Assigning Global VaR to portfolio members

Question

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:

1. 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
2. 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 1

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 2

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?