Portfolio VaR2

[Kitty]

Hello,

I have another question about portfolio VaR.

A bank has two divisions that currently have VaR of 200 and 400. The VaR of the bank as a whole will:
A. be 400
B. be 600
C. be at least 200
D. be at most 600

The answer is D. However, I'm confused that since the correlation of the two divisions should range from -1 to 1, I think portfolio VaR should range from 400-200 to 400+200 accordingly. So I don't understand why C is not correct? Are there any cases that VaR will be lower than 200?

Thanks a lot!

 

[ShaktiRathore]

Hi,
Var i think above is not logical if we do not make the assumption that correlation lies between 0 and 1 and not -1 and 1. So if you make that assumption than at correlation 0 ,min Var=sqrt(Var1^2+Var^2)=sqrt(200^2+400^2)=sqrt(20*10^4)=200*sqrt(5)~446
And max Var in that case is Var1+Var2 at corr=1=40+200=600
THus the max Var is 600 at most the Var can be this. And the least Var should be 446 rather than 200 so C rules out. So D is correct in this type of questions always make assumptions that seems valid.
thanks

 

[cdbsmith]

Hi David,

The question posted by Kitty concerns me. That is, there appears to be glaring contradictions embedded in the question. Specifically, the question refers to the combined VaR of two divisions of a bank. If we think of the bank as a just a regular widely-held business (and not as a portfolio of assets as Kitty seems to do), it may be easier to understand the following.

It is my understanding that when firms are perceived as conglomerates (i.e., large firms consisting of core and non-core business acquisitions in an apparent attempt to diversify is idiosyncratic risk in order to maximize shareholder wealth) they usually face negative reaction from the markets. That is, when firms attempt to acquire firms that are in completely different industries, their share prices usually fall. This is mainly due to the widely held belief that firms should simply focus on their core business and pass such idiosyncratic (business-specific) risks to individual shareholders as they are much more equipped to diversify such risks based on their individual requirements. The implication is that firms should not use correlation as one of the factors in decisions on acquisitions. This implies that the divisions should be uncorrelated (or independent) and simply part of the overall corporate strategy.

Based on this, at first glance, it seems illogical (or at least contradictory) to treat the two divisions of the bank (from the above question) like assets in a portfolio (i.e.. including correlation in the calculation of overall portfolio volatility). Therefore, when determining the VaR for the bank as a whole, it would seem more logical to simply add the weighted VaRs of the two divisions (this seems to make more sense as VaR is a quantile function, right?). Therefore, answer "C" or "D" would be correct. Does this make sense? If so, there is a larger point I would like to raise (see below).

I recently read Dowd Chapter 2 (Measure of Financial Risks), and it is quite fascinating. From my perspective, this reading is fundamental to the practice of financial risk management. Dowd makes some pretty strong critiques of VaR as finance risk measure. First, Dowd argues that finance risk measures should be coherent. To be coherent, a finance risk measure should satisfy four properties:

1) Monotonic
2) Subadditive
3) Positive homogeneity
4) Translational invariance

According to Dowd, subadditivity is the most important property that a coherent risk measure should satisfy. That is, a coherent risk measure must account for all possible outcomes (e.g., 100% vs 95% VaR). Dowd argues (and illustrates) the that the VaR risk measure violates the subadditivity property. Dowd argues that, in fact, the VaR is non-subadditive and is, therefore, not a coherent risk measure. Rather, Dowd argues that VaR is dangerous as a risk measure as it does not account for possible losses beyond the stated confidence level (e.g., 95% or 99%).

Now, if we can revisit the VaR question from Kitty, when considering Dowd's argument that VaR is in fact non-subadditive, this implies that non of the answers are correct. In fact, based on Dowd's aregument, the correct answer (true VaR) should be at least 600 because we should be including losses beyond the confidence level (e.g., 95% or 99%) used in the VaR calculation. That is, the true VaR is the total possible loss (loss magnitude).

