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 tail,
events.
The generalized Pareto distribution (GPD) plays a central role.
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Framework
X1 , X2 , . . . are identically distributed random variables with
continuous distribution function
F (x) = P(Xi ≤ x).
These X ’s represent losses, so we shall focus on the upper tail of
F (·).
For a level q such as 0.95 or 0.99,
VaRq (F ) = F −1 (q)
and
ESq (F ) = E [X |X > VaRq (F )] .
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Statistics and Financial Risk
Generalized Pareto Distribution
The GPD has distribution function

1
1 − 1 + ξx − ξ
β Gξ,β (x) =
1 − exp − x
β
ξ 6= 0
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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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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So for x ≥ u, F (x) can be estimated by
Nu
n − Nu
F̂ (x) =
Gξ̂,β̂ (x − u) +
n
n
−1/ξ̂
x −u
Nu
1 + ξˆ
=1−
n
β̂
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ST 810-006
Statistics and Financial Risk
Value at Risk
The approximation F̂ (x) yields, for q > Nu /n,
"
#
−ξ̂
β̂
n
dq = u +
VaR
(1 − q)
−1 .
Nu
ξˆ
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ST 810-006
Statistics and Financial Risk
Expected Shortfall
For the approximating GPD
ESq
1
β(u) − ξu
=
+
.
VaRq
1 − ξ (1 − ξ)VaRq
So we can estimate ESq by
d
ˆ
c q = VaRq + β̂ − ξu
ES
ˆ
1 − ξˆ (1 − 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 = σt Xt ,
and σt = σ(Yt−1 , Yt−2 , . . . ) satisfies a GARCH recursion.
Then for one step ahead, VaR or ES is calculated conditionally
on σt :
d q (Yt |σt ) = σt VaR
d q (X )
VaR
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Statistics and Financial Risk
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,
(
exp[−(1 + ξx)−1/ξ ] ξ 6= 0
H(x) =
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 Fréchet 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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Statistics and Financial Risk
A necessary and sufficient condition for F (·) to belong to the
domain of attraction of the Fréchet distribution (GEVD with
ξ > 0) is
1 − F (tx)
→ x −1/ξ as t → ∞.
1 − F (t)
That is, 1 − F (x) is regularly varying as x → ∞.
The same condition is therefore also necessary and sufficient for
convergence of Fu (y ) to the GPD.
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