The concept of robust optimization will now be elucidated for mean-variance port- folios, although the approach is also applicable to other kinds of optimization. The classical portfolio optimization is given by1
P𝜆=arg min
𝝎∈Ω
(1−𝜆)√
𝝎′Σ𝝎−𝜆𝝎′𝝁, (10.7)
1Most of this subsection is based upon (Schửttle 2007, Chapter 4).
k k where𝝎denotes then×1 portfolio weight vector andΩ⊂{𝝎∈ℝN|𝝎′𝟏=1}is the
set of all allowable solutions. The (expected) returns of theNassets are contained in the vector𝝁, with variance-covariance matrixΣ ∈ℝN×N (assumed to be positive definite). The parameter𝜆is allowed to take all values in the interval[0,1]and de- termines the weighting between the portfolio return and its risk. Of course, the above problem formulation can be amended by further constraints, such as a budget con- straint (𝝎′𝟏=1), a non-negativity constraint (𝝎≥0), and/or bounds on the weights for single assets or groups of assets (A𝝎≤b). The mathematical program in the above formulation includes the special cases of a minimum-variance and a maximum-return portfolio if the values for 𝜆are chosen as𝜆=0 or𝜆=1, respectively. All values of𝜆between these bounds yield portfolio solutions that lie on the feasible efficient frontier.
So far, point estimates for the unknown parameters(𝝁,Σ)have been utilized. These can be derived from classical or robust estimators, as shown in the previous section.
Incidentally, it will be shown by means of simulation in Section 10.5 that the latter class of estimators turn out to be beneficial compared to the ML estimators. However, regardless of whether the point estimates are obtained from classical or robust estima- tors, these point estimates are subject to uncertainty and in either case small deviations from the true but unknown parameter values can result in quite distinct portfolio solutions. Viewed from this angle, it would be desirable to include the parameter uncertainties directly in the formulation of the portfolio optimization. Because pa- rameter uncertainty exerts a greater impact on the portfolio composition in the case of expected returns compared to their dispersion, we will limit the following exposition to the inclusion of uncertain return parameters only and treat the parameters pertinent to portfolio risk as fixed. So far, the term “uncertainty” has been used rather vaguely.
For the purpose of applying robust optimization techniques this gap needs to be filled with a concrete definition. This is achieved by defining an uncertainty setU𝝁̂ for all allowable parameter values. In principle, three different kinds of sets are possible:
U𝝁̂ = {𝝁∈ℝN|𝝁̂1,…, ̂𝝁i,…, ̂𝝁M,i=1,…,M}, (10.8a) U𝝁̂ = {𝝁∈ℝN| |̂𝜇i−𝜇i|≤𝛿i,i=1,…,N}, (10.8b) U𝝁̂ = {𝝁∈ℝN|(𝜇− ̂𝜇)′Σ̂−1(𝜇− ̂𝜇)≤ 𝛿2
T }. (10.8c)
For the uncertainty set in (10.8a), M scenarios for the expected return vector𝝁 must be specified. These can comprise subjective expectations and/or location esti- mates derived from alternative estimators. In (10.8b) the uncertainty set is defined as intervals around the true, but unknown, return vector. These bounds can be subjec- tively set, but can also be derived from a distributional assumption. For instance, if it is assumed that each of the asset returns is normally distributed, the central fluctua- tion interval with(1−𝛼)confidence level is given as𝛿i= Φ−1(𝛼∕2)⋅̂𝜎i∕√
T), where Φ−1is the quantile function,̂𝜎ithe standard deviation of the returns for theith asset, andT denotes the sample size. The stylized fact of excess kurtosis of empirical re- turn processes can be taken into account, by utilizing a Student’stdistribution, for
k k example. In this case, an ML estimation is conducted first, to obtain an estimate of
the degrees-of-freedom parameter𝜈. Then the quantile at the desired confidence level is determined by using this empirical estimate. It is worth mentioning that in the case of an uncertainty set as in (10.8b) no dependencies between the uncertainty margins for the different assets are taken into account, but rather these are treated as being independent of each other. For the uncertainty set in (10.8c) an elliptical uncertainty set is defined. In contrast to the uncertainty set in (10.8b), it is now assumed that the uncertainties originate from a multivariate elliptical distribution; the covariances between assets’ returns are now explicitly included in the uncertainty set. Last, but not least, it should be pointed out that the first and second uncertainty sets can be combined. The scenarios ( e.g., ML- and robust-based estimators for the unknown returns) would then be given byM ∶= {𝝁̂ML, ̂𝝁MCD, ̂𝝁M, ̂𝝁MM, ̂𝝁MVE, ̂𝝁S, ̂𝝁SD, ̂𝝁OGK} and an elliptical uncertainty set can be generated as
Uest= {𝝁∈ℝN|(𝜇− ̄𝜇)′Σ̄−1(𝜇− ̄𝜇)≤ ̄𝛿2}, (10.9) with
̄𝜇= 1
|M|
∑
m∈M
m, (10.10a)
Σ =̄ diag(̄𝜎11,…, ̄𝜎NN)where ̄𝜎ii= 1
|M|−1
∑
m∈M
(mi− ̄𝜇i)2, (10.10b)
̄𝛿=arg max
m∈M
(m− ̄𝜇)′Σ̄−1(m− ̄𝜇). (10.10c)
Having introduced alternative specifications for the uncertainty sets, it will next be shown how these uncertainties can be expressed in a mathematical program. In general, and independent of the chosen form of the uncertainty set, a worst-case approach is employed for solving optimization tasks robustly. This approach is also known as a min–max approach. This tackles the question of what the optimal weight vector is given the least favorable parameter constellation, that is, the smallest portfolio return for a given risk level. Loosely speaking: expect the worst and you will at least not be disappointed.
