# Deriving conditional variance expression and CDF expansion

Question: Assume a bivariate GARCH process as follows: \begin{align} r_{mt} &= \sigma_{mt}\epsilon_{mt} \ \ \ \cdots \ \ \ \text{(1)} \\ r_{it}&=\sigma_{it}\rho_{it}\epsilon_{mt}+\sigma_{it}\sqrt{1-\rho_{it}^2}\xi_{it} \ \ \ \cdots \ \ \ \text{(2)} \\ (\epsilon_{mt}, \xi_{it}) & \sim S \end{align}

where:

$\sigma_{mt}$ is the conditional standard deviation of $r_{mt}$

$\sigma_{it}$ is the conditional standard deviation of $r_{it}$

$\rho_{it}$ is the conditional correlation between $r_{it}$ and $r_{mt}$

$(\epsilon_{mt}, \xi_{it})$ are the shocks that drive the system. The shocks $(\epsilon_{mt}, \xi_{it})$ are independent and identically distributed and have zero mean, unit variance and zero covariance. However they are not necessarily independent of each other. The distribution $S$ is unspecified.

From Eqns. $(1)$ and $(2)$ we have:

$$r_{mt} = \frac{\sigma_{mt}}{\sigma_{it}\rho_{it}}r_{it} - \frac{\sigma_{mt}\sqrt{1-\rho_{it}^2}}{\rho_{it}}\xi_{it}$$

Define $CoVaR$ as:

$$Pr\left(r_{mt} \le CoVaR \mid r_{it} = C \right)=\alpha$$

or equivalently,

$$1-Pr\left(r_{mt} \le CoVaR \mid r_{it} = C \right)=1-\alpha \\ Pr\left(\xi_{it} \le \frac{\rho_{it}}{\sigma_{mt}\sqrt{1-\rho_{it}^2}} \left(\frac{\sigma_{mt}}{\sigma_{it}\rho_{it}}C - CoVaR \right)\mid r_{it} = C \right)=1-\alpha \ \ \ \cdots \ \ \ (3)$$

Need clarification #1: The question then goes on to state: In the special case where the conditional mean function of $\xi_{it}$ is linear in $r_{it}$, the first two conditional moments of $\xi_{it}$, given $r_{it} = C$, can be expressed as:

$$\displaystyle \mathbb{E}\left(\xi_{it} \mid r_{it} = C\right) = \frac{cov(\xi_{it}, r_{it})}{\sigma_{it}^2} \times C \\ \mathbb{V}(\xi_{it} \mid r_{it} ) = \mathbb{V}(\xi_{it}) - \mathbb{V}_{r_{it}}\left[\mathbb{E}\left(\xi_{it} \mid r_{it} \right) \right] = \mathbb{V}(\xi_{it})\times\left[1-\left( \frac{cov(\xi_{it}, r_{it})}{\sigma_{it}^2}\right)^2 \sigma_{it}^2 \right]$$

Where does the expression $\displaystyle \mathbb{V}(\xi_{it})\times\left[1-\left( \frac{cov(\xi_{it}, r_{it})}{\sigma_{it}^2}\right)^2 \sigma_{it}^2 \right]$ come from? How do you derive this expression?

Need clarification #2: The question continues to state: Consider $G(\cdot)$, the conditional (location-scale) demeaned and standardized cdf of $\xi_{it}$ such that:

$$\mathbb{E}\left[\frac{1}{\rho_{it}}\left(\xi_{it} - \frac{\sqrt{1-\rho_{it}^2}}{\sigma_{it}} \times C \right) \mid r_{it} = C\right] =0 \\ \mathbb{V}\left[\frac{1}{\rho_{it}}\left(\xi_{it} - \frac{\sqrt{1-\rho_{it}^2}}{\sigma_{it}} \times C \right) \mid r_{it} = C \right] =1$$

Then Eqn. $(3)$ can be expressed as:

$$\frac{1}{\rho_{it}} \left[ \frac{\rho_{it}}{\sigma_{mt}\sqrt{1-\rho_{it}^2}} \left(\frac{\sigma_{mt}}{\sigma_{it}\rho_{it}}C - CoVaR \right) - \frac{\sqrt{1-\rho_{it}^2}}{\sigma_{it}} \times C \right] = G^{-1} \left(1-\alpha\right) \ \ \ \cdots (4)$$

Where does Eqn. $(4)$ come from? How do you derive it?