Portfolio allocation using multivariate variance gamma models

Abstract

In this paper, we investigate empirically the effect of using higher moments in portfolio allocation when parametric and nonparametric models are used. The nonparametric model considered in this paper is the sample approach; the parametric model is constructed assuming multivariate variance gamma (MVG) joint distribution for asset returns.We consider the MVG models proposed by Madan and Seneta (1990), Semeraro (2008) and Wang (2009). We perform an out-of-sample analysis comparing the optimal portfolios obtained using the MVG models and the sample approach. Our portfolio is composed of 18 assets selected from the S&P500 Index and the dataset consists of daily returns observed from 01/04/2000 to 01/09/2011.

Appendices

A. Appendix

The characteristic function (c.f.) $\varphi (t)$ for N-dimensional random vector $X$ is defined as:

$$\begin{aligned} \varphi (t):=E\left[\exp \left(i\sum _{p=1}^{N} t_{p}X_{p}\right)\right] \end{aligned}$$

where $t \in R^{N}$.

For all MVG models, the expected CARA utility can be connected with the characteristic function as follows:

$$\begin{aligned} E\left[u\left(\omega ^{\prime }X\right)\right]=E\left[-\exp \left(-\lambda \omega ^{\prime }X\right)\right]=-\varphi (i\lambda \omega ). \end{aligned}$$

In the following, the c.f. for each MVG model is derived.

1.1 MVG with common mixing density

For the MVG with common mixing density the c.f. $\varphi _{MVG}(t)$ can be easily obtained as follows:

$$\begin{aligned} \varphi _\mathrm{MVG}(t)=E\left[\exp \left(i \sum _{p=1}^{N}t_{p}\mu _{p} + iV\sum _{p=1}^{N} t_{p}\theta _{p} +i\sqrt{V} \sum _{p=1}^{N}t_{p}\sum _{h=1}^{p}a_{p,h}Z_{h}\right)\right]\!. \end{aligned}$$

Using the law of total expectation (see Billingsley 1995), we get:

$$\begin{aligned} \varphi _\mathrm{MVG}(t)=\exp \left(i \sum _{p=1}^{N}t_{p}\mu _{p} \right)E\left[E\left[\left.\exp \left(iV\sum _{p=1}^{N} t_{p}\theta _{p} +i\sqrt{V} \sum _{p=1}^{N}t_{p}\sum _{h=1}^{p}a_{p,h}Z_{h}\right)\right|V\right]\right]\!, \end{aligned}$$

since $Z_{h}$ are independent standard normals, we have:

$$\begin{aligned} \varphi _\mathrm{MVG}(t)=\exp \left(i \sum _{p=1}^{N}t_{p}\mu _{p} \right)E\left[\exp \left((i \sum _{p=1}^{N}t_{p}\theta _{p} -\frac{1}{2}\sum _{p,j}t_{j}t_{p}\sum _{h=1}^{\min {(p,j)}}a_{p,h}a_{j,h}) V\right)\right]\!. \end{aligned}$$

Being V a univariate gamma, the c.f. is

$$\begin{aligned} \varphi _\mathrm{MVG}(t)=\exp \left[i \sum _{p=1}^{N}t_{p}\mu _{p} +\alpha \ln \left(\frac{1}{1-(i \sum _{p=1}^{N}t_{p}\theta _{p} -\frac{1}{2}\sum _{p,j}t_{j}t_{p}\sum _{h=1}^{\min {(p,j)}}a_{p,h}a_{j,h})}\right)\right]\!. \end{aligned}$$

