Risk-budgeting multi-portfolio optimization with portfolio and marginal risk constraints

Abstract

Multi-portfolio optimization problems and the incorporation of marginal risk contribution constraints have recently received a sustained interest from academia and financial practitioners. We propose a class of new stochastic risk budgeting multi-portfolio optimization models that impose portfolio as well as marginal risk constraints. The models permit the simultaneous and integrated optimization of multiple sub-portfolios in which the marginal risk contribution of each individual security is accounted for. A risk budget defined with a downside risk measure is allocated to each security. We consider the two cases in which the asset universes of the sub-portfolios are either disjoint (diversification of style) or overlap (diversification of judgment). The proposed models take the form of stochastic programming problems and include each a probabilistic constraint with multi-row random technology matrix. We expand a combinatorial modeling framework to represent the feasible set of the chance constraints first as a set of mixed-integer linear inequalities. The new reformulation proposed in this paper is much sparser than previously presented reformulations and allows the efficient solution of problem instances that could not be solved otherwise. We evaluate the efficiency and scalability of the proposed method that is general enough to be applied to general chance-constrained optimization problems. We conduct a cross-validation study via a rolling-horizon procedure to assess the performance of the models, and understand the impact of the parameters and diversification types on the portfolios.

Appendices

Appendix

1.1 Appendix 1: Example for combinatorial reformulation method

Example 1

Consider the following stochastic multi-portfolio optimization problem based on formulation M1. Assets 1 and 2 (resp., 1 and 3) are considered for possible inclusion in sub-portfolio 1 (resp., 2). The random vector $\xi$ of asset returns accepts 10 equally likely realizations $l_j^k, k \in {\varOmega }= \{1,\ldots ,10\}$. The loss realizations (expressed in percentage of asset prices) and probability distribution are shown in Table 5. The notation F refers to the cumulative trivariate probability distribution of the asset returns, while $F_j, j\in J$ is the cumulative marginal probability distribution of asset j.

$$\begin{aligned} \max \quad \quad&0.001(x_{11}+x_{21}) + 0.002(x_{12}+x_{22}) + 0.003(x_{13}+x_{23}) \end{aligned}$$

(57)

$$\begin{aligned} \textit{s.t.}\quad&\mathbb {P} \begin{Bmatrix} \xi _1x_{11} \le d_{11} \\ \xi _1x_{21} \le d_{21} \\ \xi _2x_{12} \le d_{12} \\ \xi _3x_{23} \le d_{23} \end{Bmatrix} \ge 0.7 \end{aligned}$$

(58)

$$\begin{aligned}&d_{11} + d_{21} \le 0.011 \end{aligned}$$

(59)

$$\begin{aligned}&d_{12} \le 0.015 \end{aligned}$$

(60)

$$\begin{aligned}&d_{23} \le 0.012 \end{aligned}$$

(61)

$$\begin{aligned}&d_{11} + d_{12} \le 0.03 \end{aligned}$$

(62)

$$\begin{aligned}&d_{21} + d_{23} \le 0.02 \end{aligned}$$

(63)

$$\begin{aligned}&x_{11} + x_{12} = 0.40 \end{aligned}$$

(64)

$$\begin{aligned}&x_{21} + x_{23} = 0.60 \end{aligned}$$

(65)

$$\begin{aligned}&x_{11} + x_{21} \le 0.50 \end{aligned}$$

(66)

$$\begin{aligned}&x_{12} + x_{22} \le 0.40 \end{aligned}$$

(67)

$$\begin{aligned}&x_{13} + x_{23} \le 0.30 \end{aligned}$$

(68)

$$\begin{aligned}&x_{11}, x_{12}, x_{21}, x_{23} \ge 0 \end{aligned}$$

(69)

Table 5 Probability distribution

1.2 Sufficient-equivalent set of cut points and binarization

In our example, the sufficient-equivalent set $C^e(p)$ includes seven cut points.

$$\begin{aligned} C^e {=} \{ c_{11} = 0.03; c_{21} = 0.01, c_{22} = 0.02, c_{23} = 0.04; c_{31} = 0.01, c_{32} = 0.02, c_{33} = 0.03 \}. \end{aligned}$$

The binary images of the recombinations obtained with $C^e(p)$ are shown in Table 6. The truth table of the pdBf $g\left( \bar{{\varOmega }}_B^+(p), \bar{{\varOmega }}_B^-(p) \right)$ is displayed in the central part of Table 6.

