Slides toulouse

Arthur CHARPENTIER - Nonparametric quantile estimation.

Estimating quantiles
and related risk measures
Arthur Charpentier

arthur.charpentier@univ-rennes1.fr

S´minaire du GREMAQ, D´cembre 2007
e e

joint work with Abder Oulidi, IMA Angers

1


Agenda
• General introduction
Risk measures
• Distorted risk measures
• Value-at-Risk and related risk measures
Quantile estimation : classical techniques
• Parametric estimation
• Semiparametric estimation, extreme value theory
• Nonparametric estimation
Quantile estimation : use of Beta kernels
• Beta kernel estimation
• Transforming observations
A simulation based study

2


Agenda
Risk measures

3


Risk measures and price of a risk
Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth century
proposed to evaluate the “produit scalaire des probabilit´s et des gains”,
e
n
< p, x >= pi xi = EP (X),
i=1

based on the “r`gle des parties”.
e
For Qu´telet, the expected value was, in the context of insurance, the price that
e
guarantees a ﬁnancial equilibrium.
From this idea, we consider in insurance the pure premium as EP (X). As in
Cournot (1843), “l’esp´rance math´matique est donc le juste prix des chances”
e e
(or the “fair price” mentioned in Feller (1953)).

4


Risk measures : the expected utility approach

Ru (X) = u(x)dP = P(u(X) > x))dx

where u : [0, ∞) → [0, ∞) is a utility function.
Example with an exponential utility, u(x) = [1 − e−αx ]/α,
1
Ru (X) = log EP (eαX ) .
α

5


Risk measures : Yarri’s dual approach

Rg (X) = xdg ◦ P = g(P(X > x))dx

where g : [0, 1] → [0, 1] is a distorted function.
Example
– if g(x) = I(X ≥ 1 − α) Rg (X) = V aR(X, α),
– if g(x) = min{x/(1 − α), 1} Rg (X) = T V aR(X, α) (also called expected
shortfall), Rg (X) = EP (X|X > V aR(X, α)).

6


Distortion of values versus distortion of probabilities

Calcul de l’esperance mathématique

1.0
0.8
0.6
0.4
0.2
0.0

0 1 2 3 4 5 6

Fig. 1 – Expected value xdFX (x) = P(X > x)dx.

7



Calcul de l’esperance d’utilité

1.0
0.8
0.6
0.4
0.2
0.0

0 1 2 3 4 5 6

Fig. 2 – Expected utility u(x)dFX (x).

8



Calcul de l’intégrale de Choquet

1.0
0.8
0.6
0.4
0.2
0.0

0 1 2 3 4 5 6

Fig. 3 – Distorted probabilities g(P(X > x))dx.

9


Distorted risk measures and quantiles
1 −1
Equivalently, note that E(X) = 0
FX (1 − u)du, and
1 −1
Rg (X) = 0
FX (1 − u)dgu.
A very general class of risk measures can be deﬁned as follows,
1
−1
Rg (X) = FX (1 − u)dgu
0

where g is a distortion function, i.e. increasing, with g(0) = 0 and g(1) = 1.
Note that g is a cumulative distribution function, so Rg (X) is a weighted sum of
quantiles, where dg(1 − ·) denotes the distribution of the weights.

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Distortion function, VaR (quantile) − cdf Distortion function, TVaR (expected shortfall) − cdf Distortion function, cdf

1.0

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1 − probability level 1 − probability level 1 − probability level

Distortion function, VaR (quantile) − pdf Distortion function, TVaR (expected shortfall) − pdf Distortion function, pdf
1.0

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1 − probability level 1 − probability level 1 − probability level

Fig. 4 – Distortion function, g and dg

11


Agenda
Risk measures

12


Using a parametric approach
−1
If FX ∈ F = {Fθ , θ ∈ Θ} (assumed to be continuous), qX (α) = Fθ (α), and thus,
a natural estimator is
qX (α) = F −1 (α), (1)
θ

where θ is an estimator of θ (maximum likelihood, moments estimator...).

13


Using the Gaussian distribution
A natural idea (that can be found in classical ﬁnancial models) is to assume
Gaussian distributions : if X ∼ N (µ, σ), then the α-quantile is simply

q(α) = µ + Φ−1 (α)σ,

where Φ−1 (α) is obtained in statistical tables (or any statistical software), e.g.
u = −1.64 if α = 90%, or u = −1.96 if α = 95%.
Deﬁnition 1. Given a n sample {X1 , · · · , Xn }, the (Gaussian) parametric
estimation of the α-quantile is

qn (α) = µ + Φ−1 (α)σ,

14


Using a parametric models
Actually, is the Gaussian model does not ﬁt very well, it is still possible to use
Gaussian approximation
If the variance is ﬁnite, (X − E(X))/σ might be closer to the Gaussian
distribution, and thus, consider the so-called Cornish-Fisher approximation, i.e.

