Formula

[sapozan]

Mvar

 

[David Harper CFA FRM]

Hi @sapozan see the below (source: Jorion VaR 3rd, https://www.amazon.com/Value-Risk-3rd-Ed-Benchmark-ebook/dp/B004MPQFTO )

Yours is a single-asset absolute VaR (absolute: includes the drift, µ) but the derivation is assumes multi-asset (ie, multiple weights) relative VaR, so per 7.17 below, the marginal VaR just multiplying (scaling) the portfolio's marginal volatility, ∂σ(P)/∂w(i), by the deviation, α; e.g., α = 1.65 if 95% confident. I hope that's helpful,

 

[sapozan]

Hi David,

Thank you for your answer.

The only thing I cannot understand is how I can justify excluding drift from the proof ?

Thank you

 

[David Harper CFA FRM]

Hi @sapozan Yes, that's observant! It is excluded. Your formula is not in matrix format, but could represent a multi-asset portfolio such that we can think of the generalized (multi-asset version of your) Var = µ + σ ∗ z. The proof is not using this (which includes the drift and is called absolute VaR) for portfolio VaR; rather the formula is using the multi-asset version of this: Var = σ ∗ z; i.e, you are correct the drift is omitted. Because this marginal VaR is "keeping it simple" by only looking for the marginal volatility where portfolio variance is the super-convenient w^T*Σ*w where Σ is the covariance matrix and w^T is the transposed weight vectors. This does omit the drift and captures the relative VaR: the VaR relative to expected future value such that drift is omitted. So it's just for analytical simplicity, there is an analogous absolute VaR although I can't recall seeing it analytically. My hunch is that it's just easier to simulate given the drift is linear and doesn't fit in nicely.

So basically the justification is: we call it relative VaR (and keep things very simple)! If we want to include the drift, it's an absolute VaR and it's a bit more work. Thanks,

 

[sapozan]

Awesome thank you David