Distributionally Robust Portfolio Optimization

Abstract

In this paper we consider the problem of portfolio optimization involving uncertainty in the probability distribution of the assets returns. Starting with an estimate of the mean and covariance matrix of the returns of the assets, we define a class of admissible distributions for the returns and show that optimizing the worst-case risk of loss can be done in a numerically efficient way. More precisely, we show that determining the asset allocation that minimizes the distributionally robust risk can be done using quadratic programming and a one line search. Effectiveness of the proposed approach is shown using academic examples.

I. Introduction

This paper falls under the umbrella of a large body literature aimed at selection of portfolio weights to optimize a specified performance metric; see [1], [2] and references therein for a survey of results that has been written on the related portfolio selection problem. Generically speaking, the main objective of portfolio selection is the construction of portfolios that maximize expected returns while maintaining the risk of losses below a prescribed level (or equivalently minimize the risk of loss subject to minimum expected return). Usually this is accomplished using performance measures which assume the availability of expected returns and the covariance matrix of the assets.

Despite the theoretical progress over the years, Markowitz’s [3] mean-variance framework is still the major model used in practice today in portfolio selection problems. Given the estimated mean μ and covariance Σ of random returns, the Markowitz’s mean variance model considers the variance minimization in portfolio selection problem subject to minimum expected portfolio return γ. In this framework, the variance of the returns, w^TΣw, is used as surrogate for risk. More precisely, for an n-asset problem, the portfolio weights are determined by solving the following optimization problem

$$\begin{array}{ll}\underset{w\in \mathcal{W}}{\text{min}}\hfill & w^T\Sigma w\hfill \\ \text{subject to }\hfill & {\mu }^{\top }w\ge \gamma ,\hfill \end{array}$$ (1)

where $\mathcal{W}$ denotes the unit simplex.

There are many other approaches to this problem that use different ways to access/control risk. One of the popular alternative portfolio risk measures is value-at-risk (VaR) [4], [5] that gained notoriety due to its simplicity and flexibility [1]. However, VaR also has some important limitations. One of the major limitations of VaR is that it is not a coherent risk measure since it is non-subadditive [6] and it does not measure worst case losses [7]. Furthermore, VaR is non-convex and requires the knowledge of the whole distribution [8], [6]. In order to overcome these limitations, Rockafellar and Uryasev [9], [10] introduced conditional-value-at-risk (CVaR) that has addressed some of the shortcomings of VaR. Unlike VaR, it is a coherent measure of risk since it satisfies the sub-additivity property [11]. Additionally, the minimization of CVaR is a convex optimization problem [12].

One main shortcomings of these portfolio optimization methods is that resulting optimal portfolios are sensitive to the problem parameters which are estimated from historical data [13]. Recently, in order to overcome the effect of estimation errors robust portfolio selection methods have been adopted. Many formulations proposed in the field of robust portfolio optimization have focused on the worst case performance of the portfolio over admissible sets for the mean and covariance of the returns. These methods involve the use of uncertainty set, that includes the worst possible realizations of the uncertain input parameters. In addition to that significant efforts have been made towards formulating robust approaches from Markowitz based mean-variance analysis using box [14], ellipsoidal [15] and separable uncertainty sets [16], [17], [18].

The main drawbacks of the approaches to robust portfolio optimization that are available in the literature is that they either assume the distribution of random returns is normal or they minimize a surrogate measure of risk, not risk itself. This is done to “escape” one of the main problems that one is faced with when trying to optimize risk: Computing and optimizing risk, in general, requires the evaluation of multi-dimensional integrals, a numerical problem for which a “good” solution still does not exist [19].

In this paper, we provide a novel approach to the robust portfolio optimization problem aimed at addressing the above limitations. First, given the uncertainty of information that is usually available on assets, we do not assume that we know their distribution, we only assume that their distribution belongs to a class of distributions that is built on the available mean and covariance information. Second, we provide a way of exactly optimizing worst-case risk without the need for the estimation of the values of multi-dimension integrals. More precisely, we show that optimizing worst-case risk can be done using convex quadratic optimization together with a one line search.

A. Organization of the paper

In Section II-B we provide a precise statement of the robust portfolio optimization problem addressed in this paper. In Section III a reformulation of the robust portfolio optimization problem is presented and the proof of its exactness is provided in Section IV. Numerical examples showing the effectiveness of the proposed approach are shown in Section V. Concluding remarks are provided in Section VI.

