Portfolio optimization and marginal contribution to risk on multivariate normal tempered stable model

Abstract

This paper proposes a market model with returns assumed to follow a multivariate normal tempered stable distribution defined by a mixture of the multivariate normal distribution and the tempered stable subordinator. This distribution can capture two stylized facts: fat-tails and asymmetry, that have been empirically observed for asset return distributions. We discuss a new portfolio optimization method on the new market model, which is an extension of Markowitz’s mean-variance optimization. The new optimization method considers not only reward and dispersion but also asymmetry in tails. The efficient frontier is extended to a curved surface on three-dimensional space of reward, dispersion, and asymmetry in tails. We also propose a new performance measure, which is an extension of the Sharpe ratio. Moreover, we derive closed-form solutions for portfolio managers’ two important measures in portfolio construction: the marginal value-at-risk (VaR) and the marginal conditional VaR (CVaR). We illustrate the proposed model using stocks comprising the Dow Jones Industrial Average. First, perform the new portfolio optimization and then demonstrating how the marginal VaR and marginal CVaR can be used for portfolio optimization under the model. Based on this paper’s empirical evidence, our framework offers realistic portfolio optimization and tractable methods for portfolio risk management.

Appendix: Multivariate normal tempered stable distribution

Let $\alpha \in (0,2)$ and $\theta >0$, and let ${{{\mathcal {T}}}}$ be a positive random variable whose characteristic function $\phi _{{{{\mathcal {T}}}}}$ is equal to

$$\begin{aligned} \phi _{{{{\mathcal {T}}}}}(u) = \exp \left( -\frac{2\theta ^{1-\frac{\alpha }{2}}}{\alpha }\left( (\theta -iu)^{\frac{\alpha }{2}}-\theta ^{\frac{\alpha }{2}}\right) \right) . \end{aligned}$$

(12)

The random variable ${{{\mathcal {T}}}}$ is referred to as Tempered Stable Subordinator. Let $X=(X_1, X_2, \cdots , X_N)^{{\texttt {T}}}$ be a multivariate random variable given by

$$\begin{aligned} X = \mu + \beta ({{{\mathcal {T}}}}-1) + diag (\gamma ) \varepsilon \sqrt{{{{\mathcal {T}}}}} , \end{aligned}$$

where

• $\mu = (\mu _1, \mu _2, \cdots , \mu _N)^{{\texttt {T}}}\in {{\mathbb {R}}}^N$

• $\beta = (\beta _1, \beta _2, \cdots , \beta _N)^{{\texttt {T}}}\in {{\mathbb {R}}}^N$

• $\gamma = (\gamma _1, \gamma _2, \cdots , \gamma _N)^{{\texttt {T}}}\in {{\mathbb {R}}}_+^N$ with ${{\mathbb {R}}}_+=[0,\infty )$

• $\varepsilon = (\varepsilon _1, \varepsilon _2, \cdots , \varepsilon _N)^{{\texttt {T}}}$ is a N-dimensional standard normal distribution with a covariance matrix $\varSigma$. That is, $\varepsilon _n\sim \varPhi (0,1)$ for $n\in \{1,2,\ldots , N\}$ and (k, l)th element of $\varSigma$ is given by $\rho _{k,l}=\mathrm{cov}(\varepsilon _k,\varepsilon _l)$ for $k,l\in \{1,2,\ldots ,N\}$.

• ${{{\mathcal {T}}}}$ is the Tempered Stable Subordinator with parameters $(\alpha ,\theta )$, and is independent of $\varepsilon _n$ for all $n=1,2,\ldots , N$.

Then X is referred to as the N-dimensional NTS random variable with parameters $(\alpha$, $\theta$, $\beta$, $\gamma$, $\mu$, $\varSigma )$ which we denote by $X\sim NTS _N(\alpha$, $\theta$, $\beta$, $\gamma$, $\mu$, $\varSigma )$. The NTS distribution has the following properties:

• 1. The mean of X are equal to $E[X] = \mu$.

