• Robust Optimization

• Markowitz Portfolio Optimization

• Robust Maximum Return Portfolio Optimization

• Robust Minimum Variance Portfolio Optimization

• Robust Markowitz Portfolio Optimization

• Robust Sharpe Ratio Portfolio Optimization

• Summary

• A convex optimization problem is written as \[ \begin{array}{ll} \underset{\mathbf{x}}{\textsf{minimize}} & f_{0}\left(\mathbf{x}\right)\\ \textsf{subject to} & f_{i}\left(\mathbf{x}\right)\leq0\qquad i=1,\ldots,m\\ & \mathbf{h}\left(\mathbf{x}\right)=\mathbf{A}\mathbf{x}-\mathbf{b}=0 \end{array} \] where \(f_{0},\,f_{1},\ldots,f_{m}\) are convex and equality constraints are affine.

• Convex problems enjoy a rich theory (KKT conditions, zero duality gap, etc.) as well as a large number of efficient numerical algorithms guaranteed to deliver an optimal solution \(\mathbf{x}^{\star}\).

• Many off-the-shelf solvers exist in all the programming languages (e.g., R, Python, Matlab, Julia, C, etc.), tailored to specific classes of problems, namely, LP, QP, QCQP, SOCP, SDP, GP, etc.

• However, the obtained optimal solution \(\mathbf{x}^{\star}\) typically performs very poorly in practice.

• In many cases, it can be totally useless!

• Why is that?

• Recall that a problem formulation contains not only the optimization variables \(\mathbf{x}\) but also the parameters \(\boldsymbol{\theta}\).

• Such parameters define the problem instance and are typically estimated in practice, i.e., they are not exact: \(\hat{\boldsymbol{\theta}}\neq\boldsymbol{\theta}\) but hopefully close \(\hat{\boldsymbol{\theta}}\simeq\boldsymbol{\theta}\).

• The question is whether a small error in the parameters is going to be detrimental or can be ignored. That depends on each particular type of problem.

• In the case of portfolio optimization, small errors in the parameters \(\boldsymbol{\theta}=\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)\) happen to have a huge effect in the solution \(\mathbf{x}^{\star}\). To the point that most practitioners avoid the use of portfolio optimization!

• To make explicit the fact that the functions depend on parameters \(\boldsymbol{\theta}\), we can explicitly write \(f_{i}\left(\mathbf{x};\boldsymbol{\theta}\right)\) and \(h_{i}\left(\mathbf{x};\boldsymbol{\theta}\right)\).

• For example, consider an LP: \[ \begin{array}{ll} \underset{\mathbf{x}}{\textsf{minimize}} & \mathbf{c}^{T}\mathbf{x}+\mathbf{d}\\ \textsf{subject to} & \mathbf{A}\mathbf{x}=\mathbf{b}. \end{array} \]

• The parameters are \(\boldsymbol{\theta}=\left(\mathbf{A},\mathbf{b},\mathbf{c},\mathbf{d}\right)\).

• The objective function is \(f_{0}\left(\mathbf{x};\boldsymbol{\theta}\right)=\mathbf{c}^{T}\mathbf{x}+\mathbf{d}\)

• The constraint function is \(\mathbf{h}\left(\mathbf{x};\boldsymbol{\theta}\right)=\mathbf{A}\mathbf{x}-\mathbf{b}\)

• In practice, we only have an estimation \(\hat{\boldsymbol{\theta}}\). So the problem can only be formulated and solved using \(\hat{\boldsymbol{\theta}}\) obtaining the solution \(\mathbf{x}^{\star}(\hat{\boldsymbol{\theta}}\)), which is different from the desired one \(\mathbf{x}^{\star}\left(\boldsymbol{\theta}\right)\).

• The naive approach is to pretend that \(\hat{\boldsymbol{\theta}}\) is close enough to \(\boldsymbol{\theta}\) and solve the approximated problem, obtaining \(\mathbf{x}^{\star}(\hat{\boldsymbol{\theta}})\).

• For some type of problems, it may be that \(\mathbf{x}^{\star}(\hat{\boldsymbol{\theta}})\approx\mathbf{x}^{\star}\left(\boldsymbol{\theta}\right)\) and that’s it.

• For many other problems, however, that’s not the case. So we cannot really rely on the naive solution \(\mathbf{x}^{\star}(\hat{\boldsymbol{\theta}})\).

• The solution is to consider instead a robust formulation that takes into account the fact that we know we only have an estimation of the parameters.

• There are several ways to make the problem robust to parameters errors, mainly:

  • stochastic robust optimization (involving expectations)
  • worst-case robust optimization
  • chance programming or chance robust optimization.

• Stochastic optimization (SO): this includes expectations as well as chance constraints (requires probabilistic modeling of the parameter):
  📘 J. R. Birge and F. V. Louveaux. Introduction to Stochastic Programming. Springer, 2011.
  📘 A. P. Ruszczynski and A. Shapiro. Stochastic Programming. Elsevier, 2003.
  📘 A. Prekopa. Stochastic Programming. Kluwer Academic Publishers, 1995.

