Asymptotic behavior of the estimated weights and of the estimated performance measures of the minimum VaR and the minimum CVaR optimal portfolios for dependent data

Abstract

In this paper we derive the asymptotic distributions of the estimated weights and of estimated performance measures of the minimum value-at-risk portfolio and of the minimum conditional value-at-risk portfolio assuming that the asset returns follow a strictly stationary process. It is proved that the estimated weights as well as the estimated performance measures are asymptotically multivariate normally distributed. We also present an asymptotic test for the weights and a joint test for the characteristics of both portfolios. Moreover, the asymptotic densities of the estimated performance measures are compared with the corresponding exact densities. It is shown that the asymptotic approximation performs well even for the moderate sample size.

Appendix

In this section, the proofs of the theorems are given. First, we formulate an important lemma which is used for proving Theorems 1 and 2. This lemma extends the asymptotic results of Bodnar et al. (2009) obtained for a gaussian process to an arbitrary strictly stationary process.

Lemma 1

Let $\{ \mathbf{X }_t \}$ be a strictly stationary process with mean ${\varvec{\mu }}$ and cross-covariance matrix ${\varvec{\Gamma }}(h)$ at lag $h$. We assume that the infinite sum in (14) converges. Then it follows that

$$\begin{aligned} \sqrt{n}\left( \left( \begin{array}{c} \hat{R}_{GMV} \\ \hat{V}_{GMV}\\ \hat{s}\end{array}\right) -\left( \begin{array}{c} R_{GMV} \\ V_{GMV}\\ s\end{array}\right) \right) \stackrel{d}{\rightarrow } \mathcal N \left( \left( \begin{array}{c} 0 \\ 0\\ 0\end{array}\right) ,\left( \begin{array}{ccc} \sigma ^2_1 &{} \sigma _{12} &{} \sigma _{13}\\ \sigma _{12} &{} \sigma ^2_2 &{} \sigma _{23} \\ \sigma _{13} &{} \sigma _{23} &{} \sigma ^2_3 \end{array}\right) \right) \!, \end{aligned}$$

where

$$\begin{aligned} \sigma ^2_1&= ((\partial R_{GMV}/\partial {\varvec{\mu }})^{\prime } \; (\partial R_{GMV}/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime }) {\varvec{\Omega }}((\partial R_{GMV}/\partial {\varvec{\mu }})^{\prime } \; (\partial R_{GMV}/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime })^\prime \,,\\ \sigma _{12}&= ((\partial R_{GMV}/\partial {\varvec{\mu }})^{\prime } \; (\partial R_{GMV}/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime }) {\varvec{\Omega }}((\partial V_{GMV}/\partial {\varvec{\mu }})^{\prime } \; (\partial V_{GMV}/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime })^\prime \,,\\ \sigma _{13}&= ((\partial R_{GMV}/\partial {\varvec{\mu }})^{\prime } \; (\partial R_{GMV}/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime }) {\varvec{\Omega }}((\partial s/\partial {\varvec{\mu }})^{\prime } \; (\partial s/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime })^\prime \,,\\ \sigma ^2_2&= ((\partial V_{GMV}/\partial {\varvec{\mu }})^{\prime } \; (\partial V_{GMV}/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime }) {\varvec{\Omega }}((\partial V_{GMV}/\partial {\varvec{\mu }})^{\prime } \; (\partial V_{GMV}/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime })^\prime \,,\\ \sigma _{23}&= ((\partial V_{GMV}/\partial {\varvec{\mu }})^{\prime } \; (\partial V_{GMV}/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime }) {\varvec{\Omega }}((\partial s/\partial {\varvec{\mu }})^{\prime } \; (\partial s/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime })^\prime \,,\\ \sigma ^2_3&= ((\partial s/\partial {\varvec{\mu }})^{\prime } \; (\partial s/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime }) {\varvec{\Omega }}((\partial s/\partial {\varvec{\mu }})^{\prime } \; (\partial s/\partial ({\text{ vech}}{\varvec{\Gamma }}(0)))^{\prime })^\prime \end{aligned}$$

