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Evaluating Alternative Beta Strategies
By Xiaowei Kang | 24 February, 2012   

Comparing approaches to factor indexing


Evaluating Alternative Beta StrategiesSince the first market capitalisation-weighted ("cap-weighted") equity index was introduced by Standard & Poor's in 1923, cap-weighted indexing has become the dominant form of index investing. Today, cap-weighted indices account for the vast majority of assets in index-linked investment products such as ETFs and index funds, as well as trading volumes in exchange-traded and over-the-counter ("OTC") index futures, options and other derivatives.

In recent years, there has been a proliferation of alternatively weighted indices, such as fundamentally weighted indices, equal-weighted indices, and low-volatility indices. Correspondingly, there have been increased debates about the role of alternatively weighted indices (or "alternative beta") in investment portfolios.

While some proponents of alternative beta criticise market capitalisation-weighted indices as "inefficient", most market participants believe that market-cap weighting will always be the dominant form of indexing. It is not only the most representative of the market, but also has the lowest implementation cost due to its investment capacity and automatic self-rebalancing. In addition, as the cap-weighted portfolio is the only portfolio that all investors can collectively hold, it represents the ultimate benchmark, where outperformance and underperformance become a zero-sum game relative to the market.

The key to understanding alternative equity beta strategies is the extensive empirical evidence that stock returns are driven not just by the overall market factor, but also by other common risk factors that are related to the characteristics of the stocks. Notably, small-cap stocks and value stocks have historically acted differently from large-cap stocks and growth stocks, respectively, and have generated higher long-term returns.

Fama and French (1992, 1993) found that a three-factor model of market, small-cap and value factors would explain more than 90% of diversified portfolio returns, which significantly improved the explanatory power of a single-factor model, such as the Capital Asset Pricing Model (CAPM). Many studies (e.g. Jegadeesh and Titman (1993) and Carhart (1997)) identify momentum as another common equity risk factor, due to the persistency in the relative performance of past winners and past losers. Last but not least, empirical research (e.g. Haugen and Baker (1991) and Clarke, Silva, and Thorley (2006, 2010)) has shown that an equity portfolio's exposure to the volatility factor can also have significant impact on its risk and return; and, contrary to finance theory, holding high volatility stocks has not been compensated by higher long-term returns than holding low-volatility stocks. To a certain degree, this range of empirical evidence has motivated attempts to achieve better risk adjusted returns than the market capitalisation-weighted portfolio, by tilting a portfolio's exposure to certain common equity factors, such as small-cap, value, and volatility.

Figure 1 shows the historical return and volatility over the last 30 years of these most recognised equity factors for the US equity market. The market factor represents the excess return from investing in the cap-weighted US equity market. The small-cap, value, momentum and volatility factors represent the returns from portfolios that are long small stocks and short large stocks, long high book-to-market stocks and short low book-to-market stocks, long past winners and short past losers, and long high-volatility stocks and short low-volatility stocks, respectively.

It is notable that the small-cap, value and momentum factors have historically been associated with substantial positive returns. If such trends were to continue, this implies that portfolios that systematically overweight small-cap, value and momentum stocks can outperform the market. On the other hand, as the volatility factor has historically had negative returns, portfolios with a tilt to low-volatility stocks would have been better rewarded than the market.

Figure 1

Another important observation from Figure 1 is that, similar to the market factor, the small-cap, value, momentum and volatility factors have been very volatile. In other words, the potential reward from systematically tilting the portfolio towards any of these factors can vary significantly from one period to another. The results shown in Figure 1 suggest that these well-known risk factors can all have significant impacts on both the risk and return of equity portfolios.

The impressive recent performance of some alternative beta strategies, when compared with the average returns from active managers, may also have contributed to the growing interest in alternative beta strategies. For instance, Figure 2 shows that a simple equal-weighted strategy and a low-volatility strategy have significantly outperformed the S&P 500 index over the last ten years. By comparison, the average US large-cap manager has lagged the S&P 500.

In the next section, we evaluate various alternative equity beta strategies by examining the similarities and differences in their strategy objective, underlying risk drivers, portfolio construction methodology, and historical risk and return profiles.

