This section computes the first and second derivatives of CVaR for normal
mixture distributions, following [RauHasanov2004] and [Tasche1999].
General CVaR Derivatives
Let \(X = (X_1, \ldots, X_n)\) be a random vector with portfolio weights
\(w \in \mathbb{R}^n\), and define:
\[\begin{split}r_{\operatorname{VaR}_\alpha}(w) &:= \operatorname{VaR}_\alpha(w^\top X), \\
r_{\operatorname{CVaR}_\alpha}(w) &:= \operatorname{CVaR}_\alpha(w^\top X).\end{split}\]
Assumption. Let \(p(x_1 | x_2, \ldots, x_n)\) be the conditional
density of \(X_1\) given \(X_2, \ldots, X_n\). Assume:
\(y \mapsto p(y | x_2, \ldots, x_n)\) is continuous.
\((y, w) \mapsto E[p(w_1^{-1}(y - \sum_{l=2}^n w_l X_l) | X_2, \ldots, X_n)]\) is finite and continuous.
The density at the VaR quantile is strictly positive.
\((y, w) \mapsto E[X_j \, p(w_1^{-1}(y - \sum_{l=2}^n w_l X_l) | X_2, \ldots, X_n)]\) is finite and continuous.
\((y, w) \mapsto E[X_j X_k \, p(w_1^{-1}(y - \sum_{l=2}^n w_l X_l) | X_2, \ldots, X_n)]\) is finite and continuous.
First Derivatives
Under conditions 1–4 above, for \(w_1 \neq 0\):
\[\begin{split}\frac{\partial r_{\operatorname{VaR}_\alpha}}{\partial w_j}(w)
&= -\frac{E\!\left[X_j \, p\!\left(w_1^{-1}\!\left(
-r_{\operatorname{VaR}_\alpha}(w) - \sum_{l=2}^n w_l X_l\right)
\middle| X_2, \ldots, X_n\right)\right]}
{E\!\left[p\!\left(w_1^{-1}\!\left(
-r_{\operatorname{VaR}_\alpha}(w) - \sum_{l=2}^n w_l X_l\right)
\middle| X_2, \ldots, X_n\right)\right]},
\quad j = 2, \ldots, n, \\
\frac{\partial r_{\operatorname{VaR}_\alpha}}{\partial w_1}(w)
&= w_1^{-1} \left(r_{\operatorname{VaR}_\alpha}(w)
- \sum_{j=2}^n w_j
\frac{\partial r_{\operatorname{VaR}_\alpha}}{\partial w_j}(w)\right),\end{split}\]
and
(1)\[\frac{\partial r_{\operatorname{CVaR}_\alpha}}{\partial w_j}(w)
= -E[X_j \mid w^\top X \leq -r_{\operatorname{VaR}_\alpha}(w)]
= -\alpha^{-1} E\!\left[X_j \,
\mathbf{1}_{\{w^\top X \leq -r_{\operatorname{VaR}_\alpha}(w)\}}\right],\]
for \(j = 1, \ldots, n\).
Second Derivatives
Under the full assumption, for \(j, k = 2, \ldots, n\):
(2)\[\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}
{\partial w_j \partial w_k}(w)
= \frac{1}{\alpha |w_1|}
E\!\left[X_k \left(
\frac{\partial r_{\operatorname{VaR}_\alpha}}{\partial w_j}(w)
+ X_j\right) p\!\left(w_1^{-1}\!\left(
-r_{\operatorname{VaR}_\alpha}(w)
- \sum_{l=2}^n w_l X_l\right)
\middle| X_2, \ldots, X_n\right)\right],\]
and for \(j = 1, \ldots, n\):
\[\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}
{\partial w_j \partial w_1}(w)
= -w_1^{-1} \sum_{k=2}^n w_k
\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}
{\partial w_j \partial w_k}(w).\]
This second equation follows from the 1-homogeneity of CVaR:
\(\sum_{k=1}^n w_k \,
\partial^2 r_{\operatorname{CVaR}_\alpha} / \partial w_j \partial w_k = 0\).
Application to Univariate Normal Mixtures
The univariate normal mixture (1) can be viewed as a “portfolio”
with two risky assets. Define:
\[\begin{split}r_{\operatorname{VaR}_\alpha}(\mu, \gamma, \sigma) &:=
\operatorname{VaR}_\alpha(\mu + \gamma Y + \sigma \sqrt{Y} Z), \\
r_{\operatorname{CVaR}_\alpha}(\mu, \gamma, \sigma) &:=
\operatorname{CVaR}_\alpha(\mu + \gamma Y + \sigma \sqrt{Y} Z),\end{split}\]
where \(Z \sim N(0, 1)\) and \(\sigma > 0\). Denote the standard
normal density by \(\varphi\) and CDF by \(\Phi\).
