to a more normal PE ratio and lost about 83.9% of its price, a fall that in turn pushed the EURO STOXX 50 down by 34.2%. If the index had held Nokia at an equal weight of 2%, there would have been less of an impact and the index would have dropped by only 30.3%. High concentration, in short, can have a severe impact on indices as a result of idiosyncratic events or changes in valuation associated with few stocks. Table 2 summarises these three cases. Another subtle yet important point to consider is correlation. Correlation and diversification are two closely related concepts in modern portfolio theory. Conventional wisdom has it that one should not put all one s eggs in one basket, that is, that one should buy a variety of investments so that the failure of one would not result in the collapse of the whole. But what if all investment choices available are highly correlated? They could then all fail simultaneously. The benefits Table 2: Impact of the behaviour of a single stock on cap-weighted indices The following table presents cases showing the impact of stock-specific events on the cap-weighted index. The index used for comparison is the index that holds all relative weights proportional to the market cap but that changes the weight of the stock discussed here to 1/N, where N is the number of index constituents. The returns and volatility are calculated based on the period examined and are not annualised. Time Cap-weighted index Single stock Index with relevant stock downweighted to 1/N Name Return Volatility Name Return Volatility Weight at Return Volatility the peak 2010Q2-2010Q3 FTSE % 20.20% BP % 47.10% 8.00% -0.30% 20.80% 10/29/2008 DAX -0.30% N.A. Volkswagen % N.A % 15.90% N.A. 2000Q3-2001Q3 EURO STOXX % 23.60% Nokia % 73.00% 10.50% % 21.90% 12 An EDHEC-Risk Indices & Benchmarks Publication

13 1. How Diversification Generates Performance of diversification depend, in short, on the correlation of the assets. To examine the rationale for diversification in more detail and to show the diversification benefits of stocks of low correlation, we provide a simple example in which we select two stocks (Goodrich and ExxonMobil) from the S&P 500. Table 3 shows the characteristics of these two firms and their weights in simple two-stock portfolios. We construct both a cap-weighted and an efficient portfolio of two stocks. In this simplified example, only two stocks are included in both the cap-weighted and the efficiently weighted portfolio, so, insofar as they do not hold broad sets of constituent stocks, they are not indices in the usual sense. The example is merely meant to illustrate how stocks with low correlation are combined optimally and how this compares to weighting schemes, such as market capitalisation, that ignore correlation. As we can see, these companies belong to two different sectors and their market capitalisation is very different. Their returns over the period of analysis are similar. Since their correlation is low, it is possible to benefit from diversification. That these companies are not highly correlated is to be expected, of course, as Goodrich is primarily a supplier of systems and services to the aerospace and defence industries, whereas ExxonMobil is essentially an oil company. So, broadly, situations such as rises in oil prices would have a positive impact on ExxonMobil and a negative one on the aerospace industry and on Goodrich. A cap-weighted portfolio of these two companies is weighted very heavily toward ExxonMobil and fails to take advantage of the low correlation of these stocks. We compare the cap-weighted portfolio and an efficient (maximum Sharpe ratio) portfolio to assess how optimally diversified portfolios exploit the low correlation of the stocks. We run an in-sample optimisation over the time period of three years to prevent any estimation error in this simplified example. The efficient portfolio weights capture the correlation properties and provide a much less volatile portfolio and a higher Sharpe ratio. Since the stocks are not highly correlated, they tend to move independently, so, with a view to diversification, holding significant amounts of both is favourable. Table 4 presents the performance of both weighting schemes applied to this simple universe of two stocks. Table 3: A simple example of diversification benefits: characteristics of two weakly correlated stocks and their weight in efficient versus cap-weighted portfolios The following table shows the various characteristics of two major companies in the S&P 500, namely, Goodrich Corp. and Exxon Mobil Corp., which have been selected for their low correlation. The constituent weight for each index is shown for the first quarter of The analysis period runs from January 2004 to December Goodrich Corp. Exxon Mobil Corp. Sector Industrial goods Basic materials Mean return (ann. geometric mean) 23.73% 23.52% Volatility (ann.) 23.77% 19.56% Market capitalisation 3.7 billion (US dollars) 320 billion (US dollars) Correlation 0.26 Weights in cap-weighted two-stock portfolio 1.10% 98.90% Weights in efficient (maximum reward-to-risk) two-stock portfolio 46.20% 53.80% An EDHEC-Risk Indices & Benchmarks Publication 13

14 1. How Diversification Generates Performance Table 4: A simple example of diversification benefits: performance of portfolios using different weighting schemes for two weakly correlated stocks. The following table summarises the performance statistics of two different indices built using Goodrich Corp. and Exxon Mobil Corp. as components. The returns (geometric mean) and volatility are annualised and the analysis period runs from January 2004 to December The efficient portfolio is the in-sample maximum Sharpe ratio portfolio. Cap-weighted portfolio Efficient (maximum reward-to-risk) portfolio Mean return 23.57% 24.50% Volatility 19.41% 17.07% Sharpe ratio The effective number of stocks is the reciprocal of the Herfindhal index, a commonly used measure of portfolio concentration. This is a measure of concentration of its constituents and is given by the inverse of the sum of squared constituent weights. The maximum effective number of stocks that can be obtained for a fixed number of constituents will be that of an equally weighted index. It is clear that what drives the higher reward per unit of volatility in the efficient portfolio is the asynchronous behaviour of Goodrich with respect to ExxonMobil. To show how low correlation enables diversification, we divide the analysis period into quintiles based on the returns of ExxonMobil and calculate the average performance of Goodrich and the indices during these periods. The performance of Goodrich does not depend heavily on that of ExxonMobil, and the performance of the cap-weighted portfolio is nearly identical to that of ExxonMobil (as a result of the weight assigned to ExxonMobil in the cap-weighted portfolio). But the efficient portfolio seems to lend itself to a better blend of these firms and thus provides less pronounced losses over the period as a whole. In other words, by taking advantage of the low correlation of the two stocks, the efficient weighting provides a better risk-adjusted return than the cap-weighted index. To quantify the concentration of constituents in an index, we use two metrics the effective number of stocks 3 and the average weight allocated to the top ten holdings. Although these measures capture the concentration of weights in a few stocks, they do not take into account the more problematic concentration in highly correlated stocks. Therefore, it is also interesting to account for the correlation of the constituents when assessing whether or not an index is well diversified. In the following section, we present several measures of diversification and concentration. These measures can be useful in assessing the concentration and diversification of cap-weighted indices across regions and time periods. If the superior performance of efficient index can be explained by better diversification, we would expect to find more pronounced outperformance in regions (or time periods) in which the cap-weighted indices are more heavily concentrated and less well diversified. Table 5: A simple example of diversification benefits: performance of stocks and portfolios conditional on ExxonMobil stock returns The following table shows the performance of Goodrich Corp. and Exxon Mobil Corp. along with that of the indices built using these stocks. Quintiles are built by sorting the sample by the returns of ExxonMobil. The returns of Goodrich and both of the indices are shown for each of these quintiles. The analysis period runs from January 2004 to December ExxonMobil returns Exxon Mobil Corp. Goodrich Corp. Cap-weighted portfolio Efficient portfolio Low -3.4% -0.7% -3.4% -2.2% 2-0.8% -0.5% -0.8% -0.7% 3 0.5% 1.3% 0.5% 0.9% 4 1.8% 0.8% 1.8% 1.3% High 4.0% 1.6% 4.0% 2.9% 14 An EDHEC-Risk Indices & Benchmarks Publication

15 1. How Diversification Generates Performance 4 - The risk adjustment is done by leveraging or de-leveraging the reference portfolio, using a cash proxy, to the point where it has same volatility as the test portfolio. The difference between the test portfolio return and the return of this new portfolio gives GH-I values. GH-I compares portfolios with same total volatility, as opposed to Jensen s alpha, where the benchmark portfolio has the same market exposure (beta). Another important feature of GH-I is that the volatility of the risk-free asset is not necessarily assumed to be zero Where Do Efficient Indices Outperform? Our study relies on efficient indices built in eleven regions around the world (see table 16 in the appendix for a detailed description of the dataset). For each region, we use the efficient index and corresponding large-mid cap FTSE index. Weekly data, running from 2002-Q4 to 2010-Q2, is used. Moreover, to control for any impact of market structure, the indices are grouped into two sets: emerging and developed regions. The results shown in this section are average values of cross-sectional results from every quarter within the analysis period. We use the Graham-Harvey I (GH-1) measure (Graham and Harvey 1996) to measure the risk-adjusted performance of the efficient index in excess of that of the cap-weighted index. This measure is an appropriate means of representing performance in terms of returns, while making sure that the indices bear identical degrees of risk. The GH-I performance measure scales the volatility of the cap-weighted index, by combining the cap-weighted index and cash, to match the volatility of the efficient index. The measure then compares the returns of this cash-adjusted investment in the cap-weighted index and the returns of the efficient index. 4 This is a convenient representation of risk-adjusted outperformance and encompasses the benefits in terms of both volatility and average return in a single measure. It can be seen as a measure of efficiency improvement rather than just returns improvement. The first measure of concentration and diversification we use for comparison is the effective number of stocks in the cap-weighted index. A low value represents an index, termed highly concentrated, concentrated in only few stocks. Here, we separate the indices into two baskets low values for effective number of stocks and high values and show the average outperformance of each of these baskets. Figure 1: Relationship between the effective number of stocks in the cap-weighted FTSE large/mid indices for different regions and efficient index outperformance in that region The indices are subdivided into developed and emerging sets. In each of these sets the indices are ranked by effective number of stocks in the cap-weighted index and further subdivided into high and low groups. The average performance (quarterly) of efficient indices in excess of that of cap-weighted indices over the past eight years (2002-Q4 to 2010-Q2) is shown here. The average effective number of stocks is also shown over the bars. The figure shows that regions where cap-weighted indices are more highly concentrated have higher outperformance as a result of the availability of better opportunities for improvement. Efficient index outperformance (GH-1) 12% 10% 8% 6% 4% 2% 0% Efficient number of stocks (cap-weighted index) Developed Low High Emerging As expected, the outperformance of the efficient index is greater for indices with a small effective number of stocks. The results are similar for both developed and emerging regions. A weighting scheme that improves diversification is more likely to outperform in regions where cap-weighted indices are heavily concentrated. In addition, the outperformance for indices in emerging regions is greater than that of indices for developed regions. An EDHEC-Risk Indices & Benchmarks Publication 15

