Is Historical Simulation a parametric approach?

[Steve Jobs]

The HS is included in Dowd ch3 and since the next chapter, ch 4 has the title of Non-Parametric Approaches, I concluded that HS is parametric or am I wrong? but then in BT notes of Boudoukh, page 3, in the paragraph of HS Advantages, it's mentioned that HS avoids distributional assumptions. So is HS a parametric or non-parametric?

Another q, isn't the decay formula used by Boudoukh the same used in Dowd Non-Parametric ch? What's the difference between Boudoukh hybrid approach and the Age weighted HS of Dowd Non-Parametric ch?

 

[David Harper CFA FRM]

Hi @Steve Jobs

Basic (aka, simple) historical simulation is non-parametric. HS is non-parametric precisely because of the advantage you cited (so that particular advantage is tautological, I would say!): it avoids a distributional assumption (it avoids, or it can avoid, the specification of parameters such as variance or scale or shape).

But parametric versus non-parametric is not a super clean distinction (and some experts do not even approve of it):

• Dowd discusses intermediate/advanced (non basic) HS approaches that he dubs "semi-parametric;" e.g., parametric filter (GARCH) applied to the historical dataset
• Also, it's not uncommon to combine a HS for the body of the distribution with a parametric extreme tail

And it's confusing, firstly, because most approaches use historical data in some way. But with respect to the basic difference, to me, it's really a difference between:

1. Is the VaR (quantile) or ES (or whatever measure) being retrieved from a "messy" sample of data (e.g., histogram). Non-parametric approaches or components are betrayed by lots of data.
   
2. Is the measure retrieving from a "clean" distribution function with parameters. A parametric approach is betray by not having lots of data "lying around," it's a function.

The historical sim is still an empirical distribution with a mean, variance, etc, but they emerge from the data without the distributional assumptions. Conversely, we might use historical data to fit a normal distribution, but then we've specified the normal distribution (using the data to fit the distribution params) and we can "discard the data" and we've moved to a parametric approach. So the issue turns not on whether we use historical data (we mostly do regardless) but rather, how do we use the historical data. Hope that helps,

 

[Steve Jobs]

To be honest..it's still confusing but I guess only by practicing and reading more it would be more clear although the distinction might not be 100% as you mentioned.

Can you please also comment on the second q?

 

[David Harper CFA FRM]

Right, i don't think the label difference "parametric versus non-parametric (versus semi-parametric)" is as important as the substantive difference in approaches.

Yes, Dowd's age-weighted HS is the exact same approach as Boudoukh's hybrid (and is the same as Linda Allen's Hybrid). As in, "hybrid" between HS and EWMA (EWMA is the basis for age weighting and is the parametric "contribution"). Dowd refers to this age-weighted HS as semi-parametric (because EWMA produces weights) but it's quite safe (if not nearer in spirit) to call this non-parametric: the distribution has been re-weighted but it's still an empirical distribution. This (age-weighted = Boudoukh's hyprid) approach is more non-parametric than parametric, in my opinion.

 

[Steve Jobs]

Simply if the z or t values (e.g. 1.65, 1.95) are not used in calculations, then it's non-parametric! loosely correct or totally wrong?

 

[David Harper CFA FRM]

Hi @Steve Jobs Loosely correct! And, in my opinion, you've isolated on the essence with a handy shortcut (+ star for elegant isolation!). VaR is a distributional quantile. In the parametric approach, we're generally going to use a distributional lookup value like 2.33 or 1.65 (or whatever statistical lookup value according to whatever function). Thanks,

 

[Steve Jobs]

Thanks for the medal! I loved them in elementary school!

Going back to the 2nd q in the first post (Boudoukh hybrid approach and the Age weighted HS of Dowd Non-Parametric ch), even the Hull chapter on incorporating volatility... is the same volatility adjusted method mentioned in Dowd ch. It seems that both the new chapters, Boudoukh and Hull incorporating volatility... which are relevant to VaR in the section Market Risk are repeated topics, expect the first page of BT Hull incorporating volatility...where the method for calculation of VaR by banks is described which is omitted in Jorion ch.

