A sparse enhanced indexation model with chance and cardinality constraints

Abstract

Enhanced indexation aims to construct a portfolio to track and outperform the performance of a stock market index by employing both passive and active fund management strategies. This paper presents a novel sparse enhanced indexation model with chance and cardinality constraints. Its goal is to maximize the excess return that can be attained with a high probability, while the model allows a fund manger to limit the number of stocks in the portfolio and specify the maximum tolerable relative market risk. In particular, we model the asset returns as random variables and estimate their probability distributions by the Capital Asset Pricing Model or Fama-French 3-factor model, and measure the relative market risk with the coherent semideviation risk function. We deal with the chance constraint via distributionally robust approach and present a second-order cone programming and a semidefinite programming safe approximation for the model under different sets of potential distribution functions. A hybrid genetic algorithm is applied to solve the NP-hard problem. Numerical tests are conducted on the real data sets from major international stock markets, including USA, UK, Germany and China. The results demonstrate that the proposed model and the method can efficiently solve the enhanced indexation problem and our approach can generally achieve sparse tracking portfolios with good out-of-sample excess returns and high robustness.

Appendix: Proof of Theorem 1

Proof

Similar to the results in [24], the corresponding moment problem of (2.7) for each fixed w can be written as:

$$\begin{aligned}&\inf \limits _{F} \qquad \int _{z\in Z} I_{S_w}(z)dF(z)\nonumber \\&\mathrm{s.t.} \qquad \int _{z\in Z}dF(z)=1, \nonumber \\&\qquad \qquad \int _{z\in Z} z_j dF(z)=\mu _j, \quad \forall j=1,2,\ldots ,k,\nonumber \\&\qquad \qquad \int _{z\in Z} z_j^2 dF(z)=\mu _j^2+\sigma _j^2, \quad \forall j=1,2,\ldots ,k. \end{aligned}$$

(6.1)

The dual program of (6.1) is formulated by

$$\begin{aligned}&\sup \limits _{\theta ,\alpha ,\beta } \qquad \theta +\sum \limits _{j = 1}^k {\mu _j \alpha _j}+\sum \limits _{j = 1}^k (\mu _j^2+\sigma _j^2) \beta _j \end{aligned}$$

(6.2)

$$\begin{aligned}&\mathrm{s.t.}\qquad \theta +\sum \limits _{j = 1}^k {z_j \alpha _j}+\sum \limits _{j = 1}^k z_j^2 \beta _j\le I_{S_w}(z), \quad \forall z\in Z. \end{aligned}$$

(6.3)

The constraint (6.3) can be rewritten as follows:

$$\begin{aligned}&\theta +\sum \limits _{j = 1}^k {z_j \alpha _j}+\sum \limits _{j = 1}^k z_j^2 \beta _j\le 1, \quad \forall z\in Z, \end{aligned}$$

(6.4)

$$\begin{aligned}&\theta +\sum \limits _{j = 1}^k {z_j \alpha _j}+\sum \limits _{j = 1}^k z_j^2 \beta _j\le 0, \quad \forall z\in Z:y^0(w)+y(w)^T z-d<0. \end{aligned}$$

(6.5)

We first reformulate the constraint (6.4) to the following

$$\begin{aligned} \begin{array}{ll} 0\ge \max \limits _{z}\ &{} \theta -1+\alpha ^T z+\beta ^Tz^2\\ ~~~~~~~\text {s.t.} &{} l\le z\le u.\\ \end{array} \end{aligned}$$

(6.6)

By imposing $\beta <0$, we are able to give a conservative approximation for (6.6) by using the Lagrangian dual problem. Then there exist $v_{0j}$ and ${\bar{v}}_{0j}\ge 0(j=1,2,\ldots ,k)$ such that

$$\begin{aligned}&\mathop {\max }\limits _z \,\;\theta - 1 + {\alpha ^{\mathrm{T}}}z + {\beta ^{\mathrm{T}}}{z^2} + {\nu _0}^{\mathrm{T}}(z - l) - {{\overline{\nu }}} _0^{\mathrm{T}}(z - u)\\&= \mathop {\max }\limits _z \,\;\theta - 1 + \sum \limits _{j = 1}^k ( {{{\overline{\nu }}} _{0j}}{u_j} - {\nu _{0j}}{l_j}) + \sum \limits _{j = 1}^k ( {\beta _j}z_j^2 + ({\alpha _j} + {\nu _{0j}} - {{{\overline{\nu }}} _{0j}}){z_j})\\&\;\; = \mathop {\max }\limits _z \,\;\theta - 1 + \sum \limits _{j = 1}^k ( {{{\overline{\nu }}} _{0j}}{u_j} - {\nu _{0j}}{l_j}) - \sum \limits _{j = 1}^k {\frac{{{{({\alpha _j} + {\nu _{0j}} - {{{{\overline{\nu }}} }_{0j}})}^2}}}{{4{\beta _j}}}} \\&\qquad + \sum \limits _{j = 1}^k {{\beta _j}\left( {z_j} + \frac{{{\alpha _j} + {\nu _{0j}} - {{{{\overline{\nu }}} }_{0j}}}}{{2{\beta _j}}} \right) ^2}. \end{aligned}$$

