Slides univ-van-amsterdam

Arthur CHARPENTIER - Nonparametric quantile estimation.

Estimating quantiles
and related risk measures
Arthur Charpentier
arthur.charpentier@univ-rennes1.fr

Universiteit van Amsterdam, January 2008

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,
Ru (X) =

1
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

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Calcul de l’esperance mathématique

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1

2

Fig. 1 – Expected value

3

4

xdFX (x) =

5

6

P(X > x)dx.
7



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Calcul de l’esperance d’utilité

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2

3

Fig. 2 – Expected utility

4

5

6

u(x)dFX (x).
8



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Calcul de l’intégrale de Choquet

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1

2

3

Fig. 3 – Distorted probabilities

4

5

6

g(P(X > x))dx.
9


Distorted risk measures and quantiles
Equivalently, note that E(X) =
Rg (X) =

1
0

1
0

−1
FX (1 − u)du, and

−1
FX (1 − u)dgu.

A very general class of risk measures can be deﬁned as follows,
1

Rg (X) =
0

−1
FX (1 − u)dgu

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

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Distortion function, VaR (quantile) − pdf
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Distortion function, pdf

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Fig. 4 – Distortion function, g and dg
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Agenda
Risk measures
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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),
θ

(1)

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
−1

zα = Φ

2
ζ1 −1 2
ζ2 −1 3
ζ1
−1
(α)+ [Φ (α) −1]+ [Φ (α) −3Φ (α)]− [2Φ−1 (α)3 −5Φ−1 (α)],
6
24
36

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 =
and ζ2 =
− 3.
E([X − E(X)]2 )2
E([X − E(X)]2 )3/2

(3)

15


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

n

Xi and σ =
i=1

1
n−1

n
2

(Xi − µ) ,
i=1

and
2
ζ2 −1 3
ζ1
ζ1 −1 2
zα = Φ−1 (α)+ [Φ (α) −1]+ [Φ (α) −3Φ−1 (α)]− [2Φ−1 (α)3 −5Φ−1 (α)],
6
24
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(4)
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 =
and
n − 2 ( n (Xi − µ)2 )3/2
i=1

ζ2 =

n−1
(n−2)(n−3)

(n + 1)ζ2 + 6

where ζ2 =

n

(

n
4
i=1 (Xi −µ)
n
2 2
i=1 (Xi −µ)

)

− 3.

16


Parametrics estimator and error model
Density, theoritical versus empirical

Theoritical Student
Fitted lStudent
Fitted Gaussian

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Theoritical lognormal
Fitted lognormal
Fitted gamma

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Density, theoritical versus empirical

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

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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 },

Q(Y, α) = Yn−k:n +

βk
ξk

n
(1 − α)
k

− ξk

−1 ,

(5)

An alternative is to use Hill’s estimator if ξ > 0,
Q(Y, α) = Yn−k:n

n
(1 − α)
k

− ξk

1
, ξk =
k

k

log Yn+1−i:n − log Yn−k:n .

(6)

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
1
Fn (x) =
nh

n

x

k
i=1

−∞

Xi − t
h

1
dt =
n

n

K
i=1

Xi − x
h

x

where K(x) =

k(t)dt, k being a kernel and h the bandwidth.
−∞

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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
Qn (p) = Fn

i
n

= Xi:n = X[np]:n where [·] denotes the integer part.

(7)

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].

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The quantile function in R
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quantile level

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The quantile function in R

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Fig. 6 – Several quantile estimators in R.

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In order to increase eﬃciency, L-statistics can be considered i.e.
n

Qn (p) =

n

Wi,n,p Xi:n =
i=1

i
n

−1
Wi,n,p Fn
i=1

1
−1
Fn (t) k (p, h, t) dt

=

(8)

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

Qn (p) =

i
n

k
i=1

(i−1)
n

t−p
h

n

dt X(i) =

IK
i=1

i
n

−p
h

− IK

i−1
n

−p
h

X(i)
(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.

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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

Qn (p) =
i=1

i
n
(i−1)
n

Γ(n + 1)
y (n+1)p−1 (1 − y)(n+1)q−1 Xi: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.

