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Mean–variance efficient large portfolios: a simple machine learning heuristic technique based on the two-fund separation theorem

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Abstract

We revisit in this article the Two-Fund Separation Theorem as a simple technique for the Mean–Variance optimization of large portfolios. The proposed approach is fast and scalable and provides equivalent results of commonly used ML techniques but, with computing time differences counted in hours (1 min vs. several hours). In the empirical application, we consider three geographic areas (China, US, and French stock markets) and show that the Two-Fund Separation Theorem holds exactly when no constraints are imposed and is approximately true with (realistic) positive constraints on weights. This technique is shown to be of interest to both scholars and practitioners involved in portfolio optimization tasks.

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Fig. 1

Source: Wind; daily quotes of the 40 largest companies that are part of the CSI300, from 08/18/2008 to 12/31/2019. Plotted are: the Unconstrained Efficient Frontier, the capital market line (CML) and some notorious portfolios (V, N, T, C, G and G′). Computations by the authors

Fig. 2

Source: Wind; daily quotes of the 40 largest companies that are part of the CSI300, from 08/18/2008 to 12/31/2019. Plotted are: the Unconstrained Efficient Frontier, the Capital Market Line (CML) and some notorious portfolios (A, T, and BJ). Computations by the authors

Fig. 3

Source: Wind; daily quotes of the 40 largest companies that are part of the CSI300, from 08/18/2008 to 12/31/2019. Plotted are: the Unconstrained Efficient Frontier, the Efficient Frontier by Penalized Regression with low and high penalization, the Positively Constrained LASSO Efficient Frontier, the Positively Constrained Efficient Frontier by TFST, the capital market line (CML) and some notorious portfolios (\(T\), \(MAX\), \(N\), and \(V\) ), where a portfolio distinguished by a superscript “*” denotes the corresponding portfolio under positivity constraint. TFST corresponds to Two-Fund Separation Theorem, which is to build Efficient Frontiers by the combination of two pivotal portfolios: the Positively Constrained Global Minimum-Variance Portfolio (\(V\)) and the Asset with the Highest Expected Return or four pivotal portfolios: the Positively Constrained Global Minimum-Variance Portfolio, the Positively Constrained Null Index Portfolio, the Positively Constrained Tangent Portfolio and the Asset with the Highest Expected Return. Computations by the authors

Fig. 4

Source: Wind; daily quotes of the 40 largest companies that are part of the CSI300, from 08/18/2008 to 12/31/2019. Plotted are: the capital market line (CML), the Unconstrained Efficient Frontier, the Positively Constrained LASSO Efficient Frontier, the Positively Constrained Efficient Frontier by TFST with Two Pivotal Portfolios, the Positively Constrained Efficient Frontier with Four Pivotal Portfolios and some notorious portfolios (\(A\), \(MAX\), \(V\), \(T\), \(G\), \(G^{\prime}\), \(N\), \(BJ\), \(EWI\), and \(CWI\)), where EWI is an Equally-Weighted Index Portfolio and CWI is a Capitalization-Weighted Index Portfolio. A portfolio distinguished by a superscript “*” denotes the corresponding portfolio under positivity constraint. TFST corresponds to Two-Fund Separation Theorem, which is to build Efficient Frontiers by the combination of two pivotal portfolios: the Positively Constrained Global Minimum-Variance Portfolio and the Asset with the Highest Expected Return (in the blue bold line) or four pivotal portfolios (in the blue dotted bold line): the Positively Constrained Global Minimum-Variance Portfolio, the Positively Constrained Null Index Portfolio, the Positively Constrained Tangent Portfolio and the Asset with the Highest Expected Return. Computations by the authors

Fig. 5

Source: Wind; daily quotes of the 40 largest companies that are part of the CSI300, from 08/18/2008 to 12/31/2019. Plotted are: the 5%-, 50%- and 95%-quantile of the double block-bootstrapped resampled LASSO Efficient Frontier and the corresponding 50%-quantile of the Efficient Frontier by TFST. Resampled Efficient Frontiers are computed by using a double-block bootstrap method (see Michaud, 1989; Michaud & Michaud, 2008; Ledoit & Wolf, 2008, 2011) both on dates and sub-samples of stocks. TFST corresponds to Two-Fund Separation Theorem, which is used to build the Efficient Frontier by combinations of two pivotal portfolios, namely the Global Minimum-Variance Portfolio and the Asset with the Highest Expected Return. Computations by the authors

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Notes

  1. See also Markowitz (1959), Sharpe (1964), Mossin (1966), Lintner (1965a, 1965b), Black (1972), as well as Dybvig and Liu (2018), and further adaptations of the original Separation Theorem when agents do not exhibit Mean–Variance preferences or when the densities of return are not Gaussian, as in Pye (1967), Hakansson (1969), Cass and Stiglitz (1970), Harvey and Siddique (2000), Jurczenko and Maillet (2006), Dahlquist et al. (2017), or in a continuous-time stochastic setting, as in Samuelson (1967), Merton (1973), Ross (1978), and more recently in Cairns et al. (2006), Deguest et al. (2018) and Bernard et al. (2021).