• Robust optimization (RO): this includes the worst-case approach (requires definition of hard uncertainty set for the parameter):
  📘 A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski. Robust Optimization. Princeton University Press, 2009.
  📄 A. Ben-Tal and A. Nemirovski, “Selected topics in robust convex optimization”, Mathematical Programming, 112 (1), 2008.
  📄 D. Bertsimas, D. B. Brown, and C. Caramanis, “Theory and applications of robust optimization”, SIAM Review, 53 (3), 2011.
  📄 M. S. Lobo. Robust and convex optimization with applications in finance. PhD thesis, Stanford University, 2000.

• In stochastic robust optimization, one models the estimation \(\hat{\boldsymbol{\theta}}\) as a random variable that fluctuates around its true value \(\boldsymbol{\theta}\).

• Then, instead of considering the approximated function \(f(\mathbf{x};\hat{\boldsymbol{\theta}})\), it uses its expected value \(E_{\boldsymbol{\theta}}[f(\mathbf{x};\boldsymbol{\theta})]\), where \(E_{\boldsymbol{\theta}}[\cdot]\) denotes expectation over the random variable \(\boldsymbol{\theta}\).

• The random variable is typically modeled around the estimated value as \(\boldsymbol{\theta}=\hat{\boldsymbol{\theta}}+\boldsymbol{\delta}\) with \(\boldsymbol{\delta}\) following a zero-mean distribution such as Gaussian.

• For example, if the function is quadratic, say, \(f(\mathbf{x};\hat{\boldsymbol{\theta}})=(\hat{\mathbf{c}}^{T}\mathbf{x})^{2}\), and we model the parameter as \(\mathbf{c}=\hat{\mathbf{c}}+\boldsymbol{\delta}\) with \(\boldsymbol{\delta}\) zero-mean and covariance matrix \(\mathbf{Q}\), then the expected value is \[\begin{aligned} E_{\boldsymbol{\theta}}[f(\mathbf{x};\boldsymbol{\theta})] &= E_{\boldsymbol{\delta}}[((\hat{\mathbf{c}}+\boldsymbol{\delta})^{T}\mathbf{x})^{2}]\\ &= E_{\boldsymbol{\delta}}[\mathbf{x}^{T}\hat{\mathbf{c}}\hat{\mathbf{c}}^{T}\mathbf{x}+\mathbf{x}^{T}\boldsymbol{\delta}\boldsymbol{\delta}^{T}\mathbf{x}]\\ &= (\hat{\mathbf{c}}^{T}\mathbf{x})^{2}+\mathbf{x}^{T}\mathbf{Q}\mathbf{x} \end{aligned}\] where the additional term \(\mathbf{x}^{T}\mathbf{Q}\mathbf{x}\) serves as a regularizer.

• In worst-case robust optimization, the parameter is not characterized statistically. Instead, it is assumed that the true parameter lies in an uncertainty region centered around the estimated value: \(\boldsymbol{\theta}\in\mathcal{U}\).

• The uncertainty region can be chosen depending on the problem. Typical choices include:
  • sphere region: \[\mathcal{U} =\{\boldsymbol{\theta}|\parallel\boldsymbol{\theta}-\hat{\boldsymbol{\theta}}\parallel_{2})\leq\delta\}\]
  • box region: \[\mathcal{U} =\{\boldsymbol{\theta}|\parallel\boldsymbol{\theta}-\hat{\boldsymbol{\theta}}\parallel_{\infty})\leq\delta\}\]
  • elliptical region: \[\mathcal{U} =\{\boldsymbol{\theta}|(\boldsymbol{\theta}-\hat{\boldsymbol{\theta}})^{T}\mathbf{S}^{-1}(\boldsymbol{\theta}-\hat{\boldsymbol{\theta}}))\leq\delta^{2}\}\] where \(\mathbf{S}\succ\mathbf{0}\) defines the shape of the ellipsoid.

• The problem with expectations is that only the average behavior is concerned and nothing is under control about the realizations worse than the average. For example, on average some constraint will be satisfied but it will be violated for many realizations.

• The problem with worst-case programming is that it is too conservative as one deals with the worst possible case.

• Chance programming tries to find a compromise. In particular, it also models the estimation errors statistically but instead of focusing on the average it guarantees a performance for, say, 95% of the cases.

• The naive constraint \(f(\mathbf{x};\hat{\boldsymbol{\theta}})\leq0\) is replaced with \(\mathsf{Pr}_{\boldsymbol{\theta}}\left[f(\mathbf{x};\boldsymbol{\theta})\leq0\right]\geq1-\epsilon=0.95\) with \(\epsilon=0.05\).

• Chance or probabilistic constraints are generally very hard to deal with and one typically has to resort to approximations.

• The idea of the Markowitz mean-variance portfolio (MVP) (Markowitz 1952) is to find a trade-off between the expected return \(\mathbf{w}^{T}\boldsymbol{\mu}\) and the risk of the portfolio measured by the variance \(\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\): \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \mathbf{w}^{T}\boldsymbol{\mu}-\lambda\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1 \end{array}\] where \(\mathbf{w}^{T}\mathbf{1}=1\) is the capital budget constraint and \(\lambda\) is a parameter that controls how risk-averse the investor is.

• This is a convex quadratic problem (QP) with only one linear constraint which admits a closed-form solution: \[\mathbf{w}_{\sf MVP} = \frac{1}{2\lambda}\boldsymbol{\Sigma}^{-1}\left(\boldsymbol{\mu}+\nu\mathbf{1}\right),\] where \(\nu\) is the optimal dual variable \(\nu=\frac{2\lambda-\mathbf{1}^{T}\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}{\mathbf{1}^{T}\boldsymbol{\Sigma}^{-1}\mathbf{1}}\).