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Agenda
Risk measures
29


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

n

k
i=1

x − Xi
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

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

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

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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.

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Beta kernel estimators
A Beta kernel estimator of the density (see Chen (1999)) - on [0, 1] is
1
fb (x) =
n

n

x
1−x
k Xi , 1 + , 1 +
b
b
i=1

, x ∈ [0, 1],

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=0.10

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Fig. 13 – Shape of Beta kernels, diﬀerent x’s and b’s.
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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) =
1−t (u)
k
, if t ∈ [0, 2b)
 ρb (t), b


t
k b ,ρb (1−t) (u) , if t ∈ (1 − 2b, 1]
where ρb (t) = 2b2 + 2.5 −

t
4b4 + 6b2 + 2.25 − t2 − .
b

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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.
fb (x)

=
=
=
≈

1
n
1
n

x
1−x
k Xi , 1 + , 1 +
, x ∈ [0, 1]
b
b
x
1−x
k Xi , 1 + , 1 +
, x ∈ [0, 1]
b
b

Xi =0,1

n − n0 − n1
1
n
n − n0 − n1

Xi =0,1

x
1−x
k Xi , 1 + , 1 +
b
b

, x ∈ [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.

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Non-consistency of Beta kernel estimators
´
Gourieroux & Monfort (2007) proposed
(1)
fb (x)

=

fb (x)
1
0

, for all x ∈ [0, 1].

fb (t)dt

It is called macro-β since the correction is performed globally.
´
Gourieroux & Monfort (2007) proposed
1
(2)
fb (x) =
n

n

i=1

kβ (Xi ; b; x)
1
0

, for all x ∈ [0, 1].

kβ (Xi ; b; t)dt

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

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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”.

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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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Estimated density

1.4


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

41


Estimated optimal transformation

4.0


q

3.5

5

q

q

4

q

q

2.5

q
q
q
q

q
q

3

Quantile

q

q

qq

0.80

q

qq

qq

qq

qq

qq

q

q

q

q

q

q

q

1

qq

q
qq

q
qq

q
qq

q

q

q

2

2.0

q

1.5
1.0

Quantile

3.0

q

0.85

0.90
Probability level

0.95

1.00

qqq

0.80

qqq

qq
qqq

qq

0.85

qq

qq

q

q
qq

qq

qq

qq

0.90

q

q

q

q

q

q

0.95

1.00

Probability level

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

42

0.0

0.2

0.4

0.6

0.8

1.0

1.0
0.8
0.6
0.4
0.2
0.0

q
q
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q
qq
qq
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qq
qq
qq
q
qq
q
q
q


0.6
0.4
0.2
0.0


0.8

1.0


q
q
q
q
q
q
q
q
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q
q
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qq
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qq
q
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qqq
qqq
qqq
qqq
q
q
q
q
qq
qq
qq
qq
q
qq
qq
qq
q
q
q
q
q
qq
q
q
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qq
q
qq
q
q
q
q
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q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
qq
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
q
qq
qq
q
q
q
q
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q
q
q
q
q
q
q
q
q
qq
qq
qq
qq
q
q
q
q
qq
qq
qq
q
qq
qq
q
qq
q
q
q
q
q
q
qq
qq

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
1.2
1.0

q

0.6

0.8

Estimated density

1.2
1.0
0.8
0.6

Estimated density

1.4


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

4

4

q

q

q

3

q

q
q

q

3

Quantile

q

q

q
q
q

0.80

qqq

qq

q

qq

qq

q

qq

q

q

q

q

q

q

q

q

1

qqq

q
qq

q
qq

q
qq

q

q

2

2

q

1

Quantile

q

0.85

0.90
Probability level

0.95

1.00

qqq

0.80

qqq

qq

qq

q
qq

0.85

qq

qq

qq

q

q
qq

q

q

q

0.90

q

q

q

q

q

q

0.95

1.00

Probability level

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

45

0.0

0.2

0.4

0.6

0.8

1.0

1.0
0.8
0.6
0.4
0.2
0.0

q
q
q
q
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qq
q
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q