  2. We will adopt hereafter the presentation and (very similar) notations used in both of these books.

  3. For the sake of simplicity and clarification, each time we have a vector of 1 with dimensions \(\left( {N \times 1} \right)\) with \(N\) being the number of the assets, we use: \({\mathbf{1}}: = {\mathbf{1}}_{{\left( {N \times 1} \right)}}\) as a notation, whilst for a vector of 1 with dimensions \(\left( {T \times 1} \right)\), with \(T\) being the number of dates of the time-series of returns on an asset, we denote it as: \({\mathbf{1}}_{{\left( {T \times 1} \right)}}\). When we will need a \(\left( {N \times N} \right)\) matrix of 1, we will further denote it: \({\mathbf{1}}_{{\left( {N \times N} \right)}}\).

  4. See, for instance, Black and Litterman (1992) for an alternative approach with deformed expectations according to views in a Bayesian context.

  5. See Mahalanobis (1927, 1936); see also Mardia et al. (1979), Meucci (2009), Breuer et al. (2009), Breuer and Csiszár (2013), Geyer et al. (2014), Flood and Korenko (2015), McNeil et al. (2015) and De Meo (2021) for other uses of the Mahalanobis norm in financial applications.

  6. Here we follow Huang and Litzenberger (1988, p. 64) with \(A = a\), \(B = b\), \(C = c\), \(D = d\), whilst Ingersoll (1987) p. 54 uses \(B = a\), \(C = b\), \(A = c\), and \(\Delta { } = d\).

  7. Furthermore, it is not equal to zero, and positive in our financial applications—see the Online Appendix, which allows us to use the term \(d\) in some denominators below, with no risk of indetermination.

  8. We will use later on, similarly, the notations \({\mathcal{F}}^{*}\), \({\overline{\mathcal{F}}}^{*}\) and \({\underline {\mathcal{F}} }^{*}\) for the same defined sets in a constrained context.

  9. See also Gouriéroux and Jouneau (1999) for a discussion about fitted performance measures.

  10. Such as: \(S_{p} = \left[ {E(R_{p} ) - R_{f} } \right]/\sigma (R_{p} )\), with \(R_{f} \) the risk-free rate.

  11. Cf. Ingersoll (1987, p. 56) and Huang and Litzenberger (1988, p. 66).

  12. See also Corollary 2.1 in the Online Appendix A-4, when inversing the problem and setting first the targeted expected return (for Asset-Liability Management reasons for instance), we deduce the optimal related Efficient Portfolio and its linked implicit risk and just vice-versa in the Corollary 2.2 (see the Online Appendix A-5). Corollary 2.3 (see in the Online Appendix A-6) also interestingly illustrates the proper structure of some notorious Portfolios according to Proposition 2, with explicit parameters.

  13. With \(E\left( {R_{C} } \right) = a/c\) and \(\sigma \left( {R_{C} } \right) = 0\), and the two asymptotes follow a relation as such: \(E\left( {R_{p} } \right) = a/c \pm \sqrt {d/c } \sigma \left( {R_{p} } \right)\)—see A-9 for details in the Online Appendix.

  14. Similarly, Black (1972) proves that every efficient portfolio can be generated by two arbitrary portfolios \(u\) and \(v\), with \(\beta_{u} = 1\) and \(\beta_{v} = 0\), where \(\beta_{u}\) and \(\beta_{v}\) are measuring the volatility of an individual stock compared to the systematic risk.