16 1. How Diversification Generates Performance The next concentration measure we use is even more straightforward than the effective number of stocks. Instead of looking at the full weight distribution, we focus only on the top ten holdings in the index. This measure is defined as the average weight allocation to each of these ten stocks. As before, we group the indices into high/low groups and calculate with separate calculations for indices in developed regions and indices in emerging markets the average outperformance of the indices in each set. Figure 2: Relationship between the average weight of top ten holdings in the cap-weighted FTSE large/mid indices for different regions and efficient index outperformance in that region The indices are subdivided into developed and emerging sets. In each of these sets the indices are ranked by average weight allocated to the top-ten holdings (in the cap-weighted index) and further subdivided into high and low groups by this concentration measure. The average performance (quarterly) of efficient indices in excess of that of cap-weighted indices over the past eight years (2002-Q4 to 2010-Q2) is shown here. The average values for the average weights for the top holdings are also shown over the bars. The figure shows that regions where cap-weighted index is more concentrated have higher outperformance as a result of better opportunities for improvement. Efficient index outperformance (GH-1) 12% 10% 8% 6% 4% 2% 0% Average weight of top holdings (cap-weighted index) 2.5% 5.1% Developped Low High 3.6% 5.7% Emerging The average weight allocated to the top ten holdings in the cap-weighted index is positively related to the risk-adjusted outperformance of the efficient index. Figure 2 shows that efficient indexation in the regions with high concentration (where the cap-weighted index concentrates more by allocating about 5%, on average, to each of the top ten stocks) outperforms more than in regions where the cap-weighted index is less concentrated (where it has about a 3% average weight in each of the top ten stocks). We use a measure for concentration to assess the ability of an efficient index to capture diversification benefits in different regions. However, the concentration measures used above do not take into account correlation. Goetzmann, Li, and Rouwenhorst (2005) use the ratio of the variance of the portfolio of all investable assets to the weighted average variance of these investable assets as a measure of concentration that takes into account correlation properties. This measure takes into account not only the number of available assets but also the correlation properties. More specifically, a portfolio that concentrates weights in assets with high correlation will tend to have portfolio risk higher than the average standalone risk of each of its constituents. Thus it will have a high Goetzmann measure, that is, high correlation-adjusted concentration. Again, we separate the indices into developed and emerging markets and then sort them by the value of the concentration measure. The efficient index has relatively low risk-adjusted outperformance in regions where the cap-weighted index is well diversified (it has a low value for the Goetzmann correlation-adjusted concentration measure). 16 An EDHEC-Risk Indices & Benchmarks Publication

17 1. How Diversification Generates Performance Figure 3: Relationship between correlation-adjusted concentration (Goetzmann measure) of the cap-weighted FTSE large/mid indices for different regions and efficient index outperformance in that region. Results for both developed and emerging regions are shown below. The regions are subdivided into two sets, those with well diversified cap-weighted indices and those with poorly diversified cap-weighted indices (based on Goetzmann s diversification measure). The average performance (quarterly) of efficient indices in excess of that of cap-weighted indices over the past eight years (2002-Q4 to 2010-Q2) in each of these subsets is shown here. The average values for Goetzmann s diversification measure are also shown over the bars. The figure shows that regions where cap-weighted indices are less diversified have greater outperformance as a result of better opportunities for improvement. Goetzmann s diversification measure (cap-weighted index) Efficient index outperformance (GH-1) 12% 10% 8% 6% 4% 2% 0% 26.6% 37.3% Developped Low High 28.9% 64.3% Emerging On the whole, when sorting regions by the concentration of their cap-weighted indices, we find that the risk-adjusted outperformance of efficient indices over these cap-weighted indices is more pronounced in regions with highly concentrated cap-weighted indices When Do Efficient Indices Outperform? In this section we look at outperformance of the efficient index over time and assess how it is related to various diversification and concentration measures. We use the efficient index built on S&P 500 constituents for its long track record of return data. For each quarter from January 1959 to December 2008, we use three explanatory variables to analyse the outperformance (GH-1) of the efficient index. We use two different measures for concentration: the effective number of stocks and the average weight allocated to the top ten holdings. We also use a measure to quantify changes in the correlation structure to explain the outperformance of the efficient index. Apart from the concentration measures previously used, the change in correlation structure is relevant as an explanatory variable because the performance of efficient indices depends on the estimation of the correlation matrix. The more stable the correlation structure is, the better the estimates are, and hence the better the risk-adjusted performance of the efficient index. In other words, diversification is achieved by taking into account the correlation of the assets of choice. An efficient index captures this structure and makes the best possible allocation of weights to index constituents. When correlation is highly unstable, however, the benefits of achieving optimal weights based on measured correlation will be less pronounced. Our measure of correlation instability is simply the average change in correlation value (across all the stocks in the index) from one quarter to the next (see appendix A for a detailed description). We subdivided the set of data into periods with high or low concentration contingent on our measures. We do the same division into subsets using our measure of correlation stability. We find that, for periods having low changes in correlation or a low effective number of stocks in the S&P 500 or a heavy allocation to the ten top holdings, the efficient index shows more pronounced outperformance. Similar results are found for quintile subsets in which the sets of data are divided using different An EDHEC-Risk Indices & Benchmarks Publication 17

18 1. How Diversification Generates Performance explanatory variables (see figure 14 in the appendix) and when ordinary least squares regression is used to show the explanatory power of each of our variables (see table 17 in the appendix for details). Figure 4: Risk-adjusted outperformance of the US efficient index over the S&P 500 in times with low and high correlation stability, low and high effective number of stocks in the cap-weighted index, and low and high average weight in the top ten holdings in the cap-weighted index The periods (quarters) are subdivided into high and low subsets by several explanatory variables. For each of the variables, the average performance (quarterly) of efficient indices in excess of cap-weighted indices in these two sets is shown below. The analysis is done using the efficient index built on the S&P 500 over the past fifty years (January 1959 to December 2008). Outperformance (GH-1) 7% 6% 5% 4% 3% 2% 1% 0% Outperformance: high and low subsets Average change in correlation Effective number of stocks Low High Average weight of top 10 holdings Figure 4 shows a negative relationship between the outperformance of efficient indices and changes in correlation, a relationship that can be explained by the fact that the covariance estimates work best when correlations are stable. Periods when the cap-weighted index is highly concentrated offer greater potential for outperformance, so the effective number of stocks in the cap-weighted index has a negative relationship with outperformance. Similarly, when the cap-weighted index allocates high weights to a relative handful of stocks (when it is highly concentrated), it is more likely to be outperformed by the efficient index. Diversification focuses on extracting the highest possible return for a given unit of volatility. But cap-weighted indices do not provide proper diversification, as they are usually overly concentrated and the effective number of stocks is low. Index returns may be driven by the returns on very few stocks or even, as we have shown, by the returns on a single stock. Not only are cap-weighted indices highly concentrated in a few stocks but they also completely ignore the basic logic of diversification, that correlation matters. So it is of interest to find the conditions under which efficient weighting delivers results better than those of cap-weighting, and the conditions under which it struggles to do so. As expected, we find that, in the regions where cap-weighted indices are most highly concentrated or least well diversified, efficient weighting makes the greatest improvement to efficiency. Likewise, when cap-weighted indices are less highly concentrated, efficient indexation delivers less pronounced efficiency improvement. It also delivers less improvement when correlations are unstable. This is no surprise, as seeking out the benefits of diversification is harder when correlations are highly unstable. Although differences in correlations and concentration over time are of obvious relevance to the performance of diversification-based strategies, the performance of efficient indices may also depend on sector performance. Moving away from capitalisation weighting will result in differences in allocation to individual sectors and stocks. We now turn to assessing the impact of both differences. 18 An EDHEC-Risk Indices & Benchmarks Publication