• 2. The covariance between $X_k$ and $X_l$ is given by

  $$\begin{aligned} \mathrm{cov}(X_k,X_l)=\rho _{k,l}\gamma _k\gamma _l+\beta _k\beta _l\left( \frac{2-\alpha }{2\theta }\right) \end{aligned}$$

  (13)

  for $k,l\in \{1,2,\ldots ,N\}$.

• 3. The variance of $X_n$ is

  $$\begin{aligned} \mathrm{var}(X_n)=\gamma _n^2+\beta _n^2\left( \frac{2-\alpha }{2\theta }\right) \text { for } n\in \{1,2,\ldots , N\}. \end{aligned}$$

• 4. Characteristic function of $X_n$ is

  $$\begin{aligned} \phi _{X_n}(u) = \exp \left( (\mu -\beta )ui-\frac{2\theta ^{1-\frac{\alpha }{2}}}{\alpha } \left( \left( \theta -i\beta u+\frac{\gamma ^2u^2}{2}\right) ^{\frac{\alpha }{2}}-\theta ^{\frac{\alpha }{2}}\right) \right) \end{aligned}$$

Providing $\mu _n=0$ and $\gamma _n = \sqrt{1-\beta _n^2 \left( \frac{2-\alpha }{2\theta }\right) }$ with $|\beta _n|<\sqrt{ \frac{2\theta }{2-\alpha }}$ for n $\in$ $\{ 1$,2, $\cdots$,$N\}$, the N-dimensional NTS random variable X has $E[X] = (0,0,\ldots ,0)^{{\texttt {T}}}$ and $\mathrm{var}(X)$ $=$ (1,1,$\cdots$,$1)^{{\texttt {T}}}$. In this case, X is referred to as the N-dimensional standard NTS random variable with parameters $(\alpha$, $\theta$, $\beta$, $\varSigma )$ and we denote it by $X\sim stdNTS _N(\alpha$, $\theta$, $\beta$, $\varSigma )$.

For one dimensional NTS distribution, $\varSigma = 1$, we can prove the following Lemma which is changing parameterization.

Lemma 1

Let $X\sim NTS_1(\alpha$, $\theta$, $\beta$, $\gamma$, $\mu$, 1) and $\xi \sim stdNTS _1({{{\bar{\alpha }}}}$, ${{{\bar{\theta }}}}$, ${{{\bar{\beta }}}}, 1)$. Suppose ${{{\bar{\alpha }}}} = \alpha$, ${{{\bar{\theta }}}} = \theta$, and ${{{\bar{\beta }}}} = \beta /\sigma$, where $\sigma =\sqrt{\gamma ^2+\beta ^2\left( \frac{2-\alpha }{2\theta }\right) }$. Then we have $X = \mu + \sigma \xi$.

Proof

The Ch.F of X is given by

$$\begin{aligned} \phi _{X}(u)&=\exp \left( (\mu -\beta )iu-\frac{2\theta ^{1-\frac{\alpha }{2}}}{\alpha } \left( \left( \theta -i\beta u+\frac{\gamma ^2 u^2}{2}\right) ^{\frac{\alpha }{2}}-\theta ^{\frac{\alpha }{2}}\right) \right) \end{aligned}$$

(14)

By the definition of stdNTS distribution, the Ch.F of $\mu +\sigma \xi$ is equal to

$$\begin{aligned} \phi _{\mu +\sigma \xi }(u)&= E[e^{(\mu +\sigma \xi )ui}]=e^{\mu ui}E[e^{iu\sigma \xi }] \nonumber \\&=\exp \left( (\mu -{{{\bar{\beta }}}} \sigma ) iu-\frac{2{{{\bar{\theta }}}}^{1-\frac{{{{\bar{\alpha }}}}}{2}}}{{{{\bar{\alpha }}}}} \left( \left( {{{\bar{\theta }}}}-i{{{\bar{\beta }}}} \sigma u+\left( 1-{{{\bar{\beta }}}}^2\left( \frac{2-{{{\bar{\alpha }}}}}{2{{{\bar{\theta }}}}}\right) \right) \frac{\sigma ^2 u^2}{2}\right) ^{\frac{{{{\bar{\alpha }}}}}{2}}-{{{\bar{\theta }}}}^{\frac{{{{\bar{\alpha }}}}}{2}}\right) \right) . \end{aligned}$$