with

$$\begin{aligned} \frac{\partial R_{GMV}}{\partial {\varvec{\mu }}}&= \mathbf{w }_{GMV}\,, \quad \frac{\partial V_{GMV}}{\partial {\varvec{\mu }}}=\mathbf 0 \,,\quad \frac{\partial s}{\partial {\varvec{\mu }}}=2{\varvec{R}}{\varvec{\mu }}\,,\\ \frac{\partial R_{GMV}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}&= V_{GMV}\frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\Big ((\mathbf{1 }\otimes {\varvec{\mu }})-R_{GMV}(\mathbf{1 }\otimes \mathbf{1 })\Big ),\\ \frac{\partial V_{GMV}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}&= -V_{GMV}^2\frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}(\mathbf{1 }\otimes \mathbf{1 }),\\ \frac{\partial s}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}&= \frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\Big (({\varvec{\mu }}\otimes {\varvec{\mu }})+R_{GMV}^2(\mathbf{1 }\otimes \mathbf{1 })-2R_{GMV}(\mathbf{1 }\otimes {\varvec{\mu }})\Big ). \end{aligned}$$

Proof of Lemma 1

The proofs of Lemma 1 follows from the proof of Theorem 1 of Bodnar et al. (2009) and the statement that ${\hat{\varvec{\theta }}}=({\hat{\varvec{\mu }}}^\prime , {\text{ vech}}(\hat{\varvec{\Gamma }}(0))^\prime )^\prime$ is asymptotically normally distributed with mean vector ${\varvec{\theta }}$ and covariance matrix ${\varvec{\Omega }}/n$ (cf. (Brillinger (1981), Theorem 4.4.1)).

Proof of Theorem 1

It follows that

$$\begin{aligned} \mathbf{G }=\left( \begin{array}{cc} \left( \partial \mathbf{w }^{\prime }_{VaR}/\partial {\varvec{\mu }}\right) ^{\prime } &{} \left( \partial \mathbf{w }^{\prime }_{VaR}/\partial ({\text{ vech}}{\varvec{\Gamma }}(0))\right) ^{\prime }\\ \left( \partial s/\partial {\varvec{\mu }}\right) ^{\prime } &{} \left( \partial s/\partial ({\text{ vech}}{\varvec{\Gamma }}(0))\right) ^{\prime }\\ \end{array} \right) \!, \end{aligned}$$

where

$$\begin{aligned} \frac{\partial \mathbf{w }^{\prime }_{VaR}}{\partial {\varvec{\mu }}}&= \partial \mathbf{w }^{\prime }_{GMV}/\partial {\varvec{\mu }}+\partial \left( \frac{\sqrt{V_{GMV}}}{\sqrt{d_{1-\alpha }^2-s}}{\varvec{R}}{\varvec{\mu }}\right) ^{\prime }\Big /\partial {\varvec{\mu }}\\&= \mathbf 0 +\sqrt{V_{GMV}}\partial \left( \frac{{\varvec{\mu }}^{\prime }{\varvec{R}}}{\sqrt{d_{1-\alpha }^2-s}}\right) \Big /\partial {\varvec{\mu }}\\&= \sqrt{V_{GMV}}\frac{\sqrt{d_{1-\alpha }^2-s}\frac{\partial {\varvec{\mu }}^{\prime }{\varvec{R}}}{\partial {\varvec{\mu }}}- \frac{\partial \sqrt{d_{1-\alpha }^2-s}}{\partial {\varvec{\mu }}}{\varvec{\mu }}^{\prime }{\varvec{R}}}{d_{1-\alpha }^2-s}\\&= \frac{\sqrt{V_{GMV}}}{\sqrt{d_{1-\alpha }^2-s}}{\varvec{R}}+\frac{\sqrt{V_{GMV}}}{\left( d_{1-\alpha }^2-s\right) ^{3/2}}{\varvec{R}}{\varvec{\mu }}{\varvec{\mu }}^{\prime }{\varvec{R}}\end{aligned}$$

and

$$\begin{aligned} \frac{\partial s}{\partial {\varvec{\mu }}}&= \partial {\varvec{\mu }}^{\prime }{\varvec{R}}{\varvec{\mu }}/\partial {\varvec{\mu }}=2{\varvec{R}}{\varvec{\mu }}\,. \end{aligned}$$

