PQ-external Question about definition of component VaR

[no_ming]

Hi, Mr. Harper, for this question, I choose the correct answer A. But I still have a question on component VaR.

By definition on the notes, "Component VaR for position i, denoted CVaRi, is the amount a portfolio VaR would change from deleting that position in a portfolio."



In this question, the component VaR is USD 17.6 but the amount a portfolio VaR changed from deleting that position is just USD 15, is this contradict to the definition of component VaR?

 

[Deepak Chitnis]

Hi @no_ming what I understands from question is that what will be the reduction in portfolio var? We are not asked portfolio component var here. They are simply asking if you drop the asset 1 from portfolio what will be reduction in portfolio var they are not asking what will be portfolio var! If we drop asset 1 there is only one asset left (asset 2) in portfolio, so var of portfolio with both the position in 61.6 and if we drop asset 1 there is only one asset left asset 2, var of asset 2 is 46.6, so 61.6-46.6=15 is the reduction in var. Hope that helps!
Thank you!

 

[no_ming]

Hi @no_ming what I understands from question is that what will be the reduction in portfolio var? We are not asked portfolio component var here. They are simply asking if you drop the asset 1 from portfolio what will be reduction in portfolio var they are not asking what will be portfolio var! If we drop asset 1 there is only one asset left (asset 2) in portfolio, so var of portfolio with both the position in 61.6 and if we drop asset 1 there is only one asset left asset 2, var of asset 2 is 46.6, so 61.6-46.6=15 is the reduction in var. Hope that helps!
Thank you!

Hello, Deepak Chitnis, I understand what the question asks, by logic , it is easy to choose the correct answer, but refer to component VaR definition, it seems that there is a contradiction because reduction in portfolio VaR is the same as the definition of component VaR - "the amount a portfolio VaR would change from deleting that position in a portfolio.".

But in this question, component is simply USD 17.6 and the definition of component VaR cannot be applied in this question right?

 

[Deepak Chitnis]

Hi @no_ming you are right! (for more clarrification view https://forum.bionicturtle.com/threads/individual-var-vs-component-var.1373/). In this question you do not need to apply defination of component var here ( I think question wil clearly ask you to calculate component var, not sure @David Harper CFA FRM can clarrify). We are simpy calculating the reduction in portfolio var if one of asset is dropped from portfolio.
Thank you!

 

[Nicole Seaman]

Hello @no_ming

David has discussed this specific question here also: https://forum.bionicturtle.com/thre...al-value-at-risk-var-level-2.4961/#post-16574. Before posting a question in the forum, we always suggest using the search function to see if it has already been discussed. When I search just the first few words of this question, it brings up a lot of discussions

Thank you,

Nicole

 

[David Harper CFA FRM]

Hi @no_ming Nicole's link has a detailed explanation. Once you read that, hopefully you can see why (with respect to your question sample above):

• When the question asks, "If asset 1 is dropped from the portfolio, what will be the reduction in portfolio VaR?" it is the same thing as asking "What is the incremental VaR of asset 1?" such that the correct answer is given as the difference between the portfolio VaR and the individual VaR of asset 2, because that is the reduction in VaR implied by removing asset 1
  
• The "VaR contribution" column label is not a good label, it should be "Component VaR" which can be inferred by observing that these component VaRs sum to the portfolio VaR (unlike incremental VaRs which will tend to sum to less than the portfolio VaR; their summation is not meaningful). In fact, I think this question should utilize component VaR (in the exhibit) and incremental VaR (in the question) to avoid confusion, although what is written is not incorrect.
  
• In general, component VaRs will be greater than incremental VaRs (due to the fact that the sum of individual VaRs will be greater than portfolio VaRs). I hope that helps, thanks!

 

[no_ming]

Hi @no_ming Nicole's link has a detailed explanation. Once you read that, hopefully you can see why (with respect to your question sample above):

• When the question asks, "If asset 1 is dropped from the portfolio, what will be the reduction in portfolio VaR?" it is the same thing as asking "What is the incremental VaR of asset 1?" such that the correct answer is given as the difference between the portfolio VaR and the individual VaR of asset 2, because that is the reduction in VaR implied by removing asset 1
  
• The "VaR contribution" column label is not a good label, it should be "Component VaR" which can be inferred by observing that these component VaRs sum to the portfolio VaR (unlike incremental VaRs which will tend to sum to less than the portfolio VaR; their summation is not meaningful). In fact, I think this question should utilize component VaR (in the exhibit) and incremental VaR (in the question) to avoid confusion, although what is written is not incorrect.
  
• In general, component VaRs will be greater than incremental VaRs (due to the fact that the sum of individual VaRs will be greater than portfolio VaRs). I hope that helps, thanks!

