Parametric distribution

[Kaiser]

Hi,

Could you confirm why the solution says that GARCH(1,1) is not a parametric approach, specially when the question 324.1D confirms that EWMA is parametric and we know that EWMA is a special case of GARCH


323.1. B.
In regard to (C), normal GARCH(1,1) assumes conditional returns are normal but also, as a Monte Carlo Simulation, this is not a parametric approach.

Bionic Turtle FRM Practice Questions Reading 21 Allen, Understanding Market, Credit and Operational Risk: The Value at Risk Approach

323.1. Analyst Peter observes that conditional equity returns exhibit leptokurtosis (i.e., heavytail) and negative skewness. Due to time constraints, Peter must use a parametric (analytical) value at risk (VaR) model. Which of the following models is most likely able to model his conditional returns?
a) Normal VaR; i.e., basic so-called delta-normal VaR where portfolio return is a linear function of asset returns that are normal
b) Normal mixture VaR; i.e., portfolio return is a linear function of asset returns that are parametric but characterized by a mixture of normal ("normal mixture") densities
c) Normal GARCH (1,1) VaR model; i.e., Monte Carlo simulation with GARCH volatility
d) Student's t GARCH (1,1) VaR model; i.e., portfolio return is a linear function of asset returns that are characterized (parametrically) by student's t distribution

324.1. D. Implied volatility uses current prices. In regard to (A), parametric approaches tend to use the historical returns to inform (fit) the parameters, at least. In regard to (B), EWMA is parametric and hybrid is non-parametric


Rgds,

 

[Nicole Seaman]

Hello @Kaiser

Please post this question in the original forum thread. Here is the original forum thread for this practice question: https://forum.bionicturtle.com/threads/p1-t4-323-the-shape-of-asset-returns.7149/. Please make sure to ask all questions under their original post. All practice question sets include the forum links on the answer page(s). Posting in the original forum thread keeps our forum organized so there are not new or duplicate threads posted, it helps others who may have the same question to easily find answers and you may also find that your question has been answered already.

Thank you,

Nicole

 

[David Harper CFA FRM]

Hi @Kaiser It's a good observation. The reason that 324.1 says EWMA is parametric is, to be blunt, that Linda Allen (Chapter 2, assigned in Topic 4 Valuation) says EWMA and GARCH(1,1) are parametric. This is probably the best place to start; i.e., a one-day VaR based on an EWMA-updated or GARCH-updated volatility is, after all, essentially a distribution based on a few parameters (e.g., current variance) and not a set of datapoints.

However, it really depends on what do we mean by parametric exact? I think when we try to answer that, we understand why some of these methods are called semi-parametric rather than either/or ... I think Allen's distinction is good because, for starters, we can define non-parametric as any approach which determines the VaR based on a final distribution that is empirical and not itself determined by a function.

So my 323.1.(c) should maybe be more explicit. Although it implies Monte Carlo, it might be better if it explicitly read:

c) Monte Carlo VaR model with GARCH(1,1) assumption for volatility, or even
c) Multi-step Monte Carlo VaR model with GARCH(1,1) assumption for volatilty

... which shows how GARCH(1,1) can inform--and in fact would be popular dynamic to inform--a Monte Carlo model. And this begs a key difference between GARCH(1,1) and EWMA: GARCH(1,1) is not merely a generalization of EWMA. When it comes to forecasting (which is what VaR does!), EMWA has no real features (e.g., the best forecast might simply be today's covariance matrix), but GARCH(1,1) is rather built to forecast (and has many variations). Indeed, it may be the case for an n-day GARCH(1,1) estimate that we need a simulation and cannot find the result analytically. So, Linda Allen's assertion that GARCH is parametric is a low-level statement; GARCH would be a good choice in a non-parametric (or semi-parametric) simulation, too. I hope that defends an inconsistency which is a sharp observation on your part!

 

[Kaiser]

Thanks vm for shedding some light on the inconsistency. The answer makes more sense looking at GARCH this way.