0.6
0.4
0.2
0.0


0.8

1.0


q
q
q
q
q
q
q
q
q
q
q
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qqq
qqq
qqq
qqq
q
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qq
qq
qq
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qq
q
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q
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q
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q
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q
q
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q
qq
qq
q
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qq
q
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qq
q
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qq
qq
q
q
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qq
qq
qq
q
qq
qq
q
qq
q
q
q
q
q
q
qq
qq

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
1.2
1.0

q

0.6

0.8

Estimated density

1.2
1.0
0.8
0.6

Estimated density

1.4


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

10

10

12

q

q

q

8

q
q

6

Quantile

6

q

q

q

q

q

qqqqq

0.80

qq

0.85

qqq

qqq

qq

q

qq

q

q

q

q

q

2

qqq
qqqq

q
qq

q

q

4

4

q

2

Quantile

8

q

qqqqq

0.90
Probability level

0.95

1.00

0.80

qq

qqq
qqqq

0.85

qqq

qqq

qq

q

q
qq

0.90

qq

q

q

q

q

q

0.95

q

1.00

Probability level

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

48

0.0

0.2

0.4

0.6

0.8

1.0

1.0
0.8
0.6
0.4
0.2
0.0

q
qq
qq
qq
q
qq
q
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qq
q
qq
qq
qq
q
q
q
q
q
q
q


0.6
0.4
0.2
0.0


0.8

1.0


q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
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q
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qqq
qqq
qqq
qqq
q
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qq
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q
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q
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q
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qq
q
q
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qq
qq
qq
q
qq
qq
q
qq
q
q
q
q
q
q
qq
qq

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
1.2
1.0

q

0.6

0.8

Estimated density

1.2
1.0
0.8
0.6

Estimated density

1.4


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

3.5

3.5

q

q

3.0

3.0

q

q
q

q

2.0

q

qq

0.80

qq

q

qq

qq

qq

qq

qq

q

q

q

q

q

q

0.85

0.90
Probability level

q

2.0

q

q
q

q

1.0

qq

q
qq

q
qq

q

q

q

1.5

1.5

q

2.5

Quantile

2.5

q
q
q

1.0

Quantile

q

0.95

1.00

qq

qq

0.80

qq

q

q
qq

qq

qq

0.85

qq

q
qq

qq

qq

q

q

q

0.90

q

q

q

q

q

q

0.95

1.00

Probability level

−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) =
where α > 0, c ≥ 0 and M > 0.
α
α
(y + c) + (M + c) − 2cα
The associated density is then
fα,M,c (y) =

α (y + c)

α−1

α

α

((M + c) − cα )
α

((y + c) + (M + c) −

2
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)
0.30

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

qq
qq
q

HD Harrell Davis

q qqqqq
qqqqq
q qqq
qq qq
qqqq
qqq q
qqq q
qq
qq

PDG Padgett

q

PRK Park

0.15

0.20

Benchmark (R estimator)
HD (Harrell−Davis)
PRK (Park)
B1 (Beta 1)
B2 (Beta 2)

Beta1

0.10

density of estimators

0.25

qq qqqq
qqqqqq
qqqqq
q qqq
q qqq
qqq
qq
qq
qq
qq

qq
qq
q

qq qq qqqqqqqq q
qq qqq q qqq q qq q
qq qqq q qqq
qq q qq q q q
qq q
q
q q
q
q
q q
q
q
q

E Epanechnikov

q
q
q
q

qq q q q
qq qq
qq
qq
q
q
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

MACRO Beta1

q q qqq
q q qqq
q qqq
q qqq
qqq
qq
qq
qq
qq
q

0.05
0.00

q

q

q q

q

q

q q q

q
q

q

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

qqqqq
q q qq
q q qq
qqq
qqq
qqq
qqq
qqq
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 qqqqq
q qq qqq
qqqq
qq qq
qqqq
qqq q
q qq
qq
q
q

