P1.T4.VAR-HULL-DOWD_Ch2_TOPIC: Coherent Risk Measures

[gargi.adhikari]

In reference to: P1.T4.VAR-HULL-DOWD_Ch2_TOPIC: Coherent Risk Measures:-
Am missing seeing the 5.d.1 Learning spreadsheet in the study planner.. Any pointers..? Am I missing seeing something... ????

 

[brian.field]

I think it might just be a naming convention David was using and not something to worry about - but I'm not sure.

 

[gargi.adhikari]

Hi Brian, @brian.field ...this is the example to illustrate that VAR violates the Coherent Risk Measure property= sub additivity... think it is in the right context, but have questions on the example itself...and so thinking, there might be a spreadsheet 5.d.1...with the detailed workings illustrated...

 

[brian.field]

Great - what are your questions? Can you be a bit more specific?

 

[gargi.adhikari]

@brian.field am trying to figure how the with the 2nd & 3rd Bond in the portfolio, we have 0 & 1 default respectively and the VAR is $100 with the 3rd bond...?

 

[David Harper CFA FRM]

Hi @gargi.adhikari It should be in the study planner but here is my copy https://www.dropbox.com/s/x2n3k5m6w6c6uvj/5.d.1.expected_shortfall-dh.xlsx?dl=0
This mimics Dowd's example, which is maybe the cleanest example of VaR's lack of subadditivity. In the example, a single bond has PD = 2%. (If we always assume zero recovery), 95% VaR of such a bond is zero. Combine three such bonds, independently (i.i.d.), the 95% VaR is one default or the par value of one bond. Here this is about the interesting probability of a single default:

• If one bond, prob[exactly one default | n = 1, p = 0.02] = 2.0%; i.e., 1 - confidence = α = 5% > pd= 2%. If α > pd, VaR is zero because the "quantile isn't in the loss tail" so to speak.
  
• If two bonds, prob[exactly one default | n = 2, p = 0.02] = 2%*98%*C(2,1) = 3.92%; note here the 5.0% quantile still remains "outside the loss tail" and in the body of the distribution where zero defaults occur
  
• If two bonds, prob[exactly one default | n = 3, p = 0.02] = 2%*98%*C(3,1) = 5.88%; now the 5.0% quantile falls at one default. Here, α < pd. The tail includes probabilities for 2 and 3 defaults but they are only 0.12%.

So the lack of subadditivity is demonstrated by the dynamic given that cumulative PD in the tail shifts from less than alpha to greater than alpha. I hope that clarifies!

 

[gargi.adhikari]

@David Harper CFA FRM Thanks so much David ! I just saw the post that you guys are on a very well deserved vacation ! Didn't mean to inundate with questions. Please have a blesses holiday !
But thanks again for taking the time to put this explanation forth. Much Gratitude ! I'll make sure I hop on to other topics which are more familiar to me in the interim and promise not to bug ya on your time off !

 

[David Harper CFA FRM]

@gargi.adhikari thanks but your aren't inundating, we like to be helpful if we can

 

[gargi.adhikari]

@David Harper CFA FRM That's what makes you guys special..very very special !

 

[JulioFRM]

Hello, is it true that VaR & ES satisfy all the properties of coherent risk measures for normal distr., but only ES satisfies all the properties of coherent risk measures when the assumption of normality is not met?

 

[David Harper CFA FRM]

@JulioFRM Not quite. VaR and ES are both instances of (special cases) of Down's General Risk measure which is a highly flexible weighted-by-risk-aversion quantile funtion. Under certain conditions (namely, weakly increasing) the general risk measure will quality as coherent and can be called a spectral risk measure. ES meets these conditions, VaR does not (VaR is not weakly increasing: it does not incorporate information in the extreme loss tail to the end). In this way, ES and VaR are both general but only ES is spectral and therefore coherent. I mention this aside only to avoid confusion w.r.t. the spectral risk measure.

To be coherent, a risk measure must necessarily meet all four coherent conditions (monotonicity, subadditivity which is the most important/relevant, positive homogeneity, and translational invariance). Expected shortfall always meets these conditions: ES is always coherent, so it is coherent, so to speak. VaR only necessarily meets the coherence conditions if the distribution is normal (elliptical). VaR is not always (is only sometimes) coherent, so it is not coherent (so to speak). In summary:

• ES (a general and spectral measure) is coherent because it always meets all four coherent conditions
• VaR (a general but not spectral measure) is not coherent because it does not always (i.e., does not necessarily) meet all four coherent conditions, except when it happens to be normal (elliptical), so it is not coherent. I hope that clarifies a confusing concept set!

 

[JulioFRM]

Thank you