19 xxxxxxxxxxxxxxxxxxxxxx 2. Sector Exposures and Sector Attribution An EDHEC-Risk Indices & Benchmarks Publication 19

20 2. Sector Exposures and Sector Attribution 5 - See appendix for further information on the classification method. Sectors, the risk-and-return properties of which are usually familiar to investors, are the most widely used category in equity investing. Some sectors are more volatile than others. Technology stocks, for example, tend to fluctuate more strongly, whereas consumer staples are fairly stable. Moreover, different sectors often have pronounced differences in returns over different phases of the business cycle, which some investors try to exploit by betting on the relative performance of certain sectors. Since sectors are such an important category for equity investing, we analyse efficient indices from a sector perspective. We first report the sector weights of the approach and determine the share of performance accounted for by sector allocation rather than by diversification of stocks Reporting of Sector Exposures In this section, we separate the historical weight allocations for the efficient index and cap-weighted index into twelve sectors over the time period from January 1959 to December The portfolios consist of S&P 500 constituent stocks, and the sectors are those of Kenneth R. French s twelve-industry classification. 5 We compare the differences in sector weights for the two indices. This gives us further insight into the effect of different weighting methods on sector exposures and at the same time describes the variation of the sector weights over time. In addition to reporting the evolution of sector weights and outlining the main differences from the cap-weighted index, we analyse the stability of sector weights in both weighting schemes Evolution of Sector Weights The long-term evolution of weights in the cap-weighted portfolio (see figure 5) suggests a shift from a manufacturing industry base towards information technology and the financial services industry. In addition, over shorter horizons the sector trends exhibit peaks and valleys as a result of stock market booms and busts. Figure 5: Sector weights of the cap-weighted S&P 500 The following figure shows the time variation of weights allocated to different sectors in the cap-weighted S&P 500. Twelve different sectors, taken from Kenneth French s industry classifications, are used to classify the stocks. The analysis period runs from January 1959 to December An EDHEC-Risk Indices & Benchmarks Publication

21 2. Sector Exposures and Sector Attribution Figure 6: Sector weights of efficient-weighted S&P 500 constituents The following figure shows the time variation of weights allocated to different sectors in the efficient index. Twelve different sectors, taken from Kenneth French s industry classifications, are used to classify the stocks. The analysis period runs from January 1959 to December Figure 7: Sector weight differences (efficient index cap-weighted index) The following figure shows the different in weight allocation between efficient index and cap-weighted index built using S&P 500 constituents. Only specific sectors of interest are shown. The analysis period runs from January 1959 to December The sector weights for the efficient-weighted index (figure 6) cannot be interpreted as direct results of the weighting method, as this method is based not on the risk/return and correlation properties of sectors but on the risk/return and correlation properties of individual stocks. In general, sector weights follow the long-term trend of the cap-weighted sector weights. If, however, we take a closer look at the energy bubble around 1980 and the dot-com bubble of 2000, we see clearly that the efficient index does not exhibit the same trend-following behaviour as the cap-weighted index. We compare the weights in the efficient index and those in the cap-weighted index in figure 7. First, as weights in energy in the 1980s and in business equipment (technology) in the late 1990s show, the An EDHEC-Risk Indices & Benchmarks Publication 21

22 2. Sector Exposures and Sector Attribution efficient index tends to weight sectors prone to bubbles less heavily than does the cap-weighted index. This relative underweighting of bubble-prone sectors is, given the so-called herding effect that can drive the cap-weighted index, unsurprising. Before the bubble bursts, these sectors will be overpriced and in turn will have higher market capitalisation. Cap-weighted indices will, by definition, weight the more highly capitalised sectors more heavily. From 2000 to 2008, however, the overweighting of financial stocks by cap-weighted indices is less pronounced. The financial sector bubble was perhaps less striking as it expanded gradually over the period from 2000 to 2007 the huge spikes characteristic of the two other bubbles were largely absent Stability of Sector Weights We use sector-drift scores, similar to the style-drift score of Idzorek and Bertsch (2004), to provide a single measure of the variability of sectors over time. We use this score to compare the sector tilts of the efficient index and that of the cap-weighted index. It is also interesting to identify for both indices the sectors to which allocation varies most and those to which it varies least. We show these statistics along with another measure that captures long-term sector fluctuation. This long-term measure is simply the ratio of the highest weight to the lowest weight of a given sector over the full time period of fifty years. The efficient index has a lower sector-drift score, which means that its allocation to sectors is less variable. The long-term figures confirm this less variable sector allocation. When individual sectors are examined, it turns out that in the cap-weighted index the financial sector is subject to the greatest short-term fluctuations (typical variability of 0.37% per quarter). In the efficient index, the most variable sector weight is that for consumer non-durables (0.31%). The results for maximum weight change over fifty years show that the cap-weighted index displays more extreme changes in sector weights. Here, in both indices, the largest change in weight takes place in the financial sector. In the cap-weighted index, its weight increased from 0.7% in Table 6: Stability of sector weights over time This table shows several measures of the stability of sector weight allocation for the efficient index and the cap-weighted index. The method prescribed by Idzorek and Bertsch (2004) is used to calculate the sector-drift scores. For both the indices, we show the sector with most/least fluctuating allocation over time. The maximum weight change over fifty years is the ratio of the maximum weight to the minimum weight of a particular sector over the full historical period. We show the average, maximum, and minimum values for this ratio for all sectors. The analysis period runs from January 1959 to December Cap-weighted index Efficient index Short-term weight fluctuation Sector-drift score (average weight fluctuation 11.00% 10.15% across sectors) Sector with most fluctuation in weight over time Finance (0.37%) Non-durables (0.31%) Sector with least fluctuation in weight over time Shops (0.02%) Telecommunications (0.003%) Maximum weight change over 50 years Average across sectors 591% 353% Sector with highest change 2965% (Finance) 749% (Finance) Sector with lowest change 141% (Non-durables) 133% (Shops) 22 An EDHEC-Risk Indices & Benchmarks Publication

23 2. Sector Exposures and Sector Attribution 1960 to 21.1% in In the efficient index, by contrast, it changed only from 1.8% in 1960 to 15.6% in The most stable sector in the cap-weighted index is non-durable consumer goods, whereas the most stable weight in the efficient index is for the wholesale and retail sector (shops). As with the average long-term weight change across all sectors and the weight change of the most unstable sector, the weight change of the most stable sector in the index is lower for the efficient index than it is for the cap-weighted index. In addition to describing sector weights over time as well as analysing their variability, it is interesting to assess in detail whether these sector weight changes are a major driver of the risk-adjusted performance of efficient indexation. The results above show that the efficient index is less likely to make sector bets than the cap-weighted index. One may not expect such behaviour to be a major driver of better returns, though it may have obvious benefits in terms of volatility. We analyse in detail whether the sector allocations account for the higher average returns of cap-weighted indices (see below) Decomposing Outperformance: Sector-Level Effects versus Stock-Level Effects To determine whether the superior performance of efficient indexation is the result of smarter allocation to sectors or or smarter allocation to individual stocks, we run some simple tests. We later look at sectors in greater detail. These tests capture only the return attribution while ignoring the risk of the index; they do not look at risk-adjusted returns A Simple Comparison: Changing Sector Weights While Keeping Stocks Cap-Weighted within Each Sector As mentioned earlier, Amenc et al. (2010) have shown that efficient indices outperform cap-weighted indices. Figure 8 further illustrates this outperformance by showing the wealth ratio plotted from January 1960 to December The wealth ratio shows how much wealth an investor in the efficient index could have obtained for every dollar he would have obtained by investing in the cap-weighted index. This outperformance the wealth ratio reaches a peak of three around 2006 is substantial. In absolute terms, this Figure 8: Wealth ratio of efficient index and cap-weighted index The following figure shows the ratio, over time, of one dollar invested separately in an efficient index and a cap-weighted index. S&P 500 constituents are used to build both indices. The analysis period runs from January 1959 to December An EDHEC-Risk Indices & Benchmarks Publication 23

