Component versus Incremental value at risk (VaR), Level 2

[David Harper CFA FRM]

Hi, Andrew raises a good question here, with respect to GARP's sample question. The setup gives a typical two-asset portfolio with correlations and asks, "If asset 1 is dropped from the portfolio, what will be the reduction in portfolio VaR?" I think that's a bit of a mean question because it wants the incremental VaR, but doesn't use that term. So, it's understandable to think of the component VaR.

I thought it would be useful to summarize the difference. Here is the XLS for my calcs below: http://db.tt/VexMVsNR
(I personally think this is conceptually difficult. I can't speak for you, but I almost need to review the spreadsheet to begin to come to grips with these two concepts)

Here are my assumptions for a two-asset portfolio (borrowing EUR + CAD portfolio from Jorion Chapter 7, but changing the numbers to my convenience)

• Portfolio is $100 invested equally in two currencies: CAD position = $50, EUR position = $50
• Volatility (CAD) = 30%, Volatility (EUR) = 40%, correlation(CAD, EUR) = 0.
• VaR confidence in 95%, such that normal deviate = 1.645

An FRM candidate should know how to compute 95% Individual VaRs (assume normality):

• CAD = $50 * 30% * 1.645 = $24.67
• EUR = $50 * 40% * 1.645 = $32.90

Because the correlation is imperfect (less than 1.0), we expect the diversified VaR to be less than the sum of individual VaRs:

• As portfolio volatility = 25%,
• Diversified portfolio VaR = $100 * 25% * 1.645 = $41.1

First, what are the incremental VaRs?
(again, GARP's sample question L2.2.18 is asking for the Incremental VaR: "If asset 1 is dropped from the portfolio, what will be the reduction in portfolio VaR?")

• Incremental VaR (CAD) = the reduction in Portfolio VaR if we delete the CAD position = $41.1 - $32.9 = $8.22
• Incremental VaR (EUR) = the reduction in Portfolio VaR if we delete the EUR position = $41.1 - 24.67 = $16.45
• Or put another way, if we start with just a $50 CAD position, our one-asset portfolio VaR is $24.67. The incremental EUR VaR is the increase in our portfolio VaR as we add the EUR position: $24.67 + $16.45 = $41.1 for our new two-asset portfolio.

Second, what are the component VaRs? (we have several ways to compute this, see the XLS. We can use beta or correlation since they are directly related)

• Component VaR (CAD) = $50 * 25% portfolio volatility * 0.72 beta (CAD, Portfolio) * 1.645 deviate = $14.80
• Component VaR (EUR) = $50 * 25% portfolio volatility * 1.28 beta (EUR, Portfolio) * 1.645 deviate = $26.32
• By design, component VaRs sum to portfolio VaR: $14.80 + $26.32 = $41.1
• The only role of VaR here is to multiply volatility by the normal deviate (1.645, in this case). We could drop the deviates and just refer to the marginal risk contribution, that's what component VaR really is. In this example, the portfolio volatility is 25%. If we drop the deviates, we would find that the marginal contribution (CAD) to portfolio volatility is 9.0% and the marginal contribution (EUR) is 16%.
• Further, the marginal VaR is related to this component VaR as they both characterize the linear approximation
  • Marginal VaR is unitless partial derivative; i.e., the change in portfolio VaR with respect to a change in the position
    • For the CAD position, Marginal VaR[CAD] = Portfolio-VaR/Portfolio-Value*beta(CAD, Portfolio) = $41.1/$100.0 * 0.720 = 0.2961
      
    • For the EUR position, Marginal VaR[EUR] = Portfolio-VaR/Portfolio-Value*beta(EUR, Portfolio) = $41.1/$100.0 * 1.280 = 0.5264
  • Component VaR is the (unitless) Marginal VaR scaled by the position
  • For the CAD position, Component VaR[CAD] = marginal VaR[CAD] * CAD-Position = 0.2961 * $50.0 = $14.80
  • For the EUR position, Component VaR[EUR] = marginal VaR[EUR] * EUR-Position = 0.0.5264 * $50.0 = $26.32

What about question asked in the sample, "If asset 1 is dropped from the portfolio, what will be the reduction in portfolio VaR?"

• Component VaR is not a terrible answer. But Component VaR (and marginal VaR) are linear approximations (first partial derivatives). Per a grand FRM theme, the larger the component, the greater the error (see Jorion Fig 7-4 to visually see the problem: but it's just like the problem with duration). Component VaR is an approximation of incremental VaR, that loses accuracy as the position size increases
• Incremental VaR is more accurate because it's a re-pricing ("full revaluation"). The subtraction $41.1 - $32.9 = $8.22 is an illustration of full revaluation: price the portfolio, re-price the portfolio, take the difference.