  15. Or, exceptionally, on \({\overline{\mathcal{F}}}\), if the GMVP expected return is negative.

  16. See also Broadie (1993), Kempf and Memmel (2006) and the Online Appendix for their approach, and Stevens (1998) who proves that elements of \({{\varvec{\Omega}}}^{ - 1}\) are: \({{\varvec{\Omega}}}^{ - 1}_{ij} = { }\left( { - 1} \right)^{o} \times \left[ {{\upsigma }_{ii} \left( {1 - R_{i}^{2} } \right)} \right]^{ - 1} \beta_{ij}^{{}} ,\) where \({{\varvec{\Omega}}}^{ - 1}_{ij}\) is the element on the ith row and jth column of \({{\varvec{\Omega}}}^{ - 1}\), coefficient o is equal to 0 if \(j = i\) and 1 otherwise, \({\upsigma }_{ii}\) the variance of returns on the ith asset, \(\beta_{ij}^{{}}\) the sensitivities (i.e. regression coefficients) for the regression of the return of the ith asset on those of all other N–1 assets and \(R_{i}^{2}\) the multiple regression coefficient (i.e. coefficient of determination) for the same regression. As expected, we find in the hereafter empirical studies that direct estimates of the inverse Variance–CoVariance matrix, denoted \({\hat{\mathbf{\Omega }}}^{ - 1}\) (both in the unconstrained and constrained cases), correspond to the one defined according to the regression method by Stevens (1998)—See A-9 in the Online Appendix.

  17. Where, following the notations by Hastie et al. (2015), the \(\ell_{2}\)-norm is such as: \(\left\| {{\mathbf{1}}\overline{\mu }_{p} - {\mathbf{Xw}}_{p} } \right\|_{2} = \left\| {\varvec{u}} \right\|_{2} = (\mathop \sum \limits_{{t = \left[ {1, \ldots ,T} \right]}} \left| {u_{t} } \right|^{2} )^{1/2}\), where \(u_{t}\) is the tth element of the vector of residuals denoted \({\varvec{u}} = \left( {{\mathbf{1}}\overline{\mu }_{p} - {\mathbf{Xw}}_{p} } \right)\), and with \(\left\| \cdot \right\|_{2}^{2} = \left( {\left\| \cdot \right\|_{2} } \right)^{2} = \mathop \sum \limits_{{t = \left[ {1, \ldots ,T} \right]}} \left| \cdot \right|^{2}\) being the squared \(\ell_{2}\)-norm.

  18. For the sake of clarification, we use for vector of 1 with dimensions \(\left( {N \times 1} \right)\) with \(N\) the number of the assets: \({\mathbf{1}}: = {\mathbf{1}}_{{\left( {N \times 1} \right)}}\) as a notation directly, whilst for a vector of 1 with dimensions \(\left( {T \times 1} \right)\), with \(T\) the number of dates of the time-series of returns on an asset, we denote it as: \({\mathbf{1}}_{{\left( {T \times 1} \right)}}\). Furthermore, we use the notations: \({\mathbf{1}}_{{\left( {T \times 1} \right)}}\) for a \(\left( {T \times 1} \right)\) vector of 1 when observations (dates) are at stake (instead of \({\mathbf{1}}\) as used by Britten-Jones, 1999, p. 658).

  19. Or, alternatively, by Stevens (1998) or Kempf and Memmel (2006) methods with same results.

  20. See Online Appendix for more details on the OLS approach.

  21. See Online Appendix on the Kempf and Memmel (2006) approach in the specific case of the GMVP.

  22. The \(\ell_{2}\)-norm penalization term is used in the Regularization of Inverse Discrete Gradient Estimator (RIDGE), such as (see Hoerl and Kennard, 1970, p. 56): \(P\left( {{\mathbf{w}}_{p} } \right) = \left\| {{\mathbf{w}}_{p} } \right\|_{2}^{2} = \mathop \sum \limits_{i = 1}^{N} \left| {{\text{w}}_{i} } \right|^{2}\) uses the squared \(\ell_{2}\)-norm, and both the \(\ell_{1}\)-norm and the squared \(\ell_{2}\)-norm are used in Elastic Net regularization such as (see Zou and Hastie, 2005, p. 304): \(P\left( {{\mathbf{w}}_{p} } \right) = \left( {1 - \alpha } \right)\left\| {{\mathbf{w}}_{p} } \right\|_{2}^{2} + \alpha \left\| {{\mathbf{w}}_{p} } \right\|_{1}\) where \(\alpha \in \left[ {0,1} \right]\). See Hastie et al., (2015, pp. 22 and 57) and also Kremer et al., (2020, p. 8) for geometric illustrations of the effects of such norms on coefficients.