 

[David Harper CFA FRM]

Hi @Steve Jobs

Agreed, there is high overlap with respect to the non-basic historical simulation; that's why I tried to use headers like (page 7 of Hull incorporating volatility): Boudoukh et al (BRW Approach; a.k.a., hybrid, age-weighted HS). Unfortunately, that's three different names for the exact same approach (BRW = hybrid = age-weighted); I am not keen on this sort of outcome

I don't know what you mean by "except the first page of BT Hull incorporating volatility...where the method for calculation of VaR by banks is described which is omitted in Jorion ch."?

 

[Steve Jobs]

I meant this is the first time I see the steps: calculating daily VaR...square 10...multiply by 3(3 to 4) ... , presented all together as a process. This was not done in earlier chapters. I like this type of step-by-step process-wise explanations. By the way, why it's multiplied by 3? how the "3" was determined?

 

[David Harper CFA FRM]

Thanks. The multiplicative factor of 3 in Basel IMA is a controversial adjustment to introduce prudence and compensate for model risk, here is the best explain that I have in my library @ https://www.dropbox.com/s/5u8dwunond7i6z4/FRBNY_ima_scaling_factor.pdf (I don't have a good explain from BIS, if it exists, the Basel docs do not justify it very well, to my knowledge). Many call it arbitrary.

Jorion's appendix 5.a (Var 3rd edition) gives the cleverest quantitative justification that i've seen: he shows how Chebyshev's inequality guarantees 99.0% regardless of the distribution. Its so elegant and simple:

Per http://en.wikipedia.org/wiki/Chebyshev's_inequality
1% VaR = 0.5*1/k^2; 0.5* as VaR is one-tailed,
k = sqrt(0.5/1%) = 7.07
i.e., Chebyshev says 7 sigma ensures 99% VaR for even heavy tail distributions, such that 2.33*3 ~= 7 and the 3 multiplier basically gets you there(!)

 

[Steve Jobs]

Thanks David, so again it's parametric ! I don't remember the bank's name but I heard once that in one of the big banks which failed or was hurt because of financial crisis, the probability of that event was 16 sigma!

 

[David Harper CFA FRM]

Hi @Steve Jobs - Right, Hull's reading 34 (incorporating volatility) begins with a discussion of the "model building approach" (parametric) in order to contrast it with the HS (non parametric) approaches which are the focus of his paper. I never saw it called the "model building approach" before (certainly the FRM has not called it that) but Hull knows better than me. Parametric is a really loose umbrella term, and its looseness makes it comprehensive. For example, Jorion refers to the non HS approach as delta-normal approach, which is (of course) parametric but it's hardly the only parametric (aka, analytical) approach. So, for non-HS (and non-MCS), I prefer parametric or analytical (I don't like "model-building" because, um, they all seem like model building to me! ... and I'm not keen on delta-normal b/c it can support a misunderstanding that we need to assume normality)

Oh, I wonder if you are referring to when Goldman CFO referred to a 25-sigma event which is a neat chance to employ Chebyshev's; i.e., under a normal distribution the prob of a 25 sigma requires more time than the age of the universe, =NORM.S.DIST(-25, T), but relaxing the distributional assumption increases the odds greatly

 

[Steve Jobs]

Oh really 25?! it seems that the sigma is a popular excuse when things go wrong! 25 for Goldman and here is the 16 for AIG. (It's the average losses and not a particular event.)

So delta normal is the group of non HS approaches?! I should prepare a hierarchy of all the approaches and to include the akas in parentheses.

 

[David Harper CFA FRM]

thanks for the AIG link, interesting!

No, I did not mean delta-normal = non HS, I just meant that it is imprecise to imply it is the only non HS ... just because it is the most common (basic) parametric approach. Broadly, IMO, Dowd is best:

• Parametric
• Historical Sim
• Monte Carlo

The confusion engenders because parametric is often referred to by, equated with, what are sub-classes of parametric: delta-normal, analytical, "covarance-variance", even linear, but those are sub-classes; i.e.,

• Parametric
  • Delta-normal

So delta-normal is non HS (and non simulation) and parametric, but not even nearly the only parametric. Hierarchy is a good idea, I am tagging that suggestion to include in next opportunity, thanks,