# 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.

## 1 Answer

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.

Yet, in practice, this is not feasible and we need to resort to some of the estimators you brought forward.

Depending on the task at hand (risk contribution or risk increment?), your incremental ansatz will of course jump from scenario to scenario, if the investment size is material. The marginal ansatz you brought forward will, for small enough step sizes, be quite robust and interpretable as it approximates the marginal VaR contribution.

But even with this ‘stability’ of the estimation, you may still not see a stable contribution vector across all investments. In effect, you cannot solve the issues with the MC sample size, but you can solve the issue with jumping from sample to sample by using your second estimator.

• Thanks for the response. This has been my thinking as well. I don't have any experience with marginal risk measures like mVaR so I was surprised to see the large daily variance. In one case the actual stand-alone VaR changes only 5% yet the marginal contribution changes by over 400%. This also seems to be the case for our linear (normal model) so I suppose that's just part of the deal with these estimators? – Jimi Apr 29 '20 at 19:09
• Yep, that’s m understanding – Kermittfrog Apr 29 '20 at 19:14