KDE distributionally robust portfolio optimization with higher moment coherent risk

Abstract

In this paper, distributionally robust mean-HMCR (higher moment coherent risk) portfolio optimization model based on kernel density estimation (KDE) and $\phi$-divergence is proposed. In order to overcome the so-called “curse of dimensionality”, we consider the one-dimensional probability distribution of the portfolio return, rather than the joint probability distribution of the assets return vector. The two issues of “the distribution dependent on the decision variables” and “the metric-based distributional uncertainty set for the continuous distribution” are effectively addressed by using the finite dimensional KDE based probability distribution. Under the mild conditions of the kernel function and $\phi$-divergence function, the tractable reformulation of the corresponding distributionally robust optimization model is derived by Fenchel’s Duality Theorem. Moreover, the convergence of optimal value and solution set of the KDE mean-HMCR distributionally robust portfolio optimization problem to those of the corresponding stochastic optimization model with the real distribution is proved. We conduct some empirical tests with the rolling horizon approach and compare the performance of the optimal portfolio strategy obtained by the proposed model to other three strategies by four performance criteria and their cumulative wealth curves. Empirical test results show that the quality of the portfolio strategy obtained by the proposed model is better at most cases. We also conduct empirically sensitivity analysis of model parameters.

Appendix

1.1 Proof of Proposition 1

Proof

(1) It follows from Assumption 1 and $\psi _p'(x)=(G_p(x))^{\frac{1}{p}-1}G_{p-1}(x)\ge 0$ that $\frac{\partial }{\partial x}\varPsi _p(x,h)=\psi _p'(\frac{x}{h})\ge 0$.

$$\begin{aligned} \frac{\partial }{\partial h}\varPsi _p(x,h)= & {} \left( G_p\left( \frac{x}{h}\right) \right) ^{\frac{1}{p}} -\frac{x}{h}\left( G_p\left( \frac{x}{h}\right) \right) ^{\frac{1}{p}-1} G_{p-1}\left( \frac{x}{h}\right) \\= & {} \left( G_p\left( \frac{x}{h}\right) \right) ^{\frac{1}{p}-1} \left( G_{p}\left( \frac{x}{h}\right) -\frac{x}{h}G_{p-1}\left( \frac{x}{h}\right) \right) \\= & {} \left( G_p\left( \frac{x}{h}\right) \right) ^{\frac{1}{p}-1} \left( \int _{-\infty }^{x/h} \left( \frac{x}{h}-t\right) ^{p-1}(-t)k(t)dt\right) \\\ge & {} \left( G_p\left( \frac{x}{h}\right) \right) ^{\frac{1}{p}-1} \left( \int _{-x/h}^{0} \left( \frac{x}{h}-t\right) ^{p-1}(-t)k(t)dt\right. \\&+ \left. \int _{0}^{x/h}\left( \frac{x}{h}-t\right) ^{p-1}(-t)k(t)dt\right) \\= & {} \left( G_p\left( \frac{x}{h}\right) \right) ^{\frac{1}{p}-1} \left( \int _{0}^{x/h} \left( \left( \frac{x}{h}+t\right) ^{p-1} -\left( \frac{x}{h}-t\right) ^{p-1}\right) tk(t)dt\right) \ge 0, \end{aligned}$$

where the last equality holds from Assumption 2 and the last inequality holds from the monotonicity of function $f(x)=x^{p-1},x\ge 0$ for $p\ge 1$. Thus $\nabla \varPsi _p(x,h)\ge 0$

(2) By $\psi _p''(x)=(p-1)\left( G_{p}(x)\right) ^{\frac{1}{p}-2}\left[ G_{p}(x)G_{p-2}(x)-(G_{p-1}(x))^2\right] \ge 0,$ where the inequality holds from Cauchy-Schwarz inequality, and the convexity-preserving property of the operation of right multiplication (see Rockafellar 1997, p. 35), we know that $\varPsi _p(x,h)$ is a jointly convex function.