• Capital budget constraint: \[\mathbf{1}^T\mathbf{w} = 1.\]

• Long-only constraint: \[\mathbf{w} \geq 0.\]

• Dollar-neutral or self-financing constraint: \[\mathbf{1}^T\mathbf{w} = 0.\]

• Holding constraint: \[\mathbf{l}\leq\mathbf{w}\leq \mathbf{u}\] where \(\mathbf{l}\in\mathbb{R}^{N}\) and \(\mathbf{u}\in\mathbb{R}^{N}\) are lower and upper bounds of the asset positions, respectively.

• Leverage constraint: \[\left\Vert \mathbf{w}\right\Vert _{1}\leq L.\]

• Cardinality constraint: \[\left\Vert \mathbf{w}\right\Vert _{0} \leq K.\]

• Turnover constraint: \[\left\Vert \mathbf{w}-\mathbf{w}_{0}\right\Vert _{1} \leq u\] where \(\mathbf{w}_{0}\) is the currently held portfolio.

• Market-neutral constraint: \[\boldsymbol{\beta}^T\mathbf{w} = 0.\]

Markowitz’s portfolio has never been fully embraced by practitioners, among other reasons (Zhao et al. 2019) because

1. variance is not a good measure of risk in practice since it penalizes both the unwanted high losses and the desired low losses: the solution is to use alternative measures for risk, e.g., VaR and CVaR,

2. it is highly sensitive to parameter estimation errors (of covariance matrix \(\boldsymbol{\Sigma}\) and especially mean vector \(\boldsymbol{\mu}\)): solution is robust optimization (Fabozzi 2007) and improved parameter estimation,

3. it only considers the risk of the portfolio as a whole and ignores the risk diversification (i.e., concentrates risk too much in few assets, this was observed in the 2008 financial crisis): solution is the risk parity portfolio.

👉 We will here consider robust portfolio optimization against estimation errors of the parameters \(\boldsymbol{\mu}\) and \(\boldsymbol{\Sigma}\).

• The portfolio that maximizes the return (while ignoring the variance) is the linear program (LP) \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \mathbf{w}^{T}\boldsymbol{\mu}\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1. \end{array}\]

• In practice, however, \(\boldsymbol{\mu}\) is unknown and has to be estimated \(\hat{\boldsymbol{\mu}}\), e.g., with the sample mean \[\hat{\boldsymbol{\mu}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{x}_{t},\] where \(\mathbf{x}_{t}\) is the return at day \(t\).

• Unfortunately, it is well known that the estimation of \(\boldsymbol{\mu}\) is extremely noisy in practice (Chopra and Ziemba 1993).

• Instead of assuming that \(\boldsymbol{\mu}\) is known perfectly, we now assume it belongs to some convex uncertainty set, denoted by \(\mathcal{U}_{\boldsymbol{\mu}}\).

• The worst-case robust formulation is \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \underset{\boldsymbol{\mu}\in\mathcal{U}_{\boldsymbol{\mu}}}{\min}\mathbf{w}^{T}\boldsymbol{\mu}\\ \textsf{subject to} & \mathbf{1}^{T}\mathbf{w}=1. \end{array}\]

• We assume the expected returns are only known within an ellipsoid: \[\mathcal{U}_{\boldsymbol{\mu}} = \left\{\boldsymbol{\mu}=\hat{\boldsymbol{\mu}}+\kappa\mathbf{S}^{1/2}\mathbf{u}\mid\left\Vert \mathbf{u}\right\Vert _{2}\leq1\right\}\] where one can use the estimated covariance matrix to shape the uncertainty ellipsoid, i.e., \(\mathbf{S}=\hat{\boldsymbol{\Sigma}}\).

• We can solve easily the inner minimization: \[\begin{array}{ll} \underset{\boldsymbol{\mu},\mathbf{u}}{\textsf{minimize}} & \mathbf{w}^{T}\boldsymbol{\mu}\\ \textsf{subject to} & \boldsymbol{\mu}=\hat{\boldsymbol{\mu}}+\kappa\mathbf{S}^{1/2}\mathbf{u},\\ & \left\Vert \mathbf{u}\right\Vert _{2}\leq1. \end{array}\]

• It’s easy to find the minimum value using Cauchy-Schwartz’s inequality: \[\begin{aligned} \mathbf{w}^{T}\boldsymbol{\mu} &= \mathbf{w}^{T}\left(\hat{\boldsymbol{\mu}}+\kappa\mathbf{S}^{1/2}\mathbf{u}\right)\\ &= \mathbf{w}^{T}\hat{\boldsymbol{\mu}}+\kappa\mathbf{w}^{T}\mathbf{S}^{1/2}\mathbf{u}\\ &\geq \mathbf{w}^{T}\hat{\boldsymbol{\mu}}-\kappa\left\Vert \mathbf{S}^{1/2}\mathbf{w}\right\Vert _{2} \end{aligned}\] with equality achieved with \(\mathbf{u}=-\frac{\mathbf{S}^{1/2}\mathbf{w}}{\left\Vert \mathbf{S}^{1/2}\mathbf{w}\right\Vert _{2}}\).