24 2. Sector Exposures and Sector Attribution 6 - See appendix B for a description of the method. 7 - For ease of understanding and presentation, we add the third factor (interaction factor) to the stock weighting factor. outperformance amounts to an average annualised return of about 2.40%. The objective here is to analyse this outperformance against the backdrop of the sector weights presented above. For an intuitive assessment of across-sector and within-sector diversification benefits, we construct a simple benchmark in which sector weights alone change; within each sector, individual stocks are still weighted by market capitalisation. This artificial index would reflect a pure sector weighting effect. If performance differs from that of the standard cap-weighted index the difference can stem only from differences in sector allocations. In figure 9, we plot the wealth ratio of this artificial index together with the wealth ratio of the efficient index. Figure 9 shows that the index that reflects a pure sector weighting effect outperforms the cap-weighted index only slightly, as the wealth ratio is much closer to one than for the efficient index. So it can be seen that the performance of the efficient index stems mainly from sources other than differences in sector allocations Multi-period Sector Attribution Analysis To go beyond the simple exercise presented above, we take a multi-period holdingbased approach 6 (Brinson and Fachler 1985; Menchero 2004) to analyse the efficient index s outperformance; outperformance is broken down into two factors 7 one, the stock-weighting factor, accounts for the share of outperformance attributable to the ability of the index to select profitable stocks, whereas the other, the sector-weighting factor, accounts for the share attributable to the ability of the index to overweight sectors that outperform the benchmark. This approach, of course, ignores index risk. We must acknowledge, too, that the efficient index focuses on improving the Sharpe ratio, so the attribution of returns to sector effects gives a somewhat limited picture, as the initial objective of the index is to improve risk-adjusted returns rather than to improve returns alone. Nevertheless, attributing the higher returns to sector or stock-level effects is an obviously useful exercise. Figure 9: Wealth ratios of efficient index and fictional index with cap-weighted index This figure shows the wealth ratios of an efficient index and a fictional index to the cap-weighted index (S&P 500). The fictional index is built to have a pure sector weighting effect and no stock weighting effect. The analysis period runs from January 1959 to December An EDHEC-Risk Indices & Benchmarks Publication

25 2. Sector Exposures and Sector Attribution If two sectors are not highly correlated, they can, of course, be used together to obtain diversification benefits without changing the inherent weights of the stocks in these sectors. But efficient indices do much more and reweight the stocks individually. So we would expect to observe the effect of smarter allocation to stocks apart from the effect of smarter allocation to sectors. Using Menchero s performance attribution method (appendix B.2), we find that, for the entire time period from 1959 to 2008, approximately 90% of the efficient index s outperformance can be attributed to the stock weighting factor. This finding confirms that of the simple analysis above, in which we find that simply reweighting the sectors while leaving stock weights in each sector proportional to the stock s market capitalisation does not greatly improve the returns of cap-weighting. This finding is perhaps unsurprising, as the efficient index method seeks to maximise the risk/reward tradeoff by considering the properties (correlation, volatility, and return properties) of each constituent stocks individually, without any bias towards the sector. The stock weighting factor is positive for all the sectors, which confirms the ability of the efficient index to weight stocks smartly by using their returns and correlation properties. But the efficient index does have some unfavourable sector allocation in terms of performance (both positive and negative values for sector allocation factors). By avoiding bubble sectors such as technology (business equipment) and energy (energy, oil and gas), the efficient index fails to capture the upside potential related to these bubbles. Clearly, the long-term focus on efficiency has prevented the efficient index from making these bets which, over certain periods of time, pay off. Table 7: Breakdown of weighting factors for various sectors The following table shows the factors for Menchero s performance attribution (cumulative) for the efficient index. The average weight of both the efficient index and the cap-weighted index for each of the sectors is also provided. Average benchmark weights and efficient index weights are obtained by averaging their quarterly weights over the time period. Stock weighting, interaction, and sector weighting factors are calculated using equations (d) (e) (f) in appendix C. The analysis period runs from January 1959 to December Average benchmark weights Average efficient index weights Difference (efficient - benchmark) Stock weighting factor ( ) Interaction factor ( ) Sector weighting factor ( ) Non-durables 8.01% 14.62% 6.61% Durables 4.59% 3.43% -1.16% Manufacturing 19.93% 27.53% 7.59% Energy, Oil & Gas 11.93% 5.28% -6.65% Chemicals 5.94% 3.94% -2.00% Business Equipment 14.23% 7.25% -6.98% Telecommunications 6.29% 1.65% -4.64% Utilities 5.84% 10.58% 4.74% Shops 6.40% 8.29% 1.89% Healthcare 6.98% 4.38% -2.60% Finance 6.06% 6.15% 0.09% Others 3.80% 6.90% 3.10% TOTAL (8.3%) % An EDHEC-Risk Indices & Benchmarks Publication 25

26 2. Sector Exposures and Sector Attribution Figure 10: Average weight allocation The following figure shows the average weight allocation to various sectors for efficient index and cap-weighted index (S&P 500) over the period running from January 1959 to December 2008 (weights are updated quarterly). Figure 11: Outperformance attribution The following figure shows the performance of the efficient index attributable to stock weightings and sector allocation according to Menchero s performance attribution (cumulative). The analysis period runs from January 1959 to December % 92% Yet by exploiting correlation properties, and more generally the risk-and-return properties of individual stocks, the efficient index is able to outperform cap-weighting. This finding, along with the fact that more than 90% of the outperformance is attributable to stock weighting effects, confirms that diversification across securities is a stronger driver of efficient index performance than are sector tilts. As sector effects are less important than smarter stock level allocation, a more detailed analysis of how much each sector contributes to the index performance is relegated to appendix A.3. Stock weighting factor Sector weighting factor From the analysis in this section, we can conclude that, although the long-term sector exposures of the efficient index are fairly similar to those of the cap-weighted index, the efficient index is much less responsive to short-term bubble-related increases in weights. By avoiding bets on sectors, the efficient index sector exposures sometimes cause the efficient index to underperform the cap-weighted index. 26 An EDHEC-Risk Indices & Benchmarks Publication

27 3. Adjusting xxxxxxxxxxxxxxxxxxxxxx Performance for Market Conditions and Factor Exposures An EDHEC-Risk Indices & Benchmarks Publication 27

28 3. Adjusting Performance for Market Conditions and Factor Exposures Yet another way of assessing the performance of the efficient index is to look at the common economic and financial factors that influence returns. Below, we compare the performance of the cap-weighted index and that of the efficient index under various market conditions to determine the states of the market in which the efficient index outperforms or underperforms, a comparison that naturally provides insight into the drivers of outperformance. A more direct way of assessing the drivers of outperformance is to look at the factor exposures of the index. Factor analysis is a common way of focusing on a set of common drivers of returns in equity markets. Like the returns on any equity portfolio, the returns of efficient indices are highly related to such common risk drivers (to be sure, the cap-weighted broad market factor will be one of the relevant factors, as an efficient index investing in the same constituent stocks as the cap-weighted index will be exposed to a common source of market risk). Looking at the exposure to several common risk factors will provide us with useful information about the risk premia captured by the efficient index. 3.1 Conditional Performance We use standard equity risk factors to proxy for various market conditions. For each year, we calculate the cumulative return, volatility, and Sharpe ratio for each of the indices. We also calculate the average values of each of the economic factors over each year. We then sort the sample into subsamples by a particular factor value representing the prevailing economic conditions. To sort the time periods into bull and bear markets, for example, we use the returns on the broad market index (taken from Kenneth French s data library) as the indicator. The annual return on the broad market index of the NYSE, AMEX, and NASDAQ stocks is used to sort the years and form five buckets. The first bucket contains the years of the lowest broad market index returns and the fifth the years of the highest. The rest of the buckets have intermediate values. For each of these buckets, the average return, volatility, and Sharpe ratio are calculated for both the indices. We use the rest of the factors size factor (SMB), value factor (HML), and momentum factor (MOM) to divide the period into high/low factor premium sets. We also use the DEF factor, similar to that used by Schwert (1990) and Fama and French (1989), to capture credit risk and repeat the analysis. This factor is the spread between an aggregate corporate bond index and an aggregate treasury index. For the corporate bond index we use the Bank of America Merrill Lynch Corporate Master Index and for the treasury index the Bank of America Merrill Lynch Treasury Master Index, both from DataStream. Since the data for the DEF spread is available only from October 1986, the analysis for this particular case starts from this date. To compare the performance of these two indices during different market volatility regimes, we use a similar method to divide the years into five different volatility periods. The annual volatility calculated using the weekly broad market index return from Kenneth French s data library is used as the market fear gauge (Whaley 2000). In each of the subsamples constructed based on the market factor, the efficient index outperforms the cap-weighted index. The outperformance is greater in bear markets, 28 An EDHEC-Risk Indices & Benchmarks Publication