(3) For $x<0$, we have $\lim \limits _{h\rightarrow 0^+}(\varPsi _p(x,h))^p =x^p\lim \limits _{h\rightarrow 0^+}\int _{-\infty }^{x/h}\left( 1-\frac{th}{x}\right) ^pk(t)dt=0.$ For $x=0$, we have $\lim \limits _{h\rightarrow 0^+}(\varPsi _p(0,h))^p=\lim \limits _{h\rightarrow 0^+}h^p\int _{-\infty }^{0}(-t)^pk(t)dt=0.$ For $x>0$, we have

$$\begin{aligned} \lim _{h\rightarrow 0^+}(\varPsi _p(x,h))^p= & {} x^p\lim _{h\rightarrow 0^+}\int _{-\infty }^{x/h}\left( 1-\frac{th}{x}\right) ^pk(t)dt \\= & {} x^p\lim _{h\rightarrow 0^+}\left( \int _{-\infty }^{-x/h}\left( 1-\frac{th}{x}\right) ^pk(t)dt+ \int _{-x/h}^{-x/h}\left( 1-\frac{th}{x}\right) ^pk(t)dt\right) \\= & {} x^p\lim _{h\rightarrow 0^+}\Big (\int _{-\infty }^{-x/h}\left( 1-\frac{th}{x}\right) ^pk(t)dt\\&+ \int _{-x/h}^{-x/h}\left[ \sum _{n=0}^\infty \frac{p(p-1)\cdots (p-n+1)}{n!} \left( \frac{th}{x}\right) ^n\right] k(t)dt\Big )\\= & {} x^p\lim _{h\rightarrow 0^+}\sum _{n=0}^\infty \frac{p(p-1)\cdots (p-n+1)h^n}{n!x^n}\int _{-x/h}^{-x/h}t^nk(t)dt\\= & {} x^p+x^p\lim _{h\rightarrow 0^+}\sum _{n=1}^\infty \frac{p(p-1)\cdots (p-n+1)h^n}{n!x^n}\int _{-x/h}^{-x/h}t^nk(t)dt=x^p, \end{aligned}$$

where the third equality holds from Maclaurin series of $f(x)=(1+x)^p,x\in (-1,1),~p\ge 1$ and the last equality holds from Assumption 2. Thus $\lim \limits _{h\rightarrow 0^+}\varPsi _p(x,h)=\lim \limits _{h\rightarrow 0^+}h\psi _p(h^{-1}x)=[x]^+$. It follows from the definition of recession function (see Rockafellar 1997, p. 66-67) that $[x]^+$ is the recession function of $\psi _p(x)$. Moreover, for $x\ne 0$, by using L’Hospital’s rule and Assumption 2, we have

$$\begin{aligned}&\lim _{h\rightarrow 0^+}\frac{\varPsi _p(x,h)-[x]^+}{h}\\= & {} \lim _{h\rightarrow 0^+}\frac{x\left[ \int _{-\infty }^{x/h}\left( 1-\frac{th}{x}\right) ^pk(t)dt\right] ^{1/p}-[x]^+}{h} \\= & {} \lim _{h\rightarrow 0^+}\left[ \int _{-\infty }^{x/h}\left( 1-\frac{th}{x}\right) ^pk(t)dt\right] ^{1/p-1} \int _{-\infty }^{x/h}\left( 1-\frac{th}{x}\right) ^{p-1}tk(t)dt=0. \end{aligned}$$

For $x=0$, it follows from Assumption 2 that

$$\begin{aligned} \lim _{h\rightarrow 0^+}\frac{\varPsi _p(0,h)}{h} = \lim _{h\rightarrow 0^+}\frac{\left[ h^p\int _{-\infty }^{0}(-t)^pk(t)dt\right] ^{1/p}}{h} = \left( \int _{-\infty }^{0}(-t)^pk(t)dt\right) ^{1/p}\ne 0. \end{aligned}$$

$\square$

1.2 Proof of Theorem 1

Proof

Due to the fact that $f(\varvec{x})=\Vert \varvec{x}\Vert _p, \varvec{x}\in \mathbb {R}_+^n$ is nondecreasing in $x_i$ for $i=1,\ldots ,n$, problem (6) is equivalent to

$$\begin{aligned} \begin{array}{cl} \min &{}\rho \alpha -\sum _{i=1}^T\lambda _i\varvec{r}_i^\mathsf {T}\varvec{w}+ \frac{\rho \nu }{1-\gamma }\\ \mathrm {s.t.} &{}\Vert \varvec{y}\Vert _p\le \nu ,\\ &{} \lambda _i^{1/p}\varPsi _p(-\varvec{r}_i^\mathsf {T}\varvec{w}-\alpha ,h)\le y_i,~~i=1,\ldots ,T,\\ &{} \varvec{w}\in \mathcal {W},\alpha \in \mathbb {R},\nu \in \mathbb {R},\varvec{y}\in \mathbb {R}_+^T,\\ \end{array} \end{aligned}$$