Next, we calculate $\partial s /\partial ({\text{ vech}}{\varvec{\Gamma }}(0))$ and $\partial \mathbf{w }^{\prime }_{VaR}/\partial ({\text{ vech}}{\varvec{\Gamma }}(0))$. For this calculation the following identities are applied

$$\begin{aligned} \frac{\partial V_{GMV}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}&= -\frac{\partial (\mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 })}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\frac{1}{(\mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 })^2}=-V_{GMV}^2\frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}(\mathbf{1 }\otimes \mathbf{1 }),\\ \frac{\partial \mathbf{w }^{\prime }_{GMV}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}&= \frac{1}{\mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 }} \frac{\partial \mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}-\frac{1}{(\mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 })^2} \frac{\partial \mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\\&= \frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\Big ( V_{GMV}(\mathbf{1 }\otimes \mathbf{I }_k)-V_{GMV}(\mathbf{1 }\otimes \mathbf{1 })\mathbf{w }^{\prime }_{GMV}\Big ),\\ \frac{\partial ({\varvec{\mu }}^{\prime }{\varvec{R}})}{\partial ({\text{ vech}}{\varvec{\Gamma }}(0))}&= \frac{\partial {\varvec{\mu }}^{\prime }{\varvec{\Gamma }}(0)^{-1}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}- \frac{\partial \left( \frac{{\varvec{\mu }}^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 }\mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}}{\mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 }}\right) }{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}= \frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}({\varvec{\mu }}\otimes \mathbf{I }_k)\\&-\frac{1}{\mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 }}\frac{\partial {\varvec{\mu }}^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 }\mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\\&\quad +\frac{{\varvec{\mu }}^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 }}{(\mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 })^2} \frac{\partial \mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)} \mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}\\&= \frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\Big (({\varvec{\mu }}\otimes \mathbf{I }_k) +R_{GMV}(\mathbf{1 }\otimes \mathbf{1 })\mathbf{w }_{GMV}^{\prime }\Big )\\&\quad -\frac{\partial {\varvec{\mu }}^{\prime }{\varvec{\Gamma }}(0)^{-1}\mathbf{1 }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\mathbf{w }_{GMV}^{\prime } -R_{GMV}\frac{\partial \mathbf{1 }^{\prime }{\varvec{\Gamma }}(0)^{-1}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\\&= \frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\Big (({\varvec{\mu }}\otimes \mathbf{I }_k) +R_{GMV}(\mathbf{1 }\otimes \mathbf{1 })\mathbf{w }_{GMV}^{\prime }\\&\quad -R_{GMV}(\mathbf{1 }\otimes \mathbf{I }_k)-(\mathbf{1 }\otimes {\varvec{\mu }})\mathbf{w }_{GMV}^{\prime }\Big )\,, \end{aligned}$$

Hence,

$$\begin{aligned} \frac{\partial s}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}&= \frac{\partial ({\varvec{\mu }}^{\prime }{\varvec{R}})}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}{\varvec{\mu }}\\&= \frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\Big (({\varvec{\mu }}\otimes {\varvec{\mu }}) +R_{GMV}^2(\mathbf{1 }\otimes \mathbf{1 })-2R_{GMV}(\mathbf{1 }\otimes {\varvec{\mu }})\Big )\,. \end{aligned}$$

and

$$\begin{aligned} \frac{\partial \mathbf{w }^{\prime }_{VaR}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}&= \frac{\partial \mathbf{w }^{\prime }_{GMV}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}+\frac{\sqrt{d_{1-\alpha }^2-s}\frac{\partial \sqrt{V_{GMV}}{\varvec{\mu }}^{\prime }{\varvec{R}}}{\partial ({\text{ vech}}{\varvec{\Gamma }}(0))}- \frac{\partial \sqrt{d_{1-\alpha }^2-s}}{\partial ({\text{ vech}}{\varvec{\Gamma }}(0))}\sqrt{V_{GMV}}{\varvec{\mu }}^{\prime }{\varvec{R}}}{d_{1-\alpha }^2-s}\\&= \frac{\partial \mathbf{w }^{\prime }_{GMV}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}+\frac{\sqrt{V_{GMV}}}{\sqrt{d_{1-\alpha }^2-s}} \frac{\partial ({\varvec{\mu }}^{\prime }{\varvec{R}})}{\partial ({\text{ vech}}{\varvec{\Gamma }}(0))}\\&+\frac{1}{2\sqrt{d_{1-\alpha }^2-s}\sqrt{V_{GMV}}} \frac{\partial V_{GMV}}{\partial ({\text{ vech}}{\varvec{\Gamma }}(0))}{\varvec{\mu }}^{\prime }{\varvec{R}}\\&+\frac{\sqrt{V_{GMV}}}{2(d_{1-\alpha }^2-s)^{3/2}} \frac{\partial s}{\partial ({\text{ vech}}{\varvec{\Gamma }}(0))}{\varvec{\mu }}^{\prime }{\varvec{R}}. \end{aligned}$$