Q(X, α) ∼ E(X) + zα V (X), (2)

where
2
ζ1 −1 2 ζ2 −1 3 ζ1
zα = Φ (α)+ [Φ (α) −1]+ [Φ (α) −3Φ (α)]− [2Φ−1 (α)3 −5Φ−1 (α)],
−1 −1
6 24 36
(3)
where ζ1 is the skewness of X, and ζ2 is the excess kurtosis, i.e. i.e.
E([X − E(X)]3 ) E([X − E(X)]4 )
ζ1 = 2 )3/2
and ζ1 = 2 )2
− 3. (4)
E([X − E(X)] E([X − E(X)]

15


Using a parametric models
Deﬁnition 2. Given a n sample {X1 , · · · , Xn }, the Cornish-Fisher estimation of
the α-quantile is
n n
1 1 2
qn (α) = µ + zα σ, where µ = Xi and σ = (Xi − µ) ,
n i=1
n−1 i=1

and
2
−1 ζ1 −1 2 ζ2 −1 3 ζ1
zα = Φ (α)+ [Φ (α) −1]+ [Φ (α) −3Φ (α)]− [2Φ−1 (α)3 −5Φ−1 (α)],
−1
6 24 36
where ζ1 is the natural estimator for the skewness of X, and ζ2 is the natural
√
n(n − 1) n n (Xi − µ)3
i=1
estimator of the excess kurtosis, i.e. ζ1 = 3/2
and
n−2 ( n
(Xi − µ)2 ) i=1
n 4
n−1 n i=1 (Xi −µ)
ζ2 = (n−2)(n−3) (n + 1)ζ2 + 6 where ζ2 = n 2 − 3.
( i=1 (Xi −µ)2)

16


Parametrics estimator and error model

Density, theoritical versus empirical Density, theoritical versus empirical
0.8

0.3
0.6

Theoritical Student
Theoritical lognormal Fitted lStudent
Fitted lognormal

0.2
Fitted Gaussian
Fitted gamma
0.4

0.1
0.2
0.0

0.0
0 1 2 3 4 5 −4 −2 0 2 4

Fig. 5 – Estimation of Value-at-Risk, model error.

17


Using a semiparametric models
Given a n-sample {Y1 , . . . , Yn }, let Y1:n ≤ Y2:n ≤ . . .≤ Yn:n denotes the associated
order statistics.
If u large enough, Y − u given Y > u has a Generalized Pareto distribution with
parameters ξ and β ( Pickands-Balkema-de Haan theorem)
If u = Yn−k:n for k large enough, and if ξ> 0, denote by βk and ξk maximum
likelihood estimators of the Genralized Pareto distribution of sample
{Yn−k+1:n − Yn−k:n , ..., Yn:n − Yn−k:n },

βk n − ξk
Q(Y, α) = Yn−k:n + (1 − α) −1 , (5)
ξk k

An alternative is to use Hill’s estimator if ξ > 0,
− ξk k
n 1
Q(Y, α) = Yn−k:n (1 − α) , ξk = log Yn+1−i:n − log Yn−k:n . (6)
k k i=1

18


On nonparametric estimation for quantiles
−1
For continuous distribution q(α) = FX (α), thus, a natural idea would be to
−1
consider q(α) = FX (α), for some nonparametric estimation of FX .
Deﬁnition 3. The empirical cumulative distribution function Fn , based on
n
1
sample {X1 , . . . , Xn } is Fn (x) = 1(Xi ≤ x).
n i=1
Deﬁnition 4. The kernel based cumulative distribution function, based on
sample {X1 , . . . , Xn } is
n x n
1 Xi − t 1 Xi − x
Fn (x) = k dt = K
nh i=1 −∞ h n i=1
h
x
where K(x) = k(t)dt, k being a kernel and h the bandwidth.
−∞

19


Smoothing nonparametric estimators
Two techniques have been considered to smooth estimation of quantiles, either
implicit, or explicit.
• consider a linear combinaison of order statistics,
The classical empirical quantile estimate is simply

−1 i
Qn (p) = Fn = Xi:n = X[np]:n where [·] denotes the integer part. (7)
n

The estimator is simple to obtain, but depends only on one observation. A
natural extention will be to use - at least - two observations, if np is not an
integer. The weighted empirical quantile estimate is then deﬁned as

Qn (p) = (1 − γ) X[np]:n + γX[np]+1:n where γ = np − [np].

20


The quantile function in R The quantile function in R

7
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type=1 q q
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type=1 qq
type=3 qq
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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

probability level probability level

Fig. 6 – Several quantile estimators in R.

21


In order to increase eﬃciency, L-statistics can be considered i.e.
n n 1
−1 i −1
Qn (p) = Wi,n,p Xi:n = Wi,n,p Fn = Fn (t) k (p, h, t) dt (8)
i=1 i=1
n 0

where Fn is the empirical distribution function of FX , where k is a kernel and h a
bandwidth. This expression can be written equivalently
n i n i i−1
n t−p n −p n −p
Qn (p) = k dt X(i) = IK − IK X(i)
i=1
(i−1)
n
h i=1
h h
(9)
x
where again IK (x) = k (t) dt. The idea is to give more weight to order
−∞
statistics X(i) such that i is closed to pn.

22


3
2
1
0

0.0 0.2 0.4 0.6 0.8 1.0

quantile (probability) level

Fig. 7 – Quantile estimator as wieghted sum of order statistics.

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E.g. the so-called Harrell-Davis estimator is defined as
n i
n Γ(n + 1)
Qn (p) = y (n+1)p−1 (1 − y)(n+1)q−1 Xi:n ,
i=1
(i−1)
n
Γ((n + 1)p)Γ((n + 1)q)

• find a smooth estimator for FX , and then find (numerically) the inverse,
The α-quantile is defined as the solution of FX ◦ qX (α) = α.
If Fn denotes a continuous estimate of F , then a natural estimate for qX (α) is
qn (α) such that Fn ◦ qn (α) = α, obtained using e.g. Gauss-Newton algorithm.

