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ESG: The metrics jigsaw


EDHEC-Risk Efficient indexation: an alternative to cap-weighted indices

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  • EDHEC-Risk Efficient indexation: an alternative to cap-weighted indices

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

Patrice Retkowsky, Senior Research Engineer, EDHEC-Risk Institute and Deputy Head of Research & Development, EDHEC-Risk Indices & Benchmarks

Cap-weighted equity indices suffer from a variety of shortcomings that have been widely recognised. In particular, the high concentration of such indices in a small number of large cap stocks leads them to be under-diversified, thus providing investors with inefficient risk/reward properties. This inefficiency does not come as a surprise. In fact, financial theory shows that only under very unrealistic assumptions could cap-weighted indices be considered efficient investments. From a more practical perspective, cap-weighted indices have never been designed to be efficient investment supports but rather were meant to reflect broad market movements and trends. Efficient indexation provides a solution to the shortcomings of cap-weighted indices by explicitly addressing the investor's objective of holding a portfolio that provides an optimal risk-return trade-off. Rather than representing broad market movements, the objective of efficient indices is to extract the equity risk premium in an efficient way.

Rehabilitating the tangency portfolio
A potential answer to investors concern to hold well-diversified equity portfolios comes from modern portfolio theory, which precisely focuses on taking into account the stocks' risk and return characteristics in order to provide investors with optimal diversification within a given universe of constituent stocks.

Modern portfolio theory unambiguously prescribes holding the portfolio with the highest reward-to-risk ratio, also known as the tangency portfolio, or the maximum Sharpe ratio (MSR) portfolio. In an effort to improve on cap-weighted indices, we focus on designing indices with an improved Sharpe ratio. The aim of this efficient indexation approach is to provide investors with benchmarks that reflect the possible risk/reward ratio from a broadly diversified stock market portfolio, and that are thus a proxy for the normal returns of an exposure to equity risk.

To generate an investable proxy for the maximum Sharpe ratio tangency portfolio, we resort to standard mean-variance optimisation. Although our aim to maximise risk/return efficiency is fully consistent with financial theory, successful implementation of the theory depends not only on its conceptual grounds but also on the reliability of the input to the model. In the end, the results depend greatly on the quality of the parameter estimate, the covariance matrix and the expected returns of all stocks in the index.

Estimating risk parameters
The key problem in covariance matrix estimation is the curse of dimensionality, with a number of risk parameters growing more than linearly with the number of stocks under consideration. Therefore at the estimation stage, the challenge is to reduce the number of factors that come into play. Several improved estimates for the covariance matrix have been proposed, including most notably the factor-based approach. While the factor-based estimator is expected to allow for a reasonable trade-off between sample risk and model risk, the problem of choosing the "right" factor model remains. We take a somewhat agnostic approach to this question, and aim to rely as little as possible on strong theoretical assumptions by using Principal Component Analysis (PCA) to determine the underlying risk factors from the data. The PCA method is based on a spectral decomposition of the sample covariance matrix and its goal is to explain covariance structures using only a few linear combinations of the original stochastic variables which will constitute the set of (unobservable) factors. Overall, the main strength of the PCA approach at this stage is that it leads to "letting the data talk" in order to tell us the underlying risk factors that govern most of the variability of the assets at each point in time. This strongly contrasts with having to rely on the assumption that a particular factor model is the true pricing model and reduces the specification risk embedded in the factor-based approach while keeping the sample risk reduction.

Estimating expected return parameters
One outstanding challenge that remains at this stage is the estimation of expected return parameters. Instead of relying purely on statistics, which is known to generate poor expected return estimates (Merton (1980)), we use a robust estimate of expected returns that relies on the risk/reward trade-off.