1.2 Semeraro model

For the Semeraro model, using our parameterization, the c.f. can be obtained as follows:

$$\begin{aligned} \varphi _\mathrm{Sem}(t) = E\left[\exp \left(i\sum _{p=1}^{N}t_{p}\mu _{p}\!+\!i\sum _{p=1}^{N}t_{p}\theta _{p}G_{p}\!+\!i\sum _{p=1}^{N}t_{p}\sigma _{p}\sqrt{G_{p}}W_{p}\right)\right]\\ = \exp \left(\!i\sum _{p=1}^{N}t_{p}\mu _{p}\!\right)E\!\left[\exp \left(\!i\sum _{p=1}^{N}t_{p}\theta _{p}G_{p}\!\right)E\!\left[\!\left.\exp \left(i\sum _{p=1}^{N}t_{p}\sigma _{p}\sqrt{G_{p}}W_{p}\right)\right|G_{1},\ldots ,G_{N}\right]\!\right]\!. \end{aligned}$$

Since $W_{p}$ are independent standard normals, we have:

$$\begin{aligned} \varphi _{\mathrm{Sem}}(t)=\exp \left(i\sum _{p=1}^{N}t_{p}\mu _{p}\right)E\left[\exp \left(i\sum _{p=1}^{N}t_{p}\theta _{p}G_{p}-\sum _{p=1}^{N}\frac{1}{2}t^{2}_{p}\sigma ^{2}_{p}G_{p}\right)\right]\!, \end{aligned}$$

substituting $G_{p}$ we can write:

$$\begin{aligned} \varphi _{\mathrm{Sem}}(t)\!=\!\exp \left(i\sum _{p=1}^{N}t_{p}\mu _{p}\right)E\left\{ \exp \left[\sum _{p=1}^{N}\left(it_{p}\theta _{p}\!-\!\frac{1}{2}t_{p}^{2}\sigma _{p}^{2}\right)Y_{p}\!+\!\sum _{p=1}^{N}\left(it_{p}\theta _{p}\!-\!\frac{1}{2}t_{p}^{2}\sigma _{p}^{2}\right)a_{p}Z\right]\right\} . \end{aligned}$$

Being $Y_{p}$ and $Z$ independent for $p=1,\ldots ,N$, we obtain:

$$\begin{aligned} \varphi _{\mathrm{Sem}}(t) = \exp \left(i\sum _{p=1}^{N}t_{p}\mu _{p}\right)E\left\{ \exp \left[Z\sum _{p=1}^{N}\left(it_{p}\theta _{p}\!-\!\frac{1}{2}t_{p}^{2}\sigma _{p}^{2}\right)a_{p}\right]\right\} \prod _{p=1}^{N}E\left\{ \exp \left[\left(it_{p}\theta _{p}\!-\!\frac{1}{2}t^{2}_{p}\sigma ^{2}_{p}\right)Y_{p}\right]\right\} \\ = \exp \left[i\sum _{p=1}^{N}t_{p}\mu _{p}\!+\!n\ln \left(\frac{1}{1\!-\!\sum _{p=1}^{N}\left(it_{p}\theta _{p}\!-\!\frac{1}{2}t^{2}_{p}\sigma ^{2}_{p}\right)\frac{a_{p}}{k}}\right)\!+\!\sum _{p=1}^{N}l_{p}\ln \left(\frac{1}{1\!-\!(it_{p}\theta _{p}-\frac{1}{2}t_{p}^{2}\sigma _{p}^{2})\frac{1}{m_{p}}}\right)\right]. \end{aligned}$$

Since $\frac{a_{p}}{k}=\frac{1}{m_{p}}$, the c.f. of Semeraro model is

$$\begin{aligned} \varphi _{\mathrm{Sem}}(t)\!=\!\exp \left[i\sum _{p=1}^{N}t_{p}\mu _{p}\!+\!n\ln \left(\frac{1}{1\!-\!\sum _{p=1}^{N}\left(it_{p}\theta _{p}\!-\!\frac{1}{2}t^{2}_{p}\sigma ^{2}_{p}\right)\frac{1}{m_{p}}}\right)\!+\!\sum _{p=1}^{N}l_{p}\ln \left(\frac{1}{1\!-\!(it_{p}\theta _{p}\!-\!\frac{1}{2}t_{p}^{2}\sigma _{p}^{2})\frac{1}{m_{p}}}\right)\right]. \end{aligned}$$