The substitution of the above calculated derivatives leads to

$$\begin{aligned}&\frac{\partial \mathbf{w }^{\prime }_{VaR}}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\\&\quad =\frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)}\Big ( V_{GMV}(\mathbf{1 }\!\otimes \!\mathbf{I }_k)\!-\!V_{GMV}(\mathbf{1 }\!\otimes \!\mathbf{1 })\mathbf{w }^{\prime }_{GMV}\!-\!\frac{V_{GMV}^{3/2}}{2\sqrt{d_{1-\alpha }^2\!-\!s}}(\mathbf{1 }\!\otimes \!\mathbf{1 }){\varvec{\mu }}^{\prime }{\varvec{R}}\\&\qquad +\frac{\sqrt{V_{GMV}}}{\sqrt{d_{1-\alpha }^2\!-\!s}}\left( ({\varvec{\mu }}\!\otimes \!\mathbf{I }_k) \!+\!R_{GMV}(\mathbf{1 }\!\otimes \!\mathbf{1 })\mathbf{w }_{GMV}^{\prime } \!-\!R_{GMV}(\mathbf{1 }\!\otimes \!\mathbf{I }_k)\!-\!(\mathbf{1 }\!\otimes \!{\varvec{\mu }})\mathbf{w }_{GMV}^{\prime }\right) \\&\qquad +\frac{\sqrt{V_{GMV}}}{2\left( d_{1-\alpha }^2\!-\!s\right) ^{3/2}} \left( ({\varvec{\mu }}\otimes {\varvec{\mu }}) \!+\!R_{GMV}^2(\mathbf{1 }\otimes \mathbf{1 })\!-\!2R_{GMV}(\mathbf{1 }\otimes {\varvec{\mu }})\right) {\varvec{\mu }}^{\prime }{\varvec{R}}\Big )\\&\quad \!=\!\frac{\partial ({\text{ vec}}{\varvec{\Gamma }}(0)^{-1})^{\prime }}{\partial {\text{ vech}}{\varvec{\Gamma }}(0)} \Big (a_1({\varvec{\mu }}\otimes \mathbf{I }_k)\!+\!a_2(\mathbf{1 }\otimes \mathbf{I }_k) \!+\!(\mathbf{1 }\otimes {\varvec{\mu }})\mathbf{v }_1^{\prime } \!+\!(\mathbf{1 }\otimes \mathbf{1 })\mathbf{v }_2^{\prime }\!+\!({\varvec{\mu }}\!\otimes \!{\varvec{\mu }})\mathbf{v }_3^{\prime }\Big ), \end{aligned}$$

where $a_1,\,a_2,\,\mathbf{v }_1,\,\mathbf{v }_2$, and $\mathbf{v }_3$ are given in (16).

It completes the proof of Theorem 1.

Proof of Corollary 1

We get

$$\begin{aligned}&\lim _{n\rightarrow \infty }F_{\sqrt{n}\left( \hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR}\right) |0<\hat{s}<d^2_{\alpha }}(\mathbf y )\\&\quad = \lim _{n\rightarrow \infty }F_{\sqrt{n}\left( \hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR}\right) |-\sqrt{n}s<\sqrt{n}(\hat{s}-s)<\sqrt{n}\left( d^2_{\alpha }-s\right) }(\mathbf y )\\&\quad =\lim _{n\rightarrow \infty }\frac{P\left( \left\{ \sqrt{n}\left( \hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR}\right) \le \mathbf y \right\} \cap B_n\right) }{P(B_n)}\,, \end{aligned}$$

where $B_n=\{ -\sqrt{n}s<\sqrt{n}(\hat{s}-s)<\sqrt{n}(d^2_{\alpha }-s)\}$ is an increasing sequence of intervals with $\bigcup _{n=1}^\infty B_n =\Omega$. Moreover, it holds that

$$\begin{aligned} \left\{ \sqrt{n}(\hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR})\le \mathbf y \right\} \cap B_n \subset \left\{ \sqrt{n}(\hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR})\le \mathbf y \right\} \cap B_{n+1} \end{aligned}$$

and

$$\begin{aligned} \bigcup _{n=1}^\infty \left\{ \sqrt{n}(\hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR})\le \mathbf y \right\} \cap B_n&= \left\{ \sqrt{n}(\hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR})\le \mathbf y \right\} \cap \left( \bigcup _{n=1}^\infty B_n\right) \\&= \{\sqrt{n}\left( \hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR}\right) \le \mathbf y \}. \end{aligned}$$