In the case of an uncertainty set as in (10.8a), in a first step the weight vectors for theMscenarios would be determined and then the one that yields the lowest portfolio return would be selected. It should be pointed out that for this kind of uncertainty set the determination of the optimal solution can become computationally expensive and depends on the number of scenariosM. It should also be stressed that all scenarios are treated as equally likely, but that the solution is determined solely by the worst scenario. It can be deduced from this that the approach is very conservative and that the optimal outcome can be highly influenced by single outliers for the expected returns of the assets. Hence, a sound specification of the M scenarios is of the utmost importance.
Next is the description of how the problem for an uncertainty set of a symmetrical interval around the location parameters𝝁as shown in (10.8b) can be implemented.
k k The true but unknown subsequent return of theith asset is contained in the interval
𝜇i∈ [̂𝜇i−𝛿i, ̂𝜇i+𝛿i]for a given confidence level. The least favorable return for the ith asset is therefore given as𝜇i= ̂𝜇i−𝛿ifor a long position and as𝜇i= ̂𝜇i+𝛿ifor a short position. TheseNconfidence intervals form a polyhedron and can be expressed as a system of linear inequality constraints. However, it is unknown beforehand whether an asset enters into the portfolio with a positive or negative weight. To solve this problem, two slack variables,𝜔+and𝜔i, are included in the objective function:
PR𝜆=arg max
𝝎,𝝎+,𝝎−
𝝎′𝝁̂ −𝜹′(𝝎+−𝝎−) − (1−𝜆)√ 𝝎′Σ𝝎, 𝝎=𝝎+−𝝎−,
𝝎+≥0,
𝝎−≥0. (10.11)
Of course, the above problem formulation can be amended by a budget constraint 𝝎′𝟏=1 and/or other constraints. During optimization for positive weights𝜔i>0 the returns are equal to(̂𝜇i−𝛿i), and for negative weights the returns are set as(̂𝜇i+𝛿i).
In the first case, theith element of𝝎+is positive and that of𝝎−is set equal to zero according to the inequality constraint, and vice versa. Assets with a higher return uncertainty will obtain a lower portfolio weight than those for which the expected return is more certain. The uncertainty of the expected returns can also be limited to a subsetU⊂Nof the assets. In this case,𝛿i∉Uare set to zero. If long-only constraints are included as side constraints, then the formulation of the mathematical program reduces to that exhibited in (10.7), where the return vector is set as𝝁̂ −𝜹with the additional non-negativity constraint𝝎≥0.
Finally, the treatment of an elliptical uncertainty set for the returns as in (10.8c) is elucidated. Analogously to the previous approaches, this problem is also solved in the form of a worst-case approach:
PR𝜆=arg min
𝝎∈Ω arg max
𝝁∈U (1−𝜆)√
𝝎′Σ𝝎−𝜆(𝝎′𝝁). (10.12) If the returns are multivariate normally distributed, then (𝝁−𝝁)̂ ′Σ−1(𝝁−𝝁)̂ is distributed as𝜒2withNdegrees of freedom. The scalar𝛿2is then the corresponding quantile value for a given confidence level (1−𝛼). The stochastic return vector 𝝁 therefore lies in the ellipse defined by the level of confidence. The maximal distance between this uncertainty ellipsoid and the empirical location vector is now determined, such that the returns correspond to the least favorable outcome.