1.3 Wang model

In Wang model, the c.f. is

$$\begin{aligned} \varphi _\mathrm{Wang}(t)=E\left[\exp \left(i\sum _{p=1}^{N}t_{p}\mu _{p}+i\sum _{p=1}^{N}t_{p}A_{p}+i\sum _{p=1}^{N}t_{p}Y_{p}\right)\right]\!. \end{aligned}$$

Since $A_{p}$, $Y_{p}$ are independent for $p=1,\ldots ,N$, we have:

$$\begin{aligned} \varphi _\mathrm{Wang}(t)=\exp \left(i\sum _{p=1}^{N}t_{p}\mu _{p}\right)E\left[\exp \left(i\sum _{p=1}^{N}t_{p}A_{p}\right)\right]E\left[\exp \left(i\sum _{p=1}^{N}t_{p}Y_{p}\right)\right]\!. \end{aligned}$$

Being $A_{p}$ the pth component of the MVG with common mixing density and $Y_{p}$ an univariate variance gamma, we can write:

$$\begin{aligned} \varphi _\mathrm{Wang}(t)&= \exp \left[i\sum _{p=1}^{N}t_{p}\mu _{p}+\alpha _{v}\ln \left(\frac{1}{1-i\sum _{p=1}^{N}t_{p}\theta _{p}+\frac{1}{2}\sum _{p,j=1}^{N}t_{p}t{j}\sum _{h=1}^{\min {(p,j)}}a_{p,h}a_{j,h}}\right)\right]\\&\times \exp \left[\sum _{p=1}^{N}\alpha _{G_{p}}\ln \left(\frac{1}{1-it_{p}\theta _{p}+\frac{1}{2}t_{p}^{2}\sigma _{G_{p}}^{2}}\right)\right]. \end{aligned}$$

B. Appendix

The general formula of co-skewness is

$$\begin{aligned} s_{ijk}=E\left[\left(X_{i}-E(X_{i})\right)\left(X_{j}-E(X_{j})\right)\left(X_{k}-E(X_{k})\right)\right]\!. \end{aligned}$$

(29)

The general formula of co-kurtosis is

$$\begin{aligned} k_{ijkl}=E\left[\left(X_{i}-E(X_{i})\right)\left(X_{j}-E(X_{j})\right)\left(X_{k}-E(X_{k})\right)\left(X_{l}-E(X_{l})\right)\right]\!. \end{aligned}$$

(30)

1.1 MVG with common mixing density

We derive the co-skewness of the MVG with common mixing density using the definition (29) and we have:

$$\begin{aligned} s_{ijk}=E\left[\left(\theta _{i}\left(V-E(V)\right)+\sqrt{V}Y_{i}\right) \left(\theta _{j}\left(V-E(V)\right)+\sqrt{V}Y_{j}\right) \left(\theta _{k}\left(V-E(V)\right)+\sqrt{V}Y_{k}\right) \right]\nonumber \\ \end{aligned}$$

(31)

where $Y_{i}=\sum _{h=1}^{i}a_{ih}Z_{h}$ and $Y_{i}\sim {N(0,\sigma ^{2}_{i})}.$ Developing the equation, we have the following elements and their permutations w.r.t. indices $(i,j,k)$:

$$\begin{aligned} E\left[\theta _{i}(V-E(V))VY_{j}Y_{k}\right]&= \theta _{i}E\left[(V-E(V))V\right]E\left[Y_{j}Y_{k}\right]\\&= \theta _{i}\mathrm{Var}(V)\mathrm{Cov}(Y_{j},Y_{k})\\&= \theta _{i}\alpha \sigma _{j,k}\\ E\left[\theta _{i}\theta _{j}(V-E(V))^{2}\sqrt{V}Y_{k}\right]&= E\left[\theta _{i}\theta _{j}(V-E(V))^{2}\sqrt{V}\right]\underbrace{E\left[Y_{k}\right]}_{=0}=0, \end{aligned}$$