where the first constraint $\Vert \varvec{y}\Vert _p\le \nu$ can be reformulated as $\left( \varvec{y},\nu \right) \in C_p^T$ and $C_p^T=\{(\varvec{x},t)\in \mathbb {R}^T_+\times \mathbb {R}_+:\Vert \varvec{x}\Vert _p\le t\}$ is the T-dimensional p-order cone within the nonnegative orthant (see Krokhmal 2007). Since $f(\varvec{x})=\Vert \varvec{x}\Vert _p, \varvec{x}\in \mathbb {R}_+^n$ is a convex function, we know that the first constraint is a convex constraint. By Proposition 1, (1)-(2) and Lemma 1 ($m=2$), we know $\varPsi \left( u_i(\varvec{w}, \alpha ),h\right)$ is convex in $\varvec{w}$ and $\alpha$ for $i=1,\ldots , T$, where $u_i(\varvec{w},\alpha ):= -\varvec{r}_i^\mathsf {T}\varvec{w}-\alpha$ is linear in $\varvec{w}$ and $\alpha$ for $i=1,\ldots ,T$, thus the conclusion holds. $\square$

1.3 Proof of Corollary 1

Proof

The proof of this result is quite similar to that of Theorem 1 due to the fact that $h=c\Vert MR\varvec{w}\Vert$ is convex and so is omitted. $\square$

1.4 Proof of Lemma 3

Proof

By $\frac{\partial ^2 f}{\partial x^2}=p(p-1)y^{-q}x^{p-2}$, $\frac{\partial ^2 f}{\partial y^2}=q(q+1)y^{-q-2}x^{p}$ and $\frac{\partial ^2 f}{\partial x\partial x} =-pqy^{-q-1}x^{p-1}$, we have the Hessian of f(x, y)

$$\begin{aligned} \left( \begin{array}{cc} p(p-1)y^{-q}x^{p-2} &{} -pqy^{-q-1}x^{p-1} \\ -pqy^{-q-1}x^{p-1} &{} q(q+1)y^{-q-2}x^{p} \\ \end{array} \right) \succeq 0 \end{aligned}$$

and $\frac{\partial f}{\partial x}=py^{-q}x^{p-1}\ge 0$, thus the conclusion holds. $\square$

1.5 Proof of Corollary 2

Proof

For $p=\frac{t}{s}$, the first group of constraints in problem (15) can be reformulated as

$$\begin{aligned} \left\{ \begin{aligned}&z_i^t\le y^{t-s}(v_i+\varvec{r}_i^\mathsf {T}\varvec{w})^s,~~ i=1,\ldots , T,\\&v_i+\varvec{r}_i^\mathsf {T}\varvec{w}\ge 0,i=1,\ldots , T,~~i=1,\cdots ,T,\\&\varPsi _p\left( -\varvec{r}_i^\mathsf {T}\varvec{w}-\alpha ,h\right) \le z_i,~~i=1,\ldots , T,\\&\varvec{w}\in \mathcal {W},\alpha \in \mathbb {R}, u\in \mathbb {R}_+,\eta \in \mathbb {R},y\in \mathbb {R}_+,\varvec{v}\in \mathbb {R}^T,\varvec{z}\in \mathbb {R}^T. \end{aligned} \right. \end{aligned}$$

Thus the conclusion holds. $\square$

1.6 Proof of Proposition 2

Proof

We only prove problem (12) is equivalent to $\min \limits _{( w,\alpha ) \in \mathcal {W}\times \mathcal {D}}\max \limits _{\lambda \in \varLambda _\phi ^\tau } \hat{F}^{\gamma ,p,\rho }_{T}(\varvec{w},\alpha ;\varvec{\lambda })$, for some bounded and closed interval $\mathcal {D}\subset \mathbb {R}$, and omit the proof of the similar and simpler result associated with problem (3). From Assumption 3 and Proposition 1, (3), we have

$$\begin{aligned} \hat{F}^{\gamma ,p,\rho }_{T}(\varvec{w},\alpha ;\varvec{\lambda })= \rho \alpha -\sum _{i=1}^T \lambda _i\varvec{r}_i^\mathsf {T}\varvec{w} +\frac{\rho }{1-\gamma }\left( \sum _{i=1}^T \lambda _i\left( [-\varvec{r}_i^\mathsf {T}\varvec{w}-\alpha ]^+\right) ^p\right) ^{1/p}+O(h). \end{aligned}$$

