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ST 810-006 Statistics and Financial Risk
ST 810-006 Statistics and Financial Risk
Extreme Value Theory in Risk Management See McNeil, Extreme Value - - PowerPoint PPT Presentation
Extreme Value Theory in Risk Management See McNeil, Extreme Value - - PowerPoint PPT Presentation
ST 810-006 Statistics and Financial Risk Extreme Value Theory in Risk Management See McNeil, Extreme Value Theory for Risk Managers Risk management is essentially about tail events. Probabilists have developed a theory specific to extreme, or
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ST 810-006 Statistics and Financial Risk
Generalized Pareto Distribution The GPD has distribution function Gξ,β(x) = 1 −
- 1 + ξx
β
− 1
ξ
ξ = 0 1 − exp
- − x
β
- ξ = 0,
where β > 0 is a scale parameter, ξ is a shape parameter, and:
x ≥ 0 when ξ ≥ 0; 0 ≤ x ≤ −β/ξ when ξ < 0.
The GPD is heavy-tailed when ξ > 0. With ξ > 0, E(X k) < ∞ only for k < 1/ξ.
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ST 810-006 Statistics and Financial Risk
Excess Loss Distribution The conditional excess distribution function is Fu(y) = P(X ≤ u + y|X > u), y ≥ 0. That is, Fu(y) = F(y + u) − F(u) 1 − F(u) . Key result: for “nice” F(·) and for large u, Fu(y) ≈ Gξ,β(u)(y), where ξ and β(·) depend on F(·).
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ST 810-006 Statistics and Financial Risk
That is, for a high threshold u, losses over that threshold are approximately GPD. Proposal: given data X1, X2, . . . , Xn, choose a “sensible” u, and estimate ξ and β = β(u) by fitting the GPD (e.g. by Maximum Likelihood) to the Nu observed excesses {Xi : Xi ≥ u}. Note that F(x) = [1 − F(u)]Fu(x − u) + F(u) ≈ [1 − F(u)]Gξ,β(u)(x − u) + F(u) and F(u) can be estimated by (n − Nu)/n.
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ST 810-006 Statistics and Financial Risk
So for x ≥ u, F(x) can be estimated by ˆ F(x) = Nu n
- Gˆ
ξ,ˆ β(x − u) +
n − Nu n
- = 1 − Nu
n
- 1 + ˆ
ξ x − u ˆ β −1/ˆ
ξ
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ST 810-006 Statistics and Financial Risk
Value at Risk The approximation ˆ F(x) yields, for q > Nu/n,
- VaRq = u +
ˆ β ˆ ξ n Nu (1 − q) −ˆ
ξ
− 1
- .
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ST 810-006 Statistics and Financial Risk
Expected Shortfall For the approximating GPD ESq VaRq = 1 1 − ξ + β(u) − ξu (1 − ξ)VaRq . So we can estimate ESq by
- ESq =
- VaRq
1 − ˆ ξ + ˆ β − ˆ ξu (1 − ˆ ξ)
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ST 810-006 Statistics and Financial Risk
EVT and GARCH Above, losses have been modeled as the independent and identically distributed X1, X2, . . . . This ignores volatility dynamics. An EVT-GARCH approach models losses by Y1, Y2, . . . , where Yt = σtXt, and σt = σ(Yt−1, Yt−2, . . . ) satisfies a GARCH recursion. Then for one step ahead, VaR or ES is calculated conditionally
- n σt:
- VaRq(Yt|σt) = σt
VaRq(X)
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For k > 1 steps ahead, it is necessary to simulate Yt, Yt+1, Yt+k−1. We have a model only for the upper tail of X. Proposal: use the empirical distribution for the rest of the distribution (i.e., historical simulation, or bootstrap).
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ST 810-006 Statistics and Financial Risk
Extreme Value Distribution Classic extreme value theory deals with the distribution of Mn = max(X1, X2, . . . , Xn). We look for constants an > 0 and bn such that Mn − bn an converges in distribution to some limit H(·). Fisher-Tippett-Gnedenko Theorem: H(·) must be the Generalized Extreme Value Distribution; for some ξ, H(x) =
- exp[−(1 + ξx)−1/ξ]
ξ = 0 exp (−e−x) ξ = 0.
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If constants an and bn can be found, F(·) belongs to the (maximum) domain of attraction of H(·). The three cases are:
ξ > 0: the Fr´ echet distribution; ξ = 0: the Gumbel distribution; ξ < 0: the Weibull distribution.
Pickands-Balkema-de Haan Theorem: the distribution function F(·) is “nice” if and only if it belongs to the domain of attraction of H(·) for some ξ. The parameter ξ has the same value in the GEVD as in the GPD.
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ST 810-006 Statistics and Financial Risk