Integrated Portfolio Risk Measure: Estimation and Asymptotics of Multivariate Geometric Quantiles

Abstract

Portfolio management and integrated risk management are more commonly applied toward enterprise risk management, requiring multivariate risk measures that capture the dependence among many risk factors. In this paper we propose the non-parametric estimator for multivariate value at risk (MVaR) and multivariate average value at risk (MAVaR) based on the multivariate geometric quantile approach and derive the symptotic properties of the proposed estimators for MVaR. We also present their performances under both simulated data and high-frequency financial data from the New York Stock Exchange. In addition, we compare our method with the delta normal approach and order statistics approach. The overall empirical results confirm that the multivariate geometric quantile approach significantly improves the risk management performance of MVaR and MAVaR.

Appendices

Appendix A

We note that $\varvec{e}_{[j_N]}=(e_{1, [j_N]}, \ldots , e_{p, [j_N]})^T$ is an order statistic of the residuals in the linear model; see Eq. (5). When N goes to infinity, the jth sample quantile is asymptotically normal with mean $\varvec{q}$, which implies:

$$\begin{aligned} \int _{-\infty }^{q_i}f(e_i)de_i=\lambda _i, \end{aligned}$$

where $f(e_i)$ is a marginal probability density function of $e_i$ and $0<\lambda _i<1$. We denote the cumulative distribution function of $e_i$ as $F(e_i)$, and the empirical distribution as:

$$\begin{aligned} \mathbb {F}_N(e_{i,[j_N]})=\frac{1}{N}\sum _{i=1}^{N}1\{e_{i}\le e_{i,[j_N]}\}, \end{aligned}$$

where $N\mathbb {F}_N(e_{i,[j_N]})$ is binomially distributed with mean $NF(q_{ij})$. Based on the central limit theorem we have:

$$\begin{aligned} \sqrt{N}(\mathbb {F}_N(e_{i,[j_N]})-F(q_{ij}))\rightarrow N(0, F(q_{ij})(1-F(q_{ij}))). \end{aligned}$$

With the Glivenko–Cantelli theorem and the strong law of large numbers, both $\mathbb {F}_N(t) \overset{a.s.}{\rightarrow }F(t)$ and $\mathbb {F}_N(t-) \overset{a.s.}{\rightarrow }F(t-)$ for every t. Given a fixed $\varepsilon >0$, there exists a partition $-\infty =t_{0}<t_{1}<\ldots <t_{k}=\infty$ such that $F(t_\ell -)-F(t_{\ell -1})$ for every $\ell$. Therefore, for $t_{\ell -1}\le t<t_{\ell }$, we obtain:

$$\begin{aligned} \mathbb {F}_N(t)-F(t)\le & {} \mathbb {F}_N(t_\ell -)-F(t_\ell -)+\varepsilon \end{aligned}$$

and

$$\begin{aligned} \mathbb {F}_N(t)-F(t)\ge & {} \mathbb {F}_N(t_{\ell -1})-F(t_{\ell -1})-\varepsilon . \end{aligned}$$

The convergence of $\mathbb {F}_N(t)$ and $\mathbb {F}_N(t-)$ for every t is certainly uniform for t in the finite set $\{t_1,\ldots ,t_{k-1}\}$. We can almost surely conclude that $\lim \sup \Vert \mathbb {F}_N-F\Vert _{\infty }\le \varepsilon$. This is true for every $\varepsilon >0$, and hence the limit superior is zero.

With the Donsker theorem and the extension of the central limit theorem to a “uniform” or “functional”, the empirical process $G_{n,F}=\sqrt{N}(\mathbb {F}_{N}-F)$ converges in distribution in the space $D[-\infty , \infty ]$ to the F-Brownian bridge process $G_F=G_\lambda \circ F$, whose marginal distributions are zero mean normal distributions with covariance:

$$\begin{aligned} E(G_F(t_i)G_F(t_j))=F(t_i\wedge t_j)-F(t_i)F(t_j). \end{aligned}$$

The sample paths of the limit process are continuous at the points where F is continuous. The quantile function $F\mapsto F^{-1}(\lambda )$ is tangentially Hadamard-differentiable to the range of the limit process. Letting $F^{-1}=q$ be a quantile function by the functional delta method, the sequence $\sqrt{N}(e_{[j_N]_{i}}-q_i)$ is asymptotically equivalent to the derivative of the quantile function evaluated as $G_{N,F}$; that is, to $-G_{N,F}(q)/f(q)$. This is the first assertion. Next, the asymptotic normality of the sequence $\sqrt{N}(e_{[j_N]_{i}}-q_i)$ follows from the central limit theorem. We thus have:

$$\begin{aligned} \sqrt{N}(e_{[j_N]_{i}}-q_i)=-\frac{1}{N}\sum _{i=1}^{N}\frac{I(e_i\le q_i)-\lambda _i}{f(q_i)}+o(1). \end{aligned}$$

Consequently, the sequence $\sqrt{N}(e_{[j_N]_{i}}-q_i)$ is asymptotically normal with mean zero, variance:

$$\begin{aligned} \sigma _{i}^2=\frac{\lambda _i(1-\lambda _i)}{(f(q_i))^2}, \end{aligned}$$

and covariance:

$$\begin{aligned} \sigma _{ij}=\frac{\lambda _i(1-\lambda _j)}{f(q_i)f(q_j)}. \end{aligned}$$

Hence, the asymptotic covariance matrix of $\varvec{e}_{[j_N]}$ is:

$$\begin{aligned} \varvec{\lambda }(1-\varvec{\lambda })^T\Big (f_{\varvec{e}}(\varvec{q})f^T_{\varvec{e}}(\varvec{q})\Big )^{-1}. \end{aligned}$$

Appendix B

Suppose the following conditions are satisfied.