Beta2

q
q

q

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
q
q
q q q q
q
q

qqqqqqq
q q qqqq
qqqq
qqqq
qqqq
qqq
qqq
qq
q

MICRO Beta1

q
q

q qq q q qqqqq q q q q q q q q q q
q q
qq
q q q q
q qq
q
q
q
q
q
q

q

qq q
qq q
q
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
q
q q
q

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
q
q
q

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

5

10

15

20

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
q
q
q

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


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

q

1.5

q

1.5

1.5

q

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

2.0

MSE ratio, normal distribution, MACB1 (MACRO−Beta1)

2.0

MSE ratio, normal distribution, B1 (Beta1)

2.0

MSE ratio, normal distribution, HD (Harrell−Davis)

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

q

q

q
q
q q

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

q

0.0

0.2

0.4

0.6

0.8

1.0

Probability, confidence levels (p)

MSE ratio, normal distribution, B1 (Beta1)

MSE ratio, normal distribution, PRK (Park)

MSE ratio, normal distribution, MACB1 (MACRO−Beta1)

q
q

q

1.5

q

q

n= 50
n=100
n=200
n=500
q

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

q

1.5

q

2.0

q

2.0

2.0

q

q

q


q

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

q

q

q

q

q

0.2

0.4

0.6

0.8


1.0

q
q

1.0

q
q

q
q

q

q

q

0.5
0.0

0.5
0.0

0.5

q

0.0

q
q
q

q
q
q

MSE ratio

1.0

q

q

MSE ratio

1.0

MSE ratio

q

0.0

q
q

0.5
q


1.5

q

q
q

0.0

0.5
0.0

0.5

q

0.0

q
q
q

q
q
q

1.0

q

1.0

q

q

MSE ratio

q

MSE ratio

1.0

MSE ratio

q
q

q

0.0

0.2

0.4

0.6

0.8


1.0

q

0.0

0.2

0.4

0.6

0.8

1.0


Fig. 27 – Comparing MSE for 6 estimators, the normal distribution case.
55


q
q
q

2.0
1.5
q
q

q

q

q

q

q

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

q

1.0

1.5
q

q

MSE ratio

q

q

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

1.0

q
q

q

q

MSE ratio

1.5

q

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

1.0

MSE ratio

MSE ratio, Weibull distribution, MICB1 (MICRO−Beta1)

2.0

MSE ratio, Weibull distribution, MACB1 (MACRO−Beta1)

2.0

MSE ratio, Weibull distribution, HD (Harrell−Davis)

q

q
q

q

q
q
q

q

q

q
q

q

0.0

0.2

0.4

0.6

0.8

1.0

0.5
0.0

0.5
0.0

0.0

0.5

q
q

q

0.0

0.2

0.4

0.6

0.8

1.0

q

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)

2.0
q

1.0

q
q

q
q

q

q

q

1.5

q

1.5

q

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

q

q

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

q

1.0

q
q

MSE ratio

q

1.0

MSE ratio

1.5

q

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

MSE ratio

2.0

q

2.0


q

q
q

q

q
q
q

q

q

q
q

q

0.0

0.2

0.4

0.6

0.8


1.0

0.5
0.0

0.5
0.0

0.0

0.5

q
q

q

0.0

0.2

0.4

0.6

0.8


1.0

q

0.0

0.2

0.4

0.6

0.8

1.0


Fig. 28 – Comparing MSE for 6 estimators, the Weibull distribution case.
56


2.0
q

n= 50
n=100
n=200
n=500
q

q

q

1.5

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

1.5

q

1.5

1.5

q

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

MSE ratio, lognormal distribution, B1 (Beta1)

2.0

MSE ratio, lognormal distribution, MICB1 (MICRO−Beta1)

2.0

MSE ratio, lognormal distribution, MACB1 (MACRO−Beta1)

2.0

MSE ratio, lognormal distribution, HD (Harrell−Davis)

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

q

q

q

q

q

q

q
q

q

q

1.0

q

q

q

q

q

MSE ratio

q

q
q

1.0

q

q

1.0

q

q

q

q

MSE ratio

q

MSE ratio

1.0

MSE ratio

q
q
q

q
q

q

q

q

q
q

q

q

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.5

0.5
q

0.0

0.0

q

0.0

0.0

0.5
0.0

0.0

0.5

q

q

0.0

0.2

0.4

0.6

0.8

1.0

q

0.0

0.2

0.4

0.6

0.8

1.0

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)