Since $\sum \limits _{j = 1}^k {\beta _j(z_j+\frac{\alpha _j+v_{0j}-{\bar{v}}_{0j}}{2\beta _j})^2}<0$, then the inequality

$$\begin{aligned} \theta - 1 + \sum \limits _{j = 1}^k ( {{{\overline{\nu }}} _{0j}}{u_j} - {\nu _{0j}}{l_j}) - \sum \limits _{j = 1}^k {\frac{{{{({\alpha _j} + {\nu _{0j}} - {{{{\overline{\nu }}} }_{0j}})}^2}}}{{4{\beta _j}}}} \le 0 \end{aligned}$$

(6.7)

implies that

$$\begin{aligned} \mathop {\max }\limits _z \,\;\theta - 1 + {\alpha ^{\mathrm{T}}}z + {\beta ^{\mathrm{T}}}{z^2} + {\nu _0}^\mathrm{T}(z - l) - {{\overline{\nu }}} _0^{\mathrm{T}}(z - u)\le 0. \end{aligned}$$

By introducing $k_{0j}\ge 0~(j=1,2,\ldots ,k)$, we can rewrite (6.7) as

$$\begin{aligned}&\theta -1+\sum \limits _{j = 1}^k ({k_{0j}+{\bar{v}}_{0j}u_j-v_{0j}l_j})\le 0, \end{aligned}$$

(6.8)

$$\begin{aligned}&k_{0j}\ge -\frac{(\alpha _j+v_{0j}-{\bar{v}}_{0j})^2}{4\beta _j}, \quad j=1,2,\ldots ,k. \end{aligned}$$

(6.9)

The inequality (6.9) is equivalent to

$$\begin{aligned} (k_{0j}-\beta _j)^2\ge (\alpha _j+v_{0j}-{\bar{v}}_{0j})^2+(k_{0j}+\beta _j)^2, \quad j=1,2,\ldots ,k. \end{aligned}$$

(6.10)

Since $k_{0j}-\beta _j\ge 0$, the above inequality indicates that

$$\begin{aligned} (\alpha _j+v_{0j}-{\bar{v}}_{0j},k_{0j}+\beta _j,k_{0j}-\beta _j)^T\in L^3, \quad j=1,2,\ldots ,k. \end{aligned}$$

(6.11)

Combining the constraints (6.8) and (6.11) with $v_{0j},{\bar{v}}_{0j},k_{0j}\ge 0,\beta _j < 0,j=1,2,\ldots ,k$, we obtain a conservative approximation for the constraint (6.4).

Similarly, (6.5) can be reformulated by

$$\begin{aligned} \begin{array}{ll} 0\ge \max \limits _{z}\ &{} \theta +\sum \limits _{j = 1}^k {z_j \alpha _j}+\sum \limits _{j = 1}^k z_j^2\beta _j \\ ~~~~~~\text {s.t.} &{} y^0(w)+y(w)^T z-d< 0,\\ &{} l\le z\le u.\\ \end{array} \end{aligned}$$

(6.12)

Since we impose that $\beta <0$, we are able to give a conservative approximation for (6.12) by using the Lagrangian dual problem. There exist $\varphi \ge 0,v_{1j},{\bar{v}}_{1j}\ge 0(j=1,2,\ldots ,k)$ such that

$$\begin{aligned}&\max \limits _{z}\ \theta +\sum \limits _{j = 1}^k {z_j \alpha _j}+\sum \limits _{j = 1}^k z_j^2\beta _j-\varphi ( y^0(w)+y(w)^T z-d) \nonumber \\&\qquad +\sum \limits _{j = 1}^k {v_{1j} (z_j-l_j)}-\sum \limits _{j = 1}^k {{\bar{v}}_{1j} (z_j-u_j)} \nonumber \\&\quad = \max \limits _{z}\ \theta -\varphi ( y^0(w)-d)+\sum \limits _{j = 1}^k {({\bar{v}}_{1j} {u_j}-v_{1j} {l_j})}\nonumber \\&\qquad +\sum \limits _{j = 1}^k {((\alpha _j-\varphi y^j(w)+v_{1j}-{\bar{v}}_{1j})z_j+\beta _jz_j^2)} \nonumber \\&\quad = \max \limits _{z}\ \theta -\varphi ( y^0(w)-d)+\sum \limits _{j = 1}^k {({\bar{v}}_{1j} {u_j}-v_{1j} {l_j})}-\sum \limits _{j = 1}^k {\frac{(\alpha _j-\varphi y^j(w)+v_{1j}-{\bar{v}}_{1j})^2}{4\beta _j}} \nonumber \\&\qquad +\sum \limits _{j = 1}^k \beta _j\left( z_j+\frac{\alpha _j-\varphi y^j(w)+v_{1j}-{\bar{v}}_{1j}}{2\beta _j}\right) ^2 \end{aligned}$$

