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MRMM-13-parametric-approaches-extreme-value.html
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<title>Market Risk Measurement and Management | Chapter 13 | Parametric Approaches: Extreme Value</title>
<meta name="description" content="Financial Risk Manager Part 2 Study Materials">
<meta name="author" content="MacLane Wilkison">
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<section>
<h1>Chapter 13</h1>
<h3>Parametric Approaches: Extreme Value</h3>
<p>
<small>Created for <a href="http://alchemistsacademy.com">Alchemists Academy</a> by <a href="http://alchemistsacademy.com/about">MacLane Wilkison</a></small>
</p>
</section>
<section>
<h2>Introduction</h2>
<p><em>Definition: A branch of applied statistics that focuses on the distinctiveness of extreme values</em></p>
</section>
<section>
<section>
<h2>Generalized Extreme-Value Theory</h2>
<ul>
<li>H<sub>ξ,μ,σ</sub>=</li>
<ul>
<li>exp[-(1+ξ[x-μ]/σ)<sup>-1/ξ</sup>], if ξ ≠ 0</li>
<li>exp[-exp([x-μ]/σ)], if ξ = 0</li>
</ul>
<li>If ξ < 0, the GEV becomes a Frechet distribution</li>
<li>If ξ = 0, the GEV becomes a Gumbel distribution</li>
<li>If ξ < 0, the GEV becomes a Weibull distribution</li>
</ul>
<aside class="notes">
'x' satisfies the condition where 1+ξ(x-μ)/σ > 0. 'μ' = the location parameter of the limiting distribution. 'σ' = the scale parameter of the limiting distribution. 'ξ' = the tail index (an indication of the shape of the tail of the limiting distribution.
</aside>
</section>
<section>
<h2>Estimation of EV Parameters</h2>
<ul>
<li>Maximum likelihood (ML) methods</li>
<li>Regression methods</li>
<li>Semi-parametric methods</li>
</ul>
</section>
</section>
<section>
<h2>Peaks-Over-Threshold Approach: Generalized Pareto Distribution</h2>
<ul>
<li>Natural way to model exceedances over a high threshold</li>
<li>F<sub>u</sub>(x) = Pr{X-u≤x|X>u} = [F(x+u)-F(u)]/[1-F(u)]</li>
<li>Gnedenko-Pickands-Balkema-deHaan (GPBdH):</li>
<ul>
<li>If ξ≠0: G<sub>ξ,β</sub>(x) = 1-(1+ξx/β)<sup>-1/ξ</sup></li>
<li>If ξ=0: G<sub>ξ,β</sub>(x) = 1-exp(-x/β)</li>
</ul>
</ul>
<aside class="notes">
As 'u' increases, the F<sub>u</sub>(x) distribution converges to a generalized Pareto distribution given by GPBdH.
</aside>
</section>
<section>
<h2>Refinements to EV Approaches</h2>
<ul>
<li>Conditional EV</li>
<li>Dealing with dependent (non-iid) data</li>
<li>Multivariate EVT</li>
</ul>
<aside class="notes">
Conditional EV is applied when 'X' is adjusted for some dynamic structure. There are two simple solutions for dealing with non-iid data that exhibits clustering: (1) apply GEV distributions to block maxima and (2) estimate the tail of the conditional distribution rather than the unconditional one. Multivariate EVT can be used to model the tails of multivariate distributions in a theoretically appropriate way.
</aside>
</section>
<section>
<h2>Limitations of EV</h2>
<ul>
<li>Relatively few observations leads to uncertain estimates</li>
<li>Considerable model risk</li>
<li>Sensitive to small sample effects, biases, non-linearities, and other problems</li>
</ul>
</section>
<section>
<h1>THE END</h1>
<h3><a href="http://alchemistsacademy.com">AlchemistsAcademy.com</a></h3>
</section>
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