28


Agenda
Risk measures

29


Kernel based estimation for bounded supports
Classical symmetric kernel work well when estimating densities with
non-bounded support,
n
1 x − Xi
fh (x) = k ,
nh i=1
h

where k is a kernel function (e.g. k(ω) = I(|ω| ≤ 1)/2).
If K is a symmetric kernel, note that
1
E(fh (0) = f (0) + O(h)
2

30


Kernel based estimation of the uniform density on [0,1] Kernel based estimation of the uniform density on [0,1]
1.2

1.2
1.0

1.0
0.8

0.8
Density

Density
0.6

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0.4
0.2

0.2
0.0

0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 12 – Density estimation of an uniform density on [0, 1].

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Kernel based estimation for bounded supports
Several techniques have been introduce to get a better estimation on the border,
– boundary kernel (Muller (1991))
¨
– mirror image modiﬁcation (Deheuvels & Hominal (1989), Schuster
(1985))
– transformed kernel (Devroye & Gyrfi (1981), Wand, Marron &
Ruppert (1991))
– Beta kernel (Brown & Chen (1999), Chen (1999, 2000)),
see Charpentier, Fermanian & Scaillet (2006) for a survey with
application on copulas.

32


Beta kernel estimators
A Beta kernel estimator of the density (see Chen (1999)) - on [0, 1] is
n
1 x 1−x
fb (x) = k Xi , 1 + , 1 + , x ∈ [0, 1],
n i=1
b b

uα−1 (1 − u)β−1
where k(u, α, β) = , u ∈ [0, 1].
B(α, β)
If {X1 , · · · , Xn } are i.i.d. variables with density f0 , if n → ∞, b → 0, then
Bouzmarni & Scaillet (2005)

fb (x) → f0 (x), x ∈ [0, 1].

This is the Beta 1 estimator.

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Beta kernel, x=0.05 Beta kernel, x=0.10

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Beta kernel, x=0.20 Beta kernel, x=0.45
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6
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4
4

2
2
0

0.0 0.2 0.4 0.6 0.8 1.0 0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 13 – Shape of Beta kernels, diﬀerent x’s and b’s.

34


Improving Beta kernel estimators
Problem : the convergence is not uniform, and there is large second order bias
on borders, i.e. 0 and 1.
Chen (1999) proposed a modiﬁed Beta 2 kernel estimator, based on

 k t , 1−t (u) , if t ∈ [2b, 1 − 2b]
 b b

k2 (u; b; t) = k 1−t (u) , if t ∈ [0, 2b)
 ρb (t), b

k b ,ρb (1−t) (u) , if t ∈ (1 − 2b, 1]

t

t
where ρb (t) = 2b2 + 2.5 − 4b4 + 6b2 + 2.25 − t2 − .
b

35


Non-consistency of Beta kernel estimators
Problem : k(0, α, β) = k(1, α, β) = 0. So if there are point mass at 0 or 1, the
estimator becomes inconsistent, i.e.
1 x 1−x
fb (x) = k Xi , 1 + , 1 + , x ∈ [0, 1]
n b b
1 x 1−x
= k Xi , 1 + , 1 + , x ∈ [0, 1]
n b b
Xi =0,1

n − n0 − n1 1 x 1−x
= k Xi , 1 + , 1 + , x ∈ [0, 1]
n n − n0 − n1 b b
Xi =0,1

≈ (1 − P(X = 0) − P(X = 1)) · f0 (x), x ∈ [0, 1]

and therefore Fb (x) ≈ (1 − P(X = 0) − P(X = 1)) · F0 (x), and we may have
problem ﬁnding a 95% or 99% quantile since the total mass will be lower.

36


Non-consistency of Beta kernel estimators
Gourieroux & Monfort (2007) proposed
´

(1) fb (x)
fb (x) = 1 , for all x ∈ [0, 1].
0
fb (t)dt

It is called macro-β since the correction is performed globally.
Gourieroux & Monfort (2007) proposed
´
n
(2) 1 kβ (Xi ; b; x)
fb (x) = 1 , for all x ∈ [0, 1].
n i=1 kβ (Xi ; b; t)dt
0

It is called micro-β since the correction is performed locally.

37


Transforming observations ?
In the context of density estimation, Devroye and Gy¨
’orfi (1985) suggested
to use a so-called transformed kernel estimate
Given a random variable Y , if H is a strictly increasing function, then the
p-quantile of H(Y ) is equal to H(q(Y ; p)).
An idea is to transform initial observations {X1 , · · · , Xn } into a sample
{Y1 , · · · , Yn } where Yi = H(Xi ), and then to use a beta-kernel based estimator, if
H : R → [0, 1]. Then qn (X; p) = H −1 (qn (Y ; p)).
In the context of density estimation fX (x) = fY (H(x))H (x). As mentioned in
Devroye and Gyorfi (1985) (p 245), “for a transformed histogram histogram
¨
estimate, the optimal H gives a uniform [0, 1] density and should therefore be
equal to H(x) = F (x), for all x”.

