The Variance-Covariance Method

Abstract

The variance-covariance method makes use of covariances (volatilities and correlations) of the risk factors and the sensitivities of the portfolio values with respect to these risk factors with the goal of approximating the value at risk. This method leads directly to the final result, i.e., the portfolio’s value at risk; no information regarding market scenarios arises. The variance-covariance method utilizes linear approximations of the risk factors themselves throughout the entire calculation, often neglecting the drift as well. In view of Equation 21.24, we have

$$\delta {S_i}\left( t \right) \approx {S_i}\left( t \right)\left[ {{\mu _i}\delta t + \delta {Z_i}} \right] \approx {S_i}\left( t \right)\delta {Z_i}$$

((22.1))

The main idea characterizing this method, however, is that the portfolio value V is expanded in its Taylor series as a function of its risk factors S _i , i = 1, … n, and approximated by breaking off after the first or second order term. Let

$$S\left( t \right) = \left( {\begin{array}{*{20}{c}} {{S_1}\left( t \right)} \\ \vdots \\ {{S_n}\left( t \right)} \end{array}} \right)$$

denote the vector of risk factors. The Taylor expansion for the change in portfolio value δV(S) up to second order is

$$\begin{gathered} \delta V\left( {S\left( t \right)} \right) = V\left( {S\left( t \right) + \delta S\left( t \right)} \right) - V\left( {S\left( t \right)} \right) \hfill \\ \quad \quad \quad \quad \approx \sum\limits_i^n {\frac{{\partial V}}{{\partial {S_i}}}\delta {S_i}\left( t \right)} + \frac{1}{2}\sum\limits_{i,j}^n {\delta {S_i}\left( t \right)\frac{{{\partial ^2}V}}{{\partial {S_i}\partial {S_j}}}\delta {S_j}\left( t \right)} \hfill \\ \quad \quad \quad \quad = \sum\limits_i^n {{\Delta _i}\delta {S_i}\left( t \right)} + \frac{1}{2}\sum\limits_{i,j}^n {\delta {S_i}\left( t \right){\Gamma _{ij}}\delta {S_j}\left( t \right)} \hfill \\ \quad \quad \quad \quad \approx \sum\limits_i^n {{{\widetilde \Delta }_i}\left[ {{\mu _i}\delta t + \delta {Z_i}} \right]} + \frac{1}{2}\sum\limits_{i,j}^n {\left[ {{\mu _i}\delta t + \delta {Z_i}} \right]{{\widetilde \Gamma }_{ij}}\left[ {{\mu _j}\delta t + \delta {Z_j}} \right]} \hfill \\ \quad \quad \quad \quad \approx \sum\limits_i^n {{{\widetilde \Delta }_i}\delta {Z_i}} + \frac{1}{2}\sum\limits_{i,j}^n {\delta {Z_i}{{\widetilde \Gamma }_{ij}}\delta {Z_j}} \hfill \\ \end{gathered} $$

((22.2))

The first “approximately equal” sign appears due to having broken off the Taylor series of the portfolio value, the second as a result of the linear approximation of the risk factors in accordance with Equation 22.1, and finally, in the last step, because the drift has been neglected.