Figure 2

ALTERNATIVE EQUITY BETA STRATEGIES COMPARED

The authors of several recent studies have compared the increasingly numerous alternative equity beta strategies. For instance, Chow, Hsu, Kalesnik, and Little (2011) surveyed various "Heuristic-Based" (experience-based) and "Optimisation-Based" weighting strategies. Using a four-factor model of market, small-cap, value and momentum, the authors identified the sources of outperformance as exposure to the value and small-cap factors, and found no statistically significant alpha after adjustment for the factor exposures. Melas, Briand and Urwin (2011) proposed a generalised framework and characterised all "Risk-Based" and "Return-Based" strategies as special cases of mean-variance portfolio construction, subject to various assumptions for expected risk and return. Dash and Loggie (2008) suggested that all index weighting schemes can be generalised as being weighted by a certain factor raised to a power; if it is desired to amplify the influence of certain factor, an exponent can be applied.

One simple way to understand alternative beta strategies is that, while they all aim to achieve a better risk-adjusted performance than the cap-weighted portfolio, most strategies have a more specific objective, either explicitly or implicitly (see Figure 3). For instance, fundamentally weighted indices and dividend-weighted indices are essentially both value strategies that tilt portfolios towards value stocks. The minimum-variance strategy and other, non-optimised low-volatility strategies are designed with the same objective of achieving lower portfolio volatility than the market capitalisation-weighted portfolio, and have lower market beta and negative exposure to the volatility factor. The portfolio construction process (e.g. whether it is heuristic-based or optimisation based), while important, is secondary to the objective and underlying risk drivers of the strategy.

Figure 3

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Other often-cited indexing strategies that attempt to achieve more desirable risk characteristics than the market capitalisation-weighted portfolio include (but are not limited to) equal-weighted (Dash and Loggie, 2008), equal risk contribution (Maillard, Roncalli and Teiletche, 2010), and diversity-weighted (Fernholz, Garvy, and Hannon, 1998). It is important to note that these strategies differ significantly from minimum-variance or other low-volatility strategies. Low-volatility equity strategies aim to reduce portfolio volatility, primarily by taking fewer systematic risks (e.g. by holding low-beta stocks); they typically represent relatively concentrated portfolios and may therefore have higher stock-specific risks than the market portfolio. By comparison, equal-weighted, equal risk contribution and diversity-weighted strategies typically do not reduce portfolio volatility as they do not reduce systematic risks; they are designed to reduce stock-specific risks and are less concentrated than the market portfolio. We classify these three strategies that reduce stock specific risks and portfolio concentration risks into the same category, simply termed "diversification strategies".

In contrast with previous studies on the subject, our empirical analysis in this section focuses on comparing and contrasting alternative beta strategies with similar objectives and risk drivers. In particular, we focus on representative value strategies, low-volatility strategies and diversification strategies, as there are significantly different indexing strategies within each of these groups. Within each strategy group, we aim to shed some light on the differences in the portfolio construction models implied by representative strategies, as these differences may significantly impact the underlying risk exposures of the strategies. Overall, such an approach is also very helpful in highlighting the key risk drivers and characteristics that distinguish each particular group of strategies. We also review the risk and return profiles of the strategies. However, we believe that these risk/return profiles are primarily functions of the objectives, portfolio construction methodologies and risk exposures of the strategies concerned.

Alternatively Weighted Value Strategies
As discussed below in further detail, we compared three alternatively weighted value strategy indices (Figure 4). Both the FTSE RAFI (fundamental indices, Arnott, Hsu and Moore, 2005) and MSCI Value Weighted Indices (Subramanian, Kulkarni, Kouzmenko, and Melas, 2011) use weightings that are based on accounting measures of size (such as book value and sales), rather than on index constituents' market capitalisation. Asness (2006) illustrated mathematically that such fundamental indexing is precisely equivalent to a value tilt away from market capitalisation-weighted indices.