B. Notation

The sets of real numbers are denoted by $ℝ$. The Euclidean norm of column vectors $x\in {ℝ}^n$ is denoted by ∥x∥₂. The probability of a set is denoted by Prob{·}. Vectors are denoted by bold letters. We use min(x) and max(x) to denote the minimum and maximum element of the vector x respectively.

II. Problem Statement

In this paper, we concentrate on the single-period portfolio selection problem. Let the random vector

$$X={\left[X_1{X}_2\cdots X_n\right]}^T\in {ℝ}^n,$$

denote the random returns of n risky assets. We assume that the probability distribution of these assets is unknown but that its mean μ and covariance Σ has been estimated from historical data.

We use notation

$$w={\left[w_1{w}_2\cdots w_n\right]}^T\in {ℝ}^n,$$

to denote the vector of portfolio weights with w_i being fraction of the portfolio to be invested in the i-th stock. We require w_i ≥ 0 and

$$\sum _{i=1}^n{w}_i=1,$$

Hence, without loss of generality the set of all possible portfolio allocations is given by

$$\mathcal{W}=\left\{w\in {ℝ}^n:{\sum }_{i=1}^n{w}_i=1,w_i\ge 0\forall i=1,\dots ,n\right\}.$$ (2)

A. Class of Admissible Asset Distributions

Since the only information available on the probability distribution of the returns X is the mean μ and variance Σ, we propose the following class of admissible distributions which we denote by $\mathcal{F}$: Let the random vector X have the form

$$X=\mathit{\mu }+\Delta$$

where Δ is a zero mean random vector whose probability density function (pdf) f(δ) is of the form

$$f(\mathit{\delta })=g\left({\mathit{\delta }}^T{\Sigma }^{-1}\mathit{\delta }\right)$$

where $g:{ℝ}_{+}\to {ℝ}_{+}$ satisfies

1. g(y) ≥ 0 for all y ≥ 0

2. g(y) = 0 for all y > y_max

3. g(y₁) ≥ g(y₂) for all 0 ≤ y₁ ≤ y₂

and the quantity y_max represents a uniform bound on all admissible distributions. The set of all distributions of the form above is denoted by $\mathcal{F}$.

The class $\mathcal{F}$ above includes, but is not limited to, usual distributions such as truncated Gaussian with mean μ and variance Σ and uniform distributions over ellipsoids of the form

$$\mathcal{E}=\left\{x:x=\mathit{\mu }+\mathit{\delta },{\mathit{\delta }}^T{\Sigma }^{-1}\mathit{\delta }\le y\right\}$$

for any y ≤ y_max.

Remark 1: Other class of distributions [20] can be considered when formulating the robust portfolio optimization problem. To keep the presentation clear and concise, this will be discussed in future work.

B. Problem Formulation

Before we proceed with the definition of the problem addressed in this paper, the definition of risk needs to be introduced: Let α be a given threshold value. Then, the distributionally robust risk is defined as

$${\text{Risk}}_{\alpha }(w)\doteq \underset{f\in \mathcal{F}}{\text{sup\hspace{0.17em}}}{\text{Prob}}_f\left\{X:X^{T}w\le -\alpha \right\}$$ (3)

where Prob_f denotes the probability computed using pdf f belonging to the class of distributions $\mathcal{F}$ introduced above. Then, the distributionally robust portfolio allocation problem is defined as

$$\begin{array}{ll}\underset{w\in \mathcal{W}}{\text{min}}\hfill & {\text{Risk}}_{\alpha }(w)\hfill \\ \text{subject to }\hfill & {\mu }^{T}w\ge \gamma ,\hfill \end{array}$$ (4)

III. Equivalent Numerically Friendly Reformulation

Since computing probability is, in general, an NP-hard problem, in this section we present a “simple” optimization problem which i) is equivalent to the distributionally robust portfolio allocation problem (4) and ii) does not require the computation of multidimensional integrals.