It follows from the boundedness of $\varOmega$ and $\mathcal {W}$ that, for $i=1,\ldots ,T$, $\varvec{r}_i^\mathsf {T}\varvec{w}$ is bounded, and there exists a positive number B such that $\varvec{r}_i^\mathsf {T}\varvec{w}\le B$. Thus, we obtain

$$\begin{aligned} \lim _{\alpha \rightarrow +\infty }\max _{\lambda \in \varLambda _\phi ^\tau } \hat{F}^{\gamma ,p,\rho }_{T}(\varvec{w},\alpha ;\varvec{\lambda })= & {} \lim _{\alpha \rightarrow +\infty }\max _{\lambda \in \varLambda _\phi ^\tau }\left\{ \rho \alpha -\sum _{i=1}^T\lambda _i \varvec{r}_i^\mathsf {T}\varvec{w}+O(h)\right\} \\\ge & {} \lim _{\alpha \rightarrow +\infty }\left\{ \rho \alpha -B+O(h)\right\} = +\infty \end{aligned}$$

and

$$\begin{aligned} \lim _{\alpha \rightarrow -\infty }\max _{\lambda \in \varLambda _\phi ^\tau } \hat{F}^{\gamma ,p,\rho }_{T}(\varvec{w},\alpha ;\varvec{\lambda })= & {} \lim _{\alpha \rightarrow -\infty }\max _{\lambda \in \varLambda _\phi ^\tau }\left\{ -\frac{\rho \gamma \alpha }{1-\gamma }-\sum _{i=1}^T\lambda _i \varvec{r}_i^\mathsf {T}\varvec{w}+O(h)\right\} \\\ge & {} \lim _{\alpha \rightarrow -\infty }\left\{ -\frac{\rho \alpha \gamma }{1-\gamma }-B+O(h)\right\} = +\infty . \end{aligned}$$

Therefore there exists a bounded and closed interval $\mathcal {D}\subset \mathbb {R}$ such that the minimizer of problem (12) falls into the compact set $\mathcal {W}\times \mathcal {D}$, that is, problem (12) is equivalent to $\min \limits _{( w,\alpha ) \in \mathcal {W}\times \mathcal {D}}\max \limits _{\lambda \in \varLambda _\phi ^\tau } \hat{F}^{\gamma ,p,\rho }_{T}(\varvec{w},\alpha ;\varvec{\lambda })$. $\square$

1.7 Proof of Proposition 3

Proof

Denote the objective function of problem (14) by $H^{\gamma ,p,\rho }_{T}(\varvec{w}, \alpha ; u,\eta ,z)$ and using Theorem 2, we have

$$\begin{aligned} \max _{\lambda \in \varLambda _\phi ^\tau } \hat{F}^{\gamma ,p,\rho }_{T}(\varvec{w},\alpha ;\varvec{\lambda }) =\min _{u\ge 0,z\ge 0,\eta }H^{\gamma ,p,\rho }_{T}(\varvec{w}, \alpha ; u,\eta ,z). \end{aligned}$$

Therefore, if the following two inequalities can be proved, the conclusion holds.

$$\begin{aligned} \lim _{\tau \rightarrow 0}\max _{\varvec{\lambda }\in \varLambda _{\phi }^\tau } \hat{F}^{\gamma ,\rho }_{T}(\varvec{w},\alpha ;\varvec{\lambda })\ge F^{p,\rho }_{\gamma }(\varvec{w},\alpha ) \end{aligned}$$

(34)

and

$$\begin{aligned} \lim _{\tau \rightarrow 0}\min _{u\ge 0,z\ge 0,\eta } H^{\gamma ,p,\rho }_{T}(\varvec{w}, \alpha ; u,\eta ,z) \le F^{p,\rho }_{\gamma }(\varvec{w},\alpha ). \end{aligned}$$