Table 6 Set of recombinations and partially defined Boolean function

1.3 Reformulations of threshold tight minorant (27)–(29)

In our example, the system of inequalities (27)–(29) reads:

$$\begin{aligned} \lambda _{11} + \lambda _{21} + \lambda _{31}&\le 2&(27) \quad \text {for k = 11} \\ \lambda _{11} + \lambda _{21} + \lambda _{31} + \lambda _{32}&\le 2&(27) \quad \text {for k = 12} \\ \lambda _{11}&= 1&(28) \quad \text {for}\, j = 1 \\ \lambda _{21} + \lambda _{22} + \lambda _{23}&= 1&(28) \quad \text {for}\, j = 2 \\ \lambda _{31} + \lambda _{32} + \lambda _{33}&= 1&(28) \quad \text {for}\, j = 3 \\ \lambda _{jm} \in \{0,1\} \; ,\;&\forall j,m&(29) \quad \forall j,m \end{aligned}$$

1.4 Reformulations of threshold tight minorant (31)–(34)

In our example, the system of inequalities (31)–(34) reads:

$$\begin{aligned}&\gamma _{22} + \gamma _{32} \ge 1&(31) \quad \text {for}\, k = 11 \\&\gamma _{22} + \gamma _{33} \ge 1&(31) \quad \text {for}\, k = 12 \\&\gamma _{21} \ge \gamma _{22} \ge \gamma _{23}&(32) \quad \text {for}\, j = 2, m = 2, 3 \\&\gamma _{31} \ge \gamma _{32} \ge \gamma _{33}&(32) \quad \text {for}\, j = 3, m = 2, 3 \\&\gamma _{11} = \gamma _{21} = \gamma _{31} = 1&(33) \quad \text {for}\, j = 1, 2, 3 \\&\gamma _{jm} \in \{0,1\} \; , \quad \forall j,m&(34) \quad \forall j,m \end{aligned}$$

1.5 Feasible set with DMR-0 reformulation

The reformulations for the feasible set of the chance constraint defined by (50)–(54) reads:

$$\begin{aligned}&0.03y_{111} \le d_{11};&(50) \\&0.03y_{211} \le d_{21};&(50) \\&0.01y_{121} + 0.02y_{122} + 0.04y_{123} \le d_{12};&(50) \\&0.01y_{231} + 0.02y_{232} + 0.03y_{233} \le d_{23};&(50) \\&y_{111} \le 0.5\lambda _{11}; \qquad y_{211} \le 0.5\lambda _{11}&(51) \\&y_{121} \le 0.4\lambda _{21}; \qquad y_{122} \le 0.4\lambda _{22}; \qquad y_{123} \le 0.4\lambda _{23};&(51) \\&y_{221} \le 0.4\lambda _{21}; \qquad y_{222} \le 0.4\lambda _{22}; \qquad y_{223} \le 0.4\lambda _{23};&(51) \\&y_{131} \le 0.3\lambda _{31}; \qquad y_{132} \le 0.3\lambda _{32}; \qquad y_{133} \le 0.3\lambda _{33};&(51) \\&y_{231} \le 0.3\lambda _{31}; \qquad y_{232} \le 0.3\lambda _{32}; \qquad y_{233} \le 0.3\lambda _{33};&(51) \\&y_{111}, y_{121}, y_{122}, y_{123}, y_{131}, y_{132}, y_{133} \ge 0 ;&(52) \\&y_{211}, y_{221}, y_{222}, y_{223}, y_{231}, y_{232}, y_{233} \ge 0;&(52) \\&y_{111} \le x_{11};&(53) \\&y_{121} \le x_{12}; \qquad y_{122} \le x_{12}; \qquad y_{123} \le x_{12};&(53) \\&y_{131} \le x_{13}; \qquad y_{132} \le x_{13}; \qquad y_{133} \le x_{13};&(53) \\&y_{211} \le x_{21};&(53) \\&y_{221} \le x_{22}; \qquad y_{222} \le x_{22}; \qquad y_{223} \le x_{22};&(53) \\&y_{231} \le x_{23}; \qquad y_{232} \le x_{23}; \qquad y_{233} \le x_{23};&(53) \\&y_{111} \ge y_{11} + 0.5(\lambda _{11}-1); \qquad y_{211} \ge x_{21} + 0.5(\lambda _{11}-1);&(54) \\&y_{121} \ge x_{12} + 0.5(\lambda _{21}-1); \qquad y_{122} \ge x_{12} + 0.5(\lambda _{22}-1);&(54) \\&y_{123} \ge x_{12} + 0.4(\lambda _{23}-1); \qquad y_{131} \ge x_{13} + 0.4(\lambda _{31}-1);&(54) \\&y_{132} \ge x_{13} + 0.4(\lambda _{32}-1); \qquad y_{133} \ge x_{13} + 0.4(\lambda _{33}-1);&(54) \\&y_{221} \ge x_{22} + 0.5(\lambda _{21}-1); \qquad y_{222} \ge x_{22} + 0.5(\lambda _{22}-1);&(54) \\&y_{223} \ge x_{22} + 0.4(\lambda _{23}-1); \qquad y_{231} \ge x_{23} + 0.4(\lambda _{31}-1);&(54) \\&y_{232} \ge x_{23} + 0.4(\lambda _{32}-1); \qquad y_{233} \ge x_{23} + 0.4(\lambda _{33}-1);&(54) \end{aligned}$$