and the single final terms:

$$\begin{aligned} E\left[\theta _{i}\theta _{j}\theta _{k}\left(V-E(V)\right)^3\right]&= 2\alpha \theta _{i}\theta _{j}\theta _{k}\\ E\left[V\sqrt{V}Y_{i}Y_{j}Y_{k}\right]&= E\left[V\sqrt{V}\right]\underbrace{E\left[Y_{i}Y_{j}Y_{k}\right]}_{=0}=0 \end{aligned}$$

The term $E\left[Y_{i}Y_{j}Y_{k}\right]=0$ since it is the coskeness of a multivariate Normal with zero mean.

Putting together these terms into equation(31) we have the element in (11) Starting from definition (30) we derive the co-kurtosis as follows:

$$\begin{aligned} k_{ijkl}&= E\left[\left(\theta _{i}\left(V-E(V)\right)+\sqrt{V}Y_{i}\right) \left(\theta _{j}\left(V-E(V)\right)+\sqrt{V}Y_{j}\right)\right.\nonumber \\&\times \left.\left(\theta _{k}\left(V-E(V)\right)+\sqrt{V}Y_{k}\right) \left(\theta _{l}\left(V-E(V)\right)+\sqrt{V}Y_{l}\right) \right]\!. \end{aligned}$$

(32)

Following the previous procedure we obtain the elements and their permutations:

$$\begin{aligned} E\left[\theta _{i}\theta _{j}\theta _{k}(V - E(V))^{3}\sqrt{V}Y_{l}\right] = E\left[\theta _{i}\theta _{j}\theta _{k}(V - E(V))^{3}\sqrt{V}\right]\underbrace{E\left[Y_{l}\right]}_{ = 0} = 0\\ E\left[\theta _{i}\theta _{j}(V - E(V))^{2}VY_{k}Y_{l}\right] = \theta _{i}\theta _{j}\underbrace{E\left[(V - E(V))^{2}V\right]}_{ = \alpha ^2 + 2\alpha }\underbrace{E\left[Y_{k}Y_{l}\right]}_{ = \sigma _{kl}} = \theta _{i}\theta _{j}(\alpha ^2 + 2\alpha )\sigma _{kl}. \end{aligned}$$

The relation $E\left[(V-E(V))^{2}V\right]=\alpha ^2+2\alpha$ comes from the property of the gamma distribution, see Johnson and Kotz (1994).

$$\begin{aligned} E\left[\theta _{i}(V-E(V))V\sqrt{V}Y_{j}Y_{k}Y_{l}\right]=E\left[\theta _{i}(V-E(V))V\sqrt{V}\right]\underbrace{E\left[Y_{j}Y_{k}Y_{l}\right]}_{=0}=0, \end{aligned}$$

and the single final terms are given by:

$$\begin{aligned} E\left[\theta _{i}\theta _{j}\theta _{k}\theta _{l}(V-E(V))^{4}\right]=\theta _{i}\theta _{j}\theta _{k}\theta _{l}\underbrace{E\left[(V-E(V))^{4}\right]}_{=3\alpha ^{2}+6\alpha }=(3\alpha ^{2}+6\alpha )\theta _{i}\theta _{j}\theta _{k}\theta _{l} \\ E\left[V^{2}Y_{i}Y_{j}Y_{k}Y_{l}\right]=E\left[Y_{i}Y_{j} Y_{k} Y_{l}\right] \underbrace{E\left[V^{2}\right]}_{=\alpha ^{2}+\alpha }=3(\alpha ^{2}+\alpha )\sum _{h=1}^{min(i,j,k,l)}a_{ih}a_{jh}a_{kh}a_{lh}, \end{aligned}$$

where $E\left[Y_{i}Y_{j} Y_{k} Y_{l}\right]=3\sum _{h=1}^{min(i,j,k,l)}a_{ih}a_{jh}a_{kh}a_{lh}$ since it the co-kurtosis of the multivariate Normal with mean zero.