(35)

It is easy to show that

$$\begin{aligned} \lim _{T\rightarrow \infty }\max _{\varvec{\lambda }\in \varLambda _{\phi }^{\tau }} \hat{F}^{\gamma ,p,\rho }_{T}(\varvec{w},\alpha ;\varvec{\lambda }) \ge \lim _{T\rightarrow \infty }\hat{F}^{\gamma ,p,\rho }_{T}(\varvec{w},\alpha ;\varvec{\lambda }^0) =\lim _{T\rightarrow \infty }\tilde{F}^{\gamma ,p,\rho }_{T}(\varvec{w},\alpha ) \overset{\mathrm {w.p.1}}{=}F^{p,\rho }_{\gamma }(\varvec{w},\alpha ), \end{aligned}$$

where the first equality holds from Assumption 3 and Proposition 1, (3) and the last equality holds from Shapiro et al. (2014, Theorem 5.3). On the other hand,

$$\begin{aligned}&\lim _{T\rightarrow \infty }\min _{u\ge 0,z\ge 0,\eta } H^{\gamma ,p,\rho }_{T}(\varvec{w}, \alpha ; u,\eta ,z) \nonumber \\&\quad \le \min _{u\ge 0,z\ge 0, \eta }\limsup _{T\rightarrow \infty } H^{\gamma ,p,\rho }_{T}(\varvec{w}, \alpha ; u,\eta ,z) \nonumber \\&\quad \overset{\mathrm {w.p.1}}{=} \min _{u\ge 0,z\ge 0,\eta } \Bigg \{\eta + \mathbb {E}_P\left[ u\phi ^*\left( \frac{ z\left( \left[ -\varvec{\xi }^\mathsf {T}\varvec{w}-\alpha \right] ^+\right) ^p-\varvec{\xi }^\mathsf {T}\varvec{w} -\eta }{u}\right) \right] +\rho \alpha \nonumber \\&\qquad -\left( \frac{\rho }{p(1-\gamma )}\right) ^{\frac{p}{p-1}}(1-p)z^{\frac{1}{1-p}} \Bigg \} \end{aligned}$$

(36)

$$\begin{aligned}&\le \min _{z\ge 0} \Bigg \{ \rho \alpha -\frac{1}{T}\sum _{i=1}^T\varvec{r}_i^\mathsf {T}\varvec{w}+z\mathbb {E}_P\left[ \left( \left[ -\varvec{\xi }^\mathsf {T}\varvec{w}-\alpha \right] ^+\right) ^p\right] \nonumber \\&\qquad -\left( \frac{\rho }{p(1-\gamma )}\right) ^{\frac{p}{p-1}}(1-p)z^{\frac{1}{1-p}}\Bigg \} \end{aligned}$$

(37)

$$\begin{aligned}&\quad = \rho \alpha -\mathbb {E}_P\left[ \varvec{\xi }^\mathsf {T}\varvec{w}\right] +\frac{\rho }{(1-\gamma )}\mathbb {E}_P\left[ \left( \left( \left[ -\varvec{\xi }^\mathsf {T}\varvec{w}-\alpha \right] ^+\right) ^p\right) ^{1/p}\right] \nonumber \\&\quad = F^{p,\rho }_{\gamma }(\varvec{w},\alpha ), \end{aligned}$$

(38)

where (36) holds from Assumptions 3–4, Proposition 1, (3) and Shapiro et al. (2014, Theorem 5.3), and (37) holds from Assumption 5. Moreover, (38) holds from the fact that the concave conjugate function of $f(x)=\frac{x^p}{p}(x\ge 0,~0<p<1)$ is $f_*(y)=\frac{y^q}{q}(y\ge 0)$, where $1/p+1/q=1$ (see Ben-Tal et al. 2015). Combining (34) and (35), for given $\varvec{w}$ and $\alpha$, we have $\lim _{T\rightarrow \infty }\max _{\varvec{\lambda }\in \varLambda _{\phi }^\tau } \hat{F}^{\gamma ,p,\rho }_{T}(\varvec{w},\alpha ;\varvec{\lambda })\overset{\mathrm {w.p.1}}{=} F^{p,\rho }_{\gamma }(\varvec{w},\alpha )$. $\square$