1.6 Feasible set with DRM1 reformulation

The reformulations for the feasible set of the chance constraint defined by (42)–(46) reads:

$$\begin{aligned}&0.03z_{111} \le d_{11};&(42) \\&0.03z_{211} \le d_{21};&(42) \\&0.01z_{121} + 0.01z_{122} + 0.02z_{123} \le d_{12};&(42) \\&0.01z_{231} + 0.01z_{232} + 0.01z_{233} \le d_{23};&(42) \\&z_{111} \le 0.5\gamma _{11}; \qquad z_{211} \le 0.5\gamma _{11}&(43) \\&z_{121} \le 0.4\gamma _{21}; \qquad z_{122} \le 0.4\gamma _{22}; \qquad z_{123} \le 0.4\gamma _{23};&(43) \\&z_{221} \le 0.4\gamma _{21}; \qquad z_{222} \le 0.4\gamma _{22}; \qquad z_{223} \le 0.4\gamma _{23};&(43) \\&z_{131} \le 0.3\gamma _{31}; \qquad z_{132} \le 0.3\gamma _{32}; \qquad z_{133} \le 0.3\gamma _{33};&(43) \\&z_{231} \le 0.3\gamma _{31}; \qquad z_{232} \le 0.3\gamma _{32}; \qquad z_{233} \le 0.3\gamma _{33};&(43) \\&z_{111}, z_{121}, z_{122}, z_{123}, z_{131}, z_{132}, z_{133} \ge 0;&(44) \\&z_{211}, z_{221}, z_{222}, z_{223}, z_{231}, z_{232}, z_{233} \ge 0;&(44)\\&z_{111} \le x_{11};&(45) \\&z_{121} \le x_{12}; \qquad z_{122} \le x_{12}; \qquad z_{123} \le x_{12};&(45) \\&z_{131} \le x_{13}; \qquad z_{132} \le x_{13}; \qquad z_{133} \le x_{13};&(45) \\&z_{211} \le x_{21};&(45) \\&z_{221} \le x_{22}; \qquad z_{222} \le x_{22}; \qquad z_{223} \le x_{22};&(45) \\&z_{231} \le x_{23}; \qquad z_{232} \le x_{23}; \qquad z_{233} \le x_{23};&(45) \\&z_{111} \ge x_{11} + 0.5(\gamma _{11}-1); \qquad z_{211} \ge x_{21} + 0.5(\gamma _{11}-1);&(46) \\&z_{121} \ge x_{12} + 0.5(\gamma _{21}-1); \qquad z_{122} \ge x_{12} + 0.5(\gamma _{22}-1);&(46) \\&z_{123} \ge x_{12} + 0.4(\gamma _{23}-1); \qquad z_{131} \ge x_{13} + 0.4(\gamma _{31}-1);&(46) \\&z_{132} \ge x_{13} + 0.4(\gamma _{32}-1); \qquad z_{133} \ge x_{13} + 0.4(\gamma _{33}-1);&(46) \\&z_{221} \ge x_{22} + 0.5(\gamma _{21}-1); \qquad z_{222} \ge x_{22} + 0.5(\gamma _{22}-1);&(46) \\&z_{223} \ge x_{22} + 0.4(\gamma _{23}-1); \qquad z_{231} \ge x_{23} + 0.4(\gamma _{31}-1);&(46) \\&z_{232} \ge x_{23} + 0.4(\gamma _{32}-1); \qquad z_{233} \ge x_{23} + 0.4(\gamma _{33}-1);&(46) \end{aligned}$$

Appendix 2: Asset universe

See Table 7.

Table 7 Assets universe

Appendix 3: Parameters for computational study

See Table 8.

Table 8 Parameter settings