In the same way,

$$\begin{aligned} \theta -\varphi ( y^0(w)-d)+\sum \limits _{j = 1}^k {({\bar{v}}_{1j} {u_j}-v_{1j} {l_j})}- \sum \limits _{j = 1}^k {\frac{(\alpha _j-\varphi y^j(w)+v_{1j}-{\bar{v}}_{1j})^2}{4\beta _j}} \le 0 \end{aligned}$$

(6.13)

remains a conservative approximation for

$$\begin{aligned} \max \limits _{z}\ \theta +\sum \limits _{j = 1}^k {z_j \alpha _j}+\sum \limits _{j = 1}^k z_j^2\beta _j-\varphi ( y^0(w)+y(w)^T z-d) \le 0, \end{aligned}$$

since

$$\begin{aligned} \sum \limits _{j = 1}^k \beta _j(z_j+{\frac{\alpha _j-\varphi y^j(w)+v_{1j}-{\bar{v}}_{1j}}{2\beta _j})^2} \le 0. \end{aligned}$$

By introducing $k_{1j}\ge 0(j=1,2,\ldots ,k)$, we can rewrite (6.13) as

$$\begin{aligned}&\theta -\varphi ( y^0(w)-d)+\sum \limits _{j = 1}^k {(k_{1j}+{\bar{v}}_{1j} {u_j}-v_{1j} {l_j})}\le 0, \end{aligned}$$

(6.14)

$$\begin{aligned}&k_{1j}\ge -{\frac{(\alpha _j-\varphi y^j(w)+v_{1j}-{\bar{v}}_{1j})^2}{4\beta _j}}, \quad j=1,2,\ldots ,k. \end{aligned}$$

(6.15)

we rewrite the inequality (6.15) as follows

$$\begin{aligned} (k_{1j}-\beta _j)^2\ge (\alpha _j-\varphi y^j(w)+v_{1j}-{\bar{v}}_{1j})^2+(k_{1j}+\beta _j)^2, \quad j=1,2,\ldots ,k. \end{aligned}$$

(6.16)

In view of $k_{0j}-\beta _j\ge 0$, we have that

$$\begin{aligned} (\alpha _j-\varphi y^j(w)+v_{1j}-{\bar{v}}_{1j},k_{1j}+\beta _j,k_{1j}-\beta _j)^T\in L^3, \quad j=1,2,\ldots ,k. \end{aligned}$$

(6.17)

Putting together (6.14) and (6.17) with $\varphi ,v_{1j},{\bar{v}}_{1j},k_{1j}\ge 0,\beta _j \le 0,j=1,2,\ldots ,k$, we obtain a conservative approximation for the constraint (6.5).

In summary, combining (6.2), (6.8), (6.11), (6.14) and (6.17), we obtain a safe approximation for the distributionally robust chance constrained program (2.8) as

$$\begin{aligned} \begin{array}{ll} \mathop {\max }\ &{} d \\ \text {s.t.} &{} \theta +\mu ^T\alpha +(\mu ^2+\sigma ^2)^T\beta \ge 1-\varepsilon ,\\ &{} \theta -1+\sum \limits _{j = 1}^k ({k_{0j}+{\bar{v}}_{0j}u_j-v_{0j}l_j})\le 0,\\ &{} (\alpha _j+v_{0j}-{\bar{v}}_{0j},k_{0j}+\beta _j,k_{0j}-\beta _j)^T\in L^3, \quad j=1,2,\ldots ,k, \\ &{} \theta -\varphi ( y^0(w)-d)+\sum \limits _{j = 1}^k {(k_{1j}+{\bar{v}}_{1j} {u_j}-v_{1j} {l_j})}\le 0,\\ &{} (\alpha _j-\varphi y^j(w)+v_{1j}-{\bar{v}}_{1j},k_{1j}+\beta _j,k_{1j}-\beta _j)^T\in L^3, \quad j=1,2,\ldots ,k,\\ &{} s_{m}\ge r_{n+1,m}-\sum \limits _{i = 1}^n {w_i r_{i,m} },\quad m=1,2,\ldots ,M ,\\ &{} s_{m}\ge 0, \quad m=1,2,\ldots ,M,\\ &{} \sum \limits _{m= 1}^M {q_m s_m}\le \eta ,\\ &{} w\in W,\\ &{} \theta \in R^1,\alpha ,\beta \in R^k,\varphi \ge 0,v_{0j},{\bar{v}}_{0j},v_{1j},{\bar{v}}_{1j},k_{0j},k_{1j}\ge 0,\beta _j \le 0,\quad j=1,2,\ldots ,k.\\ \end{array} \end{aligned}$$

(6.18)

Assumed that $\varphi >0$ and $\theta :=\theta /\varphi ,\alpha :=\alpha /\varphi ,\beta :=\beta /\varphi ,k :=k/\varphi ,v :=v/\varphi ,{\bar{v}} :={\bar{v}}/\varphi ,\varphi :=1/\varphi$, it comes the the theorem. $\square$