The application of the continuity property of the probability leads to $\lim _{n \rightarrow \infty } P(B_n)=P\left( \bigcup _{n=1}^\infty B_n\right) =1$ and

$$\begin{aligned}&\lim _{n \rightarrow \infty } P\left( \left\{ \sqrt{n}(\hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR})\le \mathbf y \right\} \cap B_n\right) \\&\quad = \lim _{n\rightarrow \infty }P\left( \bigcup _{n=1}^\infty \left\{ \sqrt{n}(\hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR})\le \mathbf y \right\} \cap B_n\right) \\&\quad = \lim _{n\rightarrow \infty }P\left( \left\{ \sqrt{n}\left( \hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR}\right) \le \mathbf y \right\} \right) , \end{aligned}$$

and consequently

$$\begin{aligned} \lim _{n\rightarrow \infty } F_{\sqrt{n}\left( \hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR}\right) |0<\hat{s}<d^2_{\alpha }}(\mathbf y )&= \lim _{n\rightarrow \infty } P\left( \left\{ \sqrt{n}(\hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR})\le \mathbf y \right\} \right) , \end{aligned}$$

which is the marginal distribution of $\sqrt{n}(\hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR})$. Hence, $\sqrt{n}(\hat{\mathbf{w }}_{VaR}-\mathbf{w }_{VaR})$ given $0<\hat{s}<d^2_{\alpha }$ is asymptotically multivariate normally distributed with zero mean vector and covariance matrix ${\varvec{S}}_{11}$. The corollary is proved.

Proof of Theorem 2

• (a) Lemma 1 proves that

  $$\begin{aligned} \sqrt{n}\left( \left( \hat{R}_{GMV},\hat{V}_{GMV},\hat{s}\right) ^{\prime }-(R_{GMV},V_{GMV},s)^{\prime }\right) \stackrel{d}{\rightarrow }\mathcal N (\mathbf 0 ,{\varvec{\Omega }}_{EF}). \end{aligned}$$

  Thus,

  $$\begin{aligned} \sqrt{n}\left( \left( \hat{R}_{VaR},\hat{V}_{VaR},\hat{s}\right) ^\prime -(R_{VaR},V_{VaR},s)^\prime \right) \stackrel{d}{\rightarrow }\mathcal N (\mathbf 0 ,{\varvec{\Omega }}_{RVs}) \end{aligned}$$

  with ${\varvec{\Omega }}_{RVs}=\mathbf{G }_1 {\varvec{\Omega }}_{EF}\mathbf{G }_1^\prime$ where

  $$\begin{aligned} \mathbf{G }_1=\left( \begin{array}{ccc} \partial R_{VaR}/\partial R_{GMV}&{} \partial R_{VaR}/\partial V_{GMV} &{} \partial R_{VaR}/\partial s \\ \partial V_{VaR}/\partial R_{GMV} &{} \partial V_{VaR}/\partial V_{GMV} &{} \partial V_{VaR}/\partial s \\ \partial s/\partial R_{GMV} &{} \partial s/\partial V_{GMV} &{} \partial s/\partial s\\ \end{array} \right) \end{aligned}$$

  From (6) and (7) we get

  $$\begin{aligned} \mathbf{G }_1=\left( \begin{array}{ccc} 1 &{} \frac{1}{2\sqrt{V_{GMV}}}\frac{s}{\sqrt{d_{1-\alpha }^2-s}} &{} \frac{2d_{1-\alpha }^2-s}{2\left( d_{1-\alpha }^2-s\right) ^{3/2}}\sqrt{V_{GMV}} \\ 0 &{} \frac{d_{1-\alpha }^2}{d_{1-\alpha }^2-s} &{} \frac{d_{1-\alpha }^2}{\left( d_{1-\alpha }^2-s\right) ^2}V_{GMV} \\ 0 &{} 0 &{} 1\\ \end{array} \right) \end{aligned}$$