29 3. Adjusting Performance for Market Conditions and Factor Exposures Table 8: Conditional performance analysis The sample is divided into quintiles based on average yearly values of several risk factors (Kenneth French s data library). The performance (average over the years within each quintile) of both the efficient index and the cap-weighted index is shown below. The return and the volatility numbers are annualised. The analysis period for all the systematic risk factors except credit risk runs from July 1963 to December 2008 and the data is weekly. The analysis period for the credit risk factor runs from October 1986 to December 2008 and the frequency is weekly. Ann. average return Ann. volatility Sharpe ratio Efficient indexation Difference Efficient indexation Difference Efficient indexation Capweighting Capweighting Capweighting Difference Market factor Size factor Value factor Momentum factor Credit risk factor Broad market volatility Bear % % 3.33% 18.47% 19.56% -1.09% % 1.60% 2.61% 14.96% 15.14% -0.18% % 15.03% 6.37% 11.97% 11.82% 0.15% % 20.30% 0.61% 10.82% 11.32% -0.50% Bull 30.30% 30.02% 0.28% 13.15% 13.61% -0.46% Low 5.25% 10.20% -4.95% 14.07% 14.54% -0.47% % 11.06% -0.71% 13.75% 14.43% -0.68% % 7.70% 4.57% 13.40% 14.25% -0.85% % 11.04% 7.51% 15.36% 14.99% 0.37% High 19.27% 13.15% 6.12% 12.83% 13.28% -0.45% Low 12.77% 16.41% -3.64% 13.96% 14.32% -0.36% % 15.42% -0.35% 12.63% 12.53% 0.10% % 10.80% 3.96% 15.08% 15.17% -0.09% % 7.92% 4.39% 13.73% 14.50% -0.77% High 11.64% 3.66% 7.98% 13.88% 14.80% -0.92% Low 19.91% 13.46% 6.45% 15.26% 15.36% -0.10% % 6.89% 5.17% 13.02% 13.09% -0.07% % 23.35% -0.33% 11.61% 12.13% -0.52% % 0.15% 2.20% 13.20% 12.79% 0.41% High 9.45% 9.69% -0.24% 15.97% 17.63% -1.66% Low 16.03% 18.16% -2.13% 11.26% 11.80% -0.54% % 28.84% -4.25% 11.94% 12.24% -0.30% % 11.46% -2.53% 12.80% 14.12% -1.32% % 6.21% 7.74% 13.57% 15.23% -1.66% High 0.03% -3.68% 3.71% 19.06% 18.45% 0.61% Low 20.93% 18.27% 2.66% 8.35% 8.19% 0.16% % 14.23% 3.83% 11.06% 11.12% -0.06% % 14.37% 1.34% 12.99% 13.20% -0.22% % 9.05% 0.95% 15.20% 15.64% -0.44% High 2.82% -1.18% 4.00% 20.91% 22.32% -1.42% when the need for better risk-adjusted performance is acute. This outperformance is achieved while lowering volatility in all but one subsample, resulting in better Sharpe ratios for all of the quintiles. The quintiles formed with respect to other variables show that there is a general tendency for the efficient index to perform particularly well in times that could be described as bad for most stock and bond investors: the efficient index's An EDHEC-Risk Indices & Benchmarks Publication 29

30 3. Adjusting Performance for Market Conditions and Factor Exposures 8 - We can take the difference of Sharpe ratios to compare them, even though the values are negative because the volatility of the efficient index is lower than that of the cap-weighted index. 9 - We use the daily values from Kenneth French and compound them to get weekly values for the factor. outperformance of the cap-weighted index is particularly pronounced when credit spreads are high and when momentum strategies yield low returns. When the value and size premia are low, however, the cap-weighted index performs exceptionally well as a result of its large-cap growth orientation; in such market conditions, the performance of the efficient index is relatively poor. All the same, for most subsamples, the volatility of the efficient index is still lower than that of the cap-weighted index. Moreover, efficient indexation improves risk/return efficiency in different volatility regimes; from an investment perspective, the outperformance during extremely high uncertainty 8 is particularly important. To look at economic conditions in a broader sense, we do a similar analysis in which we divide the sample into two subsamples and compute performance statistics. We sort the returns into subsamples by prevailing economic conditions. To characterise economic conditions, we use a recession indicator, which we obtain from the NBER, for the US economy. Table 9 shows the results for recessionary and expansionary periods separately. The results show that during both expansion and recession efficient weighting fares much better than capitalisation weighting. In recessions, average returns are lower and volatility of returns is higher. In both stages of the business cycle, efficient indexation provides higher average returns, lower volatility, and thus higher Sharpe ratios. 3.2 Equity Factor Models By dissecting the return observations into different market conditions, the analysis above provides a feel for the factors that influence the performance of efficient indexation. This analysis, however, fails to account for the possible correlation of the conditioning variables. For an idea of the marginal impact of a factor on the performance of efficient indexation we run multiple regressions of returns on risk factors. We use the Carhart four-factor model to study the factor exposures of the efficient index. The analysis period runs from July 1963 to December As in the previous analysis, we use data at weekly frequency. The Fama-French factors, along with the momentum factor, 9 are taken from Kenneth French s data library. The returns for three-month T-bills are used as a proxy for the risk-free return. We also extend this model to add the factor for credit risk (DEF). A similar approach is taken by Fama and French (1993), who use the spread on the returns on high- and lowrated bonds as a proxy for default risk in equity markets. Since weekly data for the bond indices is available from October 1986, the analysis period for the five-factor model runs from this date to December Here, rather than the yield spread, we use the difference in the return on the corporate Table 9: Risk and return in recessions and expansions The table shows risk and return statistics computed for two subsamples. The subsamples are obtained by sorting the weekly observations based on a recession indicator for that week. The recession indicator is obtained from NBER dates for peaks and troughs of the business cycle. The results are based on weekly return data from January 1959 to December Business cycle Efficient indexation Ann. average return Ann. volatility Sharpe ratio Difference Efficient indexation Difference Efficient indexation Capweighting Capweighting Capweighting Difference Recessions 2.26% -1.64% 3.90% 22.29% 22.85% -0.56% Expansions 13.30% 11.19% 2.11% 12.92% 13.47% -0.55% An EDHEC-Risk Indices & Benchmarks Publication

31 3. Adjusting Performance for Market Conditions and Factor Exposures Table 10: Factor exposures and factor-adjusted performance of efficient indices The following table shows the regression results, using two different econometric models, for the efficient index and the S&P 500 equally weighted index. The first is the Carhart four-factor model and the second model has an additional factor for default (credit risk). t-statistics are also provided for each of the values for beta and the alpha values are annualised. The analysis period for the Carhart four-factor model runs from July 1963 to December 2008 and for the five-factor model from October 1986 to December Moreover, the results for the four-factor model, with data from October 1986 to December 2008, are provided for comparison. The risk-free return is the three-month T-bill return. Efficient index Carhart four-factor (July Dec. 2008) Carhart four-factor (Oct Dec. 2008) Five-factor (Oct Dec. 2008) Ann. alpha Market beta Size beta Value beta Momentum beta Credit beta Adj. R-squared Coefficient 1.40% % t-statistic Coefficient 1.10% % t-statistic Coefficient 1.10% % t-statistic S&P 500 equally weighted index Carhart four-factor (July 1963 Dec. 2008) Carhart four-factor (Oct Dec. 2008) Five-factor (Oct Dec. 2008) Ann. alpha Market beta Size beta Value beta Momentum beta Credit beta Adj. R-squared Coefficient 2.59% % t-statistic Coefficient 2.57% % t-statistic Coefficient 2.52% % t-statistic Figure 12: Factor exposures of the efficient index versus those of the equally weighted index The following figure shows the full period results of table 10 graphically. The analysis period is from July 1963 to December 2008 and the values of factor beta are compared for the efficient index and the S&P 500 equally weighted index. bond index and that on the treasury bond index as the conditional parameter. This factor is an ex post measure (return spread between the bond indices) rather than an ex ante measure (yield spread between the bond indices) of credit risk. We use the return spread between the bond indices as it can be interpreted as the return to An EDHEC-Risk Indices & Benchmarks Publication 31

32 3. Adjusting Performance for Market Conditions and Factor Exposures a portfolio that goes long the corporate bonds and short the Treasury bonds. We show the results for the four-factor model for the same analysis period as that for the five-factor model. The results for the four-factor model over the entire period of analysis show positive exposure to both the size and the value factors. The efficient index has negative exposure to the momentum factor. Although exposure to the credit risk factor is negative, the result is not significant and adding the factor does not increase the explanatory power of the regression (R-squared). So, after the four standard risk factors are accounted for, the DEF factor seems not to have a crucial impact on the index returns. In addition, the R-squared, the percentage of return variability of the efficient index accounted for by the variability of the factors in the regression, is still comparatively far from one, a value that underscores the inability of these factors (the broad market factor plus the usual long-short portfolio factors built up using cap-weighted portfolios) to fully capture the return properties of the efficient index portfolio. The beta of the efficient index is indistinguishable from one, a figure that shows that this index has exposure to the market similar to that of the broad cap-weighted index. To analyse the style exposures of the efficient index, we compare the results with those for the S&P 500 equally weighted index. As no information whatsoever is needed to determine the weights of this index s constituents, we would expect this index to be value/growth neutral. Equal-weighting incorporates no information on valuation characteristics (such as book-to-market ratios) or on the risk/return properties of stocks (such as past performance). So the equal-weighted index can serve as a valid benchmark for determining whether a portfolio is value/ growth neutral. The equal-weighted index used here contains only the largest stocks, S&P 500 components. So it should have zero exposure to small-cap effects. In fact, since there are no small-cap stocks in this index, economically it cannot have any small-cap bias. Of course, if one were to use an equal-weighted index that included a broad set of stocks covering all ranges of market capitalisation instead, one would ascertain a small-cap bias, as most of the stocks traded on exchange are small- or even micro-cap stocks. On the whole, the small-cap and the value exposure of the equal-weighted index should provide a good proxy for zero exposure to these factors. The efficient index and the equally weighted index are similarly exposed to the value factor; the cap-weighted index is less highly exposed. The small-cap exposure of the efficient index is similar to that of the equal-weighted S&P 500 index. Nonetheless, the equal-weighted index is highly exposed to negative momentum because it automatically rebalances away from stocks with rising prices as the weights are made equal. The exposure to negative momentum of the efficient index is much milder, in part, perhaps, because the efficient index is rebalanced not automatically but with quantitative information. For a close look at the impact of methodological choices used in efficient indexing, we attempt to use standard factors to account for the excess return. This method gives us a measure of the share of the variability of outperformance 32 An EDHEC-Risk Indices & Benchmarks Publication