• (i) $E\Vert g(\alpha , Z)\Vert <\infty$ for each $\alpha$ in a neighborhood of $\alpha _*$.

• (ii) $Q(\alpha )$ is twice differentiable at $\alpha _*$ and $D^2Q(\alpha _*)$ is positive definite.

$Q(\alpha )=E(f(\alpha ,Z)), Q(\alpha _*)=\min _{\alpha }Q(\alpha )$, $f(\alpha ,z)$ is a real function defined for $\alpha \in \mathbb {R}^p$ and $z\in \varvec{Z}$, and $g(\alpha ,z)$ is a subgradient of $f(\alpha ,z)$, such that:

$$\begin{aligned} f(\alpha , z)+(\beta -\alpha )^Tg(\alpha ,z)<f(\beta ,z), \end{aligned}$$

with $\beta \in \mathbb {R}^p$. We denote the gradient of $Q(\alpha )$ by $DQ(\alpha )$ and let $D^2Q(\alpha )$ stand for the second derivative. Following the notation of Niemiro (1992), we get:

$$\begin{aligned} S_N=\sum _{i=1}^Ng(\alpha _*,\varvec{Z}_i),~\text {and}~H=D^2Q(\alpha _*). \end{aligned}$$

With conditions (i) and (ii), we have:

$$\begin{aligned} \sqrt{N}(\alpha _n-\alpha _*)=-H^{-1}\frac{S_n}{\sqrt{N}}+o(1),\quad N\rightarrow \infty , \end{aligned}$$

in probability, and asymptotic normality of $\alpha _N$ clearly follows the central limit theorem:

$$\begin{aligned} \sqrt{N}(\alpha _N-\alpha _*)\rightarrow N(0,H^{-1}VH^{-1}),\quad N\rightarrow \infty , \end{aligned}$$

where V is the covariance matrix:

$$\begin{aligned} V=var(g(\alpha _*,Z)). \end{aligned}$$

Given $\alpha _N=\hat{\varvec{q}}_N(u)$ and $\alpha _*=\varvec{q}(u)$, we have:

$$\begin{aligned} \hat{\varvec{q}}_N(u)-\varvec{q}(u)=N^{-1}[D_1\{q(u)\}]^{-1}\times \sum _{i=1}^{N}[\Vert \varvec{y}_i-\varvec{q}(u)\Vert ^{-1}\{\varvec{y}_i-\varvec{q}(u)\}+\varvec{u}]+\varvec{R}_N(\varvec{u}), \end{aligned}$$

where as N tends to infinity, $\varvec{R}_N(\varvec{u})$ is almost surely $O(\log (N)/N)$ if $p\le 3$, and when $d=2, \varvec{R}_N(\varvec{u})$ is almost surely $o(N^{-\gamma })$ for any fixed $\gamma$ such that $0<\gamma <1$. Let $\varvec{u}_1,\ldots ,\varvec{u}_l$ be points in the open unit ball $B^{(p)}$, where l is a fixed positive integer. Following Niemiro (1992), the joint asymptotic distribution of centered and normalized geometric quantiles

$$\begin{aligned} N^{1/2}\{\hat{\varvec{q}}_N(\varvec{u}_1)-\varvec{q}(\varvec{u}_1)\}, \ldots , N^{1/2}\{\hat{\varvec{q}}_N(\varvec{u}_l)-\varvec{q}(\varvec{u}_l)\} \end{aligned}$$

will have normal distribution with mean zero and asymptotic covariance matrix

$$\begin{aligned}{}[D_1\{\varvec{q}(\varvec{u}_r)\}]^{-1}[D_2{\varvec{q}(\varvec{u}_r), \varvec{q}(\varvec{u}_s), \varvec{u}_r, \varvec{u}_s}][D_1\{\varvec{q}(\varvec{u}_s)\}]^{-1}, \end{aligned}$$

where

$$\begin{aligned} D_1(\varvec{q})= & {} \frac{\psi (\varvec{u},\varvec{y}-\varvec{q})}{\partial (\varvec{y}-\varvec{q})\partial (\varvec{y}-\varvec{q})^T} \nonumber \\= & {} E[ \Vert \varvec{Y}-\varvec{q}\Vert ^ {-1}\{ \varvec{I}_p-\Vert \varvec{Y}- \varvec{q} \Vert ^{-2}(\varvec{Y}-\varvec{q})(\varvec{Y}-\varvec{q})^T \} ], \end{aligned}$$

(12)

and

$$\begin{aligned} D_2(\varvec{q}_1, \varvec{q}_2,\varvec{u},\varvec{v})= & {} \text {cov}\Big (\frac{\psi (\varvec{u},\varvec{y}-\varvec{q})}{\partial (\varvec{y}-\varvec{q})^T}\Big )\nonumber \\= & {} E[\{\Vert \varvec{Y}-\varvec{q}_1\Vert ^{-1}(\varvec{Y}-\varvec{q}_1)+\varvec{u}\}\{\Vert \varvec{Y}-\varvec{q}_2\Vert ^{-1}(\varvec{Y}-\varvec{q}_2)+\varvec{v}\}],\nonumber \\ \end{aligned}$$

(13)

for $\varvec{q}_1, \varvec{q}_2 \in \mathbb {R}^p$, and $\varvec{u}, \varvec{v} \in B^{(p)}$.