1.3 Additional Properties
1.3.3 Self-decomposability
Suppose that (Xt) is a simple autoregressive process, i.e.
Xt= ρXt−1+ εt, t = 1, 2, . . . ,
where each εt is white noise independent of Xt−1. If (Xt) is stationary (i.e. |ρ| < 1) then Xt= Xd t+1= ρXd t+ εt+1, t = 1, 2, . . . .
If a random vector X can be represented stochastically by X = ρX + εd (ρ), ∀ ρ ∈ ]0, 1[ ,
where ε(ρ)is independent of X then it is called ‘self-decomposable’ (Barndorff-Nielsen and Shephard, 2003, Section 1.2.2). Hence, a random vector X is self-decomposable if its cha-racteristic function satisÞes the property t 7→ ϕX(t) = ϕX(ρt) ϕ(ρ)(t), ∀ ρ ∈ ]0, 1[, where ϕ(ρ)denotes the characteristic function of ε(ρ)which is considered as white noise. Note that ϕ(ρ) depends essentially on the parameter ρ. The larger ρ the smaller the contribution of the white noise and vice versa. Any self-decomposable law is inÞnitely divisible (Barndorff-Nielsen and Shephard, 2003, p. 13).
Now, consider again the characteristic generator of the sub-Gaussian α-stable distribution.
Since for every s ≥ 0,
= φsub¡ ρ2s ; α¢
· exp
−
Ã(1 − ρα)2/α
2 · s
!α/2
= φsub¡ ρ2s ; α¢
· φsub
³
(1 − ρα)2/αs ; α´ ,
any Gaussian α-stable random vector is self-decomposable. More precisely, if X is sub-Gaussian α-stable with dispersion matrix Σ then
X= ρX + (1 − ρd α)1/αε, ∀ ρ ∈ ]0, 1[ ,
where the white noise ε(ρ)= (1 − ρα)1/αε is also sub-Gaussian α-stable possessing the same dispersion matrix as X.
Not only the Gaussian and sub-Gaussian α-stable distributions are self-decomposable (and thus inÞnitely divisible) but also the family of symmetric generalized hyperbolic distributions (Barndorff-Nielsen, Kent, and Sørensen, 1982, Bingham and Kiesel, 2002). This particular elliptical distribution family is extremely useful for the modeling of Þnancial data and will be discussed in more detail in Section 3.1.
Extreme Values and
Dependence Structures
This chapter concentrates on univariate and multivariate extremes with special focus on el-liptical distributions. For a more comprehensive treatment of extreme value theory see, e.g., Bingham, Goldie, and Teugels, 1987, Coles, 2001, Embrechts, Klüppelberg, and Mikosch, 2003, Mikosch, 1999, and Resnick, 1987.
2.1 Univariate Extremes
The probability distribution of extremal events is a priori of main interest for insurance and Þnance. Extreme value theory (EVT) is a special topic of probability theory and has become standard in risk theory and management (see, e.g., Embrechts, Klüppelberg, and Mikosch, 2003). In insurance it has been used e.g. to calculate the potential severity of losses caused by natural disasters like earthquakes, hurricanes, ßoods, etc. (Haan, 1990, McNeil, 1997, McNeil and Saladin, 2000, Resnick, 1997, as well as Rootzén and Tajvidi, 1997). Calculating the value-at-risk of asset portfolios on high conÞdence-levels has retained as a typical Þnance application of the theory of extreme values (Danielsson and Vries, 2000, Frahm, 1999, and Këllezi and Gilli, 2003).
The fundamental theorem of Fisher-Tippett (Embrechts, Klüppelberg, and Mikosch, 1997, p. 121 in connection with p. 152) preludes the transition from classical statistics to EVT.
Theorem 10 (Fisher and Tippett, 1928) Let X1, . . . , Xn (n = 1, 2, . . .) be sequences of i.i.d. random variables and Mn := max {X1, . . . , Xn} the corresponding sample maximum.