Figure 4

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Despite many methodological differences, the most significant difference between the FTSE RAFI and MSCI Value Weighted Indices is that the former selects the index constituents from the whole stock universe, based on fundamental measures, while the latter only reweights all constituents of the respective cap-weighted MSCI index, without any stock selection process based on fundamental data. In and of itself, this implies that the FTSE RAFI Indices may have a stronger value tilt and a more significant exposure to small-cap stocks.

The S&P Pure Value Indices (launched in 2005) consist of value companies weighted in proportion to their relative value characteristics. This means that companies with stronger value characteristics will be assigned more weight than companies with weaker value characteristics, regardless of their size (whether measured by market capitalisation or other, accounting-based fundamental measures). This methodology gives the strategy a more significant value tilt than the fundamental indices.

We analysed the historical risk and return profiles of these three distinct alternatively weighted value strategy indices, represented by FTSE RAFI US 1000 Index, MSCI USA Value Weighted Index, and S&P 500 Pure Value Index. Using a five-factor model of market, small-cap, value, momentum and volatility, we can also gain insights into the factors that drive the performance of the strategies. The results reveal interesting similarities and differences that can be attributed to the design of the specific index strategies.

Figure 5 shows that all three value strategies delivered positive returns relative to the S&P 500 index. Not surprisingly, all three strategies have substantial and statistically significant value factor exposure. The Value Weighted strategy has the lowest level of value exposure, as well as the lowest active risk (tracking error) relative to S&P 500. It is a less aggressive value strategy, since its portfolio construction does not involve stock selection. As the Fundamental Index contains far more stocks than the S&P 500 or MSCI USA universe, it essentially has an extra allocation to mid-cap stocks outside S&P 500/MSCI USA. This also partially explains the Fundamental Index's more significant small-cap exposure than that of the Value Weighted strategy. Given that the S&P MidCap 400 outperformed the S&P 500 by 3.9% per annum in the examined period (June 1995 – October 2011), the mid-cap exposure would have improved the return of the Fundamental Index. We also notice that the Pure Value strategy has the most significant value and small-cap exposures and the highest active risk. Its stock selection and weighting mechanism based on relative value characteristics make it the most aggressive value strategy of the three in the study.

Figure 5a

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Figure 5b

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Importantly, none of the value strategies is associated with statistically significant alpha after an adjustment for factor exposures. The R-Squared of the factor regressions is above 0.95 for the two fundamentally weighted strategies, which indicates that these five well-known common equity factors explain the vast majority of the returns of the strategies. The R-Squared is lower for the Pure Value strategy, which is not surprising. The common equity factors can better explain the return of well-diversified portfolios, because such portfolios will be driven primarily by systematic factor risks. As the Pure Value strategy index is by design relatively more concentrated than the other two value strategy indices, it may display higher stock-specific risks that cannot be explained by the common factors of market, small-cap, value, momentum and volatility. Overall, the observations confirm that all these strategies are beta strategies, and their outperformance over the market comes mainly from the value and, to a lesser extent, the small-cap factors.

Low-Volatility Strategies
Low-volatility equity investing has recently attracted increased investor interest. This may be partially attributed to the turbulent markets of the past few years. The objective of constructing equity portfolios with lower overall volatility can be achieved either by using mean-variance optimisation or through a non-optimised approach (Figure 6).

Figure 6
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Mean-variance optimisation requires both estimations of stocks' expected returns and a covariance matrix. As expected returns are notoriously difficult to estimate, a typical minimum-variance strategy "simplifies" the optimisation by assuming that all stocks have the same expected returns1.

In practice, as an unconstrained optimisation may produce less representative portfolios, minimum-variance strategies typically impose some practical constraints on the optimisation, such as limiting the portfolio turnover and exposures to individual sectors.

A simpler alternative methodology is to select those stocks that have been least volatile historically, and then to weight them by the inverse of those historical volatilities. This tilts the portfolio towards low-volatility stocks. As stock volatility tends to cluster (i.e. stocks with lower volatility in the past may continue to exhibit lower volatility in the following period), such an approach can also effectively reduce portfolio volatility. By contrast with the minimum-variance strategy, the non-optimised approach does not take into account correlations between stocks2.