Putting together the elements into (32) we obtain the co-kurtosis elements given in (12).

1.2 Semeraro model

From the definition of skewness (29), we obtain:

$$\begin{aligned} s_{ijk}&= E\left[(\theta _{i}(G_{i}-E(G_{i}))+\sigma \sqrt{G_{i}}W_{i})(\theta _{j}(G_{j}-E(G_{j}))\right.\nonumber \\&\quad +\left.\,\sigma \sqrt{G_{j}}W_{j})(\theta _{k}(G_{k}-E(G_{k}))+\sigma \sqrt{G_{k}}W_{k})\right] \end{aligned}$$

(33)

Developing the equation we obtain the following elements:

$$\begin{aligned} \mathrm{term}^{a}_{ijk}&= E\left[(\theta _{i}(G_{i}-E(G_{i})))\theta _{j}(G_{j}-E(G_{j}))\theta _{k}(G_{k}-E(G_{k}))\right]\nonumber \\&= \theta _{i}\theta _{j}\theta _{k}E\left\{ \left[(Y_{i}-E(Y_{i}))+\frac{k}{m_{i}}(Z-E(Z))\right]\right.\nonumber \\&\times \left.\left[(Y_{j}-E(Y_{j}))+\frac{k}{m_{j}}(Z-E(Z))\right]\right.\nonumber \\&\times \left.\left[(Y_{k}-E(Y_{k}))+\frac{k}{m_{k}}(Z-E(Z))\right]\right\} , \end{aligned}$$

using the independence between $Y_{i},$ $Y_{j},$ $Y_{k},$ $Z$ and the property of gamma (see Johnson and Kotz 1994), we have:

$$\begin{aligned} \mathrm{term}^{a}_{iii}&= 2\theta ^{3}_{i}\frac{l_{i}+n}{m_{i}}\quad if\ \ i=j=k\\ \mathrm{term}^{a}_{iik}&= 2\theta ^{2}_{i}\theta _{k}\frac{n}{m_{i}^{2}m_{k}}\quad if\ \ i=j\ne k\\ \mathrm{term}^{a}_{ijk}&= 2n\frac{\theta _{i}\theta _{j}\theta _{k}}{m_{i}m_{j}m_{k}}\quad if\ \ i\ne j\ne k.\\ \mathrm{term}^{b}_{ijk}&= E\left[(\theta _{i}(G_{i}-E(G_{i})))\theta _{j}(G_{j}-E(G_{j}))(\sigma \sqrt{G_{k}}W_{k})\right]=0, \end{aligned}$$

since $W_{k}$ is a standard normal independent from $G_{i},$ $G_{j}$ and $G_{k}.$

$$\begin{aligned} \mathrm{term}^{c}_{ijk}=E\left[(\theta _{i}(G_{i}-E(G_{i})))\sigma ^{2}\sqrt{G_{j}}\sqrt{G_{k}}W_{k}W_{j}\right] \end{aligned}$$

we observe that if $i\ne j= k$ or $i= j= k$:

$$\begin{aligned} \mathrm{term}^{c}_{iii}&= \sigma ^2\theta _{i}\frac{l_{i}+n}{m^{2}_{i}}\quad if\ \ i=j=k\\ \mathrm{term}^{c}_{ikk}&= \sigma ^{2}\theta _{i}\frac{n}{m_{i}m_{k}} \quad if\ \ i\ne j=k \end{aligned}$$

otherwise the $\mathrm{term}^{c}_{ijk}=0.$

The last term:

$$\begin{aligned} \mathrm{term}^{d}_{ijk}=\sigma ^{3}E\left[\sqrt{G_{i}}W_{i}\sqrt{G_{j}}W_{j}\sqrt{G_{k}}W_{k}\right]=0. \end{aligned}$$