• Finally, the robust formulation becomes the SOCP \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \mathbf{w}^{T}\hat{\boldsymbol{\mu}}-\kappa\left\Vert \mathbf{S}^{1/2}\mathbf{w}\right\Vert _{2}\\ \textsf{subject to} & \mathbf{1}^{T}\mathbf{w}=1. \end{array}\]

• Recall the vanilla problem formulation was the LP \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \mathbf{w}^{T}\hat{\boldsymbol{\mu}}\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1. \end{array}\]

• So, we have gone from an LP to an SOCP.

• In general, when a problem is robustified, the complexity of the problem increases. For example:

  • LP becomes SOCP
  • QP also becomes SOCP
  • SOCP becomes SDP.

• Instead of assuming that \(\boldsymbol{\Sigma}\) is known perfectly, we now assume it belongs to some convex uncertainty set, denoted by \(\mathcal{U}_{\boldsymbol{\Sigma}}\).

• The worst-case robust formulation is \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \underset{\boldsymbol{\Sigma}\in\mathcal{U}_{\boldsymbol{\Sigma}}}{\max}\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\ \textsf{subject to} & \mathbf{1}^{T}\mathbf{w}=1. \end{array}\]

• In particular, we will assume that the estimation comes from the sample covariance matrix \(\hat{\boldsymbol{\Sigma}}=\frac{1}{T}\mathbf{X}^{T}\mathbf{X}\) where \(\mathbf{X}\) is a \(T\times N\) matrix containing the return data (assumed demeaned already).

• However, we will assume that the data matrix is noisy \(\hat{\mathbf{X}}\) and the actual matrix can be written as \(\mathbf{X}=\hat{\mathbf{X}}+\boldsymbol{\Delta}\), where \(\boldsymbol{\Delta}\) is some error matrix bounded in its norm.

• Thus, we will then model the data matrix as \[\mathcal{U}_{\mathbf{X}} =\left\{\mathbf{X}\mid\left\Vert\mathbf{X}-\hat{\mathbf{X}}\right\Vert_{F}\leq\delta_{\mathbf{X}}\right\}.\]

• Finally, we can see that both upper bounds can be actually achieved if the error is properly chosen as \[\boldsymbol{\Delta}=\delta_{\mathbf{X}}\frac{\hat{\mathbf{X}}\mathbf{w}\mathbf{w}^{T}}{\left\Vert \mathbf{w}\right\Vert _{2}\left\Vert\hat{\mathbf{X}}\mathbf{w}\right\Vert_{2}}.\]

• Thus, \[\underset{\mathbf{X}\in\mathcal{U}_{\mathbf{X}}}{\max}\;\mathbf{w}^{T}\mathbf{X}^{T}\mathbf{X}\mathbf{w}=\left(\left\Vert\hat{\mathbf{X}}\mathbf{w}\right\Vert_{2}+\delta_{\mathbf{X}}\left\Vert \mathbf{w}\right\Vert _{2}\right)^{2}.\]

• The robust problem formulation finally becomes: \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \left\Vert\hat{\mathbf{X}}\mathbf{w}\right\Vert_{2}+\delta_{\mathbf{X}}\left\Vert \mathbf{w}\right\Vert _{2}\\ \textsf{subject to} & \mathbf{1}^{T}\mathbf{w}=1 \end{array}\] which is a (convex) SOCP.

• Recall Markowitz’s formulation: \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \mathbf{w}^{T}\boldsymbol{\mu}-\lambda\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1, \quad\mathbf{w}\in\mathcal{W}, \end{array}\] where \(\mathcal{W}\) denotes some other constraints on \(\mathbf{w}\).

• Instead of assuming \(\boldsymbol{\mu}\) and \(\boldsymbol{\Sigma}\) are known perfectly, now we assume they belong to some convex uncertainty sets, denoted as \(\mathcal{U}_{\boldsymbol{\mu}}\) and \(\mathcal{U}_{\boldsymbol{\Sigma}}\), respectively.

• The worst-case formulation will consider the worst-case points within those uncertainty sets.

• A conservative and practical investment approach is to optimize the worst-case objective over the uncertainty sets (Cornuejols and Tütüncü 2006), (Fabozzi 2007): \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \underset{\boldsymbol{\mu}\in\mathcal{U}_{\boldsymbol{\mu}}}{\min}\mathbf{w}^{T}\boldsymbol{\mu}-\lambda\underset{\boldsymbol{\Sigma}\in\mathcal{U}_{\boldsymbol{\Sigma}}}{\max}\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\ \textsf{subject to} & \mathbf{w}^{T}\mathbf{1}=1,\quad\mathbf{w}\in\mathcal{W}. \end{array}\]

• The two key issues are:

  1. How to choose the uncertainty sets \(\mathcal{U}_{\boldsymbol{\mu}}\) and \(\mathcal{U}_{\boldsymbol{\Sigma}}\) so that they are meaningful in practice.
  2. To make sure the optimization problem above is still easy to solve.

• Box uncertainty set: \[\mathcal{U}_{\boldsymbol{\mu}}^{b} =\left\{\boldsymbol{\mu}\mid-\boldsymbol{\delta}\leq\boldsymbol{\mu}-\hat{\boldsymbol{\mu}}\leq\boldsymbol{\delta}\right\},\] where the predefined parameters \(\hat{\boldsymbol{\mu}}\) and \(\boldsymbol{\delta}\) denote the location and size of the box uncertainty set, respectively.

