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.
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- 1.
For all relevant confidence levels, this percentile is a negative number, see Eq. 21.13.
- 2.
This corresponds to the percentile Q 1−c of the standard normal distribution.
- 3.
This corresponds to the percentile − Q 1−c = Q c of the standard normal distribution.
- 4.
If all portfolio sensitivities are non-negative (which is often the case for instance for a private investor’s portfolio containing only long positions), then this equation can be rewritten in an alternative and quite intuitive form. Let VaRi(c) denote the value at risk of the portfolio with respect to a particular risk factor S i. Using Eq. 22.13, we can approximate this by
$$\displaystyle \begin{aligned} \mathrm{VaR}_{i}(c)\approx\left\vert \widetilde{\Delta}_{i}Q_{1-c}\sigma _{i}\sqrt{\delta t}\right\vert\;. \end{aligned}$$Thus, for the special case that none of the portfolio deltas is negative, the VaR with respect to all risk factors can be obtained by computing the square root of the weighted sum of the products of all the VaRs with respect to the individual risk factors. The weights under consideration are the respective correlations between the risk factors:
$$\displaystyle \begin{aligned} \mathrm{VaR}_{V}^{{}}(c)\approx\sqrt{\sum_{i,j=1}^{n}\mathrm{VaR}_{i} (c)\rho_{ij}\mathrm{VaR}_{j}(c)} \ \ \text{falls} \widetilde{\Delta}_{i} \geq0\forall i\in\left\{ 1,\ldots,n\right\}\;.{} \end{aligned} $$(22.16) - 5.
This goes back to a paper by Rouvinez, see [166].
- 6.
The most important results required for the analysis here receive a clear and concise treatment in [78], for example.
- 7.
\(\left ({\mathbf {O}}^{T}\mathbf {O}\right ) _{ij}=\sum _{k}(O^{T})_{ik}O_{kj}=\sum _{k}e_{k}^{i}e_{k}^{j}=\delta _{ij}\;.\)
- 8.
The notation i∉J denotes all indices i with eigenvalue λ i = 0, i.e., the set \(\left \{ 1,\ldots ,n\left \vert \lambda _{i}=0\right . \right \} \).
- 9.
Here the well-known cyclic property of the trace has been used: \(\operatorname {tr}\left ( \mathbf {ABC\,}\right ) =\operatorname {tr} \left ( \mathbf {BCA\,}\right ) \) for arbitrary matrices A,B,C.
- 10.
Recall that a normal distribution has skewness 0 and kurtosis 3, see Eq. A.59.
- 11.
Here, i denotes the imaginary number satisfying the property i 2 = −1, thus intuitively \(i=\sqrt {-1}.\)
- 12.
- 13.
It should not be forgotten that the Delta-Gamma method itself is only an approximation of the portfolio’s value obtained from the second-order Taylor series approximation.
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Deutsch, HP., Beinker, M.W. (2019). The Variance-Covariance Method. In: Derivatives and Internal Models. Finance and Capital Markets Series. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-22899-6_22
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