(15)

Hence (14)=(15) if ${{{\bar{\alpha }}}} = \alpha$, ${{{\bar{\theta }}}} = \theta$, ${{{\bar{\beta }}}} \sigma = \beta$, and $\gamma ^2 =\sigma ^2\left( 1-{{{\bar{\beta }}}}^2\left( \frac{2-\alpha }{2\theta }\right) \right)$. Since $\sigma = \beta /{{{\bar{\beta }}}}$, we have

$$\begin{aligned} \gamma ^2 = \frac{\beta ^2}{{{{\bar{\beta }}}}^2}\left( 1-{{{\bar{\beta }}}}^2\left( \frac{2-\alpha }{2\theta }\right) \right) , \end{aligned}$$

or

$$\begin{aligned} \gamma ^2{{{\bar{\beta }}}}^2 = \beta ^2\left( 1-{{{\bar{\beta }}}}^2\left( \frac{2-\alpha }{2\theta }\right) \right) = \beta ^2-\beta ^2{{{\bar{\beta }}}}^2\left( \frac{2-\alpha }{2\theta }\right) , \end{aligned}$$

and hence

$$\begin{aligned} {{{\bar{\beta }}}}^2\left( \gamma ^2+\beta ^2\left( \frac{2-\alpha }{2\theta }\right) \right) =\beta ^2. \end{aligned}$$

Therefore, we have

$$\begin{aligned} {{{\bar{\beta }}}} = \frac{\beta }{\sigma }, \end{aligned}$$

where

$$\begin{aligned} \sigma = \sqrt{\gamma ^2+\beta ^2\left( \frac{2-\alpha }{2\theta }\right) }. \end{aligned}$$

$\square$

The linear combination of NTS member variables of the NTS vector is again NTS distributed as the following proposition.

Lemma 2

Let $w = (w_1, w_2, \cdots , w_N)^{\texttt {T}}\in {{\mathbb {R}}}^N$ and $X\sim NTS _N(\alpha$, $\theta$, $\beta$, $\gamma$, $\mu$, $\varSigma )$. Then $w^{{\texttt {T}}} X \sim NTS _1(\alpha ,\theta ,{{{\bar{\beta }}}},{{{\bar{\gamma }}}},{{{\bar{\mu }}}},1)$, where

$$\begin{aligned} {{{\bar{\mu }}}}=w^{{\texttt {T}}} \mu , ~~~{{{\bar{\beta }}}}=w^{{\texttt {T}}} \beta ~~~ \text { and } ~~~ {{{\bar{\gamma }}}}=\sqrt{w^{\texttt {T}}\mathrm{diag}(\gamma )\varSigma \mathrm{diag}(\gamma ) w }. \end{aligned}$$

Proof

Since we have

$$\begin{aligned} w^{\texttt {T}}X = w^{\texttt {T}}\mu + w^{\texttt {T}}\beta ({{{\mathcal {T}}}}-1) + w^{\texttt {T}}\mathrm{diag}(\gamma )\epsilon \sqrt{{{{\mathcal {T}}}}}, \end{aligned}$$

and $w^{\texttt {T}}\mathrm{diag}(\gamma )\epsilon {\mathop {=}\limits ^{\mathrm {d}}}\sqrt{w^{\texttt {T}}\mathrm{diag}(\gamma )\varSigma \mathrm{diag}(\gamma ) w } \epsilon _0$ with $\epsilon _0\sim \varPhi (0,1)$, it is trivial. $\square$

Remark 1

In Lemma 2, the first four moments of $Y=w^{\texttt {T}}X$ are as follows [See Rachev et al. (2011) and Kim et al. (2021)]

• Mean: $\displaystyle E[Y]={{{\bar{\mu }}}}$

• Variance: $\displaystyle \mathrm{var}(Y)={{{\bar{\gamma }}}}^2+{{{\bar{\beta }}}}^2\left( \frac{2-\alpha }{2\theta }\right)$