• Easily, the worst-case mean is \[\underset{\boldsymbol{\mu}\in\mathcal{U}_{\boldsymbol{\mu}}^{b}}{\min}\mathbf{w}^{T}\boldsymbol{\mu} =\mathbf{w}^{T}\hat{\boldsymbol{\mu}}+\underset{-\boldsymbol{\delta}\leq\boldsymbol{\gamma}\leq\boldsymbol{\delta}}{\min}\mathbf{w}^{T}\boldsymbol{\gamma}=\mathbf{w}^{T}\hat{\boldsymbol{\mu}}-|\mathbf{w}|^{T}\boldsymbol{\delta},\] where \(|\mathbf{w}|\) denotes elementwise absolute value of \(\mathbf{w}\).

• Note that it is a concave function of \(\mathbf{w}\) (as it should be since it is the minimum of linear functions).

• Elliptical uncertainty set: \[\mathcal{U}_{\boldsymbol{\mu}}^{e} =\left\{\boldsymbol{\mu}\mid(\boldsymbol{\mu}-\hat{\boldsymbol{\mu}})^{T}\mathbf{S}_{\boldsymbol{\mu}}^{-1}(\boldsymbol{\mu}-\hat{\boldsymbol{\mu}})\leq\delta_{\boldsymbol{\mu}}^{2}\right\},\] where the predefined parameters \(\hat{\boldsymbol{\mu}}\), \(\delta_{\boldsymbol{\mu}}>0\), and \(\mathbf{S}_{\boldsymbol{\mu}}\succ\mathbf{0}\) denote the location, size, and the shape of the uncertainty set, respectively.

• The worst-case mean is \[\begin{aligned} \underset{\boldsymbol{\mu}\in\mathcal{U}_{\boldsymbol{\mu}}^{e}}{\min}\mathbf{w}^{T}\boldsymbol{\mu} &= \underset{\left\Vert \mathbf{S}_{\boldsymbol{\mu}}^{-1/2}\boldsymbol{\gamma}\right\Vert _{2}\leq\delta_{\boldsymbol{\mu}}}{\min}\mathbf{w}^{T}(\hat{\boldsymbol{\mu}}+\boldsymbol{\gamma})=\mathbf{w}^{T}\hat{\boldsymbol{\mu}}+\underset{\left\Vert \mathbf{S}_{\boldsymbol{\mu}}^{-1/2}\boldsymbol{\gamma}\right\Vert _{2}\leq\delta_{\boldsymbol{\mu}}}{\min}\mathbf{w}^{T}\boldsymbol{\gamma}\\ &=\mathbf{w}^{T}\hat{\boldsymbol{\mu}}+\underset{\left\Vert \tilde{\boldsymbol{\gamma}}\right\Vert _{2}\leq\delta_{\boldsymbol{\mu}}}{\min}\mathbf{w}^{T}\mathbf{S}_{\boldsymbol{\mu}}^{1/2}\tilde{\boldsymbol{\gamma}}=\mathbf{w}^{T}\hat{\boldsymbol{\mu}}-\delta_{\boldsymbol{\mu}}\left\Vert \mathbf{S}_{\boldsymbol{\mu}}^{1/2}\mathbf{w}\right\Vert _{2}. \end{aligned}\]

• Note that it is a concave function of \(\mathbf{w}\) (as it should be since it is the minimum of linear functions).

• Box uncertainty set: \[\mathcal{U}_{\boldsymbol{\Sigma}}^{b} =\left\{\boldsymbol{\Sigma}\mid\underline{\boldsymbol{\Sigma}}\leq\boldsymbol{\Sigma}\leq\overline{\boldsymbol{\Sigma}},\boldsymbol{\Sigma}\succeq\mathbf{0}\right\}.\]

• The worst-case value \(\underset{\boldsymbol{\Sigma}\in\mathcal{U}_{\boldsymbol{\Sigma}}^{b}}{\max}\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\) is given by the (convex) semidefinite problem (SDP) \[\begin{array}{ll} \underset{\boldsymbol{\Sigma}}{\textsf{maximize}} & \mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\ \textsf{subject to} & \underline{\boldsymbol{\Sigma}}\leq\boldsymbol{\Sigma}\leq\overline{\boldsymbol{\Sigma}},\\ & \boldsymbol{\Sigma}\succeq\mathbf{0}. \end{array}\]

• The equivalent dual problem is (Lobo and Boyd 2000) \[\begin{array}{ll} \underset{\overline{\boldsymbol{\Lambda}},\,\underline{\boldsymbol{\Lambda}}}{\textsf{minimize}} & \textsf{Tr}(\overline{\boldsymbol{\Lambda}}\,\overline{\boldsymbol{\Sigma}})-\textsf{Tr}(\underline{\boldsymbol{\Lambda}}\,\underline{\boldsymbol{\Sigma}})\\ \textsf{subject to} & \begin{bmatrix}\overline{\boldsymbol{\Lambda}}-\underline{\boldsymbol{\Lambda}} & \mathbf{w}\\ \mathbf{w}^{T} & 1 \end{bmatrix}\succeq\mathbf{0},\\ & \overline{\boldsymbol{\Lambda}}\geq\mathbf{0},\quad\underline{\boldsymbol{\Lambda}}\geq\mathbf{0}, \end{array}\] which is a convex SDP.