  Hence,

  $$\begin{aligned} \mathbf{G }_1{\varvec{\Omega }}_{EF}\mathbf{G }_1^{\prime }&= \left( \begin{array}{ccc} 1 &{} \frac{1}{2\sqrt{V_{GMV}}}\frac{s}{\sqrt{d_{1-\alpha }^2-s}} &{} \frac{2d_{1-\alpha }^2-s}{2\left( d_{1-\alpha }^2-s\right) ^{3/2}}\sqrt{V_{GMV}} \\ 0 &{} \frac{d_{1-\alpha }^2}{d_{1-\alpha }^2-s} &{} \frac{d_{1-\alpha }^2}{\left( d_{1-\alpha }^2-s\right) ^2}V_{GMV} \\ 0 &{} 0 &{} 1\\ \end{array} \right) \times \left( \begin{array}{ccc} \sigma _1^2 &{} \sigma _{12} &{} \sigma _{13} \\ \sigma _{12} &{} \sigma _2^2 &{} \sigma _{23} \\ \sigma _{13} &{} \sigma _{23} &{} \sigma _3^2 \\ \end{array} \right) \\&\times \left( \begin{array}{ccc} 1 &{} 0 &{} 0\\ \frac{1}{2\sqrt{V_{GMV}}}\frac{s}{\sqrt{d_{1-\alpha }^2-s}} &{} \frac{d_{1-\alpha }^2}{d_{1-\alpha }^2-s} &{} 0\\ \frac{2d_{1-\alpha }^2-s}{2\left( d_{1-\alpha }^2-s\right) ^{3/2}}\sqrt{V_{GMV}} &{} \frac{d_{1-\alpha }^2}{\left( d_{1-\alpha }^2-s\right) ^2}V_{GMV} &{} 1 \\ \end{array} \right) =\left( \begin{array}{ccc} \tilde{\sigma }_1^2 &{} \tilde{\sigma }_{12} &{} \tilde{\sigma }_{13}\\ \tilde{\sigma }_{12} &{} \tilde{\sigma }_2^2 &{} \tilde{\sigma }_{23}\\ \tilde{\sigma }_{13} &{} \tilde{\sigma }_{23} &{} \tilde{\sigma }_3^2\\ \end{array} \right) \!, \end{aligned}$$

  where

  $$\begin{aligned} \tilde{\sigma }_1^2&= \sigma _1^2+2\sigma _{12}\frac{1}{2\sqrt{V_{GMV}}}\frac{s}{\sqrt{d_{1-\alpha }^2-s}}\\&+2\sigma _{13}\frac{2d_{1-\alpha }^2-s}{2\left( d_{1-\alpha }^2-s\right) ^{3/2}} \sqrt{V_{GMV}}+\sigma _2^2\frac{1}{4V_{GMV}}\frac{s^2}{d_{1-\alpha }^2-s}\\&+2\sigma _{23}\frac{s\left( 2d_{1-\alpha }^2-s\right) }{4\left( d_{1-\alpha }^2-s\right) ^2}+ \sigma _3^2\frac{\left( 2d_{1-\alpha }^2-s\right) ^2}{4\left( d_{1-\alpha }^2-s\right) ^3}V_{GMV}\,,\\ \tilde{\sigma }_{12}&= \sigma _{12}\frac{d_{1-\alpha }^2}{d_{1-\alpha }^2-s}+\sigma _2^2\frac{1}{2\sqrt{V_{GMV}}}\frac{sd_{1-\alpha }^2}{\left( d_{1-\alpha }^2-s\right) ^{3/2}}+\sigma _{13}\frac{d_{1-\alpha }^2}{\left( d_{1-\alpha }^2-s\right) ^2}V_{GMV} \\&+\sigma _{23}\left( \frac{d_{1-\alpha }^2\left( 2d_{1-\alpha }^2-s\right) }{2\left( d_{1-\alpha }^2-s\right) ^{5/2}}\sqrt{V_{GMV}}+\frac{sd_{1-\alpha }^2}{2\left( d_{1-\alpha }^2-s\right) ^{5/2}}\sqrt{V_{GMV}}\right) \\&+\sigma _3^2\frac{d_{1-\alpha }^2\left( 2d_{1-\alpha }^2-s\right) }{2\left( d_{1-\alpha }^2-s\right) ^{7/2}}V_{GMV}^{3/2}=\sigma _{12}\frac{d_{1-\alpha }^2}{d_{1-\alpha }^2-s}\\&+\sigma _2^2\frac{1}{2\sqrt{V_{GMV}}}\frac{sd_{1-\alpha }^2}{\left( d_{1-\alpha }^2-s\right) ^{3/2}}+ \sigma _{23}\frac{d_{1-\alpha }^4}{\left( d_{1-\alpha }^2-s\right) ^{5/2}}\sqrt{V_{GMV}}\\&+\sigma _{13}\frac{d_{1-\alpha }^2}{\left( d_{1-\alpha }^2-s\right) ^2}V_{GMV}+\sigma _3^2\frac{d_{1-\alpha }^2\left( 2d_{1-\alpha }^2-s\right) }{2\left( d_{1-\alpha }^2-s\right) ^{7/2}}V_{GMV}^{3/2}\,,\\ \tilde{\sigma }_{13}&= \sigma _{13}+\sigma _{23}\frac{1}{2\sqrt{V_{GMV}}}\frac{s}{\sqrt{d_{1-\alpha }^2-s}}+ \sigma _3^2\frac{2d_{1-\alpha }^2-s}{2\left( d_{1-\alpha }^2-s\right) ^{3/2}}\sqrt{V_{GMV}}\,,\\ \tilde{\sigma }_2^2&= \sigma _2^2\frac{d_{1-\alpha }^4}{\left( d_{1-\alpha }^2-s\right) ^2}+2\sigma _{23}\frac{d_{1-\alpha }^4}{\left( d_{1-\alpha }^2-s\right) ^3}V_{GMV}+ \sigma _3^2\frac{d_{1-\alpha }^4}{\left( d_{1-\alpha }^2-s\right) ^4}V_{GMV}^2\,,\\ \tilde{\sigma }_{23}&= \sigma _{23}\frac{d_{1-\alpha }^2}{d_{1-\alpha }^2-s}+ \sigma _3^2\frac{d_{1-\alpha }^2}{\left( d_{1-\alpha }^2-s\right) ^2}V_{GMV}\,,\\ \tilde{\sigma }_3^2&= \sigma _3^2 \,. \end{aligned}$$

