## VaR Attribution using the Component VaR Approach

Calculating the variance-covariance value at risk of a portfolio is great but being able to identify how much each component of the portfolio contributes to the total portfolio VaR is even better. The Component VaR is one way to achieve that analysis.

# Calculation of Parametric Variance-Covariance Value at Risk

The VaR over d days at a confidence level c is calculated as
$VaR(c,d) = F(c,d)\sigma$
where
$F(c,d) = \mathcal{N}^{-1}(c,0,1)\sqrt{d}$ is a scaling factor
$\sigma$ is the daily volatility
Note: $\sigma$ can be scaled back from a weekly, monthly or annualised volatility by applying the following:
$\sigma_daily = \frac{\sigma_weekly}{\sqrt{5}} = \frac{\sigma_monthly}{\sqrt{21}} = \frac{\sigma_yearly}{\sqrt{260}}$
$\sigma = \sqrt{\omega^t\Gamma\omega}$
where $\omega = \begin{bmatrix}\omega_i \end{bmatrix}$ are the weights of the securities in the portfolio,
and $\Gamma = \begin{bmatrix} \gamma_{i,j} \end{bmatrix}$ is the covariance matrix.

# Component VaR

## Concept

The idea behind Component VaR (CVaR) is to find a way of calculating the contribution to the total VaR of a portfolio of any sub-portfolios, including individual positions. Asusming $(P_i)_{i=1,n}$ is a partition of the total portfolio, i.e. the union of all the sub-portfolios is exactly equal to the total portfolio, the CVaR has the following properties:

• $\sum^n_{i=1}{CVaR(P_i)} = VaR(P)$, the sum of the CVaRs is the total VaR
• $VaR(P\setminus P_i) \approx VaR(P) - CVaR(P_i)$, if a sub-portfolio is removed, the impact on total VaR is approximately its CVaR
• if a sub-portfolio is a hedge, its CVaR should be negative

## Definition

VaRDelta is defined by [Garman] as $\Delta=\frac{1}{\sigma}\Gamma\omega$
VaRDelta has the following interesting property:
$^t\omega\Delta = \frac{^t\omega\Gamma\omega}{\sqrt{^t\omega\Gamma\omega}} = \sqrt{^t\omega\Gamma\omega} = \sigma$

The Component VaR of the security i is then defined as $CVaR_i(c,d) = {^t\omega}_{\delta_{i}}\Delta F(c,d)$,
where $\omega_{\delta_{i}} = \begin{bmatrix}\omega_k \delta_{ik} \end{bmatrix}$ and $\delta_{ik}$ is the kronecker symbol.