More specifically, both common sense and asset pricing theory suggest that expected return parameters should be positively related to risk parameters. The academic literature has generated a wealth of insights on the risk-return relationship, starting with the classic view that there should be a linear positive relationship between the excess expected return on a stock and the stock's beta with a variety of systematic risk factors. Such multi-factor explanations of differences in expected returns are supported by the Arbitrage Pricing Theory (Ross (1976)). More recently it has been recognised that specific risk may also be rewarded in equilibrium (Merton (1987)) in the event that investors are unwilling or unable to fully diversify their portfolios. In fact, a number of papers have found that aggregating both systematic and specific risk in simple measures of downside risk allows the cross-section of expected returns to be explained (see Amenc, Goltz, Martellini and Retkowsky (2010) for more details and references to this literature).

Building on this rich literature, we suggest using a downside risk measure as a proxy for excess expected returns. We further introduce minimum and maximum weight constraints in the design of the maximum Sharpe ratio portfolio to help further increase the robustness of the methodology (see Jagannathan and Ma (2003) for a formal analysis of portfolio constraints).

Performance of efficient indices
We use constituent data for the S&P 500 index to construct tangency portfolio proxies based on the same set of stocks as these cap-weighted indices. Overall, our efficient indices obtain both higher average returns and lower volatility than do cap-weighted indices. However, portfolios rebalanced every quarter are subject to high turnover. We reduce turnover by limiting rebalancing; only when significant new information arrives do we rebalance our optimal weights.

This approach leads to significantly less turnover yet maintains high Sharpe ratios. Annual turnover in excess of the cap-weighted index is less than 20%. Over the long term, our efficient index increases the Sharpe ratio of the S&P 500 cap-weighted index by more than 70%. Interestingly, this improved risk/reward trade-off does not come at the cost of an increase in extreme risk, and it holds when conditioning on business cycles or implied volatility. When performance over several 10-year periods is analysed, the efficient indexation strategy had lower Sharpe ratios only during the bull markets of the 1990s, although volatility was still lower than that of the cap-weighted indices.

In addition to the long-term US analysis reported above, it is interesting to test the approach in other markets around the world. Figure 2 shows results based on five regional markets where we use the corresponding FTSE indices for large- and mid-cap stocks to constitute the universe. The figure shows risk and return statistics computed for efficient indexation and cap-weighting applied to these stock market index constituents.

The results show that risk/return efficiency in terms of the Sharpe ratio is improved considerably for all five indices. In addition, the improvement is actually quite similar across the five indices, with Sharpe ratios approximately 0.15 higher than those of the cap-weighted index. In general, analysis of international data suggests that our results are not specific to US data, as the method yields similar results in stock markets around the world.

Designing better equity benchmarks

Cap-weighted indices weight stocks by the footprint they leave on the stock market. Recently introduced characteristics-based indices weight stocks by their footprint in the economy (Arnott, Hsu and Moore (2005)). Our approach weights instead stocks by the "risk/return footprint" they have on the investor's portfolio. Obviously, investors want to have a high weight in stocks that contribute positively to the portfolio's Sharpe ratio and a low weight in stocks that contribute less to increasing the Sharpe ratio. Our research shows that it is possible to design an index construction methodology that explicitly takes this investor objective into account.

For more information on the efficient indices, please visit

Amenc, N., F. Goltz, L. Martellini, and P. Retkowsky, 2010. Efficient Indexation: An Alternative to Cap-Weighted Indices, EDHEC Risk Institute Publication.
Arnott, R., J. Hsu, and P. Moore, 2005. Fundamental Indexation, Financial Analysts Journal, 61, 2.
Jagannathan and Ma, 2003. Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps, Journal of Finance, 58, 4, 1651-1684.
Merton, R., 1980. On Estimating the Expected Return on the Market: An Exploratory Investigation, Journal of Financial Economics, 8, 1-39.
Merton, R., 1987. A Simple Model of Capital Market Equilibrium with Incomplete Information, Journal of Finance, 42, 483-509.
Ross, S., 1976. The Arbitrage Theory of Capital Asset Pricing, Journal of Economic Theory, 13, 3, 341-360.

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