• The constraints are jointly convex in the inner dual variable variables \(\overline{\boldsymbol{\Lambda}}\) and \(\underline{\boldsymbol{\Lambda}}\) and the outer variable \(\mathbf{w}\).

• Elliptical uncertainty set: \[\mathcal{U}_{\boldsymbol{\Sigma}}^{e} =\left\{\boldsymbol{\Sigma}\mid\left(\mathsf{vec}(\boldsymbol{\Sigma})-\mathsf{vec}(\hat{\boldsymbol{\Sigma}})\right)^{T}\mathbf{S}_{\boldsymbol{\Sigma}}^{-1}\left(\mathsf{vec}(\boldsymbol{\Sigma})-\mathsf{vec}(\hat{\boldsymbol{\Sigma}})\right)\leq\delta_{\boldsymbol{\Sigma}}^{2},\boldsymbol{\Sigma}\succeq\mathbf{0}\right\}\] where \(\hat{\boldsymbol{\Sigma}}\) denotes the location, \(\delta_{\boldsymbol{\Sigma}}\) denotes the size, and \(\mathbf{S}_{\boldsymbol{\Sigma}}\) determines the shape.

• The worst-case value \(\underset{\boldsymbol{\Sigma}\in\mathcal{U}_{\boldsymbol{\Sigma}}^{e}}{\max}\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\) is given by the (convex) SDP \[\begin{array}{ll} \underset{\boldsymbol{\Sigma}}{\textsf{maximize}} & \mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\ \textsf{subject to} &\left(\mathsf{vec}(\boldsymbol{\Sigma})-\mathsf{vec}(\hat{\boldsymbol{\Sigma}})\right)^{T}\mathbf{S}_{\boldsymbol{\Sigma}}^{-1}\left(\mathsf{vec}(\boldsymbol{\Sigma})-\mathsf{vec}(\hat{\boldsymbol{\Sigma}})\right)\leq\delta_{\boldsymbol{\Sigma}}^{2},\\ & \boldsymbol{\Sigma}\succeq\mathbf{0}. \end{array}\]

• Since the problem is convex, strong duality holds (zero duality gap).

• Thus, the maximum objective value equals the minimum objective value of its dual problem: \[\begin{array}{ll} \underset{\mathbf{Z}}{\textsf{minimize}} & \mathsf{Tr}\left(\hat{\boldsymbol{\Sigma}}\left(\mathbf{w}\mathbf{w}^{T}+\mathbf{Z}\right)\right)+\delta_{\boldsymbol{\Sigma}}\left\Vert \mathbf{S}_{\boldsymbol{\Sigma}}^{1/2}\left(\mathsf{vec}(\mathbf{w}\mathbf{w}^{T})+\mathsf{vec}(\mathbf{Z})\right)\right\Vert _{2}\\ \textsf{subject to} & \mathbf{Z}\succeq\mathbf{0}. \end{array}\]

• We can now plug in this inner problem in the original problem: \[\begin{array}{ll} \underset{\mathbf{w},\mathbf{Z}}{\textsf{maximize}} & \begin{array}{r} \underset{\boldsymbol{\mu}\in\mathcal{U}_{\boldsymbol{\mu}}}{\min}\mathbf{w}^{T}\boldsymbol{\mu}-\lambda\left(\mathsf{Tr}\left(\hat{\boldsymbol{\Sigma}}\left(\mathbf{w}\mathbf{w}^{T}+\mathbf{Z}\right)\right)\right.\hspace{2cm}\\ \left.+\delta_{\boldsymbol{\Sigma}}\left\Vert \mathbf{S}_{\boldsymbol{\Sigma}}^{1/2}\left(\mathsf{vec}(\mathbf{w}\mathbf{w}^{T})+\mathsf{vec}(\mathbf{Z})\right)\right\Vert _{2}\right) \end{array}\\ \textsf{subject to} & \mathbf{w}^{T}\mathbf{1}=1,\quad\mathbf{w}\in\mathcal{W}\\ & \mathbf{Z}\succeq\mathbf{0}. \end{array}\]

• However, now this problem contains a complicated term with the composition of \(\mathsf{vec}(\mathbf{w}\mathbf{w}^{T})\) and the norm \(\left\Vert \cdot\right\Vert _{2}\).

• We can further include a new variable as \(\mathbf{X}=\mathbf{w}\mathbf{w}^{T}\) so that the objective function becomes nicer.

• But this constraint is not convex!

• Luckily we can instead use \(\mathbf{X}\succeq\mathbf{w}\mathbf{w}^{T}\) (because we can easily show that at the optimal point it will be achieved with equality), which can be further expressed as \(\left[\begin{array}{cc} \mathbf{X} & \mathbf{w}\\ \mathbf{w}^{T} & 1 \end{array}\right]\succeq\mathbf{0}\).

