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Historical Simulation..Parametric or Nonparametric Appoach?

Thread starter #1
Hi @David Harper CFA FRM, I am revising some areas in Market Risk and came across a point (not in your notes), which suggest that HS is a parametric Approach. I have always been under the impression, until I saw this point, that HS was a nonparametric approach. In your notes you cited a number of advantages of HS, to include the ability of HS to deal with non-normal data and not be constrained by the restrictive assumptions of a normal distribution. Can you please shed some light on this issue.
 
#2
This is a very basic question on Quantifying Volatility in VaR models but I just want a definitive answer.
Do the non-parametric approaches make any assumptions about the distribution? Ie does it have to be normally distributed returns when using the MDE or Historical approaches?
 

David Harper CFA FRM

David Harper CFA FRM
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#3
Hi @Hermz29 Good question! Basic historical simulation has always been considered a non-parametric approach in the FRM (and elsewhere to my knowledge; e.g., definitely in Carol Alexander) per the longstanding (~ 10 years!) Dowd Chapter 4 assignment. So I am very confident your impression is correct ;) (here is a prior conversation about this https://www.bionicturtle.com/forum/threads/is-historical-simulation-a-parametric-approach.7662).

I think a minor challenge here is just the definition of "parametric" given the context. This looks correct to me: https://en.wikipedia.org/wiki/Parametric_statistics. If I want to be as crude as possible, I'd wqy that a parametric approach is one that employs an off-the-shelf probability (statistical) function, even as we expected data to inform the parameters of the distribution. So, when we use historical data to compute a mean and variance of returns, the historical data is itself is "non parametric," (although retrieving the variance of a historical series is not a VaR approach...) but when we go to retrieve a 95% VaR with -µ + σ*1.645, this is parametric because we are using the normal probability function. As opposed to retrieving the VaR by sorting and finding the 5% quantile among the dataset (aka, non parametric).

The bigger challenge is that the major approaches can be blended. VaR has three approaches: parametric (aka, analytical), historical simulation and monte carlo (which i think of as forward simulation). As Dowd shows, there can be a parametric "add-on" to a non-parametric approach, which occasionally in his text he calls semi-parametric. Easily the most common (in FRM) is the hybrid (aka, age-weighted) where the historical returns are adjusted by the age weighting function. Now the data (itself non-parametric) is altered by a function (parametric). There are discussions here on the forum about his. My own view is that age-weighted HS is essentially non-parametric (because it's ultimately still "looking up" the quantile of dataset, rather than solving for it via a function), but technically Dowd calls this semi-parametric (but do the labels matter if we understand? When you go to code/XLS this, the labels aren't so important because the non-parametric approach is so obviously about a huge batch of data!). I hope that's helpful, good luck revising!
 

David Harper CFA FRM

David Harper CFA FRM
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#4
Hi @sharman.jamie I was just asked a similar question, see above.
  • Re: Do the non-parametric [VaR] approaches make any assumptions about the distribution? No, they do not, and the question answers itself because it's almost accurate to say that the definition of a non-parametric approach is that it does not make any assumptions about the [probability] distribution [of the loss data, and therefore of course does not assume it is normal, lognormal, etc]
  • Ie does it have to be normally distributed returns when using the MDE or Historical approaches? Similarly, no. MDE is essentially non-parametric and makes no assumption about the distribution of the dataset, however, it does condition (filter) the data in a way not greatly unlike the age-weighted (hybrid) so MDE could be called semi-parametric. Thanks!
 
#7
Hi David,

I am so confused. Why do non-parametric methods work for VaR if they do not make distributional assumptions? Isn't VaR premised on the mean-variance framework that assumes normality? Do non-parametric methods work for VaR estimation because of CLT? Please help!!

Thanks in advance
 

David Harper CFA FRM

David Harper CFA FRM
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#8
Hi @Laely It's important to understand that, no, VaR is most definitely not premised on the mean-variance framework that assumes normality. CLT tells us the average of a sequence of i.i.d. variables tends to be normal, so it might apply: for example, it is a decent justification for assuming daily P/L is approximately normal, but importantly CLT assumes independent and identical variables which is often unrealistic.

VaR is the quantile of a distribution; e.g., where does the 5.0% tail "start" in the distribution, such that 5.0% of the time the loss is worse (what is that 0.05 quantile on the distribution)? It has no requirement as to how the distribution is generated, it can be a function (parametric), a histogram based on collected data (non-parametric) or a simulation. VaR is just a property of the distribution. Instead of asking, what is the median (which is the 50%), VaR asks, what is the %ile?

Parametric versus non-parametric refers to the method by which we generate the distribution, and consequently, how it appears. If we use a function (e.g., normal, mixture, Poisson), that's parametric and it appears as a "coherent" pattern, like a bell shaped curve or a unimodal binomial. Non-parametric methods collect a historical window of data, which is "messy;" this is what it means that non-parametric methods do not make distributional assumption. Instead, they generate a possibly incoherent histogram. Unlike a normal which is well-behaved because it's a function, a non-parametric histogram can have all sort of shapes (like a shoreline on a map).

A normal distribution is parametric and there are many other parametric distributions, normal is just popular for learning and sometimes justified by CLT. VaR is a feature of any valid probability distribution, which itself can be parametric, non-parametric or some combination.

All of this has a world of further depth. I hope that's helpful.
 
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