# Incremental/marginal contribution to VaR in a simulation setting

Estimating marginal contributions to VaR in a simulation setting is apparently quite difficult (see e.g. this blog post) due to issues with sampling variability. My question is whether the following approach for incremental (where a position is removed in entirety) has the same issues. In practice I am seeing a lot of variability in the figures, hence my question.

Let $$P$$ be a portfolio on $$n$$ assets $$X_1, X_2, \dots, X_n$$. Suppose also that we are in a simulation setting and so that we have, for some $$k$$ scenarios $$1,2,\dots, k$$, the returns for the portfolio $$P$$ under scenario $$j$$ given by $$R^j = \sum_{i=1} R_i^j$$ Where $$R_i^j$$ denotes the return of asset $$i$$ under scenario $$j$$. The $$\mathrm{VaR}_\alpha(P)$$ for portfolio $$P$$ is then simply the $$\lfloor (1-\alpha)k \rfloor$$ smallest element of the vector $$R_P = (R^1, R^2, \dots, R^k)$$.

I wish to calculate the incremental VaR, given by $$\mathrm{iVaR}_\alpha(P_i) = \mathrm{VaR}_\alpha(P) - \mathrm{VaR}_\alpha(P - P_i)$$

To calculate the second term in the above expression I simply subtract the component vector $$R_{P_i} = (R_i^1, R_i^2, \dots, R_i^k)$$ from $$R_P$$ and find the new $$\lfloor (1-\alpha)k \rfloor$$ smallest element.

My question is: is this a sound approach? I am seeing quite a lot of variability in the iVaR figures and so I worry that this approach have the same statistical issues.

If this approach is indeed not problematic, then surely $$\frac{\mathrm{VaR}_\alpha(P) - \mathrm{VaR}_\alpha(P - hP_i)}{h}$$ Should be a decent approximation to the marginal var, i.e. $$\partial \mathrm{VaR}_\alpha/\partial P_i$$?

Apologies if these questions are basic - I am new in the quant scene and google has unfortunately failed me.

As I see it, in both, a (MC) simulation or a historical simulation, risk estimators (VaR, iVaR, mVaR) suffer from the instability of the quantile. If we had a sufficiently “dense” set of observations around the $$(1-\alpha)$$ quantile strip, we could compute a weighted average around that quantile and find risk factor and instrument contributions.