• skewness: $\displaystyle S (Y)= \frac{{{{\bar{\beta }}}}\,\left( 2-\alpha \right) \,\left( 6\,{{{{\bar{\gamma }}}}}^2\,\theta -\alpha {{{\bar{\beta }}}}^2+4{{{\bar{\beta }}}}^2\right) }{\sqrt{2\theta }\,{\left( 2\,{{{{\bar{\gamma }}}}}^2\,\theta -\alpha {{{\bar{\beta }}}}^2+2{{{\bar{\beta }}}}^2\right) }^{3/2}}$

• Excess kurtosis:

  $\displaystyle K (Y){=}\frac{\left( 2{-}\alpha \right) \,\left( {\alpha }^2{{{\bar{\beta }}}}^4{-}10\,\alpha {{{\bar{\beta }}}}^4{-}12\,\alpha {{{\bar{\beta }}}}^2\,{{{{\bar{\gamma }}}}}^2\,\theta {+}24\beta ^4+48{{{\bar{\beta }}}}^2\,{{{{\bar{\gamma }}}}}^2\,\theta {+}12\,{{{{\bar{\gamma }}}}}^4\,{\theta }^2\right) }{2\,\theta \,{\left( 2\,{{{{\bar{\gamma }}}}}^2\,\theta {-}\alpha {{{\bar{\beta }}}}^2+2{{{\bar{\beta }}}}^2\right) }^2}$

In general, higher moments can be obtained by the cumulants $c_n(Y)$:

$$\begin{aligned} c_n(Y) = \frac{1}{i^n}\left[ \frac{d^n}{du^n}\log (\phi _Y(u))\right] _{u=0}, n = 1,2,\cdots . \end{aligned}$$

Finally we can provide the proof of Proposition 1.

1.1 Proof of Proposition 1

Proof of Proposition 1

By (1) and (2), we have

$$\begin{aligned} R&= \mu + \mathrm{diag}(\sigma )(\beta ({{{\mathcal {T}}}}-1)+\mathrm{diag}(\gamma )\epsilon \sqrt{{{{\mathcal {T}}}}})\\&= \mu + \mathrm{diag}(\sigma )\beta ({{{\mathcal {T}}}}-1)+\mathrm{diag}(\sigma )\mathrm{diag}(\gamma )\epsilon \sqrt{{{{\mathcal {T}}}}} \\ {}&\sim NTS _N (\alpha , \theta , \mathrm{diag}(\sigma ) \beta , \mathrm{diag}(\sigma ) \gamma , \mu , \varSigma ), \end{aligned}$$

where $\gamma =(\gamma _1, \gamma _2, \cdots , \gamma _N)^{{\texttt {T}}}$ with $\gamma _n = \sqrt{1-\beta _n^2 \left( \frac{2-\alpha }{2\theta }\right) }$ and ${{{\mathcal {T}}}}$ is the tempered stable subordinator with parameter $(\alpha ,\theta )$. By Lemma 2,

$$\begin{aligned} R_P(w) = w^{\texttt {T}}R \sim \mathrm{NTS}_1(\alpha , \theta , w^{\texttt {T}}\mathrm{diag}(\sigma )\beta , \sqrt{w^{\texttt {T}}\mathrm{diag}(\sigma )\mathrm{diag}(\gamma ) \varSigma \mathrm{diag}(\gamma )\mathrm{diag}(\sigma )w}, w^{\texttt {T}}\mu , 1). \end{aligned}$$

By Lemma 1, we have

$$\begin{aligned} R_P(w) = w^{\texttt {T}}\mu + {{{\bar{\sigma }}}}(w) \varXi \end{aligned}$$

where

$$\begin{aligned} {{{\bar{\sigma }}}}(w) = \sqrt{w^{\texttt {T}}\mathrm{diag}(\sigma )\mathrm{diag}(\gamma ) \varSigma \mathrm{diag}(\gamma )\mathrm{diag}(\sigma )w+(w^{\texttt {T}}\mathrm{diag}(\sigma )\beta )^2\left( \frac{2-\alpha }{2\theta }\right) } \end{aligned}$$

and

$$\begin{aligned} \varXi \sim \mathrm{stdNTS}_1(\alpha , \theta , w^{\texttt {T}}\mathrm{diag}(\sigma )\beta /{{{\bar{\sigma }}}}(w), 1). \end{aligned}$$