38


Transforming observations ? a monte carlo study
Assume that sample {X1 , · · · , Xn } have been generated from Fθ0 (from a familly
F = (Fθ , θ ∈ Θ). 4 transformations will be considered
– H = Fθ (based on a maximum likelihood procedure)
– H = Fθ0 (theoritical optimal transformation)
– H = Fθ with θ < θ0 (heavier tails)
– H = Fθ with θ > θ0 (lower tails)

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Fig. 14 – F (Xi ) versus Fθ (Xi ), i.e. P P plot.
ˆ

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Estimated density

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Fig. 15 – Nonparametric estimation of the density of the Fθ (Xi )’s.
ˆ

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Estimated optimal transformation Estimated optimal transformation
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0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00

Probability level Probability level

−1
Fig. 16 – Nonparametric estimation of the quantile function, Fθ (q).
ˆ

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qq
q q
q
qq qq
qq
q
0.4

0.4
q
qqqq
qq
q
q
qq
q
q
qq
q q
q
qq
q q
qq
q
qq
qq q
q
q
qqq
qq
q
q
q
q
q
q
qq
q
qq
q qq
q
q
q
q
q qqq
qq
q
q
qq
q
q
q
qq
q
qqq
qq
qq
qq
q q
qq
q
qq
qq qq
qq
0.2

0.2
q qq
q
q
qqq
qq
qq
q q
q
qq
q
q q
qq
q
q
qq
qq
qq qq
qq
qq
qq
q qq
q
q
qq qq
qq
qq
qq
qqqq
qq
qq qq
q
qq
qq
qq qq
qq
qq
q
qq
qq
qq
q
qq
qq
q
qq
qq
qq qq
q
qq
q
q q
q
q
0.0

0.0
q qqq
q qq
q
q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 17 – F (Xi ) versus Fθ0 (Xi ), i.e. P P plot.

43


1.4

1.4
1.2

1.2
Estimated density

Estimated density
1.0

1.0
q
0.8

0.8
0.6

0.6
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 18 – Nonparametric estimation of the density of the Fθ0 (Xi )’s.

44



q q
4

4
q q

q q
Quantile

Quantile
q q
3

3
q q
q q
q q
q q
q q
q q
q q
q
2

q
q q

2
q q
q q
q q
q qq q
q
q
q qq qq
qq q
qq qq
q qq
q qq q qq
qq qq
qq
q qq
qq
qqq qqq
1

qqq qqq

1
0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00


−1
Fig. 19 – Nonparametric estimation of the quantile function, Fθ0 (q).

45


1.0

1.0
q qq
q
q q
qq
q
q
qq qq
q
q
qqq
qq
q q
q
q
q
qq
q
qq qq
q
qq
qq
qq
qq
q q
q
q
q
qq
qq
qq
qq qq
q
q
q
q
qq
qq
q
q qq
qq
q
q
q
qq
q
q
q qq
qq
0.8

0.8
qq qq
qq
q
qqq
qq
q qq
qq
qq
q
q
qq q
q
qq
q
qq q
qq
qq
qqq qq
qq
q

q
qq
qq qq
q
q
qq
qq q
q
qq
q
qqq
qq
q
q
q qq
qq
qq
qq
q
q
q qq
q
q
q
qq
q
q q
qq
qq
q qq
qq
qq
qq
qq
qq
q
q
0.6

0.6
q
q q
q
qq
qq
qq
q
q
qq
q
q
q
qq
q q
qq
qq
qqq qqq
qqq
qqq
qqq
qq
qq
qq
qq q
q
qq
qq
q
q
q
q qq
qq
qq
qq
q
q
q qq
qq
q
qq
qq
qq
qq
q
q q
q
qqq
qq qq
qq
q
qq
qqq
qqq
qqq
q
q
qq qq
qq
qq
0.4

0.4
q
qq q
q
qqq
q
q
q
q q
q
q
q
qq
qq
q
qq qq
qq
qq
qqq
q
q
qq
q
q
qq
q
q
q
q qq
q
q
q
q qqq
qq
q
q
qq
qq
q qq
qq
qq
q
qqq
q q
qq
qq
q
q qq
q qq
0.2

0.2
qq qq
q
q
q
qq q
q
qq
qqqq
qq
qq q
qq
q
qq
q qq
qq
qq
qq q
q
q
qq
qq
qq
qq qq
qq
qq
qq
qq qq
q
qq
qq
qq
qq
q
q qq
qq
qq
q
q
q
q
q
q qq
qq
qq
q
q
q
qq q
q
q
0.0

0.0
q qqq
q qq
q
q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 20 – F (Xi ) versus Fθ (Xi ), i.e. P P plot, θ < θ0 (heavier tails).

46


1.4

1.4
1.2

1.2
Estimated density

Estimated density
1.0

1.0
q
0.8

0.8
0.6

0.6
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 21 – Estimation of the density of the Fθ (Xi )’s, θ < θ0 (heavier tails).

47


12

q q

12
10

10
q q
8

8
Quantile

Quantile
q q
6

q q

6
q q
q q
q q
4

q q

4
q q
q q
q q
q q
q q
q qq q qq
q qq q qq
qq qq
2

qqq qqq

2
qqq qqq qqq
qqq
qq qqqq qq qqqq
qqqqq qqqqq

0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00


−1
Fig. 22 – Estimation of quantile function, Fθ (q), θ < θ0 (heavier tails).