Consider the optimization problem

$$\begin{array}{ll}\underset{w\in \mathcal{W}}{\text{min}}\hfill & \frac{{\Vert {\Sigma }^{1/2}w\Vert }_2}{\alpha +{\mathit{\mu }}^{T}w}\hfill \\ \text{subject to }\hfill & {\mu }^{T}w\ge \gamma .\hfill \end{array}$$ (5)

The following result shows the equivalence between the above problem and the distributionally robust portfolio allocation problem.

Theorem 1: Let $\mathit{\mu }\in {ℝ}^n$, $\Sigma \in {ℝ}^{n\times n}$, α > 0 and $\mathcal{F}$ be given. Then, Problem (5) is equivalent to Problem (4) in the following sense: w* achieves the optimum of the Problem (5) if and only if w* achieves the optimum of the Problem (4).

Proof: See Section IV. ■

Remark 2: The result above shows that, in the provided setup, minimizing risk and minimizing variance is different, as it is expected. Minimizing risk is equivalent to minimizing a trade-off between variance and expectation of the portfolio return. However, if expected return is fixed then risk minimization and variance minimization are equivalent and the distributionally robust portfolio allocation problem is, in this case, equivalent to Markowitz approach.

Remark 3: When μ^Tw = β is fixed, the Problem (5) becomes a convex quadratic optimization problem, i.e., minimization of a convex quadratic function subject to convex constraints. Hence, this problem can be numerically solved using quadratic optimization together with a one line search; see Algorithm 1.

IV. Proof of Theorem 1

Consider the problem

$$\begin{array}{ll}\underset{w\in \mathcal{W}}{\text{max}}\hfill & \underset{f\in \mathcal{F}}{\text{inf\hspace{0.17em}}}{\text{Prob}}_f\left\{X:X^{T}w\ge -\alpha \right\}\hfill \\ \text{subject to }\hfill & {\mu }^{T}w\ge \gamma ,\hfill \end{array}$$ (6)

which can be seen to be equivalent to the risk minimization problem (4) by noting that

$$\underset{f\in \mathcal{F}}{\text{inf\hspace{0.17em}}}{\text{Prob}}_f\left\{X:X^{T}w\ge -\alpha \right\}=1-{\text{Risk}}_{\alpha }(w).$$

By exploiting this equivalence, we consider the problem (6) in the proof of Theorem 1. Before presenting the proof of the theorem we give the following result that is essential part of our proof.

Lemma 1: Let the random vector X is of the form X = μ+Δ where the distribution f for Δ is taken to be unknown but assumed to belong to the class $\mathcal{F}$ and

$$\mathit{\mu }\in \left\{X:X^{T}w\ge -\alpha \right\}$$

Then

$$\underset{f\in \mathcal{F}}{\text{inf\hspace{0.17em}}}{\text{Prob}}_f\left\{X:X^{T}w\ge -\alpha \right\}$$

is achieved when Δ has a uniform distribution over the set

$$\mathcal{R}=\left\{\Delta \in {ℝ}^n:{\Delta }^T{\Sigma }^{-1}\Delta \le y_{\text{max}}\right\}$$ (7)

where y_max represents the uniform bound on the support of the distribution of Δ.

Proof: In [20], it is proven that

$$\underset{f\in \mathcal{F}}{\text{inf\hspace{0.17em}}}{\text{Prob}}_f\left\{X:X^{T}w\ge -\alpha \right\}$$

is achieved when Δ has a truncated uniform distribution, i.e., Δ is uniformly distributed over the set

$${\mathcal{E}}_t=\left\{\Delta \in {ℝ}^n:{\Delta }^T{\Sigma }^{-1}\Delta \le t\right\}$$

for some 0 ≤ t ≤ y_max.

To simplify rest of the proof, we apply volume-preserving linear map from an ellipsoid to a hypersphere. We can always find a linear transformation such that

$$\tilde{\Delta }=\frac{1}{\text{det}\left({\Sigma }^{-1/2}\right)}{\Sigma }^{-1/2}\Delta$$

where Σ^−1/2 symmetric matrix such that Σ⁻¹ = Σ^−1/2Σ^−1/2. So that, the ellipsoid ε_t becomes

$${\tilde{\mathcal{E}}}_t=\left\{\tilde{\Delta }\in {ℝ}^n:{\tilde{\Delta }}^T\tilde{\Delta }\le \tilde{t}\right\}$$