To gain more insight into these two low-volatility strategies, we chose to analyse the MSCI Minimum Volatility Indices (Nielsen and Aylursubramanian, 2008), which are representative of the minimum-variance strategy, and the S&P Low Volatility Indices (Soe, 2011) which are representative of the non-optimised low-volatility strategy. Figure 7a presents the historical performance of the two strategies. An important observation is that both strategies effectively lowered annual portfolio volatility from 16.2% (S&P 500) to about 12%, a reduction of about 25% in relative terms. Interestingly, although the minimum-variance strategy should in theory achieve lower volatility as it is the result of an optimisation, both strategies have achieved almost the same realised volatility in the examined period. One possible explanation may be that the theoretical aim of achieving minimum risk may be dampened by the optimisation constraints imposed in practice.

Figure 7a

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Figure 7b reveals the primary risk drivers that distinguish low-volatility strategies from other alternative beta strategies: both strategies have a market beta significantly below 1, as well as strong negative exposure to the volatility factor. Another critical observation is that, when compared with the minimum-variance strategy, the non-optimised low-volatility strategy exhibits significantly higher active risk, lower market beta, and a more significant exposure to the volatility factor. This may be explained mainly by two differences in the designs of the strategies. First, the MSCI Minimum Volatility index imposes active constraints on sectors and other risk factors, while the S&P Low Volatility index does not involve active risk constraints. Secondly, the MSCI Minimum Volatility index is rebalanced on a semi-annual basis with a turnover constraint, while the S&P low-volatility index is rebalanced on a quarterly basis and may have higher portfolio turnover. These differences make the S&P index a more aggressive low-volatility/low-beta strategy.

Figure 7b

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After adjustment for factor exposures, the alpha of the strategies was not statistically significant during the examined period3. The five-factor model explains over 90% of the return variation of the minimum-variance strategy, but has less explanatory power for the returns of the non-optimised low-volatility strategy. As noted earlier, this may in part be due to relatively high stock-specific risks.

It is worth noting that low-volatility equity strategies can reduce not only portfolio volatility, but also downside risks. Figure 7c compares the maximum drawdown of the low-volatility strategies with S&P 500. Both the minimum-variance strategy and the non-optimised low-volatility strategy effectively reduced downside risk during the IT bubble and the recent financial crisis.

Figure 7c

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Diversification Strategies
The above analysis of minimum-variance and non-optimised low-volatility strategies confirms that the strategies reduce portfolio volatility primarily by reducing beta to the market and by a negative exposure to the volatility factor. By contrast, equal-weighted, equal risk contribution and diversity-weighted strategies aim to reduce stock-specific risks by holding portfolios that are less concentrated than the cap-weighted portfolio. In addition, unlike the low-volatility strategies, the three diversification strategies do not apply any stock selection screen, but only reweight all the stocks in the universe.

The equal-weighted strategy allocates the same portfolio weighting to each stock and is the simplest strategy to reduce concentration risk. This approach is also known as naïve diversification. Between January 1991 and October 2011, the S&P 500 Equal Weight Index outperformed the S&P 500 by 2.5% per annum, albeit with slightly higher volatility (16.9% versus 15.1%). The equal risk contribution approach (Maillard, Roncalli and Teiletche, 2010) takes into account not only stocks' weights but also their marginal risk contributions, so that each stock contributes equally to the total risk of the portfolio. Fernholz, Garvy, and Hannon (1998) measure the level of market concentration by "diversity". By taking the market-cap weights for individual securities and raising individual weights to a power of between zero and one, the diversity-weighted portfolio essentially represents a middle ground between cap-weighted and equal-weighted portfolios. Due to its closer proximity to the cap-weighted portfolio, the diversity-weighted strategy has lower turnover and active risk than the equal-weighted strategy.

Figure 8a and 8b present the historical risk and return profiles and factor exposures for equal-weighted, diversity-weighted, and equal risk contribution strategies. The analysis confirms that these strategies are very distinct from the low-volatility strategies. First, unlike low-volatility strategies, all three diversification strategies do not reduce volatility, and have a market beta that is fairly close to 1. Secondly, the R-Squared of the five-factor regression is strikingly close to 1 for all three strategies, which indicates that they represent very well diversified portfolios that are driven almost completely by the common risk factors. By contrast, low-volatility strategies typically represent more concentrated portfolios with more significant stock-specific risks.

