Individual VaR vs. Component VaR

[hsuwang]

Hi David,
What is the difference between the VaR for individual position in a portfolio and the component VaR for that position?

The component VaR is the marginal VaR multiplied by the dollar weight in a position, so how would this differ from the individual VaR and what is actually making the difference here?

Thank you.

 

[David Harper CFA FRM]

Hi Jack

http://www.bionicturtle.com/learn/article/value_at_risk_var_2007_frm_part_11_incremental_component_var
(this just replicates Jorion's Chapter 7 example)

I have recently blogged two practice questions that explore the difference:
http://forum.bionicturtle.com/viewthread/1360/
http://forum.bionicturtle.com/viewthread/1400/

The individual VaR has no awareness of the portfolio; the component VaR is the change in portfolio VaR it the position were to be removed from the portfolio (based on a linear approximation!). So, component VaR is an imperfect *approximation* of the position's contribution to portfolio VaR. At one extreme, if the position has zero correlation with the portfolio (actually, impossible because the portfolio includes the position), then the component VaR, reflecting a *contribution* to portfolio risk, would be zero/near zero; at the other extreme, a large position will, be definition, have large correlation with the portfolio, and its component VaR will approach its individual VaR as correlation tends to 1.0 (Component VaR = Individual VaR * correlation between component & portfolio; a correlation that in practice, has a hard time reaching either 0.0 or 1.0 since the portfolio includes the component). Typical is: position imperfectly correlated to portfolio, such that "risk diversification benefits" imply component VaR < Individual VaR but typically not dramatically less...

importantly: sum of component VaRs = portfolio VaR but sum of individual VaRs > portfolio VaR

David

 

[hsuwang]

Hi David,
Thank you for the clarification.
Another question is if both the component VaR and incremental VaR uses marginal VaR for estimation, then would they be the same (for example for calculating the decrease in VaR if a position were to be taken out from the portfolio)?

I see that usually incremental VaR is calculated using full-revaluation, and I think that would make a difference with the component VaR, but if we use marginal VaR to estimate incremental VaR, then I guess it would be the same as Component VaR?

Thanks!

 

[David Harper CFA FRM]

Hi Jack,

Yes I agree exactly with your statements. Jorion's Incremental VaR requires a full-revaluation (to capture non-linearities) so it is the "ideal."
But as a practical matter, he tends to approximate it by using marginal VaR, in which case it is equivalent to component VaR; e.g., if you look at all of his VaR mapping examples, they rely on the linear approximation and (by way of convenience) are calculating component VaRs.

I think there are potentially two differences:
1. Approximate versus Precise: The component VaR is, by definition, a linear *approximation* (i.e., function of correlation matrix/beta). Although components sum to portfolio VaR, we have no expectation these are precise. The incremental VaR implies the full revaluation required especially to the extent the position is (i) large and (ii) with nonlinear risk contribution. Component VaR used to approximate incremental VaR is only a convenience.
2. Full position or just a trade: Less critical, the component VaR implies the entire position; i.e., what is the VaR contribution of the entire position? Incremental VaR can simply be a trade; i.e., if add to position or reduce position, what is incremental impact? So, they converge only (i) the trade is the full position and (ii) when we, by convenient approximation, settle for linear approximation

David

 

[ajsa]

Hi David,

"Incremental VaR can simply be a trade; i.e., if add to position or reduce position, what is incremental impact? "

So if we approximate incremental VAR by using marginal VaR, is it true the only difference between incremental VAR and marginal VaR is that marginal VaR is per unit change? and what is their mathmatical relationship under this approximation? Can we use some similar formula like component VAR?

Thanks,

 

[David Harper CFA FRM]

Hi asja,

Re: "is it true the only difference between incremental VAR and marginal VaR is that marginal VaR is per unit change?"
Yes, the difference here is merely units. Marginal VaR is unitless (like correlation), so under Jorion's approximation (7.23, p 169)...
Incremental VaR ~ marginal VaR' * dollar trade (a)
...i think it is a useful meditation b/c, once again, we are dealing with a first derivative: marginal VaR = dVaR/dPosition, so you can see how this:
incremental VaR ~ dVaR/dPosition * dollar change in position = an estimated change in (portfolio) VaR

both component VaR and the *approximation* of incremental VaR are using marginal VaR, and they are the same of the trade is to reduce the position/component by 100%; i.e., component VaR = the approximation of incremental VaR if a = -100%.

but the approximation is not really the point of incremental VaR. Hence, there is no formula. Incremental VaR, unlike its approximation, really implies full revaluation. Otherwise, i don't think there would need to be a different term for incremental/component; "component" implies use of first-deriviative (marginal Var), "incremental" implies accuracy.

This whole cluster here is analogous to bond duration:
marginal VaR = dollar duration (=optoin delta) = slope of tangent line; ergo, using it is always an approximation and inherently inaccurate

estimating bond price chage with duration (i.e., -duration * 1% yield shock) is like estimating portfolio VaR change with marginal VaR * position trade (a)
In both cases, the underlying dynamic is (i) multifactor and (ii) nonlinear, so we know we are rough....
in both cases, to get the true new price/value, we have to (i) reprice the bond and (ii) reprice the portfolio to find the incremental VaR

Hope that helps, David