which is a hypersphere with a radius $\tilde{t}$. Now, let $\tilde{h}$ denote the distance of the hyperplane $\left\{\tilde{X}:{\tilde{X}}^{T}w=-\tilde{\alpha }\right\}$ to the origin. Then, $\tilde{t}-\tilde{h}$ corresponds the height of the hyperspherical cap. The n-dimensional volume of this hyperspherical cap V_C of height $\tilde{t}-\tilde{h}$ and radius $\tilde{t}$ is given by [21]

$$V_C=\frac{{\pi }^{\frac{n-1}{2}}{\tilde{t}}^n}{\Gamma \left(\frac{n+1}{2}\right)}{\int }_0^{{\text{cos}}^{-1}\left(\frac{\tilde{h}}{\tilde{t}}\right)}{\text{sin}}^n(w)dw$$

where Γ is the gamma function. Then, the probability of this hyperspherical cap, $\text{Prob}\left\{C\right\}=\text{Prob}\left\{\tilde{X}:{\tilde{X}}^{T}w\le -\tilde{\alpha }\right\}$, is given by

$$\text{Prob}\left\{C\right\}=\frac{V_C}{V_{{\tilde{\mathcal{E}}}_t}}=M{\int }_0^{{\text{cos}}^{-1}\left(\frac{\tilde{h}}{\tilde{t}}\right)}{\text{sin}}^n(w)dw$$

where $V_{{\tilde{\mathcal{E}}}_t}$ denotes the volume of the hypersphere ${\tilde{\mathcal{E}}}_t$ and M is constant that does not depend on $\tilde{t}$.

It is clear that the probability of this hyperspherical cap is monotonic with respect to ${\text{cos}}^{-1}(\tilde{h}/\tilde{t})$ which is an increasing function of $\tilde{t}$. This implies the probability

$$\text{Prob}\left\{\tilde{X}:{\tilde{X}}^{T}w\ge -\tilde{\alpha }\right\}=1-\text{Prob}\left\{C\right\}$$

is decreasing function of $\tilde{t}$. Hence, it immediately implies that the minimum is attained at $\tilde{t}=y_{\text{max}}$ ■

Before proceeding with the proof of the theorem, we give some definitions that we use in our reasoning.

We first introduce the concept of floating body in a similar fashion to the definition given in [22]. Given a probability level 0 < ϵ < 1/2, we define the floating body R_ϵ of the probability distribution is a convex symmetric set for which each supporting hyperplane cuts off a set of probability ϵ. More precisely, the floating body is defined as:

Definition 1: (Floating body) Let 0 < ϵ < 1/2 be given and consider a set $\mathcal{R}$. For any direction u ∈ Rⁿ, ∥u∥₂ = 1, let H(u) be the supporting hyperplane of R_ϵ normal to u. Also, let ${\mathcal{H}}^{+}\left(u\right)$ be the half-space defined by H(u) which does not contain the origin. Then, ${\mathcal{R}}_{ϵ}$ is the floating body associated with ϵ of the uniform probability measure over $\mathcal{R}$ if, given any direction ∥u∥₂ = 1,

$$\text{Prob}\left({\mathcal{H}}^{+}(u)\right)=ϵ.$$

It is proven in [22] that the convex floating body ${\mathcal{R}}_{ϵ}$ exists as long as the support set $\mathcal{R}$ of the uniform distribution is convex and symmetric.

Definition 2: (Dual norm) Let ∥·∥ be a norm in ${ℝ}^n$. Then the dual norm is defined as

$$\Vert x{\Vert }_{*}\triangleq \text{max}\left\{x^{T}y:\Vert y\Vert \le 1\right\}.$$

Definition 3: (Minkowski functional) Let the set ${\mathcal{R}}_{ϵ}\subset {ℝ}^n$. Then, the Minkowski functional associated with the set ${\mathcal{R}}_{ϵ}$, denoted by ∥x∥_ϵ, is given by

$$\Vert x{\Vert }_{ϵ}\triangleq \text{inf\hspace{0.17em}}\left\{t>0:\frac{1}{t}x\in {\mathcal{R}}_{ϵ}\right\}$$

for all $x\in {ℝ}^n$. Note that the expression above defines a norm when the set ${\mathcal{R}}_{ϵ}$ is compact, convex and symmetric.

