EDHEC-Risk Alternative weighting schemes: conditions for optimality

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  • EDHEC-Risk Alternative weighting schemes: conditions for optimality

Felix Goltz, Head of Applied Research, EDHEC-Risk Institute and Director of Research & Development, EDHEC-Risk Indices & Benchmarks

Lionel Martellini, Professor of Finance, EDHEC Business School, Scientific Director, EDHEC-Risk Institute, and Scientific Advisor, EDHEC-Risk Indices & Benchmarks

Introduction: shortcomings of cap-weighted indices

Cap-weighted stock market indices are widely used by investors and asset managers as investment benchmarks. In the past few years, however, this well-established paradigm has been subject to increasing criticism. On the one hand, a number of papers have offered convincing empirical evidence that market cap-weighted indices exhibit a poor risk-adjusted performance because they are too concentrated in a limited number of stocks, while other studies have questioned the validity of utilising market cap as a proxy for a company size and economic influence.

The combination of these empirical and theoretical developments has significantly weakened the case for market cap-weighted indices, and a consensus is slowly but surely emerging regarding the inadequacy of market cap-weighted indices as investment vehicles. This attack on cap-weighted indices, which have been shown to be neither representative nor efficient, has however left investors with a void. As a result, a host of questions remain regarding what alternative weighting scheme should be used by investors, who certainly could use some conceptual guidance to reposition the new offerings within the context of portfolio theory so as to assess which new form of efficient index may make sense and add value in the long run.

Building more representative benchmark portfolios
In a quest for a more representative weighting scheme, recently launched characteristic-based indices, also known as fundamental-weighted indices, have proposed to weight stocks by firm characteristics, such as earnings or book value (see for example Arnott, Hsu, and Moore (2005)). More specifically, these indices attempt to be more representative than cap-weighted indices by introducing a different measure of firm size. The idea behind such indices is not to optimise the risk/reward trade-off but to have measures of firm size that are more reliable than market capitalisation. As a result, conditions under which a fundamental weighting scheme would be optimal are not entirely clear. As an example, it would be optimal if risk parameters are identical and expected returns are proportional to the particular mix of fundamental variables used for the weighting.

Building more efficient benchmark portfolios
As opposed to pursuing the goal of designing a representative portfolio, a number of index providers and asset managers have instead focused on efficiency - ie, achieving the highest risk/reward ratio - arguably a more relevant objective in the eye of the investor. We review below five of the most popular methodologies that have been proposed to design better diversified portfolios than market cap-weighted indices.

Naïve approach to building better diversified portfolios: equal dollar contribution
Equal-weighted indices simply attribute the same weight to each of their constituents. While such indices avoid the concentration and trend-following of cap-weighted indices and typically lead to higher Sharpe ratios, accepting equally-weighted indices as investment benchmarks really means that input parameter estimation is so hopeless that we agree to give up on any form of fundamental or statistical analysis. In fact, the equal-weighted portfolio would have the highest possible Sharpe ratio if and only if pairwise correlations, volatilities and expected returns were identical for all stocks.

Semi-naïve approach to building better diversified portfolios: equal risk contribution
The starting point in this approach consists of recognising that contribution to risk is not proportional to dollar contribution. To correct for these imbalances, Qian (2005) and Maillard, Roncalli and Teiletche (2010) suggest forming so-called equal risk portfolios - ie, portfolios that are better diversified in the sense that the constituents of the portfolio exhibit a balanced contribution to risk. While attractive from an intuitive standpoint, this approach is purely ad-hoc and heuristic. In an attempt to analyse under which conditions ERC portfolios would be optimal Maximum Sharpe Ratio (MSR) portfolios, Maillard, Roncalli and Teiletche (2010) show that this would hold if and only if all Sharpe ratios are identical for all stocks, and if correlations are identical for all pairs of stocks, which is obviously a very restrictive assumption.

Statistical approach to building better diversified portfolios: maximum diversification ratio portfolios
Diversification is not about generating a low volatility portfolio based on low volatility stocks, but instead about starting with high volatility constituents, and optimally mixing them so as to generate a low volatility portfolio by taking advantage of the structure of pairwise correlations. In fact, one may introduce a measure of portfolio diversification, known as the diversification ratio, and defined in terms of distance between portfolio volatility the volatility of individual components. This diversification index has been used by Choueifaty and Coignard (2008) in a portfolio optimisation context. While achieving an optimal risk-reward ratio is not the explicit focus in this approach, it is straightforward to see that maximum diversification portfolios would actually coincide with maximum Sharpe ratio portfolios if all Sharpe ratios were identical for all stocks.