0.4

0.6

0.8


1.5

1.5
q

q

q q

1.0

q

q

q

q

q

0.2

0.4

0.6

0.8


1.0

q
q
q

q

q

q

q

q

q

q
q

0.0

0.0

q

0.0

1.0

1.0

q

q
q

MSE ratio

q

q

q

0.5

q

MSE ratio

q
q

0.5
0.2

q

n= 50
n=100
n=200
n=500
q

0.0

0.5
q

0.0

q

0.5

1.0

q

q

q

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

q

q

0.0

MSE ratio

q

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

1.0

q
q

MSE ratio

1.5

q

1.5

q

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

2.0


2.0


2.0


2.0


q

0.0

0.2

0.4

0.6

0.8


1.0

q

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, MICB1 (MICRO−Beta1)

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

q

1.5

q

1.5

1.5

q

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

MSE ratio, mixture distribution, B1 (Beta1)

2.0

2.0

2.0

q

2.0

MSE ratio, mixture distribution, MACB1 (MACRO−Beta1)

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

q

1.5

MSE ratio, mixture distribution, HD (Harrell−Davis)

n= 50
n=100
n=200
n=500
q

q

q

q

q

q

q
q

q

q
q

1.0

q

q

MSE ratio

q

1.0

q

q

MSE ratio

q

q

1.0

q

q

MSE ratio

1.0

MSE ratio

q

q

q

q

q q

q

q

q

q

q

q

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.5

0.5
q

0.0

0.0

q

0.0

0.0

0.5
0.0

0.0

0.5

q

q

0.0

0.2

0.4

0.6

0.8

1.0

q

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)

q

q

q

q

q

q

q
q

q

q

q

q

q
q

q

1.0

q

q

q

q

1.0

q

MSE ratio

q

1.0

MSE ratio

1.0

q

q

q

q

q

q
q
q

q

0.2

0.4

0.6

0.8


1.0

0.2

0.4

0.6

0.8


1.0

0.5

0.5
q

0.0

0.0

q

0.0

0.0

0.0

0.5

q

0.5
0.0

MSE ratio

q

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

q
q

MSE ratio

q

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

1.5

q

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

1.5

q

1.5

1.5

q
q

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

2.0


2.0


2.0


2.0


q

0.0

0.2

0.4

0.6

0.8


1.0

q

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)}
 u.c. E(ω X) ≥ η and ω 1 = 1

nonconvex


 ω ∗ ∈ argmax{E(ω X)}
 u.c. VaR(ω X, α) ≤ η ,ω 1 = 1

59


Empirical data, Eurostocks
0.05

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FTSE (4)

0.10

Fig. 31 – Scatterplot of log-returns.
60


Value−at−Risk (75%) on the grid

0
−1
−2

on
t)

−3

ati

asse

loc

n (4
th

Al

catio

(3

rd
a

ss

et)

Quantile
Allo

Allocation (4th asset)

1

2

Value−at−Risk (75%) on the grid

−3

−2

−1

0

1

2

Allocation (3rd asset)

Value−at−Risk (97,5%) on the grid

t)

0
−1
−3

on
ati

asse

Al
loc

catio
n (4
th

−2

(3

rd

as
s

et)

Quantile
Allo

Allocation (4th asset)

1

2

Value−at−Risk (97.5%) on the grid

−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


q
q
q
q
q

q

q
q
q
q
q
q
q
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q
q
q
q
q
q

97.5%

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

95%

92.5%

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

90%

87.5%

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
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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
q
q
q
q
q
q
q
q
q

q
q
q
q
q
q
q
q
q
q
q
q

85%

82.5%

80%

77.5%

75%

q
q
q
q
q
q
q
q
q

0.0 0.5 1.0 1.5 2.0

q

−1.0

q
q
q
q
q
q
q
q
q
q
q
q
q

Optimal allocation (asset 2)

weight of allocation

0.0 0.5 1.0 1.5 2.0
−1.0



q
q
q
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q

q

q

q

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q

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q
q
q
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q
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q

q
q
q
q
q
q
q
q
q

97.5%

95%

92.5%

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
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q
q
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q

q
q
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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