A. Proof of the Theorem 1

As a direct application of Lemma 1 and under the assumptions, the maximization problem (6) can be recast as

$$\begin{array}{ll}\underset{w\in \mathcal{W}}{\text{max}}\hfill & \text{Prob}\left\{X:X^{T}w\ge -\alpha \right\}\hfill \\ \text{subject to }\hfill & {\mu }^{\top }w\ge \gamma ,\hfill \end{array}$$ (8)

where Δ has a uniform distribution over the set $\mathcal{R}$ defined in (7). Without loss of generality, let 0 < ϵ < 1/2 and suppose that

$$\text{Prob}\left\{X^{T}w\ge -\alpha \right\}\ge 1-ϵ$$

where X = μ + Δ. So, plugging X in the above equation yields

$$\text{Prob}\left\{-{\left(\mathit{\mu }+{\Delta }_x\right)}^{T}w\le \alpha \right\}\ge 1-ϵ$$

Since the set $\left\{\Delta \in {ℝ}^n:-{(\mathit{\mu }+\Delta )}^{T}w\le \alpha \right\}$ is a halfspace, by the definition of the floating body, satisfying the above inequality is equivalent to satisfy the inequality given by

$$-{(\mathit{\mu }+\Delta )}^{T}w\le \alpha$$

for all $\Delta \in {\mathcal{R}}_{ϵ}$, where ${\mathcal{R}}_{ϵ}$ is the floating body of the probability distribution of Δ.

Then, by symmetry of the floating body ${\mathcal{R}}_{ϵ}$, the inequality can be equivalently rewritten as

$${\Delta }^{T}w\le \alpha +{\mathit{\mu }}^{T}w.$$ (9)

Thus, given the definition of dual norm and the Minkowski functional, this is equivalent to

$$\Vert w{\Vert }_{ϵ,*}\le \alpha +{\mathit{\mu }}^{T}w.$$

where ∥w∥_ϵ,_* denotes the dual norm of the Minkowski functional associated with its floating body ${\mathcal{R}}_{ϵ}$.

So, if we can determine the quantity ∥w∥_ϵ,_*, then we can numerically solve the resulting problem.

Since Δ has an elliptical density function of the form described in Section II-A, given 0 < ϵ < 1/2, the floating body ${\mathcal{R}}_{ϵ}$ is of the form

$${\mathcal{R}}_{ϵ}=\left\{\Delta \in {ℝ}^n:{\Delta }^T{\Sigma }^{-1}\Delta \le r^2(ϵ)\right\}.$$

where r(ϵ) > 0.

It follows from [23] that letting

$${\Delta }_y=\frac{1}{r(ϵ)}{\Sigma }^{-1/2}\Delta$$

leads to $\Delta \in {\mathcal{R}}_{ϵ}$ if and only if ∥Δ_y∥₂ ≤ 1 where the symmetric matrix Σ^−1/2 is such that Σ⁻¹ = Σ^−1/2Σ^−1/2. Then, the inequality (9) can be rewritten as

$$r(ϵ){\Sigma }^{1/2}{\Delta }_y\le \alpha +{\mathit{\mu }}^{T}w$$

for all ∥Δ_y∥₂ ≤ 1. Since the dual norm of 2-norm is the 2-norm, we have

$${\Vert r(ϵ){\Sigma }^{1/2}w\Vert }_2\le \alpha +{\mathit{\mu }}^{T}w,$$ (10)

which leads to

$$r(ϵ)\le \frac{\alpha +{\mathit{\mu }}^{T}w}{{\Vert {\Sigma }^{1/2}w\Vert }_2},$$ (11)

Hence, risk minimization, the problem (8), is equivalent to maximizing the radius r(ϵ) of the floating body ${\mathcal{R}}_{ϵ}$.

The r(ϵ) is easily verified to be non-increasing function of ϵ with r(ϵ) > 0 for 0 < ϵ < 1/2. Thus, r(ϵ) ≤ r(ϵ*) implies 1/r(ϵ*) ≤ 1/r(ϵ) so that maximizing the r(ϵ) is equivalent to minimizing 1/r(ϵ). Then the problem becomes

$$\underset{w\in \mathcal{W}}{\text{min}}\frac{{\Vert {\Sigma }^{1/2}w\Vert }_2}{\alpha +{\mathit{\mu }}^{T}w}$$ (12)

where Σ is covariance matrix of the distribution x.