33 3. Adjusting Performance for Market Conditions and Factor Exposures Table 11: Factor analysis for excess return of efficient index over two benchmarks The following table shows the results of factor analysis for the excess return of the efficient index over the cap-weighted index and over the equally weighted index. A four-factor Carhart model is used in this analysis and weekly returns and factors over the period from July 1963 to December 2008 are used. The coefficient of regression, R-squared, and t-statistics of regression are provided. Given its weak impact in the regressions above, we do not consider the credit risk factor here we focus instead on the standard equity factors. Excess return of efficient index over S&P 500 cap-weighted index Excess return of efficient index over S&P 500 equally weighted index Market beta Size beta Value beta Momentum beta Adj. R-squared Coefficient % t-statistic Coefficient % t-statistic that can be explained by standard equity risk factors. Here, we use the cap-weighted index and the equally weighted index as the benchmarks against which we measure outperformance. We do a four-factor model analysis for the excess return of the efficient index over these two benchmarks the results are as follows. It is clear that, in both cases, only about 50% of the excess returns can be accounted for by standard factors, a finding that again strongly suggests that these factors do not fully capture the return properties of the efficient index portfolio. Moreover, the size and value exposures of the excess return over the equal-weighted index are close to zero. For two reasons, only about half of the variability of the outperformance of efficient indexation over the cap-weighting or equal-weighting can be explained by exposure to common equity factors. First, the method used for efficient indexation takes into consideration the correlation of its constituents, correlation that cannot be captured by mere static exposure to these factors. Second, the efficient weighting method is applied to individual securities rather than to pre-constructed factor portfolios, so the efficient weights can be chosen with much greater flexibility than in an allocation across factor portfolios. An EDHEC-Risk Indices & Benchmarks Publication 33

34 3. Adjusting Performance for Market Conditions and Factor Exposures 34 An EDHEC-Risk Indices & Benchmarks Publication

35 4. When xxxxxxxxxxxxxxxxxxxxxx Do Efficient Indices Underperform? A Focus on the Run-up to the Tech Bubble An EDHEC-Risk Indices & Benchmarks Publication 35

36 4. When Do Efficient Indices Underperform? A Focus on the Run-up to the Tech Bubble The risk/return profile of the efficient index is superior to that of the cap-weighted index; its sector and factor exposure is very different from that of the cap-weighted index as well. But our detailed analysis shows that it does not outperform the cap-weighted index under all market conditions. During the run-up to the tech bubble, for example, the efficient index underperforms the cap-weighted index (see figure 8). So it is interesting to focus on this period and look at the sector and factor exposures in detail. This detailed examination provides insight into the possible causes of the underperformance of the efficient index. We examine the period from 1995 to 2000, the period during which efficient indexation is most badly outperformed by capitalisation weighting. Table 12 shows that, although the efficient index underperforms the cap-weighted index, its volatility is still lower than that of the cap-weighted index and it thus has a slightly better Sharpe ratio. So the performance of the cap-weighted index comes at the cost of much higher volatility. Moreover, during this period the cap-weighted index returned more than 20% per year, so it was perhaps not the most painful period in which to underperform. One of the reasons for such pronounced underperformance during this period is the sector allocation. In line with our earlier Table 12: Performance of the efficient index and the cap-weighted index in the run-up to the tech bubble The table shows the annual return, volatility, and Sharpe ratio of both the efficient index and the cap-weighted index during the tech bubble. We have used data from January 1995 to December The returns are geometric averages and are annualised. Tech bubble Efficient indexation Cap-weighting Difference Annual average return 19.16% 21.51% -2.35% Annual volatility 13.46% 16.58% -3.13% Sharpe ratio Table 13: Breakdown of performance attribution in the run-up to the tech bubble The following table shows the factors for Menchero s performance attribution (cumulative) for the efficient index during the tech bubble. The average weight of both the efficient index and the cap-weighted index for each of the sectors is also provided. Average weights for the benchmark and efficient index are obtained by averaging the quarterly weights over the time period. Stock weighting, interaction, and sector allocation factors are calculated using equations (d) (e) (f) in appendix C. The analysis period runs from January 1995 to December Average benchmark weights Average efficient index weights Difference (efficient/ benchmark) Stock weighting factor ( ) Interaction factor ( ) Sector allocation factor ( ) Non-durables 8.70% 19.31% 10.61% Durables 2.57% 4.20% 1.63% Manufacturing 21.85% 24.28% 2.43% Energy, Oil & Gas 6.23% 4.79% -1.43% Chemicals 4.36% 4.59% 0.23% Business Equipment 17.71% 5.55% % Telecommunications 5.08% 1.65% -3.43% Utilities 3.47% 14.10% 10.63% Shops 6.34% 7.58% 1.24% Healthcare 9.65% 3.36% -6.29% Finance 10.59% 3.79% -6.80% Others 3.45% 6.80% 3.35% An EDHEC-Risk Indices & Benchmarks Publication

37 4. When Do Efficient Indices Underperform? A Focus on the Run-up to the Tech Bubble discussion of sector weight differences, the efficient index tends to assign overvalued sectors relatively low weights, leading to underperformance during bubbles. Table 13 is a replica of table 7 but for the period from 1995 to The underperformance over the tech bubble period is driven much more by sector effects than is the case for the outperformance over the entire period under analysis. Figure 13, in short, shows that sector effects are quite prominent during the tech bubble. Figure 13: Performance attribution in the run-up to the tech bubble The following figure shows the efficient index s performance that can be attributed to stock section and sector allocation according to Menchero s performance attribution (cumulative). The values shown here are the factors for underperformance. The interaction factor is ignored and only stock and sector weighting factors are shown, as it is these two quantities that we are interested in. The analysis period runs from January 1995 to December % 44% Stock weighting factor Sector weighting factor We also run a regression analysis for this particular period to look at the factor exposures of the efficient index. The results are shown in table 14. During the tech bubble, there is less exposure to the momentum factor and greater exposure to the value factor than there is over the entire period of analysis (see table 10). For this reason, the efficient index, which seeks to avoid concentration and is less likely to lead to trend following than is the cap-weighted index, may underperform when, as during the tech bubble, growth stocks outperform greatly and the market is borne ever skyward by positive momentum. In addition, although the long-term market beta of the efficient index strategy is almost indistinguishable from one, during the tech bubble, the beta of the efficient index was slightly lower, as it did not make the same bets on momentum in large-cap growth stocks as the cap-weighted index did. When the unfavourable factor exposures over the tech bubble are taken into account, the efficient index outperformed its factor benchmark by approximately 1.5% (annualised). Moreover, the factor benchmark accounts for only about 90% of efficient index returns, as the factors fail to capture all of the risk/ return properties of a portfolio that focuses mainly on improving diversification across constituent stocks. We also show the factor exposure for the S&P 500 equally weighted index during the tech bubble period. As explained above, the equal-weighted index is, by design, a style-neutral benchmark. Like the equalweighted index, the efficient index has a stronger negative exposure to momentum and stronger positive exposure to value during the tech bubble than it does during the entire period of analysis (see table 10). The cap-weighted index, in short, became more concentrated in momentum-growth stocks during the tech bubble and thus outperformed the efficient index (more neutral weighting scheme), which resists high concentration in these stocks. An EDHEC-Risk Indices & Benchmarks Publication 37

38 4. When Do Efficient Indices Underperform? A Focus on the Run-up to the Tech Bubble Table 14: Factor exposures and factor-adjusted performance of efficient indices in the run-up to the tech bubble ( ) The following table shows the regression results, using two different econometric models, for the efficient index and the S&P 500 equally weighted index. The first model uses the Carhart four-factor model and the second model uses the Carhart four-factor model with an additional factor for default (credit risk). t-statistics are provided for each value for beta and the alpha values are annualised. The analysis period for both the models runs from January 1995 to December The risk-free return used in these models is the three-month T-bill return. Efficient Index Carhart four-factor Five-factor model Ann. alpha Market beta Size beta Value beta Momentum beta Credit beta Adj. R-squared Coefficient 1.51% % t-statistic Coefficient 1.48% % t-statistic S&P 500 equally weighted index Ann. alpha Carhart four-factor Five-factor model Market beta Size beta Value beta Momentum beta Credit beta Adj. R-squared Coefficient 2.46% % t-statistic Coefficient 2.33% % t-statistic An EDHEC-Risk Indices & Benchmarks Publication