As all three diversification strategies systematically overweight small-cap stocks by reference to the cap-weighted portfolio, they have exposure to the small-cap factor. By construction, equal-weighted has the most significant small-cap exposure, while diversity-weighted has the least aggressive small-cap tilt. Another important observation is that all three strategies have a statistically significant negative exposure to momentum, which results from their disciplined rebalancing away from stocks with higher past returns. This observation confirms that these strategies have an inherent contrarian bias.

Figure 8a

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Figure 8b

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IMPLEMENTING ALTERNATIVE BETA STRATEGIES

In this section, we aim to build upon the empirical analysis of alternative equity beta strategies undertaken in the previous section by discussing some key considerations and potential risks of alternative beta strategies.

Key Considerations
Our comparison of alternative beta strategies suggests that although such strategies aim to achieve a better risk-adjusted performance than the cap-weighted portfolio, they are constructed with more specific objectives and take very distinct risks. It is the specific strategy objective and risk drivers that define the expectation of the strategy performance in different market circumstances. For example, as low-volatility strategies take fewer systematic risks, they may be expected to deliver a smoother ride through market cycles than the cap-weighted portfolio, with less impressive performance in bull markets but better downside protection in bear markets. And if the investment objective is to reduce the potential impacts of stock-specific events (e.g. the Enron scandal; BP's disastrous oil spill), diversification strategies such as equal-weighted and equal risk contribution can be applied. The starting point for evaluating an alternative beta strategy may therefore be to examine its strategy objective and underlying risk drivers, and to ascertain whether they are consistent with the investor's investment objectives and preferences for risk taking.

Our empirical analysis of representative value strategies, low-volatility strategies and diversification strategies indicates that different portfolio construction methodologies can have pronounced impacts on the risk and return profile of strategies with similar objectives and risk drivers. For instance, the significance of the factor tilt of the strategy can be impacted by a number of considerations: whether the strategy employs a stock selection screen based on the target characteristics; whether the strategy employs active constraints; the strategy's rebalancing policy, and so on. Factor exposure analysis is therefore useful in evaluating alternative beta strategies. Such analysis can give insights not only into the risk factors driving the strategy, but also into how aggressive the strategy is.

Some industry participants have highlighted implementation cost as one key criterion for evaluating alternative beta strategies4. Cost is certainly a key consideration for all passive strategies. Especially when evaluating alternative beta strategies with similar objectives and risk drivers, implementation cost, which is impacted by portfolio turnover, liquidity and investment capacity, should be a key criterion. However, the simplicity and transparency of the strategy should be an equally important evaluation criterion. Simplicity and transparency not only make the strategy more replicable, but also make the outcome of the strategy more defined and easily interpretable. In addition, simplicity and transparency may also drive down the cost by guarding against potentially higher fees charged by more complex and proprietary strategies that may ultimately deliver similar beta exposures.

Potential Risks Of Alternative Beta Strategies
The alternative beta strategies we have discussed are subject to overall market risk similar to the cap-weighted portfolio. In fact, with the exception of low-volatility strategies, the alternative beta strategies typically have a market beta that is close to 1. However, as they aim to achieve a better risk-adjusted return than the cap-weighted portfolio, the alternative beta strategies are also subject to significant active risks relative to the market. Figure 9 shows the historical performance of representative alternative beta strategies relative to the cap-weighted portfolio. Although these alternative beta strategies have outperformed the market over the examined period, they have all experienced periods of significant underperformance.

