# CVaR Derivatives for Normal Mixture Distributions

This section computes the first and second derivatives of CVaR for normal
mixture distributions, following {ref}`RauHasanov2004 <rauhasanov2004>` and {ref}`Tasche1999 <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:

```{math}
r_{\operatorname{VaR}_\alpha}(w) &:= \operatorname{VaR}_\alpha(w^\top X), \\
r_{\operatorname{CVaR}_\alpha}(w) &:= \operatorname{CVaR}_\alpha(w^\top X).
```

**Assumption.** Let $p(x_1 | x_2, \ldots, x_n)$ be the conditional
density of $X_1$ given $X_2, \ldots, X_n$. Assume:

1. $y \mapsto p(y | x_2, \ldots, x_n)$ is continuous.
2. $(y, w) \mapsto E[p(w_1^{-1}(y - \sum_{l=2}^n w_l X_l) | X_2, \ldots, X_n)]$ is finite and continuous.
3. The density at the VaR quantile is strictly positive.
4. $(y, w) \mapsto E[X_j \, p(w_1^{-1}(y - \sum_{l=2}^n w_l X_l) | X_2, \ldots, X_n)]$ is finite and continuous.
5. $(y, w) \mapsto E[X_j X_k \, p(w_1^{-1}(y - \sum_{l=2}^n w_l X_l) | X_2, \ldots, X_n)]$ is finite and continuous.

### First Derivatives

Under conditions 1--4 above, for $w_1 \neq 0$:

```{math}
\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),
```

and

```{math}
:label: cvar-gradient

\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 $j, k = 2, \ldots, n$:

```{math}
:label: cvar-hessian

\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 $j = 1, \ldots, n$:

```{math}
\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 {eq}`nm-def` can be viewed as a "portfolio"
with two risky assets. Define:

```{math}
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),
```

where $Z \sim N(0, 1)$ and $\sigma > 0$. Denote the standard
normal density by $\varphi$ and CDF by $\Phi$.

### First Derivatives

```{math}
\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),
```

and

```{math}
:label: cvar-nm-grad

\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).
```

### Second Derivatives

```{math}
:label: cvar-nm-hessian

\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}.
```

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 {eq}`nm-portfolio` 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:

```{math}
\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:

```{math}
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}.
```