It is evident from (10.12) that the uncertainty is confined to the returns only and that the variance-covariance matrix is taken as given. Therefore, in a first step the maximal distance can be determined by utilizing a Lagrange approach, where P̂ = 1
TΣ̂is the variance-covariance matrix of the empirical returns and𝛾 denotes the Lagrangian multiplier:
L(𝝁, 𝛾) =𝝎′𝝁̂−𝝎′𝝁− [𝛾
2(𝝁̂−𝝁)′P̂−1(𝝁̂−𝝁) −𝛿2]
. (10.13)
k k The optimal solution is then found by taking the partial derivatives of (10.13) with
respect to𝝁and𝛾 and setting these to zero. This yields a system of two equations, which is solved for𝝁:
𝝁=𝝁̂− (√ 𝛿
𝝎′P̂𝝎P𝝎 )
. (10.14)
After left-multiplication by𝝎′one obtains, for the portfolio returns, 𝝎′𝝁=𝝎′𝝁̂−𝛿√
𝝎′P̂𝝎
=𝝎′𝝁̂−𝛿||P̂12𝝎||. (10.15) It is evident from this equation that the portfolio return in the case of a robust optimization with elliptical uncertainty is smaller than the classical solution by the term 𝛿√
𝝎′P𝝎. The square root of the quantile value can be interpreted as a risk aversion parameter with respect to the uncertainty of the estimates. Substituting the inner solution from equation (10.15) into the robust optimization specification as in (10.12), one obtains
PR𝜆=arg min
𝝎∈Ω arg max
𝝁∈U (1−𝜆)√
𝝎′Σ𝝎−𝜆(𝝎′𝝁)
=arg min
𝝎∈Ω (1−𝜆)√
𝝎′Σ𝝎−𝜆(𝝎′𝝁) +𝜆 𝛿√ T
√𝝎′Σ𝝎̂
=arg min
𝝎∈Ω
(
1−𝜆+𝜆 𝛿√ T
)√
𝝎′Σ𝝎−𝜆𝝎′𝝁.̂ (10.16)
Equation (10.16) has the following implications:
•The efficient frontier of a portfolio when optimized robustly under elliptical uncertainty is, except for a shortening factor, the same as the efficient frontier of a classical mean-variance portfolio.
•The optimal weight vector for a minimum-variance portfolio is the same for both kinds of optimization. This must be true by definition, because the uncer- tainty has been limited to returns only.
With respect to the first point, the risk–return trade-off parameter in (10.16) is now expressed as𝜃. The equivalent trade-off parameter𝜆in the problem formulation of (10.7) is then given by
𝜆∶= 𝜃
1+𝜃√𝛿
T
. (10.17)
The defined interval for𝜃∈ [0,1]is now bounded above for the equivalent classical mean-variance portfolios:𝜆∈ [0,1∕(1+𝛿∕√
T)].
k k But the solution is also conservative in the case of an elliptical uncertainty set, in
the sense that the optimal portfolio weights correspond to a situation where for all assets the least favorable returns are realized. The mathematical problem formulated as in (10.16) can be expressed in the form of a second-order cone program (SOCP).
SOCPs are solved by means of interior-point methods. More precisely, a mathemat- ical program is expressed as an SOCP if it can be written as
arg min
x
f′x
subject to||Ajx+bj||≤c′jx+djforj=1,…,J, (10.18) where J denotes the number of cone constraints. The SOCP formulation includes linear programs (the Aj would be null matrices and thebj would be null vectors), quadratic programs, and problems with hyperbolic constraints, as well as problems that can contain sums and maxima of vector norms (see Lobo et al. 1998). As already implied by the name of this formulation, a quadratic cone optimization requires a cor- responding cone—also known as a Lorenz or ice-cream cone–with the property that the first element is at least as great as the Euclidean norm of its remaining elements.
Analytically, the cone is defined as:
C= {x= (x1,…,xN) ∈ℝN∶x1≥||x2,…,xN||}. (10.19) The SOCP in (10.18) can also be expressed in its dual form:
arg max
z,w −
∑J
j=1
(b′jzj+dj𝑤j)
subject to
∑J
j=1
(A′jzj+cj𝑤j) =f, (10.20)
||zj||≤𝑤jforj=1,…,J.
Here,z∈ℝnj−1andw∈ℝJdenote the optimization variables. Like its primal form, this is a convex program, since the objective function to be maximized is concave and the constraints are convex. Further details about these kinds of optimization can be found, for instance, in Nesterov and Nemirovsky (1994) or Boyd and Vanden- berghe (2004).
Now, with regard to the task of bringing the problem formulation in (10.12) into the SOCP form, one defines as slack variable for the unknown worst-case portfolio returnt=𝝁′𝝎. The vectorxin (10.18) and (10.19) is thenx= (t,𝝎)′and the problem can be written as
arg min
t,𝝎 t
subject tot≤𝝁̂′𝝎−𝛿𝛼||P12𝝎||, (10.21) 𝜎max≥||Σ12𝝎||.
k k The first inequality constraint is the cone constraint and the second is a quadratic
constraint with respect to the portfolio risk. Of course, the problem specification in (10.21) can be amended by other linear constraints, such as budget, non-negativity, and/or bound and group constraints.