Figure 9b

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The active risks in alternative beta strategies are often primarily the direct results of those strategies' factor exposures. As factor returns can be volatile over time and difficult to predict, a strategy's active exposures to common factors such as value, small-cap, momentum and volatility can have significant implications for the strategy's performance relative to the overall market. For instance, we have shown in Figure 1 that, over the last 30 years, the value factor was associated with an annualised return of 3.6%, while the volatility factor was associated with an annualised return of -1.9%. However, Figures 10a and 10b demonstrate that both value and volatility factor returns varied greatly during this period. Notably, the value factor was associated with substantial negative returns in 1998/1999 during the IT bubble, as well as in 2007/2008 during the credit crisis. Not surprisingly, Figure 9 shows that the fundamentally weighted strategy significantly underperformed the market during these two periods, as this strategy has significant value exposure.

Another observation is that the volatility factor returned over 50% in 1999 and about 40% in 2009, as high volatility/high beta stocks outperformed the market in these two years. Correspondingly, low-volatility strategies significantly underperformed the cap-weighted portfolio in 1999 and 2009 (Figure 9). The implication from these observations is that, despite the potential of alternative beta strategies to deliver better risk-adjusted performance than the market over the long term, investors may need to be prepared for periods of significant underperformance.

Figure 10

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Potential Benefits Of Combining Alternative Beta Strategies
Another important observation from Figure 9 is that the periods of underperformance of individual alternative beta strategies do not always coincide with each other. For example, when the fundamentally weighted, equal-weighted and low-volatility strategies all underperformed during the 1998-1999 IT bubble, the momentum strategy outperformed. When the fundamentally weighted and equal-weighted strategies underperformed in 2008 amid the credit crisis, the low-volatility strategy outperformed during that same time.

After all, as we noted earlier, the active risks of value strategies, low-volatility strategies, diversification strategies and momentum strategies may be driven by different risk factors. As the common equity risk factors may not be highly correlated, combining alternative beta strategies may potentially reduce active risks. Figure 11 provides the correlations between the common risk factors, and shows that some factors are in fact negatively correlated with each other. In short, the observations from Figure 9 and Figure 11 indicate the potential to diversify the active risk by combining alternative beta strategies that are driven by different sets of risk factors.

Figure 11

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Figure 12 shows the historical risk and return profiles of fundamentally weighted, equal-weighted, low-volatility and momentum strategies over the last 20 years. We tested the idea of combining alternative beta strategies by constructing a composite strategy that equally weights all four strategies. The composite strategy achieved a higher return than the S&P 500 index with slightly lower volatility (14.2% versus 15.1%). This indicates that diversifying via different alternative beta strategies has limited potential to reduce the total risk of the equity portfolio (a more effective way to reduce portfolio volatility is via low-volatility strategies). However, the more important observation is that the active risk of the composite strategy is significantly reduced to 3.4%, compared with an average of 7.2% for individual strategies. This also results in an information ratio that is significantly higher than any individual strategies. This confirms the potential to manage active risks and partially reduce the possibility of significant underperformance by combining alternative beta strategies.

Figure 12

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CONCLUSIONS

Although alternative beta strategies aim to achieve better risk-adjusted performance than the cap-weighted portfolio, they are often constructed with more specific objectives in mind. These objectives include achieving a systematic value tilt, lowering portfolio volatility, or reducing stock-specific risks. While the risk and return profiles of the alternative beta strategies examined in this paper are to a large degree driven by the well-known equity risk factors (market, value, small-cap, momentum and volatility), the primary factor drivers of individual strategies are often distinct, and in turn may define the essence of the strategy. When evaluating an alternative beta strategy, a starting point for investors may therefore be to examine its objective and risk drivers, in the context of those investors' own investment objectives and preferences for risk taking.

When it comes to implementation, our analysis suggests that portfolio construction methodologies can have significant implications for the risk and return profiles of alternative beta strategies, and should therefore be examined carefully. Our findings suggest that implementation costs, as well as simplicity and transparency, may also be considered important evaluation criteria.

We caution that alternative beta strategies often take substantial active risks, which are largely driven by their factor exposures. As factor returns can be volatile over time, all alternative beta strategies may experience periods of significant underperformance relative to the cap-weighted market portfolio. However, as common equity risk factors may not be correlated, combining alternative beta strategies that are driven by distinct sets of risk factors may help to reduce the active risk and improve the information ratio.

