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# Difference between Marginal and incremental VAR

#### bpdulog

##### Active Member
Thanks @David Harper CFA FRM

Can you please explain how do you arrive at rho (hij,klm).

Regards.
You solve for rho since portfolio SD is given:

EDIT: I am actually getting a correlation of .5126

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Ali Ehsan Abbas I used the formula to which @bpdulog refers such that sqrt[(140/300)^2*0.20^2 + (160/300)^2*0.12^2 + 2*(140/300)*(160/300)*0.20*0.120*0.50] = 13.7042%. If I uses sovler, ρ = 0.499042585684933, so I think they were assuming 0.50 Thanks!

#### Kamakshi

##### New Member
Just to add, how i think of it: Incremental VaR is the exact (fully simulated) answer to the change in VaR resulting from removal of the position. Marginal VaR, as a partial derivative, informs an linear approximation to removal--or just a change--in the position; i.e., marginal VaR gives us Component VaR which is an approximation. Marginal VaR is the convenience approximation; incremental VaR is an actual answer. A weak analogy would be: using duration to approximate a price change in bond; duration is the linear approximation (analogous to marginal VaR), whereas re-pricing would give us the accurate, exact answer (analogous to incremental VaR). Thanks,
@David Harper CFA FRM
The reading says both incremental and component VaR refers to the change in VaR due to deletion of a position/component. So what exactly is the difference between Incremental and Component VaR?

#### emilioalzamora1

##### Well-Known Member
I don't quite understand why every single piece (in particular qualitative stuff) needs to be explained. There is a vast amount of literature where you can get the information/proof yourself. And this is sth. which is the minimum what can be expected from a well rounded candidate.

Component VaR (CVaR): it is simply the marginal VaR times the wealth (W) invested in a particular asset and shows how the portfolio VaR would change if the component was deleted from the portfolio. Porfolio volatility changes in a non-linear fashion due to different underlying components which drive the portfolio volatility (VaR). Component VaR is only useful when we have 1.) a large portfolio (high level of wealth invested) and 2.) having small individual positions (components). This ensures that the portfolio VaR changes more or less (approx.) proprionately if a component would be deleted from the portfolio.
CVaR can be sort of interpreted as risk measure to reflect correlations because part of the CVaR calculation (formula) depends on the correl coefficient. Having a negative correl and therefore a negative beta, the position acts a hedge against higher portfolio VaR (in other words, it has a positive effect yielding to lower portfolio VaR).

Remember:

CVaR(asset A) = alpha * sigma(p) * W(A) * beta(A) = CVaR(A)/VaR(p) = w(a) * beta(A)

where the two on the right-hand side, yield the %-contribution of Asset(A) to total Portfolio VaR
1. CVaR(A)/VaR(p)
2. w(a) * beta(A)

• alpha denotes the z-value (will either be 1,65 at 95% or 2,33 at 99%)
• sigma (p) stands for portfolio volatility

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#### WhizzKidd

##### Member
Hi @David Harper CFA FRM

Say, an index halted trading for 5 weeks, during that 5 week, would the VaR of a portfolio replicating the index be 0 or unchanged from before the halt in trading?
Is it correct in saying here, that VaR cannot be zero, so either unchanged or lower?
Then later would the VaR just after trading has started again now be higher, unchanged or lower?
This would also be lower, as the 25 trading day period is used in the VaR calc?

#### Srilakshmi

##### Member
Is the beta given in the question with respect to the portfolio? If so, should the beta of the portfolio with respect to itself should be 1 right?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Srilakshmi yes, exactly. That's why this table in Q60 cannot be correct. The marginal VaR of HIJ is informed by β(HIJ, Portfolio) such that, while the β(HIJ) + β(KLM) <> 1.0, it nevertheless should show portfolio beta of 1.0 to reflect that all three betas are with respect to the portfolio.

#### GARP2015

##### New Member
@Srilakshmi yes, exactly. That's why this table in Q60 cannot be correct. The marginal VaR of HIJ is informed by β(HIJ, Portfolio) such that, while the β(HIJ) + β(KLM) <> 1.0, it nevertheless should show portfolio beta of 1.0 to reflect that all three betas are with respect to the portfolio.

Dear David,
what was the GARP feedback about this question that the betas are not internally consistent? will they consider this issue in FRM exam? and why they don't publish a correction to this question?

Finally, the sum of weighted betas should equal 1.0. For example 140/300 * B(HIJ) + 160/300*B(KLM) = 1.0. This can be used as a check !!

Regards,

Jamal

#### Nicole Seaman

Staff member
Subscriber
@GARP2015

I'm not sure if @David Harper CFA FRM received anything further on this, but this is the response that we received from GARP regarding this question:

"Regarding Q60 in the Part II Practice Exam, we will be putting out a revised version of that question soon that should address the issue of consistency amongst the inputs numbers, calculations, and outputs."

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thank you @Nicole Seaman that is exactly the relevant email: we only received it this week. We've been driving this point for years actually, so I am very happy to report that GARP finally understands the importance of a correctly specified portfolio VaR. Jamal (@GARP2015 ) you are exactly correct about an easy test for the betas; below is the specification that I sent GARP for this question. Notice that indeed (140/300) * 1.3347 + (160/1300) * 0.7071 = 1.00. Thank you!