39 Conclusion xxxxxxxxxxxxxxxxxxxxxx An EDHEC-Risk Indices & Benchmarks Publication 39

40 Conclusion Looking at the outperformance of efficient indices from various perspectives reconfirms that efficient indices, on average, provide a better risk/return tradeoff than do cap-weighted indices. High concentration, as we have seen, can cause a handful of stocks to drive the risks and returns of cap-weighted indices. But cap-weighted indices do not suffer only from concentrating in few stocks. They also fail to take advantage of the diversification benefits of exploiting the correlation of these stocks. Comparing the efficient index and the cap-weighted index on different measures of concentration shows that the efficient index outperforms the cap-weighted index across different regions and over time largely as a result of better diversification. Risk-adjusted outperformance is in fact more pronounced when the level of diversification in the cap-weighted index is low. When the share of performance attributable to sector allocation is determined, it is clear that, over the longer term, sector allocation has very little to do with the performance of the efficient index. The source of most outperformance is better diversification using individual stocks. The cap-weighted index usually makes large bets on individual sectors as bubbles expand, whereas the efficient index does not. large-cap and growth stocks are doing particularly well. On the whole, it is largely its exploitation of the potential for diversification of the constituent universe of a broad equity market index that enables the efficient index to outperform. At the same time, our results show that, in certain market conditions, efficient indices tend to underperform, as they did in the run-up to the tech bubble, during which the efficient index mechanism to its relative detriment avoided placing heavy bets on the technology sector and, more generally, on growth or momentum. It is our hope, then, that this paper, with its analyses of diversification effects, sector allocations, market conditions, and factor exposures, provides a sense of the ways efficient indices will behave under different scenarios and a better understanding of the causes of their outperformance of standard cap-weighted indices. We find that the efficient index has better performance in both bear and bull markets, but that outperformance is more pronounced in bear markets. More generally, the performance of efficient indexation actually seems strongest in bad times for investors, such as times with high credit spreads and low momentum returns. Cap-weighting, however, tends to outperform efficient indexation when 40 An EDHEC-Risk Indices & Benchmarks Publication

41 Appendix An EDHEC-Risk An Indices EDHEC-Risk & Benchmarks Institute Publication 41

42 Appendix Appendix A Diversification Measures and Definitions 1. Goetzmann s Diversification Measure Goetzmann s diversification measure (Goetzmann, Li, and Rouwenhorst 2005) is the ratio of the index s variance to the weighted sum of its constituents variances. It is given by the following equation: It is a ratio of systematic to total risk. A small ratio means that portfolio risk (when constituents are combined) is lower than the average total risk of constituents (when constituent risk is assessed in isolation), which suggests a well-diversified index. 2. Effective number of stocks The effective number of stocks is given by the inverse of sum of squared constituent weights. 3. Average Change in Correlation To calculate this measure, we consider only the stocks present in the index for consecutive quarters. The past two years of weekly return data are used to calculate the correlation matrix. The absolute value of the difference between these two matrices is then calculated. The value of this measure is the average value of these differences in correlation over all the stocks. Mathematically, Where n is the size of the correlation matrix and the measure is calculated at quarter t. Appendix B Performance Attribution Method 1. Single-Period Sector-Based Performance Decomposition (Brinson, Hood and Beebower 1986) Brinson s holdings-based performanceattribution approach attempts to break an index s outperformance of a benchmark, the cap-weighted index in this case, down into three main components: a) stock weighting effect, b) sector allocation effect, c) interaction effect, Where and are benchmark (cap-weighted) index returns, sector i returns and sector i weights, whereas and are the efficient index i sector returns and sector i weights over quarterly time periods t. Quadrant (I) -> Return on benchmark portfolio Quadrant (II) -> Return on sector allocation Quadrant (II) -> Return on stock weighting Quadrant (IV) -> Return on efficient index Here, outperformance (IV) (I) is broken down into three parts. First, the sector weighting factor is obtained by keeping the benchmark Table 15: Brinson model Sector allocation effect Efficient index (p) Benchmark index (B) Stock weighting effect Efficient index (p) Benchmark index (B) Adapted from Portfolio Theory and Performance Analysis (Amenc and Le Sourd 2003, 214) 42 An EDHEC-Risk Indices & Benchmarks Publication

43 Appendix stock weighting effect constant. This factor is given by (II) (I). In a similar fashion, the stock weighting factor is obtained from (III) (I). Lastly, the interaction term is given by (IV) (III) (II) + (I). 2. Multi-period Sector-Based Performance Decomposition (Menchero 2004) Although the Brinson model is straightforward, intuitive, and easy to understand, computing multi-period sectorbased performance attribution is harder. The problem that arises when multi-period performance attribution is looked at is that arithmetic attribution effects cannot be added together as a result of the geometric effect of compounding in several periods. A few methods of dealing with this problem have been proposed in the literature. One such method is the optimised linking algorithm proposed by Menchero (2004). This algorithm minimises the deviations of the linking coefficients from some value, A, that corresponds to a natural scaling factor, B, from a single-period to multi-period space. With a Lagrange multiplier approach, the following solutions are derived: From equations (a), (b), and (c), we obtain a new measure for stock, sector, and interaction factors by multiplying each component with a scaling factor B. d) e) f) Where And Where T is the total number of periods. As such, we can also obtain an intuitive measure, closely related to the Brinson model, by summing up the factors (d), (e), and (f) across time and thus decomposing the outperformance as: g) Table 16: Efficient indices used for regional analysis Region Canada Dev. Asia ex Japan Dev. Europe ex Eurobloc ex UK Developed Euro bloc Japan United Kingdom United States Asia Pacific Europe Emerging Latin America Middle East and Africa Ticker EDHCND EDHDAXJ EDHDEXEU EDHEBLOC EDHJPN EDHUK EDHUSA EDHEAP EDHEEU EDHELAM EDHEMEA Table 17: Correlation and OLS results for different control variables with outperformance The following table shows the correlation and regression results for different explanatory variables with outperformance of the S&P 500 index over time. The analysis periods runs from January 1959 to December 2008 and the explanatory variables and outperformance are calculated every quarter. Weekly data for past two years is used for each quarter to calculate the correlation matrix. Average change in correlation Effective number of stocks Average weight allocated to top 10 holdings Correlation with outperformance -10% -15% 11% Regression with Slope outperformance t-statistic An EDHEC-Risk Indices & Benchmarks Publication 43

44 Appendix Figure 14: Outperformance of top and bottom quintiles The set of quarters is subdivided into quintiles based on various explanatory variables. The same variables are used here as in the regression analysis. The average outperformance (quarterly) of the efficient index over the cap-weighted index in the top and bottom quintiles is shown below, for each of the control variables. The analysis is done on efficient index built on the S&P 500 over the past fifty years (January 1959 to December 2008). Outperformance (GH-1) Outperformance of top and bottom quintiles Average change in correlation Effective number of stocks Quintile 1 Quintile 5 Average weight of top 10 holdings Appendix C: Which Sectors Contributed Most to Performance? For an intuitive feel for the composition of the efficient index, the index s average returns are broken down by its sector, ranked, and reported. The results below are obtained by averaging across time and are highly time dependent. Nevertheless, they show that both manufacturing and non-durables account for about 40% of the index returns. After all, until the 1990s, these sectors had a relatively larger number of stocks, allowing efficient indexation to exploit their correlation properties. The efficient index also reaps a big part of its return from non-durables. This sector is usually stable over various market conditions. This leads to better estimates for the inputs used to construct the efficient index and hence provides near optimal returns for the index. Table 18: Average sector contribution to efficient index returns The return of the efficient index is broken down into sectors and the following table shows the average values of these returns over time and the returns that can be attributed to each sector. In the last two columns, these returns are shown as a percentage of the index s overall return. Weekly stock returns are compounded to quarterly frequency to coincide with quarterly rebalancing periods. Average returns are obtained by averaging over 200 quarterly periods from January 1959 to December Arithmetic average returns (annualised) Geometric average returns (annualised) Arithmetic average returns (annualised) as a percentage of overall index return Geometric average returns (annualised) as a percentage of overall index return Manufacturing 3.83% 3.68% 27.24% 26.16% Non-durables 2.13% 2.09% 15.17% 14.88% Utilities 1.25% 1.23% 8.86% 8.76% Shops 1.22% 1.20% 8.65% 8.55% Others 1.04% 1.03% 7.43% 7.34% Business equipment 1.02% 1.00% 7.23% 7.08% Energy, oil & gas 0.81% 0.80% 5.73% 5.68% Finance 0.78% 0.77% 5.54% 5.47% Healthcare 0.69% 0.68% 4.87% 4.85% Chemicals 0.55% 0.55% 3.91% 3.89% Durables 0.52% 0.51% 3.69% 3.66% Telecommunications 0.24% 0.24% 1.68% 1.68% 44 An EDHEC-Risk Indices & Benchmarks Publication