48


1.0

1.0
qq
qq
qq
q qq
q
qq
q
q qq
qq
q
qqq
qqq
qqq q
q
q
q
q
q
q qq
qqq
qq
qq
qq
q
q
q
q
qq qq
q
q
q q
q
q
q
qq
q qq
q
q
q
q
qq q
qqq
qq
q
qq
qq
q
q
q qq
qq
0.8

0.8
q
q
q qq
qq
q
q
q
q qqq
qq
qq
q
q q
q
qq
q
qq
q
q q
q
q
q qq
qq

q
q
q
q
q
qqq
q
q
q
q
q
q q
q
q qq
qq
q
qq qq
qq
qq
qq
qq
qq
qq qq
q
q
q
q
qq q
q
q
qq
q q
qq
q
qq q
0.6

0.6
q
qqq
qq
q q
q
q
qq qq
q
q
qq
q
q
q q
qq
qq
q
qq
q qqq
qqq
qqq
qqq
qq
q q
q
q
qq
q
q
q
qq qq
qq
qq
qq
q
q
q
q
q
q q qq
qq
q
qq
q
q q
q
q
q
q qqq
qq
q
q q
q
q qq
qq
q
0.4

0.4
q
qq q
q
qq q
q
qqq
qq
qq
q
q
q
q
q
q
qqq
q
qq q
q
q
q
qq
q
q q
qq
q qq
qq
qq q
q
q
q qq
qq
qq
qq
q qq
qq
q
q
q
qq qq
q
qq
qq
qq
qq
q
qqq
q
q
qq
q
q q
qq
qq
qq
qq
qq qq
qq
0.2

0.2
qq
q
qq q
qqq
qq
q
qq
qqq
q
qq
q
q
q
q
q
q qq
qq
qq
q qq
q
qq
q
q
q
q qq
qq
qq
qq
qq
qq
qq
qq qq
q
qq
qq
qqq qq
qq
qq
q
qq
q
qq qq
qq
q
qqq
qq
qqq q
qqq
q
q q
0.0

0.0
q
q qqq
q
q
q qq
q
q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 23 – F (Xi ) versus Fθ (Xi ), i.e. P P plot, θ > θ0 (lighter tails).

49


1.4

1.4
1.2

1.2
Estimated density

Estimated density
1.0

1.0
q
0.8

0.8
0.6

0.6
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 24 – Estimation of density of Fθ (Xi )’s, θ > θ0 (lighter tails).

50



q q
3.5

3.5
q q
3.0

3.0
q q

q q
2.5

2.5
Quantile

Quantile
q q

q q
q q
q q
2.0

q q

2.0
q q
q q
q q
q q
q q
q q
q q
q q
1.5

q q

1.5
qq qq
q qq q qq
qq qq
qq qq
qq qq
q qq q qq
q qq q qq
1.0

qq qq

1.0
qq qq
qq qq

0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00


−1
Fig. 25 – Estimation of quantile function, Fθ (q), θ > θ0 (lighter tails).

51


A universal distribution for losses
BNGB considered the Champernowne generalized distribution to model
insurance claims, i.e. positive variables,
α
(y + c) − cα
Fα,M,c (y) = α α where α > 0, c ≥ 0 and M > 0.
(y + c) + (M + c) − 2cα
The associated density is then
α−1 α
α (y + c) ((M + c) − cα )
fα,M,c (y) = α α 2 .
((y + c) + (M + c) − 2cα )

52


A Monte Carlo study to compare those nonparametric
estimators
As in ....., 4 distributions were considered
– normal distribution,
– Weibull distribution,
– log-normal distribution,
– mixture of Pareto and log-normal distributions,

53


Box−plot for the 11 quantile estimators

Density of quantile estimators (mixture longnormal/pareto)
R benchmark qqqqqq
qq qqq
qqqqq
q qqqq
q qq q
qq
qq
qq
qq
qq q qqq qqqqq qq
qqq q q qq q qq qq
qqq q q qq qqq q
qqq q q q q q q
qqq q
qq q
q
q qq q q
qq
q qq
qq
q q
q q q
0.30

E Epanechnikov qq qqqq
qqqqqq
qqqqq
q qqq
q qqq
qqq
qq
qq
qq
qq qq qq qqqqqqqq q
qq qqq q qqq q qq q
qq qqq q qqq
qq q qq q q q
qq q
qq q q q
q q
qq
q qq
qq
q q
q q q q q
0.25

HD Harrell Davis q qqqqq
qqqqq
q qqq
qq qq
qqqq
qqq q
qqq q
qq
qq qqqqq qqq q q q q qq q q q
qqqqqq qq q qq q q qq qqq
qq q q qq qq q q q q q
qq q q q
qq q
qq q q q q
q q q q q q qq q q

Benchmark (R estimator)
HD (Harrell−Davis) PDG Padgett
0.20

q q q q qqqq
q q qqqqq
q qqq qq
q qq qqq
q qq qqq
q qq qq
q q qq
q q
density of estimators

PRK (Park)
B1 (Beta 1)
PRK Park qqqqqqq
q q qqqq
qqqq
qqqq
qqqq
qqq
qqq
qq
q qq q q q
qq qq
qq
qq
q qq q qq q q qqqqq q q q q q q q q q q
q q
qqq q q q
q qq
q q q qq q q