Also, by (13), we have

$$\begin{aligned} \left( {{{\bar{\sigma }}}}(w)\right) ^2&= \sum _{k=1}^N\sum _{l=1}^N w_k w_l \gamma _k\gamma _l\sigma _k\sigma _l \rho _{k,l} + \left( \sum _{k=1}^N w_k\sigma _k\beta _k\right) ^2\left( \frac{2-\alpha }{2\theta }\right) \\&= \sum _{k=1}^N\sum _{l=1}^N w_k w_l \sigma _k\sigma _l \left( \gamma _k\gamma _l\rho _{k,l}+\beta _k\beta _l\left( \frac{2-\alpha }{2\theta }\right) \right) \\&= \sum _{k=1}^N\sum _{l=1}^N w_k w_l \sigma _k\sigma _l\mathrm{cov}(X_k,X_l). \end{aligned}$$

Hence, we have

$$\begin{aligned} \left( {{{\bar{\sigma }}}}(w)\right) ^2 = \sum _{k=1}^N\sum _{l=1}^N w_k w_l \mathrm{cov}(R_k, R_l) = w^{\texttt {T}}\varSigma _R w \end{aligned}$$

where $\varSigma _R$ is the covariance matrix of R. $\square$

1.2 Proof of Proposition 2

Proof of Proposition 2

By (2), we have

$$\begin{aligned} R_P(w) = {{{\bar{\mu }}}}(w) + {{{\bar{\sigma }}}}(w) \varXi ~~~\text { for }~~~ \varXi \sim stdNTS _1(\alpha , \theta , {{{\bar{\beta }}}}(w), 1), \end{aligned}$$

where

$$\begin{aligned} {{{\bar{\mu }}}}(w)&= w^{\texttt {T}}\mu = \sum _{k=1}^N w_k\mu _k, \\ {{{\bar{\beta }}}}(w)&= \frac{w^{\texttt {T}}\mathrm{diag}(\sigma )\beta }{{{{\bar{\sigma }}}}(w)}= \frac{\sum _{k=1}^N w_k\sigma _k\beta _k}{{{{\bar{\sigma }}}}(w)},\\ {{{\bar{\sigma }}}}(w)&=\sqrt{w^{\texttt {T}}\varSigma _R w}=\sqrt{\sum _{k=1}^N\sum _{l=1}^N w_k w_l \mathrm{cov}(R_k, R_l)}. \end{aligned}$$

Hence, the first derivative of ${{{\bar{\beta }}}}(w)$ and ${{{\bar{\sigma }}}}(w)$ are obtained as follows:

$$\begin{aligned} \frac{\partial {{{\bar{\beta }}}}(w)}{\partial w_n}&= \frac{\partial }{\partial w_n} \left( \frac{\sum _{k=1}^N w_k \sigma _k \beta _k}{{{{\bar{\sigma }}}}(w)}\right) \nonumber \\&=\frac{\sigma _n\beta _n{{{\bar{\sigma }}}}(w) - \left( \sum _{k=1}^N w_n \sigma _k \beta _k\right) \frac{\partial {{{\bar{\sigma }}}}(w)}{\partial w_n}}{\left( {{{\bar{\sigma }}}}(w)\right) ^2} \nonumber \\&=\frac{\sigma _n\beta _n}{{{{\bar{\sigma }}}}(w)} - \frac{{{{\bar{\beta }}}}(w)}{{{{\bar{\sigma }}}}(w)} \frac{\partial {{{\bar{\sigma }}}}(w)}{\partial w_n} \end{aligned}$$

(16)

and

$$\begin{aligned} \frac{\partial {{{\bar{\sigma }}}}(w)}{\partial w_n}&= \frac{1}{2{{{\bar{\sigma }}}}(w)} \frac{\partial }{\partial w_n}\sum _{k=1}^N\sum _{l=1}^N w_k w_l \mathrm{cov}(R_k, R_l) \\&= \frac{\sum _{k=1}^Nw_k\mathrm{cov}(R_n, R_k)}{{{{\bar{\sigma }}}}(w)}. \end{aligned}$$