If there exist norming constants an> 0, bn ∈ IR and a non-degenerate c.d.f. H such that Mn− bn
an
−→ H,d n −→ ∞, then there exist parameters σ > 0 and µ, ξ ∈ IR such that
H (x) = Hξ
µx − µ σ
¶
=
exp³
−¡
1 + ξ · x−µσ
¢−1ξ´
, ξ 6= 0, exp¡
− exp¡
−x−µσ ¢¢
, ξ = 0,
with support
IDξ,σ,µGEV ≡
x > µ − σξ, ξ > 0, x ∈ IR, ξ = 0, x < µ − σξ, ξ < 0.
19
The limit law H is referred to as ‘generalized extreme value distribution’ (GEV).
Proof. Resnick, 1997, pp. 9-10, using the Jenkinson-von-Mises representation of the ex-treme value distributions (Embrechts, Klüppelberg, and Mikosch, 1997, p. 152).
The limit theorem of Fisher-Tippett can be interpreted as a ‘sample-maxima analogue’ to the classical central limit theorem which is based on sample sums (Embrechts, Klüppelberg, and Mikosch, 1997, p. 120). But rather than the probabilistic property of a sum of i.i.d.
variables the stochastic behavior of the maximum plays the key role when investigating the tail of a distribution.
DeÞnition 3 (MDA) The c.d.f. F of X (or roughly speaking the random variable X) belongs to the maximum domain of attraction (MDA) of the GEV Hξ if the conditions of the Fisher-Tippett theorem hold for F (for X). We write F ∈ MDA (Hξ) or X ∈ MDA (Hξ), respectively.
The parameter ξ is constant under affine transformations so that both the scale parameter σ and the location parameter µ can be neglected.
Theorem 11 (Tail behavior of MDA (H0)) Let F be a c.d.f. with right endpoint xF ≤ ∞ and F := 1 −F its survival function. F ∈ MDA (H0) if and only if a constant v < xF exists, such that
F (x) = γ (x) · exp
−
Zx
v
1 f (t) dt
, v < x < xF,
where γ is a measurable function with γ (x) → γ > 0, x %xF, and f is a positive, absolutely continuous function with f0(x) → 0, x%xF.
Proof. Resnick (1987), Proposition 1.4 and Corollary 1.7.
Theorem 11 implies that every tail of a distribution F ∈ MDA (H0) with xF = ∞ may be approximated by the exponential law
F (x) ≈ γ (v) · exp µ
−x − v f (v)
¶
, ∀ x > v,
provided v is a sufficiently high threshold. We say that the distribution F is ‘light-’, ‘thin-’, or ‘exponential-tailed’. This is given e.g. for the normal-, lognormal-, exponential-, and the gamma-distribution. The following theorem can be found in Embrechts, Klüppelberg, and Mikosch (2003, p. 131).
Theorem 12 (Tail behavior of MDA (Hξ>0)) F ∈ MDA (Hξ>0) if and only if F (x) = λ (x) · x−1ξ, x > 0,
where λ is a slowly varying function, i.e. λ is a measurable function IR+→ IR+ satisfying λ (tx)
λ (x) −→ 1, x −→ ∞, ∀ t > 0.
Proof. Embrechts, Klüppelberg, and Mikosch, 2003, pp. 131-132.
In the following α := 1/ξ is called the ‘tail index’ of F . For ξ = 0 we deÞne α = ∞. A measurable function f : IR+→ IR+ with
f (tx)
f (x) −→ t−α, x −→ ∞, ∀ t > 0,
is called ‘regularly varying (at ∞) with tail index α ≥ 0’ (Mikosch, 1999, p. 7). Note that a slowly varying function is regularly varying with tail index α = 0.
A survival function F is regularly varying if and only if F ∈ MDA (Hξ>0) (Resnick, 1987, p. 13). Thus a regularly varying survival function exhibits a power law, that is to say
F (x) ≈ λ (v) · x−1ξ = λ (v) · x−α, ξ, α > 0, ∀ x > v > 0,
for a suffiently high threshold v. Now the c.d.f. F is said to be ‘heavy-’, ‘fat-’, or ‘power-tailed’. This is the case e.g. for the Pareto-, Burr-, loggamma-, Cauchy-, and Student’s t-distribution. Clearly a power tail converges slower to zero than an exponential tail and therefore this class is of special interest to risk theory.