B2 (Beta 2)
0.15

Beta1 qqqqq
q q qq
q q qq
qqq
qqq
qqq
qqq
qqq
q qq qqqqq q q qqqq
q q qqqqqq qqq q
q q qq qq q q q q q
q q qq q qq
q q qq q
q q qq q
q
q
q qq q
q q q qq q
qq q
q
q
0.10

MACRO Beta1 q q qqq
q q qqq
q qqq
q qqq
qqq
qq
qq
qq
qq
q qqqqqq q q q
qqqqq q qqq
qq qqq q q
qq qq q
qq qqq
qq q q
qq q q
qq q

MICRO Beta1 q qqqqq
q qq qqq
qqqq
qq qq
qqqq
qqq q
q qq
qq
q
q qqqq qqqqq q qq q
qqqq qqqqq q q qq q qq
qqqq qqqqq
q qq q qqq qq
qqq qqqq
q qq q q
q q
q q qq q q
0.05

Beta2 qq qqqqq
q qqqqq
qqqqq
qqqq
qqqq
qq
qq
qq
qq
q qqqq q qq q q q qq
q qq q qqq q
q qqq qq q q
q qqq qq
q qqq qq
q q
q q
q q qq q
0.00

MACRO Beta2 q qqqq
q qqqq
q qqqq
q qqqq
qqq
qqq
qq
qq
q qqqqqqqqqq
qqqqqqqq q
qqq qq q q q
qqq qqq q q
qqq q q
qqq q q
qqq
q q
q q q

q

MICRO Beta2 qqqqq
q qqqq
q qqqq
qqqq
qqq
qqq
qq q
qqq
qq
q qqq qqqq qqq q q
qqqq qqq qq
qq qqqq q qq
qq qqq qq
qqqq q q
q qq
q
q
q q qq q

5 10 15 20 q

Estimated value−at−risk 5 10 15 20

Fig. 26 – Distribution of the 95% quantile of the mixture distribution, n = 200,
and associated box-plots.

54


MSE ratio, normal distribution, HD (Harrell−Davis) MSE ratio, normal distribution, B1 (Beta1) MSE ratio, normal distribution, MACB1 (MACRO−Beta1)

2.0

2.0

2.0
n= 50 n= 50 n= 50
n=100 n=100 n=100
n=200 n=200 n=200
n=500 n=500 n=500
1.5
q

1.5
q

1.5
q

q
q
q
MSE ratio

MSE ratio

MSE ratio
q
1.0

1.0

1.0
q q q
q qq q
q q q q q q q q q
q q q q
q q
q q
q
q
0.5

0.5

0.5
0.0

0.0

0.0
q q q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Probability, confidence levels (p) Probability, confidence levels (p) Probability, confidence levels (p)

MSE ratio, normal distribution, PRK (Park) MSE ratio, normal distribution, B1 (Beta1) MSE ratio, normal distribution, MACB1 (MACRO−Beta1)
q q
2.0

2.0

2.0
q q
q
q
n= 50 n= 50 n= 50
n=100 n=100 n=100
n=200 n=200 n=200
n=500 n=500 n=500
1.5

q
1.5
q

1.5
q

q q q
q
q
MSE ratio

MSE ratio

MSE ratio
q
1.0

1.0

1.0
q q
q qq q
q q
q q q
q q
q q
q q
q
0.5

0.5

0.5
0.0

0.0

0.0
q q q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0


Fig. 27 – Comparing MSE for 6 estimators, the normal distribution case.

55


MSE ratio, Weibull distribution, HD (Harrell−Davis) MSE ratio, Weibull distribution, MACB1 (MACRO−Beta1) MSE ratio, Weibull distribution, MICB1 (MICRO−Beta1)

2.0

2.0

2.0
n= 50 n= 50 n= 50
n=100 n=100 n=100
n=200 n=200 n=200
n=500 n=500 n=500
1.5
q

1.5
q

1.5
q
MSE ratio

MSE ratio

MSE ratio
q q
q
q q
1.0

1.0

1.0
qq q q
q q q q q
q q q q q
q q q
q q q
q q
q
q
q
0.5

0.5

0.5
0.0

0.0

0.0
q q q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0


MSE ratio, Weibull distribution, PRK (Park) MSE ratio, Weibull distribution, MACB1 (MACRO−Beta1) MSE ratio, Weibull distribution, MICB1 (MICRO−Beta1)
q
2.0

2.0

2.0
n= 50 q n= 50 n= 50
q
n=100 q n=100 n=100
n=200 n=200 n=200
q
n=500 n=500 n=500
1.5

q
1.5
q

1.5
q
MSE ratio

MSE ratio

MSE ratio
q
q q
q q
1.0

1.0

1.0
q q q
q
q q q q
q q q
q q
q q
q
q
q
0.5

0.5

0.5
0.0

0.0

0.0
q q q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0


Fig. 28 – Comparing MSE for 6 estimators, the Weibull distribution case.

