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Correlated Risk Diversification

Our goal is to relate the quantile adverse move when there are several positions to the quantile adverse move for each position estimated separately.

The PnL $P$ due to a $n$ factor gaussian market move $x$ is linear, given by: $$ P = \delta . x $$ The problem of finding the PnL quantile $q$ is that of finding the $\bar{x}$ corresponding to the most adverse move $$\max \delta . x$$ subject to $$ |x|_{\Sigma^{-1}} = \Phi^{-1}(q)$$ The joint normal density of $x$ implies that quantiles are geometrically on an ellipsoid. These are given by the Mahalinobis distance defined by the scalar product $M=\Sigma^{-1}$, where $\Sigma$ is the covariance of $x$.

Solution with Lagrangian

This optimisation problem has the lagrangian $$ L(x,\lambda) = \delta . x + \lambda (x' \Sigma^{-1} x - p^2) $$ where $p$ isthe normal inverse cumulative of the quantile $q, p := \Phi^{-1}(q)$.

The lagrangian gradient is: $$ \frac{\partial L }{\partial x_i }= \delta_i + \lambda (\sum_j m_{ij} x_j + m_{ii} x_i) = 0 $$ $$ \frac{\partial L }{\partial \lambda}= (\sum_{ij} m_{ij} x_jx_i -p^2) = 0 $$ where $M := (m_{ij}) = \Sigma^{-1}$, and $d(M) = (m_{ij} \delta_{ij})$ is the diagonal matrix made from $M$.

The optimal $\bar{x},\bar{\lambda}$ is given by: $$ \bar{x} = - \frac{1}{\bar{\lambda}} [M + d(M)]^{-1} \delta $$ $$ \bar{\lambda} = - \frac{1}{p} |[M + d(M)]^{-1} \delta|_M $$

From Univariate quantiles to Multivariate adverse quantile

We therefore obtain a relation between the most adverse quantile pnl move when each position is considered separately: $\bar{P}(i) = \delta_i p \sigma_{ii}^{1/2}$ and the most adverse quantile PnL move when positions are correlated using $\bar{P} = \delta . \bar{x}$.

That relation is based on the covariance of $x$ $\Sigma$.