45 Appendix Table 19: Average sector contribution to cap-weighted index returns The return of the cap-weighted index (S&P 500) is broken down into sectors and the table shows the average values of these returns over time and the returns that can be attributed to a particular sector. In the last two columns, these returns are shown as a percentage of the index s overall return. Weekly stock returns are compounded to quarterly frequency to coincide with quarterly rebalancing periods. Average returns are obtained by averaging over 200 quarterly periods from January 1959 to December Arithmetic average returns (annualised) Geometric average returns (annualised) Arithmetic average returns (annualised) as a percentage of overall index return Geometric average returns (annualised) as a percentage of overall index return Manufacturing 0.56% 0.54% 19.55% 19.34% Business equipment 0.40% 0.39% 14.11% 13.92% Energy, oil & gas 0.37% 0.36% 12.97% 12.98% Non-durables 0.29% 0.28% 10.03% 10.12% Healthcare 0.22% 0.22% 7.68% 7.74% Shops 0.19% 0.19% 6.77% 6.81% Utilities 0.15% 0.15% 5.30% 5.36% Chemicals 0.15% 0.15% 5.29% 5.34% Finance 0.15% 0.15% 5.28% 5.27% Telecommunications 0.15% 0.15% 5.25% 5.29% Durables 0.12% 0.12% 4.28% 4.31% Others 0.10% 0.10% 3.48% 3.51% The cap-weighted index is unable to benefit dynamically from the trends in the sectors. Average returns are also less highly dispersed. In addition, the efficient index extracts the return from the sector in a way very different from that of the cap-weighted index, which might suggest that there is a lot of stock-level impact in each of these sectors. Non-durables Consumer Durables Manufacturing SIC Codes and Classification Energy Oil, Gas Chemicals and Allied Products Business Equipment Telephone and Television Transmission Utilities Shops Healthcare, Medical Equipment, and Drugs Money Finance An EDHEC-Risk Indices & Benchmarks Publication 45

46 Appendix 46 An EDHEC-Risk Indices & Benchmarks Publication

47 xxxxxxxxxxxxxxxxxxxxxx References An EDHEC-Risk Indices & Benchmarks Publication 47

48 References Amenc, N., F. Goltz, L. and V. Le Sourd The performance of characteristics-based indices-super-1. European Financial Management 15 (2): Amenc, N., F. Goltz, and L. Martellini Improved beta? Journal of Indices (forthcoming). Amenc, N., F. Goltz, L. Martellini, and P. Retkowsky Efficient indexation: An alternative to cap-weighted indices. Journal of Investment Management (forthcoming). Amenc, N., and V. Le Sourd Portfolio theory and performance analysis. Chichester: Wiley. Brinson, G. P., and N. Fachler Measuring non-us equity portfolio performance. Journal of Portfolio Management 11 (3): Brinson, G. P., L. R. Hood, and G. L. Beebower Determinants of portfolio performance. Financial Analysts Journal (July-August): Fama, E. F., and K. R. French Business conditions and expected returns on stocks and bonds. Journal of Financial Economics 25 (1): Fama, E. F., and K. R. French Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33 (1): Graham, J. R., and C. R. Harvey Market timing ability and volatility implied in investment newsletters' asset allocation recommendations. Journal of Financial Economics 42 (3): Goetzmann, W. N., L. Li, and K. G. Rouwenhorst Long-term global market correlations. Journal of Business 78 (1): Idzorek, T. M., and F. Bertsch The style drift score. Journal of Portfolio Management 31 (1): Menchero, J Multiperiod arithmetic attribution. Financial Analysts Journal 60 (4): Schwert, G. W Stock returns and real activity: A century of evidence. Journal of Finance 45 (4): Whaley, R The investor fear gauge. Journal of Portfolio Management 26 (3): An EDHEC-Risk Indices & Benchmarks Publication

49 About xxxxxxxxxxxxxxxxxxxxxx EDHEC-Risk Indices & Benchmarks An EDHEC-Risk Indices & Benchmarks Publication 49

50 About EDHEC-Risk Indices & Benchmarks Founded in 1906, EDHEC is one of the foremost European business schools. Accredited by the three main international academic organisations, EQUIS, AACSB, and Association of MBAs, EDHEC has for a number of years been pursuing a strategy for international excellence that led it to set up EDHEC-Risk Institute in With sixty-four professors, research engineers, and research associates, this institute has the largest asset management research team in Europe. While EDHEC-Risk Institute makes important public contributions to the advancement of academic financial research and its application towards the improvement of industry practices, it also employs its expertise to conduct proprietary research and develop new products with business partners. As such, the insights drawn from EDHEC-Risk s Indices & Benchmarking, ALM and Asset Management and Derivatives and Asset Management research programmes over the past several years have led to a series of products that provide more efficient or more academic-based solutions to investors needs than the indices and benchmarks currently available on the market. In order to clearly identify this type of activity and distinguish it from the fundamental research activities, EDHEC-Risk Institute created a spin-off in 2010, EDHEC-Risk Indices & Benchmarks, which aims to be one of the leading beta designers for the investment industry. FTSE EDHEC-Risk Efficient Index Series FTSE Group, the award-winning global index provider, and EDHEC-Risk Institute launched the first set of FTSE EDHEC-Risk Efficient Indices in early Offered for a full global range, including All World, All World ex US, All World ex UK, Developed, Emerging, USA, UK, Eurobloc, Developed Europe, Developed Europe ex UK, Japan, Developed Asia Pacific ex Japan, Asia Pacific, Asia Pacific ex Japan, and Japan, the index series aims to capture equity market returns with an improved risk/ reward efficiency compared to capweighted indices. The weighting of the portfolio of constituents achieves the highest possible return-to-risk efficiency by maximising the Sharpe ratio (the reward of an investment per unit of risk). In order to maximise the Sharpe ratio, the Index Ann. average return (compounded) Ann. standard deviation Long-Term Robust Outperformance: Risk and Return Sharpe Ratio (compounded) Information ratio Tracking error Efficient index 11.63% 14.65% % Cap-weighted 9.23% 15.20% % Difference 2.40% -0.55% (efficient minus cap-weighted) p-value for difference 0.14% 6.04% 0.04% - - The table shows risk and return statistics for portfolios constructed with the same set of constituents as the cap-weighted index. Rebalancing is quarterly subject to an optimal control of portfolio turnover (by setting the reoptimisation threshold to 50%). Portfolios are constructed by maximising the Sharpe ratio given an expected return estimate and a covariance estimate. The expected return estimate is set to the median total risk of stocks in the same decile when sorting by total risk. The covariance matrix is estimated using an implicit factor model for stock returns. Weight constraints are set so that each stock's weight is between 1/2N and 2/N, where N is the number of index constituents. P-values for differences are computed using the paired t-test for the average, the F-test for volatility, and a Jobson-Korkie test for the Sharpe ratio. The results are based on weekly return data from 01/1959 to 12/ An EDHEC-Risk Indices & Benchmarks Publication

51 About EDHEC-Risk Indices & Benchmarks methodology seeks to reliably estimate two essential inputs needed for portfolio optimisation: the expected returns of each stock, which are calculated indirectly by the riskiness of each stock; and the covariance matrix of returns for all stocks, which is calculated using statistical factor models that describe the co-movement of stock prices through their exposure to common risk factors. These indices provide investors with an enhanced risk-adjusted strategy in comparison to cap-weighted indices, which have been the subject of numerous critiques, both theoretical and practical, over the last few years. The index series is based on all constituent securities in the FTSE All-World Index Series. Constituents are weighted in accordance with EDHEC- Risk s portfolio optimisation, reflecting their ability to maximise the rewardto-risk ratio for a broad market index. The index series is rebalanced quarterly at the same time as the review of the underlying FTSE All-World Index Series. The performances of the EDHEC-Risk Efficient Indices are published monthly on Using factor analysis techniques, the EDHEC Alternative Indexes are built as the best one-dimensional summaries of the information conveyed by competing indices for a given style. The EDHEC composites are thus able to capture a very large fraction of the information contained in the competing indices while implicitly minimising their various biases. Consequently, the EDHEC Alternative Indexes tend to be very stable over time and thus are easily replicable. The thirteen EDHEC Alternative Indexes are published monthly on and are freely available to managers and investors. EDHEC-Risk Alternative Indexes The different hedge fund indices available on the market are computed from different data, and in accordance to diverse fund selection criteria and index construction methods; unsurprisingly, they tell very different stories. Challenged by this heterogeneity, investors cannot rely on competing hedge fund indices to obtain a true and fair view of performance and are at a loss when selecting benchmarks. To address this issue, EDHEC-Risk launched the first composite hedge fund strategy indices. The EDHEC IEIF Commercial Property (France) Index Institutional investors allocate considerable shares of their portfolios to real estate. They would like to use indexbased products for this purpose; however, real estate indexing has proven challenging. It has been challenging largely because real estate features such characteristics rarely found in other asset classes as high unit values and indivisibility, limited liquidity, great heterogeneity; active property management is also required. An EDHEC-Risk Indices & Benchmarks Publication 51