The Variance — Covariance Method

• Hans-Peter Deutsch
Part of the Finance and Capital Markets Series book series (FCMS)

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 19.24, we have

$$\matrix{ {\delta {S_i}\left( t \right) \approx {S_i}\left( t \right)\,\left[ {\mu i\delta t\, + \,\delta {Z_i}} \right]\, \approx \,{S_i}\left( t \right)\delta {Z_i}} \cr }$$
(20.1)

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 Si, j = 1,…s, n, and approximated by breaking off after the first or second order term. Let

$$S\left( t \right)\, = \,\left( {\matrix{ {{S_1}\left( t \right)} \cr \vdots \cr {{S_n}\left( t \right)} \cr } } \right)$$

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

$$\matrix{ {\delta V\left( {{\mathop{\rm S}\nolimits} \left( t \right)} \right)} \hfill & {V\left( {S\left( t \right)\, + \,\delta {\rm{S}}\left( t \right)} \right)\, - \,V\left( {S\left( t \right)} \right)} \hfill & {} \hfill \cr {} \hfill & { \approx \sum\limits_i^n {{{\partial V} \over {\partial {S_i}}}\delta {S_i}\left( t \right)\, + \,{1 \over 2}\sum\limits_{i,j}^n {\delta S\int_{}^{} i \left( t \right){{{\partial ^2}V} \over {\partial {S_i}\partial {S_j}}}\delta {S_j}\left( t \right)} } } \hfill & {} \hfill \cr {} \hfill & { = \sum\limits_i^n {{\Delta _i}\delta {S_i}\left( t \right)\, + \,{1 \over 2}\sum\limits_{i,j}^n {\delta {S_i}\left( t \right){\Gamma _{ij}}\delta {S_j}\left( t \right)} } } \hfill & {} \hfill \cr {} \hfill & { \approx \sum\limits_i^n {{{\tilde \Delta }_i}\left[ {{\mu _i}\delta t\, + \,\delta {Z_i}} \right]} \, + \,{1 \over 2}\sum\limits_{i,j}^n {\left[ {{\mu _i}\delta t\, + \,\delta {Z_i}} \right]{{\tilde \Gamma }_{ij}}\left[ {{\mu _j}\delta t\, + \,\delta {Z_j}} \right]} } \hfill & {} \hfill \cr {} \hfill & { \approx \sum\limits_i^n {{{\tilde \Delta }_i}\delta {Z_i}\, + \,{1 \over 2}\sum\limits_{i,j}^n {\delta {Z_i}{{\tilde \Gamma }_{ij}}\delta {Z_j}} } } \hfill \cr}$$
(20.2)

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 20.1, and finally, in the last step, because the drift has been neglected. The last line in 20.2 is referred to as the delta-gamma approximation. An analogous approach leads to the delta approximation, for which the Taylor series in the above derivation is taken up to linear order only, resulting in an approximation solely consisting of the first of the two sums appearing in the last equation in 20.2.

Keywords

Internal Model Moment Generate Function Individual Risk Factor Short Position Single Risk Factor
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.