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# Historical Simulation..Parametric or Nonparametric Appoach?

#### Hermz29

##### Member
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.

#### sharman.jamie

##### Member
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
Staff member
Subscriber
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
Staff member
Subscriber
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!

#### Laely

##### New Member
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!!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
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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#### MR GATE

##### New Member
hi David can you shade some light on some of the challenges that are faced in quantifying MODEL RISK?