Since we have

$$\begin{aligned} \mathrm{VaR}_\eta (R_p(w)) = -{{{\bar{\mu }}}}(w) - {{{\bar{\sigma }}}}(w) F_\mathrm{stdNTS}^{-1}(\eta , \alpha , \theta , {{{\bar{\beta }}}}(w)) \end{aligned}$$

we obtain

$$\begin{aligned}&\frac{\partial \mathrm{VaR}_\eta (R_p(w))}{\partial w_n} \nonumber \\&\quad = -\frac{\partial {{{\bar{\mu }}}}(w)}{\partial w_n} - \frac{\partial {{{\bar{\sigma }}}}(w)}{\partial w_n} F_\mathrm{stdNTS}^{-1}(\eta , \alpha , \theta , {{{\bar{\beta }}}}(w)) - {{{\bar{\sigma }}}}(w)\frac{\partial F_\mathrm{stdNTS}^{-1}(\eta , \alpha , \theta , {{{\bar{\beta }}}}(w))}{\partial w_n} \nonumber \\&\quad =-\mu _n - \frac{\partial {{{\bar{\sigma }}}}(w)}{\partial w_n}F_\mathrm{stdNTS}^{-1}(\eta , \alpha , \theta , {{{\bar{\beta }}}}(w)) + {{{\bar{\sigma }}}}(w)\frac{\partial F_\mathrm{stdNTS}^{-1}(\eta , \alpha , \theta , \beta )}{\partial \beta }\Big |_{\beta = {{{\bar{\beta }}}}(w)}\frac{\partial {{{\bar{\beta }}}}(w)}{\partial w_n} \end{aligned}$$

(17)

By substituting (16) into (17), we obtain (4).

Since there is $\delta >0$ such that $|\phi _\varXi (-u+i\delta )|<\infty$ for all $u\in {{\mathbb {R}}}$, we have

$$\begin{aligned} \frac{\partial \mathrm{CVaR}_\eta (\varXi )}{\partial \beta }\Big |_{\beta ={{{\bar{\beta }}}}(w)}=\frac{\partial }{\partial \beta }\mathrm{CVaR}_\mathrm{stdNTS}(\eta , \alpha , \theta , \beta )\Big |_{\beta ={{{\bar{\beta }}}}(w)}. \end{aligned}$$

By (3), we have

$$\begin{aligned}&\frac{\partial }{\partial \beta }\mathrm{CVaR}_\mathrm{stdNTS}(\eta , \alpha , \theta , \beta ) \\&\quad =-\frac{\partial }{\partial \beta }F_\mathrm{stdNTS}^{-1}(u,\alpha ,\theta , \beta ) \\&\qquad -\frac{1}{\pi \eta }\mathrm{Re} \int _0^\infty e^{(iu+\delta )F_\mathrm{stdNTS}^{-1}(u,\alpha ,\theta , \beta )}\frac{\phi _\mathrm{stdNTS}(-u+i\delta , \alpha , \theta , \beta )}{(-u+i\delta )^2}\\&\times \left( (\delta +iu)\frac{\partial }{\partial \beta }F_\mathrm{stdNTS}^{-1}(u,\alpha ,\theta , \beta ) +\frac{\partial }{\partial \beta }\log \phi _\mathrm{stdNTS}(-u+i\delta , \alpha , \theta , \beta )\right) du. \end{aligned}$$

By setting $\psi (z, \alpha , \theta , \beta )=\frac{\partial }{\partial \beta }\log \phi _\mathrm{stdNTS}(z, \alpha , \theta , \beta )$, we can simplify

$$\begin{aligned} \frac{\partial }{\partial \beta }\mathrm{CVaR}_\mathrm{stdNTS}(\eta , \alpha , \theta , \beta )&=-\frac{\partial }{\partial \beta }F_\mathrm{stdNTS}^{-1}(u,\alpha ,\theta , \beta ) \\&-\frac{1}{\pi \eta }\mathrm{Re} \int _0^\infty e^{(iu+\delta )F_\mathrm{stdNTS}^{-1}(u,\alpha ,\theta , \beta )}\frac{\phi _\mathrm{stdNTS}(-u+i\delta , \alpha , \theta , \beta )}{(-u+i\delta )^2}\\&\times \left( (\delta +iu)\frac{\partial }{\partial \beta }F_\mathrm{stdNTS}^{-1}(u,\alpha ,\theta , \beta ) +\psi (-u+i\delta , \alpha , \theta , \beta )\right) du. \end{aligned}$$