Using these terms, we obtain the element of the co-skewness as reported in (19).

$$\begin{aligned} k_{ijkl}&= E\left[(\theta _{i}(G_{i}-E(G_{i}))+\sigma \sqrt{G_{i}}W_{i})(\theta _{j}(G_{j}-E(G_{j}))+\sigma \sqrt{G_{j}}W_{j})\right.\nonumber \\&\times \left.(\theta _{k}(G_{k}-E(G_{k}))+\sigma \sqrt{G_{k}}W_{k})(\theta _{l}(G_{l}-E(G_{l}))+\sigma \sqrt{G_{l}}W_{l})\right]\!.\qquad \end{aligned}$$

(34)

As above, we have the following terms:

$$\begin{aligned} \mathrm{term}^{a}_{ijkl}=\theta _{i}\theta _{j}\theta _{k}\theta _{l}E\left[(G_{i}-E(G_{i}))(G_{j}-E(G_{j}))(G_{k}-E(G_{k}))(G_{l}-E(G_{l}))\right], \end{aligned}$$

using the definition of $G_{i}=Y_{i}+\frac{k}{m_{i}}Z$, we can write:

$$\begin{aligned} \mathrm{term}^a_{ijkl}&= \theta _{i}\theta _{j}\theta _{k}\theta _{l}E\left\{ \left[Y_{i}-E(Y_{i})+\frac{k}{m_{i}}(Z-E(Z))\right]\left[Y_{j}-E(Y_{j})+\frac{k}{m_{j}}(Z-E(Z))\right]\right.\\&\times \left.\left[Y_{k}-E(Y_{k})+\frac{k}{m_{k}}(Z-E(Z))\right]\left[Y_{l}-E(Y_{l})+\frac{k}{m_{l}}(Z-E(Z))\right]\right\} . \end{aligned}$$

Applying the properties of gamma distribution and the independence between $Y_{i},$ $Y_{j},$ $Y_{k},$ $Y_{l}$ and $Z$ we obtain:

$$\begin{aligned} \mathrm{term}^{a}_{ijkl}&= \frac{\theta _{i}\theta _{j}\theta _{k}\theta _{l}}{m_i m_j m_k m_l}\left(3+\frac{6}{n}\right)n^2 \ \quad \ if \ \ i \ne j \ne k \ne l \\ \mathrm{term}^{a}_{iikl}&= n\frac{\theta _{i}^2\theta _{k}\theta _{l}}{m_i^2 m_k m_l}\left[l_{i}+\left(3+\frac{6}{n}\right)n\right] \ \quad \ if \ \ i = j \ne k \ne l \\ \mathrm{term}^{a}_{iill}&= \frac{\theta ^2_i \theta ^2_l}{m^2_i m^2_l}l_i [l_l +n]+\frac{\theta ^2_i \theta ^2_l}{m^2_i m^2_l}\left[nl_l+\left(3+\frac{6}{n}\right)n^2\right] \ \quad \ if \ \ i = j \ne k = l \\ \mathrm{term}^{a}_{iiil}&= 3\frac{\theta ^3_i \theta _l}{m_i^3 m_l}l_i n+\frac{\theta ^3_i \theta _l}{m_i^3 m_l}\left(3+\frac{6}{n}\right)n^2 \ \quad \ if \ \ i = j = k \ne l \\ \mathrm{term}^{a}_{iiii}&= \frac{\theta _{i}^4}{m^4_i}\left[3 l^2_i+6 l_i+ 6 l_i n+ 3 n^2+ 6 n\right] \ \quad \ if \ \ i = j = k = l. \end{aligned}$$

The following term and its permutations are always zero

$$\begin{aligned} \mathrm{term}^b_{ijkl}=\theta _{i}\theta _{j}\theta _{k}\sigma _{l}E\left[(G_{i}-E(G_{i}))(G_{j}-E(G_{j}))(G_{k}-E(G_{k}))(\sqrt{G_{l}}W_{l})\right]\!, \end{aligned}$$

since $W$ is independent from $G$.