Endnotes And References

  1. Chopra and Ziemba (1993) discussed the effects of estimation errors on mean-variance optimisation.
  2. The implication is that a volatile stock which has low correlation with the rest of the portfolio may be included in the minimum-variance portfolio, due to its low marginal contribution to portfolio risk. By contrast, the non-optimised approach ignores correlation and will exclude such a stock as a result of its high volatility.
  3. As the S&P 500 Low Volatility index shows a substantial alpha of 3% per annum during the examined period, we further tested using the available index data going back to 1990, and found an annual factor-adjusted alpha of 1.2%, without statistical significance.
  4. For instance, Chow, Hsu, Kalesnik, and Little (2011).
  • Arnott, Robert D., Jason Hsu and Philip Moore. 2005. "Fundamental Indexation." Financial Analysts Journal, Vol. 61, No. 2, March/April: pp. 83-99.
  • Asness, Clifford. 2006. "The Value of Fundamental Indexing." Institutional Investor, Vol. 40, No. 10 (2006), pp. 94-99.
  • Bender, Jennifer, Remy Briand, Frank Nielsen, and Dan Stefek. 2010. "Portfolio of Risk Premia: A New Approach to Diversification." Journal of Portfolio Management, Vol. 36, No 2: pp. 17-25.
  • Carhart, Mark M. 1997. "On Persistence in Mutual Fund Performance." Journal of Finance, Vol. 52, No. 1. (Mar., 1997), pp. 57-82.
  • Chopra, Vijay K., and William T. Ziemba. 1993. "The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice." Journal of Portfolio Management, Vol. 19, No. 2 (Winter): pp. 6–11.
  • Chow, Tzee-man, Jason Hsu, Vitali Kalesnik, and Bryce Little. 2011. "A Survey of Alternative Equity Index Strategies." Financial Analyst Journal, Vol. 67, No. 5.
  • Clarke, Roger G., Harindra de Silva, and Steven Thorley. 2006. "Minimum-Variance Portfolios in the U.S. Equity Market." Journal of Portfolio Management, Vol. 33, No. 1 (Fall): pp. 10–24.
  • Clarke, Roger G., Harindra de Silva, and Steven Thorley. 2010. "Know Your VMS Exposure." Journal of Portfolio Management, Winter 2010, Vol. 36, No. 2: pp. 52-59.
  • Dash, Srikant. and Keith Loggie. 2008. "Equal weight indexing—Five years later." Standard & Poor's Index Research Paper.
  • Fama, Eugene F. and Kenneth R. French. 1992. The Cross-Section of Expected Stock Returns, Journal of Finance, VOL XLVII, No 2.
  • Fama, Eugene F. and Kenneth R. French. 1993. Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics, 33, pp. 3-56.
  • Fernholz, Robert, Robert Garvy, and John Hannon. 1998. "Diversity-Weighted Indexing." Journal of Portfolio Management, Vol. 24, No. 2 (Fall): pp. 74-82.
  • Goldman Sachs Asset Management. 2010. "Is Indexing the Optimal Equity Strategy?" GSAM Quantitative Investment Strategies.
  • Haugen, Robert, and Nardin Baker. 1991. "The Efficient Market Inefficiency of Capitalization-Weighted Stock Portfolios." The Journal of Portfolio Management, Spring 1991, pp. 35-40.
  • Jegadeesh, Narasimhan and Sheridan Titman. 1993. "Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency." Journal of Finance, Vol 48, No.1: pp. 65-91.
  • Maillard, Sébastien, Thierry Roncalli , and Jérôme Teïletche. 2010. "The Properties of Equally Weighted Risk Contribution Portfolios." Journal of Portfolio Management, Vol. 36, No. 4: pp. 60.
  • Melas, Dimitris, Remy Briand, and Roger Urwin. 2011. "Harvesting Risk Premia with Strategy Indices." MSCI Research Insight.
  • Soe, Aye. 2011. "S&P 500 Low Volatility Index." Standard & Poor's Index Research Paper.
  • Subramanian, Madhusudan, Padmakar Kulkarni, Roman Kouzmenko, and Dimitris Melas. 2011. "Capturing the Value Premium". MSCI Research Insight.

 



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