The characteristic function $\phi _\mathrm{stdNTS}(u, \alpha , \theta , \beta )$ is equal to

$$\begin{aligned}&\phi _\mathrm{stdNTS}(u, \alpha , \theta , \beta ) \\&\quad =\exp \left( -\beta ui-\frac{2\theta ^{1-\frac{\alpha }{2}}}{\alpha } \left( \left( \theta -i\beta u+\left( 1-\frac{\beta ^2(2-\alpha )}{2\theta }\right) \frac{u^2}{2}\right) ^{\frac{\alpha }{2}}-\theta ^{\frac{\alpha }{2}}\right) \right) , \end{aligned}$$

hence we have

$$\begin{aligned}&\psi (z, \alpha , \theta , \beta ) \\&\quad =\frac{\partial }{\partial \beta } \left( -\beta zi-\frac{2\theta ^{1-\frac{\alpha }{2}}}{\alpha } \left( \left( \theta -i\beta z+\left( 1-\frac{\beta ^2(2-\alpha )}{2\theta }\right) \frac{z^2}{2}\right) ^{\frac{\alpha }{2}}-\theta ^{\frac{\alpha }{2}}\right) \right) \\&\quad =-zi +\left( 1-\frac{iz\beta }{\theta } +\left( 1-\frac{\beta ^2(2-\alpha )}{2\theta } \right) \frac{z^2}{2\theta } \right) ^{\frac{\alpha }{2}-1} \left( zi +\frac{\beta (2-\alpha )}{2\theta }z^2 \right) . \end{aligned}$$

As VaR case, CVaR for $R_P(w)$ is calculated using $\mathrm{CVaR}\eta (\varXi )$ that

$$\begin{aligned} \mathrm{CVaR}_\eta (R_p(w)) = -{{{\bar{\mu }}}}(w) +{{{\bar{\sigma }}}}(w) \mathrm{CVaR}_\eta (\varXi ). \end{aligned}$$

Therefore, we have

$$\begin{aligned} \frac{\partial \mathrm{CVaR}_\eta (R_p(w))}{\partial w_n}&= -\mu _n + \frac{\partial {{{\bar{\sigma }}}}(w)}{\partial w_n} \mathrm{CVaR}_\eta (\varXi ) + {{{\bar{\sigma }}}}(w)\frac{\partial \mathrm{CVaR}_\eta (\varXi )}{\partial w_n} \\&= -\mu _n + \frac{\partial {{{\bar{\sigma }}}}(w)}{\partial w_n} \mathrm{CVaR}_\eta (\varXi ) + {{{\bar{\sigma }}}}(w)\frac{\partial \mathrm{CVaR}_\eta (\varXi )}{\partial \beta }\Big |_{\beta ={{{\bar{\beta }}}}(w)}\frac{\partial {{{\bar{\beta }}}}(w)}{\partial w_n}, \end{aligned}$$

and, by (16), we obtain

$$\begin{aligned}&\frac{\partial \mathrm{CVaR}_\eta (R_p(w))}{\partial w_n} \nonumber \\&\quad = -\mu _n + \frac{\partial {{{\bar{\sigma }}}}(w)}{\partial w_n} \mathrm{CVaR}_\eta (\varXi ) + \left( \sigma _n\beta _n - {{{\bar{\beta }}}}(w) \frac{\partial {{{\bar{\sigma }}}}(w)}{\partial w_n}\right) \frac{\partial \mathrm{CVaR}_\eta (\varXi )}{\partial \beta }\Big |_{\beta = {{{\bar{\beta }}}}(w)} \end{aligned}$$

(18)

By substituting $\mathrm{CVaR}_\eta (\varXi )=\mathrm{CVaR}_\mathrm{stdNTS}(\eta ,\alpha ,\theta ,{{{\bar{\beta }}}}(w))$ into (18), we obtain (5). $\square$