V. Numerical results

In this section, we present numerical results comparing the solution of a risk minimization (R-M) problem (5) to the solution of the variance minimization (V-M) problem (1) for different choices of minimum expected returns γ. Our portfolio weights are restricted to lie in the set $\mathcal{W}$ which is defined in (2). The threshold value α is set as (α = 0), that means we aim at minimizing the risk of losses.

In the results, $w_{R-M}^{*}$ and $w_{V-M}^{*}$ correspond to the optimal portfolios of (R-M) and (V-M) problem respectively. Given these optimal portfolios, we use the Monte Carlo simulations to estimate the worst-case risk which is defined as ${\text{sup}}_{f\in \mathcal{F}}$ Prob_f {X : X^T w* ≤ 0} where X = μ + Δ.

As we can see from the results in Section IV, the worst-case distribution of Δ is the uniform distribution over the ellipsoid with largest radius. Hence, in our simulations, distributionally robust risk is estimated using Monte Carlo simulations with Δ being uniformly distributed over the set

$$\mathcal{R}=\left\{\Delta \in {ℝ}^n:{\Delta }^T{\Sigma }^{-1}\Delta \le y_{\text{max}}\right\}.$$

Furthermore, randomly generated estimates of mean and covariance of random returns were used in the provided examples. Our results were obtained using CVX, a package for specifying and solving convex programs [24], [25].

Example 1: In this example, we consider a portfolio involving n = 5 assets where the mean is randomly generated as μ = [0.34, 0.68, 0.40, 0.28, 0.46]^⊤.

Example 2: In this example, we consider a portfolio involving n = 7 assets where the mean is randomly generated as μ = [0.98, 0.33, 0.35, 0.54, 0.60, 0.51, 0.05]^⊤.

Tables I and II show the optimal portfolios and corresponding worst-case risk values of (R-M) and (V-M) problems. These results indicates that when the requested expected return γ is low the two approaches give different allocations as expected. Actually, the requirement on expected return is not binding for low levels of risk for the risk minimization scheme. In those cases the average return is higher than the lower bound imposed. For mean-variance minimization scheme, the allocation calculated yields the minimum allowed average return for all cases. Moreover, these two schemes leads to same results when high expected return is requested.

TABLE I:

Comparison of R-M and V-M with 5 assets

γ	0.2		0.4		0.6
	R – M	V – M	R – M	V – M	R – M	V – M
w	0.140	0.207	0.140	0.207	0.009	0.000
	0.382	0.190	0.382	0.190	0.670	0.653
	0.196	0.185	0.196	0.185	0.125	0.084
	0.080	0.211	0.080	0.211	0.000	0.000
	0.203	0.206	0.203	0.206	0.196	0.263
μT w	0.503	0.430	0.503	0.430	0.600	0.600
Risk	0.180	0.227	0.180	0.227	0.202	0.202

TABLE II:

Comparison of R-M and V-M with 7 assets

γ	0.60		0.80		0.95
	R – M	V – M	R – M	V – M	R – M	V – M
w	0.345	0.149	0.561	0.563	0.912	0.912
	0.076	0.137	0.000	0.000	0.000	0.000
	0.091	0.143	0.000	0.000	0.000	0.000
	0.148	0.150	0.136	0.139	0.000	0.000
	0.189	0.140	0.204	0.189	0.088	0.088
	0.151	0.141	0.102	0.109	0.000	0.000
	0.000	0.140	0.000	0.000	0.000	0.000
μT w	0.669	0.489	0.800	0.800	0.950	0.950
Risk	0.079	0.229	0.095	0.095	0.139	0.139

VI. Conclusion

In this paper, we provide a novel approach to the problem of portfolio optimization when the distribution of the returns of the assets is not known. More precisely, we start with estimates of the mean and covariance matrix of the asset returns and define a class of distributions that contains common distributions compatible with this a-priori information. With this class of admissible distributions at hand, we show that minimizing the worst-case risk of loss with respect to the admissible class of distributions can be done in an efficient way. More precisely, we show that determining the optimal asset allocation can be done with quadratic optimization together with a one line search. Effort is now being put in obtaining similar results for other classes of distributions and studying the case where there is uncertainty in the estimation of mean and variance of the returns.