Combining equilibrium, resampling, and analyst’s views in portfolio optimization

Abstract

Portfolio optimization methodologies play a central role in strategic asset allocation (SAA), where it is desirable to have portfolios that are efficient, diversified, and stable. Since the development of the traditional mean-variance approach of Markowitz (1952), many improvements have been made to overcome problems such as lack of diversification and strong sensitivity of optimal portfolio weights to expected returns.The Black and Litterman (1992) model (BL) is among the most used approaches. The idea behind this model is that expected returns are the result of two important sources of information: the first is market information in the form of equilibrium returns (implicit returns that clear out the outstanding market allocation), and the second is analysts' views, which tilt the market portfolio to another diversified portfolio compatible with investor beliefs. In this fashion, portfolio managers get an intuitive but formal model to generate optimal allocation. However, while the BL model offers a very useful and intuitive approach to deal with asset allocation, the inputs considered for the calculation of equilibrium returns are subject to estimation error, and thus expected returns will also contain estimation error. Michaud (1998) proposed the use of a statistical tool known as resampling to deal with estimation error, which is an important source of lack of diversification in mean-variance portfolios. This technique considers that data come from a stochastic process instead of being a deterministic input as in Markowitz (1952). This paper proposes the use of a portfolio optimization methodology which combines features of both the BL and resampling methodologies. This novel methodology allows the combination of equilibrium and investor's views as in BL, and at same time deals with estimation risk as in Michaud (1998). Thus, it generates robust and diversified optimal allocations which are desirable properties for long-term investors such as central banks and sovereign wealth funds. We empirically test the new methodology using a sample of fixed income and equity indices, achieving very supportive results. We find strong evidence supporting the use of resampling techniques to improve standard models like BL and Markowitz, and this result is more pronounced for medium levels of risk. In general, our proposed methodologies, both with and without views, generated very competitive portfolios compared to the other methodologies, considering the three evaluation dimensions: financial efficiency, diversification, and allocation stability. For medium levels of risk, our methodologies are markedly better than others.

Strategic asset allocation by mixing shrinkage, vine copula and market equilibrium

“…Risk predictability is assessed by the difference between in‐sample and out‐of‐sample skewness and the difference between in‐sample and out‐of‐sample excess kurtosis. The allocation efficacy comprises the allocation diversification and stability, and these are evaluated respectively by the mean Herfindahl index, given by the sum of the squared asset weights as suggested in Barros Fernandes et al (), and the average turnover given by the sum of changes of each asset between two consecutive years divided by the value of the portfolio.…”

“…They recognize the potential of the Black-Litterman for allocation efficacy and combine market equilibrium with investors' opinions. Barros Fernandes, Haas Ornelas, and Martínez Cusicanqui (2012) use the Black-Litterman plus resampling techniques to deal with the estimation error. However, the resampling method is less intuitively appealing and less theoretically founded than the Bayesian method used in Jorion (1991) and others for estimation error.…”

Asset allocation with multiple analysts’ views: a robust approach

“…For example, Bertsimas et al (2012) consider inverse optimisation for the BL method. Fernandes et al (2012) combine the re-sampling method with the BL method.…”

“…Our second strand of research is the robust counterpart approach. Recent studies reveal that parameter estimates based on historical market data are subject to sampling errors (Chopra and Ziemba, 1993;Schöttle and Werner, 2009;Fernandes et al, 2012;de Jong, 2018). The robust counterpart approach is a useful alternative because it includes a wide range of possible input parameter values without complicated adjustment to the original optimisation framework (Ben-Tal and Nemirovski, 1998;Gregory et al, 2011).…”

Strategic Asset Allocation for Central Bank's Management of Foreign Reserves: A New Approach

“…Based on these equilibrium returns, the specific views of each investor (which can be regarded as additional information or further insights) are combined with the prior to generate the posterior distribution of asset returns. Barros Fernandes et al (2012) combine the B-L approach with the resampling approach of Michaud and Michaud (2008) to generate a portfolio optimisation for central banks' strategic asset allocation. Petrovic (2010) applies the Black-Litterman model to central banking practice.…”