The Smart Beta 2.0 Approach - EDHEC-Risk

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38 An ERI Scientific Beta Publication — Smart Beta 2.0 — April 2013 Copyright © 2013 ERI Scientific Beta. All rights reserved. Please refer to the disclaimer at the end of this document. Appendix: Overview of Smart Beta Equity Portfolios (this is sometimes called the 1/N rule). For example, an equal-weighted version of the S&P 500 would use the same 500 constituents as the cap-weighted version of the index but it attributes an equal weight of 0.2% to each constituent. These equal-weighted benchmarks are attractive because they avoid the concentration and trend-following of capweighted indices and typically lead to higher Sharpe ratios compared to their cap-weighted counterparts (DeMiguel, Garlappi and Uppal, 2009, Platen and Rendek, 2010). However, for investors to adopt an equal weighting scheme is an extreme stance since it implies that they would accept to give up on any form of fundamental or statistical analysis to gather useful insights about the stocks’ characteristics. 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. Equal-weighting is a simple way of “de-concentrating” a portfolio and allows us to benefit from systematic rebalancing back to fixed weights. Depending on the universe and on whether additional implementation rules are used, the rebalancing feature of equal-weighting can be associated with relatively high turnover and liquidity problems. Maximum Deconcentration can be perceived as a generalisation of a simple equal weighting scheme, the aim being to maximise the effective number of stocks defined as the inverse of the Herfindahl index (see footnote four): Maximum Deconcentration minimises the distance of weights from the equal weights subject to constraints on turnover and liquidity. In addition, investors may have the option to add constraints on tracking error, sector weight deviations or country weight deviations with respect to the capweighted reference index. In the absence of any constraints, Maximum Deconcentration coincides with the equal-weighting (also known as the “1/N” weighting scheme), which owes its popularity mainly to its robustness and it has been shown to deliver attractive performance despite highly unrealistic conditions of optimality, even when compared to sophisticated portfolio optimisation strategies (De Miguel, Garlappi and Uppal, 2009).
An ERI Scientific Beta Publication — Smart Beta 2.0 — April 2013 Copyright © 2013 ERI Scientific Beta. All rights reserved. Please refer to the disclaimer at the end of this document. 39 Appendix: Overview of Smart Beta Equity Portfolios Risk Parity (RP, also known as Equal Risk Contribution or ERC) and Diversified Risk Parity (DRP) Portfolios The starting point in this approach consists of recognising that contribution to risk is not proportional to dollar contribution. To see this, we use the following decomposition for the portfolio volatility: where wi is the portfolio weight and σ p the portfolio volatility. Hence, we define the contribution to risk as: To correct for these imbalances, and to generate portfolios that are better diversified in the sense of exhibiting a more balanced contribution to risk by the constituents of the portfolio, Qian (2005) and Maillard, Roncalli and Teiletche (2010) suggest to form so-called equal risk portfolios by choosing the portfolio weight w i so that for all i, j: (see Qian, 2005, or Maillard, Roncalli and Teiletche, 2010, for more details). It should be noted that no analytical solution is available to this program, which therefore needs to be solved numerically. However, in a recent paper Clarke et al. (2013) provide a semi-analytical solution. The RP portfolio weights can be written as: Here the beta β =( β 1 ,..., β N ) is the vector of betas with respect to the RP portfolio, hence portfolio weights for the Risk Parity portfolio appear on both sides of the equation, which needs to be solved numerically. In an attempt to rationalise equal-risk contribution portfolios (also known as Risk Parity portfolios), Maillard, Roncalli and Teiletche (2010) show that risk-parity portfolios would be optimal Maximum Sharpe Ratio (MSR) portfolios if all Sharpe ratios are identical for all stocks, and if correlations are identical for all pairs of stocks, obviously a very restrictive assumption.
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