• (b) The proof of part (b) is similar to the proof of part (a). It holds that

  $$\begin{aligned}&\sqrt{n}\left( \left( \hat{R}_{VaR},\hat{M}_{VaR},\hat{s}\right) ^\prime -(R_{VaR},M_{VaR},s)^\prime \right) \stackrel{d}{\rightarrow }\mathcal N (\mathbf 0 ,{\varvec{\Omega }}_{RMs})\\&\quad =\mathcal N (\mathbf 0 ,\mathbf{G }_1 {\varvec{\Omega }}_{EF}\mathbf{G }_1^\prime )\,, \end{aligned}$$

  where

  $$\begin{aligned} \mathbf{G }_2&= \left( \begin{array}{ccc} \partial R_{VaR}/\partial R_{GMV}&{} \partial R_{VaR}/\partial V_{GMV} &{} \partial R_{VaR}/\partial s \\ \partial M_{VaR}/\partial R_{GMV} &{} \partial M_{VaR}/\partial V_{GMV} &{} \partial M_{VaR}/\partial s \\ \partial s/\partial R_{GMV} &{} \partial s/\partial V_{GMV} &{} \partial s/\partial s\\ \end{array} \right) \\&= \left( \begin{array}{ccc} 1 &{} \frac{1}{2\sqrt{V_{GMV}}}\frac{s}{\sqrt{d_{1-\alpha }^2-s}} &{} \frac{2d_{1-\alpha }^2-s}{2\left( d_{1-\alpha }^2-s\right) ^{3/2}}\sqrt{V_{GMV}} \\ -1 &{} \frac{1}{2\sqrt{V_{GMV}}}\sqrt{d_{1-\alpha }^2-s} &{} -\frac{1}{2\sqrt{d_{1-\alpha }^2-s}}\sqrt{V_{GMV}} \\ 0 &{} 0 &{} 1\\ \end{array} \right) \end{aligned}$$