Hello, Mr. Harper, the link is very very useful, thx a lot. I conclude the following, can you help me to check whether my concept is right?

1. Component VaR is linear approximation of Incremental VaR;
2. If there is 3 assets and the question DOES NOT PROVIDE the portfolio VaR after deducting one of asset(e.g. asset 1), we can use component VaR instead of incremental VaR to measure the change in Portfolio VaR, but its only a approximation;
3. If there is 3 assets and the question PROVIDE the portfolio VaR after deducting one of asset(e.g. asset 1), we should use incremental VaR for the more accurate calculation (e.g. portfolio VaR before deduction of asset 1 - portfolio VaR after deduction of asset 1)
4. The answer in (2) & (3) will have difference and the larger the component of deducted asset, the greater the difference.

Many Thanks

 

[David Harper CFA FRM]

Hi @no_ming Glad it was helpful In regard to your follow-up questions:

1. Yes, exactly! A key relationship is given by: Component VaR = (marginal VaR)*$Position. But marginal VaR is effectively a slope; i.e., ΔPortfolio-VaR/ΔPosition, such that Component VaR is really just a special case of: Linearly estimated $Portfolio-VaR ~= marginal-VaR*Δ$Position. For example, using the above GARP example, the reduction in position of Asset 2 from 100 to 50 implies an estimated $50*0.440 = $22.0 reduction in portfolio VaR. And you can see that (despite a different internal inconsistency with their numbers) the exhibit does show a correlation relationship: Component VaR of 44.0 = $100 position * 0.440 marginal VaR. In this way, marginal VaR is analogous to bond (dollar) duration and option delta; they are each first partial derivatives. So, it's true that Component VaR estimates Incremental VaR, but it's a linear estimate and (as shown by Jorion's Fig 7-4) in a typical portfolio situation, we expect it to be greater than the more accurate incremental VaR (in a manner similar to our expectation that a bond loss estimated by duration is greater than the actual loss because convexity is omitted).
2. Yes, true!
3. Yes, true!
4. Yes, true, and further: our expectation is that the incremental VaR (in your #3) will be less than the component VaR (in your #2) due to the linear nature of the estimate. I hope that is helpful!

 

[no_ming]

Hi @no_ming Glad it was helpful In regard to your follow-up questions:

1. Yes, exactly! A key relationship is given by: Component VaR = (marginal VaR)*$Position. But marginal VaR is effectively a slope; i.e., ΔPortfolio-VaR/ΔPosition, such that Component VaR is really just a special case of: Linearly estimated $Portfolio-VaR ~= marginal-VaR*Δ$Position. For example, using the above GARP example, the reduction in position of Asset 2 from 100 to 50 implies an estimated $50*0.440 = $22.0 reduction in portfolio VaR. And you can see that (despite a different internal inconsistency with their numbers) the exhibit does show a correlation relationship: Component VaR of 44.0 = $100 position * 0.440 marginal VaR. In this way, marginal VaR is analogous to bond (dollar) duration and option delta; they are each first partial derivatives. So, it's true that Component VaR estimates Incremental VaR, but it's a linear estimate and (as shown by Jorion's Fig 7-4) in a typical portfolio situation, we expect it to be greater than the more accurate incremental VaR (in a manner similar to our expectation that a bond loss estimated by duration is greater than the actual loss because convexity is omitted).
2. Yes, true!
3. Yes, true!
4. Yes, true, and further: our expectation is that the incremental VaR (in your #3) will be less than the component VaR (in your #2) due to the linear nature of the estimate. I hope that is helpful!

Mr. Harper, this question is above diversified VaR, but I doubt whether the replacement of new common stock can result in the decrease of diversified VaR as it just tell us the % of variance in the fund decrease from 85% to 65%. If the new common stock is positive correlated to other stocks in the fund, there is still a chance to have a higher diversified VaR with 65% variance accounted for the fund, do you agree?

 

[David Harper CFA FRM]

Hi @no_ming I think the question is trying to suggest the new stock position adds uncorrelated factors to the diversified portfolio; ie, on the assumption that less PCA variance is captures in the orthogonal residual error, which would be true if there were only the principal components and the residual. Given I don't see why we can make such an assumption, I tend to agree with you. But the question gives me a headache. You'd never see this on the FRM exam for several reasons; the FRM doesn't give a candidate sufficient understanding of PCA to fully grok this question, nevermind the FRM would not refer to an "orthogonal error" as that's a vector concept really that has never been explored by the FRM. I would be interested in the answer to see if our intuition is correct and the question maybe suffers from an imprecision? Or maybe I am missing something when I tend to agree with you Thanks,