We know that the VaR does not provide information regarding the total magnitude of potential losses. But, Dowd's critique goes further and really details the perils of using VaR as a finance risk measure.

Clearly, GARP is fully aware of this issue with VaR. So, why are they allowing writing questions with answers like the one from Kitty that are contradictory (or at least technically incorrect)? These questions just seem to be causing avoidable confusion.

Any thoughts?

Anyway, that's my rant for today.

Thanks,

Charles

 

[ShaktiRathore]

Hi,
I may like to pass some thoughts in addition to David if you don't mind. Such questions assumes that the returns for the divisions follow a normal distribution and whenever we calculate Var for a portfolio at least in FRM exam we assume normality assumption for portfolio returns. Seeing from the exam point of view it seems ok but unless and otherwise stated we always assume normality assumption. Furthermore in reality this might not be the case where returns might follow non-normal distribution and can give wrong Var for portfolios but this is not taken for granted in the exam. Exam has well laid down aims which focus on normal distributed returns when calculating portfolio Var. I have not faced any question during my frm preparation where we assume non-normality condition so there should arise no confusion.
Var is of course one measure that is not sub-aaditive but that is the case when we violate the normality assumption but hardly there is any focus on this non-normality assumption but we always assume by default the normality so Var is always sub-additive under this assumption, unless and otherwise specifically stated in question we cannot deny that we have to assume by default the normality assumption.
As far as the Question is concerned, if we assume the normality assumption then Var is sub-additive which implies that Var can vary from 0 to 600. Because Bank has divisions its possible that the other division with var does not take any risk and Var =0 and the other division Var is also 0 for not taking the risk hence overall Var minimum possible is 0(only current var is given otherwise Var can go downside to 0) and Max is possible when correlation is 1 that is 200+400= 600 so at most Var can go is 600, but the option C is totally nonsense when we consider the normality assumption because unless and otherwise we have non normal returns how can Var exceed the 600 value?? But that is 600 max right.But option D does makes some sense as i stated.
thanks

 

[David Harper CFA FRM]

I agree, Charles, absolutely.

ShaktiRathore: these are bank divisions, not (eg) P&L portfolios. There is no good reason to assume normal (elliptical) distributions; quite the contrary, part of the point of non-market-risk VaRs like operational and credit risk and ERM distributions is that they are explicitly not normal. I understand why you would make that assumption to "save" the question, but we can't make that assumption. Maybe if it's market risk, but not for divisions. If the question needs to assume normality here, it is wrong for omitting it.

Charles is right: on the literal question, there is every reason to allow for a combined VaR greater than 600 due to the lack of sub-additivity. Really great insight! The question needs to add qualifiers.

But, if normality is assumed, the question still has a problem: if the minimum is 446, then (C) is TRUE (Kitty is correct, too!). Note that by using these VaR formulas, which employ the (Pearson's) linear correlation, we implicitly assume normality; but we do that without any endorsement from the question: the use of linear correlation is another way of implicitly assuming elliptical distribution.

Re: "why are they allowing writing questions with answers like the one from Kitty that are contradictory (or at least technically incorrect)?"

• I do not recognize the question, sorry. I did not assume it is GARP's question? What is the source? If it is, I will surely submit it to them as problematic. please advise. Thanks!

 

[cdbsmith]

David,

I really appreciate your insight on this issue. Dowd's chapter on measures of financial risk is absolutely fantastic in my view. While the reading makes extremely valid critiques of the mean-variance and VaR approaches to measuring financial risk, it also does an excellent job of outlining a minimum standard set of properties that good financial risk measures should satisfy.

This reading has really enhance my understanding and perspective of financial risk management.

Regarding the question on why GARP would allow such questions, I made the assumption that Kitty got it from a GARP practice exam. Sorry for the confusion.

Thanks again for confirming my thought process. I really respect your opinions as you have shown yourself to be true authority in this this area.

Warm Regards,

Charles