56


MSE ratio, lognormal distribution, HD (Harrell−Davis) MSE ratio, lognormal distribution, MACB1 (MACRO−Beta1) MSE ratio, lognormal distribution, MICB1 (MICRO−Beta1) MSE ratio, lognormal distribution, B1 (Beta1)
2.0

2.0

2.0

2.0
n= 50 n= 50 n= 50 n= 50
n=100 n=100 n=100 n=100
n=200 n=200 n=200 n=200
n=500 n=500 n=500 n=500
1.5

q

1.5
q

1.5
q

1.5
q
q
q
q
q
MSE ratio

MSE ratio

MSE ratio

MSE ratio
q
q q q
1.0

1.0

1.0

1.0
q qq q q
q q q q q q q q q
q q q q q
q q q q q q q q q
q
q q
q
0.5

0.5

0.5

0.5
0.0

0.0

0.0

0.0
q q q q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Probability, confidence levels (p) Probability, confidence levels (p) Probability, confidence levels (p) Probability, confidence levels (p)

MSE ratio, lognormal distribution, PRK (Park) MSE ratio, lognormal distribution, MACB2 (MACRO−Beta2) MSE ratio, lognormal distribution, MICB2 (MICRO−Beta2) MSE ratio, lognormal distribution, B2 (Beta2)
2.0

2.0

2.0

2.0
q

n= 50 n= 50 n= 50 n= 50
n=100 n=100 n=100 n=100
n=200 n=200 n=200 n=200
n=500 n=500 n=500 n=500
1.5

q
1.5

q

1.5
q

1.5
q
q
q

q q q
MSE ratio

MSE ratio

MSE ratio

MSE ratio
1.0

1.0

1.0

1.0
q q q q q q
q
q q q q q
q q
q q q q q q q q
q q q q q q
q q
q
q
0.5

0.5

0.5

0.5
0.0

0.0

0.0

0.0
q q q q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0


Fig. 29 – Comparing MSE for 9 estimators, the lognormal distribution case.

57


q
MSE ratio, mixture distribution, HD (Harrell−Davis) MSE ratio, mixture distribution, MACB1 (MACRO−Beta1) MSE ratio, mixture distribution, MICB1 (MICRO−Beta1) MSE ratio, mixture distribution, B1 (Beta1)
q
2.0

2.0

2.0

2.0
n= 50 n= 50 n= 50 n= 50
n=100 n=100 n=100 n=100
n=200 n=200 n=200 n=200
n=500 n=500 n=500 n=500
1.5

q

1.5
q

1.5
q

1.5
q

q
q
q
MSE ratio

MSE ratio

MSE ratio

MSE ratio
q q
1.0

1.0

1.0

1.0
q q q q
q q q q q q
q
q q
q q q q
q q q q
q q q
q
q q
q
0.5

0.5

0.5

0.5
0.0

0.0

0.0

0.0
q q q q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0


MSE ratio, mixture distribution, PRK (Park) MSE ratio, mixture distribution, MACB2 (MACRO−Beta2) MSE ratio, mixture distribution, MICB2 (MICRO−Beta2) MSE ratio, mixture distribution, B2 (Beta2)
2.0

2.0

2.0

2.0
n= 50 n= 50 n= 50 n= 50
n=100 n=100 n=100 n=100
n=200 n=200 n=200 n=200
n=500 n=500 n=500 n=500
1.5

q
1.5

q

1.5
q

1.5
q
q
q
q q
q
q
q
MSE ratio

MSE ratio

MSE ratio

MSE ratio
q q q q
q q q
q q q
1.0

1.0

1.0

1.0
q q q q q q q q
q q q q q

q q
q
q q
0.5

0.5

0.5

0.5
0.0

0.0

0.0

0.0
q q q q

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0


Fig. 30 – Comparing MSE for 9 estimators, the mixture distribution case.

58


Portfolio optimal allocation
Classical problem as formulated in Markowitz (1952), Journal of Finance,
 
 ω ∗ ∈ argmin{ω Σω}  ω ∗ ∈ argmax{ω µ}
convex
⇔
 u.c. ω µ ≥ η and ω 1 = 1  u.c. ω Σω ≤ η and ω 1 = 1

Roy (1952), Econometrica,“the optimal bundle of assets (investment) for
investors who employ the safety ﬁrst principle is the portfolio that minimizes the
probability of disaster”.

 
 ω ∗ ∈ argmin{VaR(ω X, α)}  ω ∗ ∈ argmax{E(ω X)}
nonconvex

 u.c. E(ω X) ≥ η and ω 1 = 1  u.c. VaR(ω X, α) ≤ η ,ω 1 = 1

59


Empirical data, Eurostocks

−0.05 0.05 −0.05 0.05
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−0.05 0.05 −0.10 0.00 0.10

Fig. 31 – Scatterplot of log-returns.