The next term (with its permutation) is given by:

$$\begin{aligned} \mathrm{term}^c_{ijkl}&= \theta _{i}\theta _{j}\sigma _{k}\sigma _{l}E\left[(G_{i}-E(G_{i}))(G_{j}-E(G_{j}))(\sqrt{G_{k}}W_{k})(\sqrt{G_{l}}W_{l})\right]\\&= \theta _{i}\theta _{j}\sigma _{k}\sigma _{l}E\left[(G_{i}-E(G_{i}))(G_{j}-E(G_{j}))\sqrt{G_{k}}\sqrt{G_{l}}\right]E\left[W_{k}W_{l}\right]\!. \end{aligned}$$

This term is zero except when $k=l$

$$\begin{aligned} \mathrm{term}^{c}_{ijll}&= n\frac{ \theta _i \theta _j \sigma _l^2}{m_i m_j m_l}(2+n+l_l) \ \quad \ if \ \ i \ne j \ne k = l \\ \mathrm{term}^{c}_{iill}&= \frac{ \theta _i^2 \sigma _l^2}{m_i^2 m_l }[l_i(l_l+n)+n l_l+n(n+2)] \ \quad \ if \ \ i = j \ne k = l \\ \mathrm{term}^{c}_{illl}&= \frac{\theta _{i} \theta _{l} \sigma _{l}^2}{m_l^2 m_i}[n(n+2)+l_l n] \ \quad \ if \ \ i \ne j = k = l \\ \mathrm{term}^{c}_{llll}&= \frac{\theta _l^2 \sigma _l^2}{m_l^3}\left[l_l (l_l+2)+n(n+2)+2 l_l n \right] \ \quad \ if \ \ i = j = k = l. \end{aligned}$$

The following term and its permutation are zero:

$$\begin{aligned} \mathrm{term}^d_{ijkl}=\theta _{i}\sigma _{j}\sigma _{k}\sigma _{l}E\left[(G_{i}-E(G_{i}))(\sqrt{G_{j}}W_{j})(\sqrt{G_{k}}W_{k})(\sqrt{G_{l}}W_{l})\right]=0, \end{aligned}$$

since $E[W_{j}W_{k}W_{l}]=0$ for all possible combination of indeces.

The last term is

$$\begin{aligned} \mathrm{term}^d_{ijkl}=\sigma _{i}\sigma _{j}\sigma _{k}\sigma _{l}E\left[\sqrt{G_{j}}\sqrt{G_{j}}\sqrt{G_{k}}\sqrt{G_{l}}\right]E\left[W_{i}W_{j}W_{k}W_{j}\right] \end{aligned}$$

and we have the following cases:

$$\begin{aligned} \mathrm{term}^d_{iill}&= \frac{\sigma ^2_i \sigma ^2_l}{m_i m_l}\left[l_i l_l + l_i n + l_l n+(n+n^2)\right] \ \quad \ if \ \ i = j \ne k = l \\ \mathrm{term}^d_{llll}&= 3\frac{\sigma ^4_l}{m^2_l}[(l_l + 1)l_l + (n+1)n +2 l_l n] \ \quad \ if \ \ i = j = k = l \end{aligned}$$

With these terms, we obtain the elements of co-kurtosis as reported in (20).

1.3 Wang model

The co-skewness and co-kurtosis elements in Wang model can be easily obtained (using the same procedure as previously), since the Wang model is the sum of the MVG with common mixing density and an independent component with VG distribution.

C. Appendix