A random variable X is said to be regularly varying with tail index α > 0 if both X ∈ MDA¡
H1/α¢
and −X ∈ MDA¡ H1/α¢
(Mikosch, 2003, p. 23). Then the tail index α has a nice property: X has no Þnite moment of orders larger than α. Conversely, for α = ∞ every moment exists and is Þnite. Hence the smaller the tail index the bigger the weight of the tail. Therewith both ξ and α are well suited to characterize the tailedness of the underlying distribution (Embrechts, Klüppelberg, and Mikosch, 1997, p. 152):
ξ > 0 ⇔ 0 < α < ∞ : Fréchet class,
ξ = 0 ⇔ α = ∞ : Gumbel class,
ξ < 0 ⇔ −∞ < α < 0 : Weibull class.
DeÞnition 4 (GPD) The distribution function
Gξ
µx − µ σ
¶
=
1 −¡
1 + ξ ·x−µσ
¢−1ξ
, ξ 6= 0, 1 − exp¡
−x−µσ ¢
, ξ = 0,
σ > 0,
with support
IDGP Dξ,µ,σ ≡
x ≥ µ, ξ ≥ 0,
µ ≤ x < µ − σξ, ξ < 0, is referred to as ‘generalized Pareto distribution’ (GPD).
The second crucial theorem of EVT is the following one.
Theorem 13 (Pickands, 1975) F ∈ MDA (Hξ∈IR) if and only if a positive function σ exists with
vlim%xF
sup
0≤y<xF−v
¯¯
¯¯Fv→(y) − Gξ
µ y σ (v)
¶¯¯¯¯ = 0, where
Fv→(y) := P (X − v ≤ y | X > v) . Proof. Embrechts, Klüppelberg, and Mikosch, 2003, pp. 165-166.
The c.d.f. Fv→is referred to as ‘excess distribution (over the threshold v)’. Thus the GPD is the limit distribution of an arbitrary excess distribution Fv→ provided its threshold v is sufficiently high. Theorem 13 implies that the excess distribution depends asymptotically only on two parameters, a scale parameter σ and a shape parameter ξ, respectively. The scale parameter is determined by the selected threshold whereas the shape parameter is constant through all thresholds because it only depends on the MDA of F . So the probability of extremal events may be quantiÞed without knowing the underlying distribution family and
the corresponding parameters may be estimated in a semi-parametric fashion allowing for heavy tails.
The advantage of non-parametric methods is that no information about the distribution law is required and unbiased estimates can be attained. That is to say the resulting estimates are robust regarding an error in the interpretation of the true law. The trade off is large variance of the estimates. If the true law were known clearly the corresponding parametric approach would be more efficient. But biased estimates appear if the predicted distribution law is wrong, i.e. parametric methods bear an awkward model risk. With semi-parametric methods one can take advantage of the particular attributes of both, the parametric and the non-parametric approaches. So EVT is a proper compromise between bias (parametric methods) and variance (non-parametric methods) provided the intention to quantify the probability of extremal events.
Several empirical studies (see, e.g., Haan, 1990, McNeil, 1997, and Rootzén and Tajvidi, 1997) in the context of insurance show that loss distributions belong signiÞcantly to the Fréchet class. So classical loss distributions like gamma and lognormal are not suitable for modeling extreme losses. There is also a huge number of empirical studies showing that log-returns of Þnancial assets usually are heavy tailed, too (see, e.g., Eberlein and Keller, 1995, Embrechts, Klüppelberg, and Mikosch, 2003, Section 7.6, Fama, 1965, Mandelbrot, 1963, and Mikosch, 2003, Chapter 1). From our point of view we are interested in the class of distributions which can be properly applied for modeling the generating variate of elliptically distributed log-returns. Especially it is worth to know the consequences regarding the dependence structure of elliptical distributions.