• The final problem is \[\begin{array}{ll} \underset{\mathbf{w},\mathbf{X},\mathbf{Z}}{\textsf{maximize}} & \begin{array}{r} \underset{\boldsymbol{\mu}\in\mathcal{U}_{\boldsymbol{\mu}}}{\min}\mathbf{w}^{T}\boldsymbol{\mu}-\lambda\left(\mathsf{Tr}\left(\hat{\boldsymbol{\Sigma}}\left(\mathbf{X}+\mathbf{Z}\right)\right)\right.\hspace{2cm}\\ \left.+\delta_{\boldsymbol{\Sigma}}\left\Vert \mathbf{S}_{\boldsymbol{\Sigma}}^{1/2}\left(\mathsf{vec}(\mathbf{X})+\mathsf{vec}(\mathbf{Z})\right)\right\Vert _{2}\right) \end{array}\\ \textsf{subject to} & \mathbf{w}^{T}\mathbf{1}=1,\quad\mathbf{w}\in\mathcal{W}\\ &\left[\begin{array}{cc} \mathbf{X} & \mathbf{w}\\ \mathbf{w}^{T} & 1 \end{array}\right]\succeq\mathbf{0}\\ & \mathbf{Z}\succeq\mathbf{0}. \end{array}\]

• Finally, the worst-case robust formulation can be written as the (convex) SDP: \[\begin{array}{ll} \underset{\mathbf{w},\overline{\boldsymbol{\Lambda}},\underline{\boldsymbol{\Lambda}}}{\textsf{maximize}} & \mathbf{w}^{T}\hat{\boldsymbol{\mu}}-|\mathbf{w}|^{T}\boldsymbol{\delta}-\lambda\left(\textsf{Tr}(\overline{\boldsymbol{\Lambda}}\,\overline{\boldsymbol{\Sigma}})-\textsf{Tr}(\underline{\boldsymbol{\Lambda}}\,\underline{\boldsymbol{\Sigma}})\right)\\ \textsf{subject to} & \mathbf{w}^{T}\mathbf{1}=1,\quad\mathbf{w}\in\mathcal{W},\\ & \begin{bmatrix}\overline{\boldsymbol{\Lambda}}-\underline{\boldsymbol{\Lambda}} & \mathbf{w}\\ \mathbf{w}^{T} & 1 \end{bmatrix}\succeq\mathbf{0},\\ & \overline{\boldsymbol{\Lambda}}\geq\mathbf{0},\quad\underline{\boldsymbol{\Lambda}}\geq\mathbf{0}. \end{array}\]

• This problem does not have a closed-form solution, but it is an SDP that can be easily solved with an off-the-shelf SDP solver.

📘 F. J. Fabozzi. Robust Portfolio Optimization and Management. Wiley, 2007.

📄 D. Goldfarb and G. Iyengar, “Robust portfolio selection problems”, Mathematics of Operations Research, vol. 28, no. 1, pp. 1–38, 2003.

📄 R. H. Tutuncu and M. Koenig, “Robust asset allocation”, Annals Operations Research, vol. 132, no. 1, pp. 157–187, 2004.

📄 L. El Ghaoui, M. Oks, and F. Oustry, “Worst-case value-at-risk and robust portfolio optimization: A conic programming approach”, Operations Research, pp. 543–556, 2003.

📄 Z. Lu, “Robust portfolio selection based on a joint ellipsoidal uncertainty set”, Optimization Methods & Software, vol. 26, no. 1, pp. 89–104, 2011.

• We will consider one particular example based on modeling the returns via a factor model: \[\mathbf{r}=\boldsymbol{\mu}+\mathbf{V}^{T}\mathbf{f}+\boldsymbol{\epsilon}\] where \(\mathbf{f}\) denotes the random factors distributed according to \(\mathbf{f}\sim\mathcal{N}\left(\mathbf{0},\mathbf{F}\right)\) and \(\boldsymbol{\epsilon}\) denotes the a random residual error with uncorrelated elements distributed according to \(\boldsymbol{\epsilon}\sim\mathcal{N}\left(\mathbf{0},\mathbf{D}\right)\) with \(\mathbf{D}=\mathsf{diag}\left(\mathbf{d}\right)\).

• The returns are then distributed according to \(\mathbf{r}\sim\mathcal{N}\left(\boldsymbol{\mu},\mathbf{V}^{T}\mathbf{F}\mathbf{V}+\mathbf{D}\right)\) and the obtained return using portfolio \(\mathbf{w}\) has mean \(\boldsymbol{\mu}^{T}\mathbf{w}\) and variance \(\mathbf{w}^{T}\left(\mathbf{V}^{T}\mathbf{F}\mathbf{V}+\mathbf{D}\right)\mathbf{w}\).

• We will consider uncertainty in the knowledge of \(\boldsymbol{\mu}\), \(\mathbf{V}\), and \(\mathbf{D}\), while \(\mathbf{F}\) is assumed know; in fact, we consider \(\mathbf{F}=\mathbf{I}\).