 

[mcarthur724]

Thanks for the explanation. One quick follow-up to clarify...
Regarding the statement "Incremental VaR (CAD) = the reduction in Portfolio VaR is we delete the CAD position = $41.1 - $32.9 = $8.22" It appears that this is actually deleting the EUR position (EUR individual VAR = $32.9) Is this the correct method to delete the EUR VAR to calculate the CAD Incremental VAR?

Thanks again

 

[David Harper CFA FRM]

Hi Andrew,

Yes (although my "is" should read "if") because it is the reduction:

• Diversified Portfolio VaR = $41.1
• Individual VaR of EUR of $32.90

If we delete the CAD position, we reduce the portfolio VaR by the difference between $41.1 (the two-asset portfolio) and $32.90 (the one-asset portfolio consisting only of the remining EUR).

Note that is consistent with GARP's sample question, they asked:
"If asset 1 is dropped from the portfolio, what will be the reduction in portfolio VaR?" (aka, incremental VaR of Asset 1)
... and their answer 2 subtracts the individual VaR of asset 2:
Portfolio VaR of USD 61.6 - USD 46.6 [the individual VaR of asset 2]= USD 15.0.

Thanks, David

 

[mcarthur724]

Thanks, I've got it now!

 

[FRM Candidate 2011]

Just a quick remark:

Since Component Var is basically partial derivative * Notional, it is pretty useless, since by definition the partial derivative is linear (while VaR is mostly not) and approximates only at the point taken. So IMO you either use marginal var to get the derivative or incremental var to get the full non-linearity.

 

[sl]

"If asset 1 is dropped from the portfolio, what will be the reduction in portfolio VaR?" (aka, incremental VaR of Asset 1)

Just to clear my understanding, shouldn't it be' incremental VaR of Asset 2' and not Asset 1, because we compute the incremental VaR of the new asset being added (in this case EUR) and subtract its VaR from the original VaR. Moreover, the example doesn't actually say that we are adding a new position, so where is the increment happening?

 

[David Harper CFA FRM]

Hi laxmsun, no I don't thinks so, did you read comment above (http://forum.bionicturtle.com/threa...al-value-at-risk-var-level-2.4961/#post-13451 )? Incremental, by default, is the deduction of the entire position. In a two-asset (Jorion's example) portfolio consisting of EUR + CAD, the incremental VaR (EUR) = Portfolio VaR [i.e., EUR + CAD] - Individual VaR(CAD); and incremental VaR (CAD) = Portfolio VaR [i.e., EUR + CAD] - Individual VaR(EUR). Thanks,

 

[sl]

Thank you, David. That cleared my confusion

 

[Ruolin]

David, you mentioned a few times (in your study notes, in the focus review and in the comments above" that "by construction, component VaRs sum to portfolio VaR". However, given Component VaR= (portfolio VaR)*beta*weight, shouldn't the saying be (Component VaR/beta) sum to portfolio VaR, i.e. "normalized Component VaRs sum to portfolio VaR"?

I am asking this because I looked at GARP 2010 P.2.23 (your T8 Focus Review page 26). In this example, certainly sum of component VaR's does not equal the portfolio VaR. (19,477+0.1169*600,000=89,617, not the portfolio VaR of 97,384).

 

[David Harper CFA FRM]

Hi Roulin,

No, the statements in the notes and above are correct:

• Component VaRs are expressed in dollar terms and, by definition, sum to diversified Portfolio VaR (which is rarely called DVaR, but often just referred to as portfolio VaR; i.e., portfolio VaR connotes diversified VaR, it does not connot undiversifed VaR. Undiversifed VaR is atypical such that it has the burden of needing a label.).
  In the example above: component VaRs sum to portfolio VaR: $14.80 + $26.32 = $41.1
• Normalized VaRs are percentages that sum to 100% (see Jorion 7.30), but this has never appeared on the exam, to my knowledge. In the example above, normalized VaR(CAD) = 14.80/41.1 = 36% = (weight)*beta(CAD,Portfolio) = 50%*0.72; and normalized VaR(EUR) = 26.320/41.1 = 64% = (weight)*beta(EUR,Portfolio) = 50%*1.28..

But your observation is excellent! Few have noticed that the 2010.P2.23 contains an implicit error (internal inconsistency), we reported this to GARP in 2010. See http://forum.bionicturtle.com/threads/question-23-component-var-invest.2172/The betas cannot be as shown, ie,

Please note that 23.3 shows the betas cannot be 0.5 and 1.2. But if we use correct betas of beta (A, portfolio) ~= 0.3665 and beta (B, portfolio) ~= 1.42, then:
Component VaR (A) ~= $97.3 * 0.3665 * 40% = $14.265, and
Component VaR (B) ~= $97.3 * 1.42 * 60% = $83.05
and $14.265 + $83.05 ~= $97.3

 

[Aleksander Hansen]

Just a quick remark:

Since Component Var is basically partial derivative * Notional, it is pretty useless, since by definition the partial derivative is linear (while VaR is mostly not) and approximates only at the point taken. So IMO you either use marginal var to get the derivative or incremental var to get the full non-linearity.

