Allocating Portfolio VaR with Component Risk Contributions
Summary
The document asks how to assign a parametric portfolio Value at Risk (VaR) across three assets. It contrasts allocating VaR by portfolio weights, which ignores differences in volatility, with allocating it in proportion to each asset’s standalone VaR, which ignores correlations. The replies recommend component VaR and distinguish it from marginal and incremental VaR, noting that the appropriate calculation depends on the VaR method used.
For a variance-covariance model, one reply describes component contributions using portfolio VaR, asset weights, and beta-like risk sensitivities, so contributions reflect portfolio interactions rather than standalone risk alone. The intended property is that component contributions sum to portfolio VaR under the stated model. The exchange is brief and its displayed notation is difficult to interpret, so it does not provide a full derivation or general implementation procedure. It also cautions that VaR is not generally additive and that other estimation methods may not have the same linear decomposition.
Key ideas
- Component VaR assigns portfolio risk to assets while accounting for their relationship to the portfolio.
- Weight-only allocation omits differences in asset volatility.
- Allocating in proportion to standalone VaR omits correlation effects.
- Marginal, incremental, and component VaR are distinct ways to describe asset contributions.
- The decomposition depends on the VaR estimation method, and VaR is not generally additive.
Tags
Full text
# Assigning Global VaR to portfolio members
# Assigning Global VaR to portfolio members
Assuming that I calculate a parametric VaR of a portfolio with 3 assets, and I need to assign the amount each asset (equity) contributes to the VaR.
Lets say that:
- $C$: Is the correlation matrix
- $w$: Is a vector of the weight of each asset
- $s$: Is the vector of the standard deviation of each asset (volatility)
- $VaR_t$: Total VaR of the combined portfolio
- $VaR_i$: Is the individual VaR of the Asset $i$
- $VaR_i'$: Is the part of the $VaR_t$ assigned to the Asset $i$
What is the best way to do it?
I thought 2 different ways but I don't feel comfortable with them:
- Using the w vector so each VaR will be $$VaR_i=VaR_t \cdot w_i.$$ The problem is that I don't take into account the volatility
- Using the individual VaR of each asset, so $$VaR_1'=\frac{VaR_t \cdot VaR_1}{VaR_1+VaR_2+VaR_3}.$$ The problem is that I don't take into account the correlation.
## Answer by sen_saven (score 2)
https://quant.stackexchange.com/a/38505
You wrote: "I need to assign the amount each asset (equity) contributes to the VaR"
There are a few flavours of VaR that focus on this: marginal VaR, incremental VaR, component VaR, etc.
I would have a look at this paper for further info: Decomposing Portfolio Value-at-Risk: A General Analysis, by Winfried G. Hallerbach
## Answer by MCM (score 1)
https://quant.stackexchange.com/a/38511
I would go for the component VaR. But this depends on your method how you receive the VaR (historical Simulation (exp vs equally weighted), variance covariance or monte carlo. You have to be careful since not ever method is linear. The main point is that VaR is not additive. For Variance Covariance it would look like the following:
$$ 𝑪𝑽𝒂𝑹_{i} =(\Delta VaR_{i})*w_{i}P = VaR*\beta_{i}*w_{i} $$
The sum would give you the portfolio VaR:
$$ \sum_{i=1}^{N} 𝑪𝑽𝒂𝑹_{i} = VaR\left(\sum_{i=1}^{N}\beta_{i}*w_{i}\right) = VaR_{p} $$
the correlation are determined as:
$$ 𝑪𝑽𝒂𝑹_{i} = VaR_{p}*w_{i}*\beta_{i} = (\alpha*\sigma_{i}*P) \beta_{i}*w_{i} =VaR_{i}*\rho_{i} \Rightarrow p(i) = \frac{𝑪𝑽𝒂𝑹_{i}}{VaR} = \beta_{i}*w_{i} $$
Hope that helps a little bit?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.