90%

87.5%

85%

82.5%

80%

q
q
q
q
q
q
q
q
q
q
q

77.5%

q
q
q
q
q
97.5%
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

92.5%

90%

87.5%

85%

q
q
q
q
q
q
q
q
q
q
q
95%
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

82.5%

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
80%

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

77.5%

75%
q

Probability level (97.5%−75%)

0.0 0.5 1.0 1.5 2.0

q
q
q
q
q
q
q
q
q
q

q
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q

q

q
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q

q
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q

q
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q

q
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q

q
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q

q
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q

q
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q
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q

q
q
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q
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q

q
q
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q
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q

q
q
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q
q
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q

q
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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
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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

−1.0

−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
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q
q
q


0.0 0.5 1.0 1.5 2.0

q

q

75%


q

q
q
q
q
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q
q
q
q
q
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q

q
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q





q
q
q
q
q
q
q
q

97.5%

95%

92.5%

90%

87.5%

85%

82.5%

80%

77.5%

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
62
to the fourth asset (bottom right).


mean

75%

77.5%

80%

82.5%

85%

87.5%

90%

92.5%

95%

97

variance
asset 1

0.2277

0.222
(0.253)

0.206
(0.244)

0.215
(0.259)

0.251
(0.275)

0.307
(0.276)

0.377
(0.241)

0.404
(0.243)

0.394
(0.224)

0.402
(0.214)

0.
(0.

asset 2

0.5393

0.550
(0.141)

0.558
(0.136)

0.552
(0.144)

0.530
(0.152)

0.500
(0.154)

0.460
(0.134)

0.444
(0.135)

0.448
(0.124)

0.441
(0.121)

0.
(0.

asset 3

−0.2516

−0.062
(0.161)

−0.083
(0.176)

−0.106
(0.184)

−0.139
(0.187)

−0.163
(0.215)

−0.196
(0.203)

−0.228
(0.163)

−0.253
(0.141)

−0.310
(0.184)

−0
(0.

asset 4

0.4846

0.289
(0.162)

0.319
(0.179)

0.339
(0.191)

0.357
(0.204)

0.357
(0.221)

0.359
(0.205)

0.380
(0.175)

0.410
(0.153)

0.466
(0.170)

0.
(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


4. Hill’s estimator, with k = [n/5], Q(Y, α) = Yn−k:n
1
ξk =
k

k

log
i=1

n
(1 − α)
k

−ξk

, where

Yn+1−i:n
(assuming that ξ > 0),
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

λi (α)F −1 (i/n).

distribution function, i.e. Q(Y, α) =
i=1

64


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

Est. 4
Hill

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

Est. 5
Kernel

q
q
q
q
q
q
q
q

q

−2

q
q

q
q
Est. 3
q
Gaussian
q
q

q

1

q

Est. 2
mixture

Est. 1
raw

2

Est. 5
Kernel

0

q
q
q
q
q
q
q
q
q
q
q
q

Est. 4
Hill

−1

q
q
Est. 3
q
Gaussian


q
q
q
q
q
q
q
q
q

−1

0

q
q
q
q
q
q
q
q
q
q
q
q
q
q

−2


q
q
q
q
q
q

Est. 2
mixture
q

1

2

Est. 1
raw

q

−3

−3

q

q

Quantile estimator

Quantile estimator

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

2
1

1
0
−1
−2


q
q
q
q
q

q
q
q
q
q
q
q
q
q
q
q

Est. 1
raw

Est. 2
mixture

q
q

q

q


Est. 5
Kernel

q
q
q
q
q
q
q
q
q
q

0

q

Est. 4
Hill

−1

Est. 3
Gaussian

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

Est. 3
q
Gaussian
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

Est. 4
Hill

Est. 5
Kernel

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

−2

Est. 2
mixture


2

Est. 1
raw

Quantile estimator

−3

−3

q

q

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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