60


Value−at−Risk (75%) on the grid Value−at−Risk (75%) on the grid

2
1
Allocation (4th asset)
Quantile

0
−1
t)
se
as

−2
rd
(3
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catio

on
n (4
th

ati
asse

−3
loc
t)

Al
−3 −2 −1 0 1 2

Allocation (3rd asset)

Value−at−Risk (97.5%) on the grid Value−at−Risk (97,5%) on the grid

2
1
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Quantile

0
−1
et)
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as

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rd
(3

Allo
catio
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n (4
th
ati

asse

−3
loc

t)
Al

−3 −2 −1 0 1 2

Allocation (3rd asset)

Fig. 32 – Value-at-Risk for all possible allocations on the grid G (surface and level
curves), with α = 75% on the left and α = 97.5% on the right.
61


Optimal allocation (asset 1) Optimal allocation (asset 2)

0.0 0.5 1.0 1.5 2.0

0.0 0.5 1.0 1.5 2.0
weight of allocation

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97.5% 95% 92.5% 90% 87.5% 85% 82.5% 80% 77.5% 75% 97.5% 95% 92.5% 90% 87.5% 85% 82.5% 80% 77.5% 75%

Probability level (97.5%−75%) Probability level (97.5%−75%)

0.0 0.5 1.0 1.5 2.0

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Probability level (97.5%−75%) Probability level (97.5%−75%)

Fig. 33 – Optimal allocations for diﬀerent probability levels (α =
75%, 77.5%, 80%, ..., 95%, 97.5%), with allocation for the ﬁrst asset (top left) up
to the fourth asset (bottom right). 62


mean 75% 77.5% 80% 82.5% 85% 87.5% 90% 92.5% 95% 97
variance
asset 1 0.2277 0.222 0.206 0.215 0.251 0.307 0.377 0.404 0.394 0.402 0.
(0.253) (0.244) (0.259) (0.275) (0.276) (0.241) (0.243) (0.224) (0.214) (0.
asset 2 0.5393 0.550 0.558 0.552 0.530 0.500 0.460 0.444 0.448 0.441 0.
(0.141) (0.136) (0.144) (0.152) (0.154) (0.134) (0.135) (0.124) (0.121) (0.
asset 3 −0.2516 −0.062 −0.083 −0.106 −0.139 −0.163 −0.196 −0.228 −0.253 −0.310 −0
(0.161) (0.176) (0.184) (0.187) (0.215) (0.203) (0.163) (0.141) (0.184) (0.
asset 4 0.4846 0.289 0.319 0.339 0.357 0.357 0.359 0.380 0.410 0.466 0.
(0.162) (0.179) (0.191) (0.204) (0.221) (0.205) (0.175) (0.153) (0.170) (0.

Tab. 1 – Mean and standard deviation of estimated optimal allocation, for diﬀerent
quantile levels.

1. raw estimator Q(Y, α) = Y[α·n]:n
n
2. mixture estimator Q(Y, α) = λi (α)Yi:n , which is the standard quantile
i=1
estimate in R (see [?]),
3. Gaussian estimator Q(Y, α) = Y + z1−α sd(Y , where sd denotes the empirical
standard deviation,

63


n −ξk
4. Hill’s estimator, with k = [n/5], Q(Y, α) = Yn−k:n (1 − α) , where
k
k
1 Yn+1−i:n
ξk = log (assuming that ξ > 0),
k i=1
Yn−k:n
5. kernel based estimator is obtained as a mixture of smoothed quantiles,
derived as inverse values of a kernel based estimator of the cumulative
n
distribution function, i.e. Q(Y, α) = λi (α)F −1 (i/n).
i=1

64


q q
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q Est. 4 Est. 5 Est. 1 Est. 2 Est. 3
q Est. 4 Est. 5

2

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q Hill Kernel
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0

0
q
q
q q
q
q q
q q
q q q
q
q q q q
q
q q
q
q q
q
q
q q
q q
q q q q
q
q
q
q q
q q
q
q
q q q
q q
q q q
q
q q
q
−1

−1
q q

q
q
−2

−2
q

q
−3

−3
q

Quantile estimator Quantile estimator

Optimal allocation (asset 3)
q Optimal allocation (asset 4)

Est. 1 Est. 2 Est. 3 Est. 4 Est. 5 Est. 1 Est. 2 Est. 3
q Est. 4 Est. 5
2

2
raw mixture Gaussian Hill Kernel raw mixture Gaussian Hill Kernel
q q
q q
q
q
q
q
q
q q
q q
q q
q
q q q q
1

1
q q q
q
q q
q q
q q q
q
q
q q
q q q
q
q
q q q q q

q q q q q
q q q q q
q
q
q
q q q
q
q q q q q
q
q q
q q
q q
q q
q q
q q
q q
q q q q q
q
q q
q q q
0

0
q
q q
q q q
q q
q q
q
q
q q q
q
q q
q q q
q q q
q q
q
q
q
q q
q
q q
q q
q q
q
q q
q q q q
q
q q
q q
q
q q q q
q
q q
q q
q
q q q q
−1

−1
q
q q
q
q
q
q
q
q q
q
q
−2

−2
q

q
−3

−3
q

Quantile estimator Quantile estimator

Fig. 34 – Optimal allocations for diﬀerent 95% quantile estimators, with allocation
for the ﬁrst asset (top left) up to the fourth asset (bottom right).

65


Some references
Charpentier, A. & Oulidi, A. (2007). Beta Kernel estimation for
Value-At-Risk of heavy-tailed distributions. in revision Journal of Computational
Statistics and Data Analysis.
Charpentier, A. & Oulidi, A. (2007). Estimating allocations for
Value-at-Risk portfolio optimzation. to appear in Mathematical Methods in
Operations Research.
Chen, S. X. (1999). A Beta Kernel Estimator for Density Functions.
Computational Statistics & Data Analysis, 31, 131-145.
Gourieroux, C., & Montfort, A. 2006. (Non) Consistency of the Beta
´
Kernel Estimator for Recovery Rate Distribution. CREST-DP 2006-32.

66

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