  Hence,

  $$\begin{aligned} \mathbf{G }_2{\varvec{\Omega }}_{EF}\mathbf{G }_2^{\prime }&= \left( \begin{array}{ccc} 1 &{} \frac{1}{2\sqrt{V_{GMV}}}\frac{s}{\sqrt{d_{1-\alpha }^2-s}} &{} \frac{2d_{1-\alpha }^2-s}{2\left( d_{1-\alpha }^2-s\right) ^{3/2}}\sqrt{V_{GMV}} \\ -1 &{} \frac{1}{2\sqrt{V_{GMV}}}\sqrt{d_{1-\alpha }^2-s} &{} -\frac{1}{2\sqrt{d_{1-\alpha }^2-s}}\sqrt{V_{GMV}} \\ 0 &{} 0 &{} 1\\ \end{array} \right) \times \left( \begin{array}{ccc} \sigma _1^2 &{} \sigma _{12} &{} \sigma _{13} \\ \sigma _{12} &{} \sigma _2^2 &{} \sigma _{23} \\ \sigma _{13} &{} \sigma _{23} &{} \sigma _3^2 \\ \end{array} \right) \\&\times \left( \begin{array}{ccc} 1 &{} -1 &{} 0\\ \frac{1}{2\sqrt{V_{GMV}}}\frac{s}{\sqrt{d_{1-\alpha }^2-s}} &{} \frac{1}{2\sqrt{V_{GMV}}}\sqrt{d_{1-\alpha }^2-s}&{}0\\ \frac{2d_{1-\alpha }^2-s}{2(d_{1-\alpha }^2-s)^{3/2}}\sqrt{V_{GMV}} &{} -\frac{1}{2\sqrt{d_{1-\alpha }^2-s}}\sqrt{V_{GMV}}&{}1\\ \end{array} \right) = \left( \begin{array}{ccc} \breve{\sigma }_1^2 &{} \breve{\sigma }_{12} &{} \breve{\sigma }_{13}\\ \breve{\sigma }_{12} &{} \breve{\sigma }_2^2 &{} \breve{\sigma }_{23}\\ \breve{\sigma }_{13} &{} \breve{\sigma }_{23} &{} \breve{\sigma }_3^2\\ \end{array} \right) \!, \end{aligned}$$

  where

  $$\begin{aligned} \breve{\sigma }_1^2&= \tilde{\sigma }_1^2=\sigma _1^2+2\sigma _{12}\frac{1}{2\sqrt{V_{GMV}}}\frac{s}{\sqrt{d_{1-\alpha }^2-s}}+2\sigma _{13}\frac{2d_{1-\alpha }^2-s}{2(d_{1-\alpha }^2-s)^{3/2}} \sqrt{V_{GMV}}\\&+\sigma _2^2\frac{1}{4V_{GMV}}\frac{s^2}{d_{1-\alpha }^2-s}+2\sigma _{23}\frac{s\left( 2d_{1-\alpha }^2-s\right) }{4(d_{1-\alpha }^2-s)^2}+ \sigma _3^2\frac{\left( 2d_{1-\alpha }^2-s\right) ^2}{4\left( d_{1-\alpha }^2-s\right) ^3}V_{GMV}\,,\\ \breve{\sigma }_{12}&= -\sigma _1^2+\sigma _2^2\frac{s}{4V_{GMV}}-\sigma _3^2\frac{2d_{1-\alpha }^2-s}{4\left( d_{1-\alpha }^2-s\right) ^2}V_{GMV} +\sigma _{12}\frac{1}{2\sqrt{V_{GMV}}}\frac{d_{1-\alpha }^2-2s}{\sqrt{d_{1-\alpha }^2-s}}\\&-\sigma _{13}\frac{3d_{1-\alpha }^2-2s}{2\left( d_{1-\alpha }^2-s\right) ^{3/2}}\sqrt{V_{GMV}}+\sigma _{23}\frac{1}{2}\,,\\ \breve{\sigma }_{13}&= \sigma _{13}+\sigma _{23}\frac{1}{2\sqrt{V_{GMV}}}\frac{s}{\sqrt{d_{1-\alpha }^2-s}}+ \sigma _3^2\frac{2d_{1-\alpha }^2-s}{2\left( d_{1-\alpha }^2-s\right) ^{3/2}}\sqrt{V_{GMV}}\,,\\ \breve{\sigma }_2^2&= \sigma _1^2-2\sigma _{12}\frac{1}{2\sqrt{V_{GMV}}}\sqrt{d_{1-\alpha }^2-s}+ 2\sigma _{13}\frac{1}{2\sqrt{d_{1-\alpha }^2-s}}\sqrt{V_{GMV}}\\&+\sigma _2^2\frac{1}{4V_{GMV}}\left( d_{1-\alpha }^2-s\right) -2\sigma _{23}\frac{1}{4}+\sigma _3^2\frac{1}{4\left( d_{1-\alpha }^2-s\right) }V_{GMV}\,,\\&-2\sigma _{23}\frac{1}{4}+\sigma _3^2\frac{1}{4\left( d_{1-\alpha }^2-s\right) }V_{GMV}\,,\\ \breve{\sigma }_{23}&= -\sigma _{13}+\sigma _{23}\frac{1}{2\sqrt{V_{GMV}}}\sqrt{d_{1-\alpha }^2-s}-\sigma _3^2\frac{1}{2\sqrt{d_{1-\alpha }^2-s}}\sqrt{V_{GMV}}\,,\\ \breve{\sigma }_3^2&= \sigma _3^2\,. \end{aligned}$$