Scientific approach to building well-diversified low risk portfolios: global minimum variance portfolios
One particular point on the efficient frontier is the Global Minimum Variance (GMV) portfolio, the only efficient portfolio for the estimation of which expected return estimates are not needed. More precisely, the GMV portfolio is the optimal risk/reward portfolio if one assumes that all stocks have the same expected returns, which is hardly a neutral or reasonable choice. Another outstanding problem with minimum variance benchmarks is their lack of performance. In a recent paper, DeMiguel, Garlappi and Uppal (2009) evaluate the out-of-sample performance of GMV portfolios and find that they do not perform consistently better than the equally-weighted counterparts rule in terms of Sharpe ratio. In other words, GMV portfolios appear to be low risk but also low return portfolios; they are typically highly concentrated in stocks with low volatility. Risk reduction is achieved not so much by diversification but rather by a bias towards stocks with low volatility.

Scientific approach to building well-diversified high risk-reward portfolios: maximum Sharpe ratio portfolios
Recognising that the focus of investors is precisely on using investment benchmarks that achieve the highest risk-adjusted returns, Amenc et al. (2010) have proposed to focus on efficient-weighted indices, which are explicitly designed to improve the Sharpe ratio compared to cap-weighted indices by weighting stocks by their impact on portfolio risk and reward. Efficient indices focus directly on risk/reward properties. Their greater efficiency thus results from the construction method, as long as robust parameter estimates for expected returns, correlations, and volatilities are used. To obtain parameter estimates for the stocks' return comovements, an equity factor model is used to estimate common return drivers. Academic research also suggests that stocks that bear a high risk of losses for investors should also reward them with high expected returns. Using this robust relationship to find efficient portfolio weights leads to promising results, and equity indices constructed using this approach have substantially higher Sharpe ratios compared to their cap-weighting counterparts.

Recent track records of alternative equity indices
It is useful to take a look at the recent performance generated by published indices that use the above mentioned weighting schemes. We conduct such an analysis for the four following weighting schemes that have been used by the main index providers to propose alternatives to market cap-weighted indices: efficient indices (FTSE), fundamental indices (FTSE), minimum-volatility indices (MSCI), and equal-weighted indices (S&P). These indices are fully transparent in that the respective index providers provide publicly available returns data and detailed index construction rules. Figure 1 shows differences in average returns, in volatility, and in Sharpe ratios between each US index and the cap-weighted S&P 500. The statistics are based on 11 years of weekly data from 8 January 1999 to 1 January 2010. Differences that are statistically significant at the 5% level are indicated in bold.

It can be seen from Figure 1 that the non-cap-weight indices lead to significantly higher Sharpe ratios, while the minimum volatility approach lowers the volatility but does not increase the Sharpe ratio significantly. However, it should also be noted that on top of the performance numbers presented in the figure, it is important that investors judge the various alternative indexation forms by the implicit or explicit assumptions they make. As track records only provide a way of assessing the past, looking beyond track records and into the conceptual groundings of each indexation methodology is critical.

Conclusion: concept selection versus concept diversification
The main objective of cap-weighted indices is to represent the stock market, thus neglecting the need for the most efficient risk-return trade-off. A number of alternatives exist that can allow investors to gain access to more representative and/or better diversified portfolios. Track records for all these non-cap-weighted indices look rather impressive, with strong outperformance compared to cap-weighted counterparts on a risk-adjusted basis.

As opposed to relying solely on track records, which by definition always look impressive, investors should perhaps focus on the conceptual assumptions underpinning these efforts. Some investors have decided to adopt one of these alternative non-cap-weighted indices as their investment benchmark, while others have instead decided to allocate to various non-cap-weighted benchmarks. Diversification may ultimately appear to be the best protection against concept uncertainty as much as it is a protection against return uncertainty!

Amenc, N., F. Goltz, P. Retkowsky, and L. Martellini, 2010. Efficient indexation: An alternative to cap-weighted indices, Journal of Investment Management, forthcoming.
Arnott, R., J. Hsu, and P. Moore, 2005. Fundamental indexation. Financial Analysts Journal 60 (2): 83-99.
Choueifaty, Y. and Y. Coignard, 2008, Toward Maximum Diversification, Journal of Portfolio Management, 35, 1, 40-51.
DeMiguel, V., L. Garlappi, and R. Uppal, 2009. Optimal versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?, Review of Financial Studies, 22, 5, 1915-53.
Maillard S., T. Roncalli and J. Teiletche, 2010, On the Property of Equally-weighted Risk Contribution Portfolios, Journal of Portfolio Management, 36, 4, 60-70.
Qian E., 2005, Risk Parity Portfolios: Efficient Portfolios through True Diversification, Panagora Asset Management, September.


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