• We can then formulate the robust mean-variance problem as \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \underset{\mathbf{V}\in\mathcal{S}_{\mathbf{V}},\mathbf{D}\in\mathcal{S}_{\mathbf{D}}}{\textsf{max}} \mathbf{w}^{T}\left(\mathbf{V}^{T}\mathbf{V}+\mathbf{D}\right)\mathbf{w}\\ \textsf{subject to} & \underset{\boldsymbol{\mu}\in\mathcal{S}_{\boldsymbol{\mu}}}{\textsf{min}}\mathbf{w}^{T}\boldsymbol{\mu}\geq\beta\\ & \mathbf{1}^{T}\mathbf{w}=1. \end{array}\]

• We will define the uncertainty as follows: \[\begin{aligned} \mathcal{S}_{\boldsymbol{\mu}} &= \left\{ \boldsymbol{\mu}\mid\boldsymbol{\mu}=\boldsymbol{\mu}_{0}+\boldsymbol{\delta},\,\left|\delta_{i}\right|\leq\gamma_{i},\,i-1,\ldots,M\right\}\\ \mathcal{S}_{\mathbf{V}} &= \left\{ \mathbf{V}\mid\mathbf{V}=\mathbf{V}_{0}+\boldsymbol{\Delta},\,\left\Vert \boldsymbol{\Delta}\right\Vert _{F}\leq\rho\right\}\\ \mathcal{S}_{\mathbf{D}} &= \left\{ \mathbf{D}\mid\mathbf{D}=\mathsf{diag}\left(\mathbf{d}\right),\,d_{i}\in\left[\underline{d}_{i},\overline{d}_{i}\right],\,i-1,\ldots,M\right\} . \end{aligned}\]

• Let’s now elaborate on each of the inner optimizations…

• First, consider the worst case mean return: \[\underset{\boldsymbol{\mu}\in\mathcal{S}_{\boldsymbol{\mu}}}{\textsf{min}}\;\boldsymbol{\mu}^{T}\mathbf{w}=\boldsymbol{\mu}_{0}^{T}\mathbf{w}+\underset{\left|\delta_{i}\right|\leq\gamma_{i}}{\textsf{min}}\boldsymbol{\delta}^{T}\mathbf{w}=\boldsymbol{\mu}_{0}^{T}\mathbf{w}-\boldsymbol{\gamma}^{T}\left|\mathbf{w}\right|\] which is a concave function.

• Second, let’s turn to the second term of the worst-case variance: \[\underset{\mathbf{D}\in\mathcal{S}_{\mathbf{D}}}{\textsf{max}} \;\mathbf{w}^{T}\mathbf{D}\mathbf{w}=\underset{d_{i}\in\left[\underline{d}_{i},\overline{d}_{i}\right]}{\textsf{max}} \sum_{i=1}^{M}d_{i}w_{i}^{2}=\sum_{i=1}^{M}\overline{d}_{i}w_{i}^{2}=\mathbf{w}^{T}\overline{\mathbf{D}}\mathbf{w}.\]

• Finally, let’s focus on the first term of the worst-case variance: \[\underset{\mathbf{V}\in\mathcal{S}_{\mathbf{V}}}{\textsf{max}}\; \mathbf{w}^{T}\mathbf{V}^{T}\mathbf{V}\mathbf{w}\equiv\underset{\left\Vert \boldsymbol{\Delta}\right\Vert _{F}\leq\rho}{\textsf{max}} \left\Vert \mathbf{V}_{0}\mathbf{w}+\boldsymbol{\Delta}\mathbf{w}\right\Vert =\left\Vert \mathbf{V}_{0}\mathbf{w}\right\Vert +\rho\left\Vert \mathbf{w}\right\Vert.\]

• Recall the maximum Sharpe ratio portfolio formulation in convex form: \[\begin{array}{ll} \underset{\tilde{\mathbf{w}}}{\textsf{minimize}} & \tilde{\mathbf{w}}^{T}\boldsymbol{\Sigma}\tilde{\mathbf{w}}\\ \textsf{subject to} & \tilde{\mathbf{w}}^{T}(\boldsymbol{\mu}-r_{f}\mathbf{1})=1\\ & \mathbf{1}^{T}\tilde{\mathbf{w}}\geq0\\ & \tilde{\mathbf{w}}\geq\mathbf{0}. \end{array}\]

• The portfolio is then obtained with the correct scaling factor as \[\mathbf{w}=\tilde{\mathbf{w}}/(\mathbf{1}^{T}\tilde{\mathbf{w}}).\]

• In order to obtain a worst-case robust formulation, we first relax the equality \(\tilde{\mathbf{w}}^{T}(\boldsymbol{\mu}-r_{f}\mathbf{1})=1\) to inequality \(\tilde{\mathbf{w}}^{T}(\boldsymbol{\mu}-r_{f}\mathbf{1})\geq1\) since optimality is always achieved with equality anyway.

• The worst-case robut Sharpe ratio problem can be formulated as \[\begin{array}{ll} \underset{\tilde{\mathbf{w}}}{\textsf{minimize}} & \underset{\boldsymbol{\Sigma}\in\mathcal{U}_{\boldsymbol{\Sigma}}}{\max}\tilde{\mathbf{w}}^{T}\boldsymbol{\Sigma}\tilde{\mathbf{w}}\\ \textsf{subject to} & \underset{\boldsymbol{\mu}\in\mathcal{U}_{\boldsymbol{\mu}}}{\min}\tilde{\mathbf{w}}^{T}(\boldsymbol{\mu}-r_{f}\mathbf{1})\geq1\\ & \mathbf{1}^{T}\tilde{\mathbf{w}}\geq0\\ & \tilde{\mathbf{w}}\geq\mathbf{0}. \end{array}\]

• Now we can use our favorite uncertainty sets \(\mathcal{U}_{\boldsymbol{\mu}}\) and \(\mathcal{U}_{\boldsymbol{\Sigma}}\) and proceed as before. Easy!