It is possible to carry out the Component VaR decomposition through non parametric methodologies?

Question

I would like to know if it is possible to break down the VaR risk (using historical and Monte Carlo methodology) among the portfolio assets just like the Component VaR concept in the parametric methodology.

For example lets say we have a parametric VaR of 1000 dollars with 95% confidence interval. From those 1000, 250 correspond to asset A, 500 to asset B and 250 to asset C.

What I am looking forward to know is whether this same breakdown into Component VaRs can be obtain through non-parametric methodologies such as historical VaR or Monte Carlo VaR.

Answer 1

Yes, it is possible in general to allocate portfolio VaR to single positions in a portfolio. This is based on a general result on homogeneous risk measures called the Euler allocation. This allocation provides the marginal change in your risk measure when the size of one position changes (for an infinitesimally) small amount. For VaR this gradient can be expressed as follows: Let $L=\sum L_i$ denote your portfolio's P&L then the marginal increase of VaR in direction $L_i$ will be $$ \frac{d \text{VaR}(L + h L_i)}{dh} = E[L_i | L=VaR].$$ This equality is non-trivial for more details see the papers by Tasche, start here.

No assumptions (except mild technical ones) on the distribution of $L$ or the $L_i$ are necessary to derive this result. But if you make normal ("parametric") assumptions for your P&L the conditional expectation on the RHS can be readily calculated and leads to the well known formulas for parametric Component VaR.

In case of historical or Monte Carlo estimates, i.e. estimates based on samples, the conditional expectation poses a challenge since $L=VaR$ will generally be a set of measure zero.

In the Monte Carlo case this can be solved by broadening the condition or interpolation (i.e. you condition on $VaR-\epsilon<L<VaR + \epsilon$ and interpolate between quantiles). Another possibility is to use kernel estimates to estimate the conditional probability. Again the paper cited above provides more details.

Theoretically the same applies for historical observations, but of course the data/sample situation is even worse. Unless you have plenty of observations at the relevant quantiles this is probably fruitless.

Answer 2

For historical simulation you need a vector of losses $L_{i,t}$ for each asset $i$ in the portfolio and each day $t$ in the look back period. The portfolio loss is the sum of the asset losses: $$L_{t} = \sum_{i} L_{i,t}$$

In order to calculate the $VaR$ we determine the date $t^{\star}$ with the n'th highest portfolio loss, where n is determined by your number of days and your significance level (e.g. 95%)

$$VaR = L_{t^{\star}} = \sum_{i}{L_{i,t^{\star}}}$$

From here it seems natural to define the $component$ $VaR$ of asset $i$ as the contribution of that asset to the above sum: $$VaR_{i}^{comp}=L_{i,t^{\star}}$$

Be aware, that $L_{i,t^{\star}}$ is not the standalone $VaR_{i}$ for asset $i$, since the $t^{\star}$ has been chosen considering the whole portfolio. Also to derive the exact relationship between stand alone $VaR_{i}$ and $VaR^{comp}_{i}$ you need to assume something about the involved distributions.