  23. With \(\left\| \cdot \right\|_{q}^{k}\) denoting the power k of the \(\ell_{q}\)-norm operator, such as: \(\left\| {x_{i} } \right\|_{q}^{k} = \left( {\left\| {x_{i} } \right\|_{q} } \right)^{k} = \left( {\mathop \sum \limits_{{i \in {\mathbb{N}}}} \left| {x_{i} } \right|^{q} } \right)^{k/q}\). As mentioned before, we use in this equation the squared \(\ell_{2}\)-norm as: \(\left\| {{\mathbf{1}}\overline{\mu } - {\mathbf{Xw}}_{p} } \right\|_{2}^{2} = \left\| {\varvec{u}} \right\|_{2}^{2} = \left( {\mathop \sum \limits_{{t = \left[ {1, \ldots ,T} \right]}} \left| {u_{t} } \right|^{2} } \right)\), where \(u_{t}\) is the tth element of the vector of residuals denoted \({\varvec{u}} = \left( {{\mathbf{1}}\overline{\mu } - {\mathbf{Xw}}_{p} } \right)\)—see Eq. (22) for instance.

  24. As mentioned previously, the \(\ell_{1}\)-norm penalization term in LASSO regression writes in Eq. (22) as \(P\left( {{\mathbf{w}}_{p} } \right) = \left\| {{\mathbf{w}}_{p} } \right\|_{1} = \mathop \sum \nolimits_{i = 1}^{N} \left| {w_{i} } \right|\) as in Brodie et al. (2009), p. 12,268.

  25. In order to avoid concentrated and time-varying portfolios, or to deal with liquidity problem as in Vieira and Filomena (2020—see Sect. 2.2.2, p. 1060), or limit transaction costs (e.g., Lobo et al., 2007) or impose cardinality constraints (e.g., Anagnostopoulos & Mamanis, 2011; Chang et al., 2000; Woodside-Oriakhi et al., 2011).

  26. Equals here to 4000 in our restricted Chinese illustration sample.

  27. Equals here to 7800 in our illustrative sample.

  28. This relies on the expected return of the risk-free asset, where we find that the distance between Portfolio (\(N\)) and (\(T\)) in the Chinese and US market (with a higher return on the risk-free asset) is significantly higher than that in the French market (with a very low expected return of the risk-free asset).

  29. Whilst the same result does not hold in all markets: the Capitalized-Weighted Index in the French market is not optimal for instance.

  30. For instance, the computation of the whole Efficient Frontier on the high-dimensional data in the Chinese stock market with the Positively Constrained LASSO technique takes approximately 35 h on a 64 cores machine, whilst it takes less than a minute with a one core-computer when using the extended 4 fund TFST (with a related efficiency ratio of approximately 130,000).

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Acknowledgements

We thank Philippe Bernard, François Quittard-Pinon for previous collaborations on this topic, as well as Jean-Charles Garibal for preliminary research assistance, and Jean-Luc Prigent and Thierry Roncalli for their positive suggestions when writing first drafts of this article. We also sincerely appreciate the constructive discussion by Dionisis Philippas during the FEM2021 conference (Paris, June 2021). We are also grateful to the participants of the XXXVth AFFI Conference, of the XXXVth JMA conference, as well as Referees of the IFC2020, the ISFBI2021 and the FEM2021 conferences, as well as the Editor of ANOR in charge and the two anonymous Referees for their fair comments and suggestions. Resources linked to this article are available on: www.performance-metrics.eu. The usual disclaimer applies.

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Authors and Affiliations

  1. Ca’ Foscari University of Venice, Venice, Italy

    Michele Costola

  2. Leibniz Institute for Financial Research SAFE, Frankfurt, Germany

    Michele Costola

  3. emlyon business school (AIM QUANT Research Center), Paris, France

    Bertrand Maillet & Zhining Yuan

  4. Univ. La Reunion, Saint-Denis, France

    Bertrand Maillet

  5. Variances, Paris, France

    Bertrand Maillet

  6. Coactis, Univ. of Lyon, Lyon, France

    Zhining Yuan

  7. School of Finance/Institute of Big Data Research, Southwestern University of Finance and Economics (SWUFE), Chengdu, China

    Xiang Zhang

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  1. Michele Costola
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  2. Bertrand Maillet
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  4. Xiang Zhang
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Correspondence to Xiang Zhang.

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Costola, M., Maillet, B., Yuan, Z. et al. Mean–variance efficient large portfolios: a simple machine learning heuristic technique based on the two-fund separation theorem. Ann Oper Res 334, 133–155 (2024). https://doi.org/10.1007/s10479-022-04881-3

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