First Derivatives
\[\begin{split}\frac{\partial r_{\operatorname{VaR}_\alpha}}{\partial \mu} &= -1, \\
\frac{\partial r_{\operatorname{VaR}_\alpha}}{\partial \gamma}
&= -\frac{E\!\left[\sqrt{Y} \, \varphi\!\left(
\frac{-r_{\operatorname{VaR}_\alpha} - \mu - \gamma Y}
{\sigma \sqrt{Y}}\right)\right]}
{E\!\left[\frac{1}{\sqrt{Y}} \, \varphi\!\left(
\frac{-r_{\operatorname{VaR}_\alpha} - \mu - \gamma Y}
{\sigma \sqrt{Y}}\right)\right]}, \\
\frac{\partial r_{\operatorname{VaR}_\alpha}}{\partial \sigma}
&= \sigma^{-1} \left(r_{\operatorname{VaR}_\alpha} + \mu
- \gamma \frac{\partial r_{\operatorname{VaR}_\alpha}}
{\partial \gamma}\right),\end{split}\]
and
(3)\[\begin{split}\frac{\partial r_{\operatorname{CVaR}_\alpha}}{\partial \mu} &= -1, \\
\frac{\partial r_{\operatorname{CVaR}_\alpha}}{\partial \gamma}
&= -\alpha^{-1} E\!\left[Y \, \Phi\!\left(
\frac{-r_{\operatorname{VaR}_\alpha} - \mu - \gamma Y}
{\sigma \sqrt{Y}}\right)\right], \\
\frac{\partial r_{\operatorname{CVaR}_\alpha}}{\partial \sigma}
&= \sigma^{-1} \left(r_{\operatorname{CVaR}_\alpha} + \mu
- \gamma \frac{\partial r_{\operatorname{CVaR}_\alpha}}
{\partial \gamma}\right).\end{split}\]
Second Derivatives
(4)\[\begin{split}\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}{\partial \mu^2}
= \frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}
{\partial \mu \, \partial \gamma}
= \frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}
{\partial \mu \, \partial \sigma} &= 0, \\
\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}{\partial \gamma^2}
&= \frac{1}{\alpha \sigma} E\!\left[\sqrt{Y} \, \varphi\!\left(
\frac{-r_{\operatorname{VaR}_\alpha} - \mu - \gamma Y}
{\sigma \sqrt{Y}}\right)
\left(\frac{\partial r_{\operatorname{VaR}_\alpha}}{\partial \gamma}
+ Y\right)\right], \\
\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}
{\partial \gamma \, \partial \sigma}
&= -\frac{\gamma}{\sigma}
\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}{\partial \gamma^2}, \\
\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}{\partial \sigma^2}
&= -\frac{\gamma}{\sigma}
\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}
{\partial \gamma \, \partial \sigma}.\end{split}\]
All derivatives can be computed via Monte Carlo by generating i.i.d. samples
of the mixing variable \(Y\).
Portfolio CVaR Gradient and Hessian
For a portfolio \(w\), using (2) we can write
\(r_{\operatorname{CVaR}_\alpha}(w) =
r_{\operatorname{CVaR}_\alpha}(w^\top \mu, w^\top \gamma,
\sqrt{w^\top \Sigma w})\). The chain rule gives:
\[\frac{\partial r_{\operatorname{CVaR}_\alpha}}{\partial w_j}(w)
= -\mu_j + \gamma_j \frac{\partial r_{\operatorname{CVaR}_\alpha}}
{\partial \gamma}
+ \frac{(\Sigma w)_j}{\sqrt{w^\top \Sigma w}}
\frac{\partial r_{\operatorname{CVaR}_\alpha}}{\partial \sigma},\]
where the partial derivatives on the right are evaluated at
\((w^\top \mu, w^\top \gamma, \sqrt{w^\top \Sigma w})\).
The Hessian matrix is:
\[\begin{split}H_{r_{\operatorname{CVaR}_\alpha}}(w)
&= \gamma \gamma^\top
\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}{\partial \gamma^2}
+ (w^\top \Sigma w)^{-1/2}
(\gamma w^\top \Sigma + \Sigma w \, \gamma^\top)
\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}
{\partial \gamma \, \partial \sigma} \\
&\quad + (w^\top \Sigma w)^{-1}
\Sigma w \, w^\top \Sigma
\frac{\partial^2 r_{\operatorname{CVaR}_\alpha}}{\partial \sigma^2} \\
&\quad + (w^\top \Sigma w)^{-3/2}
(\Sigma \, w^\top \Sigma w - \Sigma w \, w^\top \Sigma)
\frac{\partial r_{\operatorname{CVaR}_\alpha}}{\partial \sigma}.\end{split}\]