Component VaR can be very useful for risk reporting.

 

[Ruolin]

Thank you, David, for explaining (particularly for pointing out the question has an error). From your explaination, I understand that by beta's construction, sum of (beta*weight)=1. This is to say that while sum of weights=1, by beta's contrcution, sum of (beta*weight)=1 as well.

Thank you, David. Your reply enhanced my understanding to beta.

 

[Aleksander Hansen]

Thank you, David, for explaining (particularly for pointing out the question has an error). From your explaination, I understand that by beta's construction, sum of (beta*weight)=1. This is to say that while sum of weights=1, by beta's contrcution, sum of (beta*weight)=1 as well.

Thank you, David. Your reply enhanced my understanding to beta.

Pay close attention to David's, "sum to 100%," rather than 1 so people don't get confused.

 

[best_seller249]

As far as my understandings,

Incremental VaR follows the before and after approach (full revaluation) to consider the impact of adding new position (long or short) to the portfolio toward portfolio VaR.

Component VaR = Marginal VaR follows partial derivatives (approximation approach) to decompose the total risk of the portfolio toward its component positions. Thanks to this, we can approximate changes in total portfolio VaR when adjusting some portfolio positions. Since, it's a linear approximation, it will only be correct with insignificant changes in asset weights of the portfolio. Component VaR => the term emphasizes on additive characteristic and Marginal VaR => the term emphasizes on partial derivatives characteristics. In practice, marginal VaR can be computed by applying DelVaR vector.

David, please correct me if I'm wrong. Thanks

 

[Johnny Firpo]

Hi,

the link to the Excel-File is not working any more. Can you please post the current link!

Thanks in advance!

 

[JFZ2014]

Thanks David. Your explaination is very clear.
But I still have two questions:
1. What is the beta here?
2. Can a component VaR be negative? Say, I am long EUR but short CAD?

 

[Nicole Seaman]

Hi,

the link to the Excel-File is not working any more. Can you please post the current link!

Thanks in advance!

Hello @Johnny Firpo

Can you please be more specific about where the link is that you are referring to (i.e. Study Notes, Question Set, forum thread)? This makes it easier for me to fix any broken links more quickly, rather than having to search through all of the materials to find the broken link that you are referring to.

Thank you,

Nicole

 

[David Harper CFA FRM]

(@Nicole Manley I agree, I first thought @Johnny Firpo was referring to the XLS link above, but it's working so I don't know ....)

@JFZ2014 Thank for liking the explanation! I do recommend looking at the XLS, which implements this example, to better understand. It's here @ https://www.dropbox.com/s/mu06c2mspwoerd1/Component_vs_Incl_111111.xlsx?dl=0 (same file as above, just the longer link)

1. Beta is the same beta as usual, but the key idea here is that it's the beta between the individual position and the portfolio which contains the individual position. So Beta(i, Port) = covariance(i,Port)/ Variance (P).
   
2. Yes, exactly positions which hedge each other would be, I think, necessary but not sufficient condition for negative component (i.e., as I play with the model now, I am able to hedge but still imply positive component vars for both positions). A similar effect can be achieved is the positions simply have a negative correlation (but, in mean variance, this is effectively the same as long/short per your idea). For example, if I change (in the example above) the correlation between CAD and EUR from zero to -0.80, then I get component VaR [CAD] of -$2.05 and component VaR[EUR] of $21.86. This makes sense: a simple expression of Component VaR is Individual [aka, Position i] VaR * correlation[i, Portfolio] such that negative correlation[i, P] will imply negative component VaR. However, as the portfolio contains the position, it's requires aggressive assumption. I hope that explains,

 

[JFZ2014]

Thank you David! I looked at the XLS and it works perfectly. It makes many things clear.
It is so good. I really like your way of explaining.

Yes. You explained it well. I also played with the XLS, with different combinations of
CAD and EUR. It works fine except when I make EUR to be -$50 such that the total
amount is $0. In that case, several cells have non-numbers. Have you looked at it in
that way?

 

[David Harper CFA FRM]

Hi @JFZ2014 I had not considered the case of 100/100 long/short; indeed, it produces #DIV/0! errors. One (the?) problem is the portfolio value, under that special case, is zero (which is used as a divisor). I don't have a quick fix. It's an interesting problem, my instinct is that it's a "bug" that can be fixed, but I don't have current time to address it (sorry). Good catch, however!