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Arthur CHARPENTIER - Nonparametric quantile estimation.
Estimating quantilesand related risk measures
Arthur Charpentier
Seminaire du GREMAQ, Decembre 2007
joint work with Abder Oulidi, IMA Angers
1
Arthur CHARPENTIER - Nonparametric quantile estimation.
Agenda
β’ General introduction
Risk measures
β’ Distorted risk measuresβ’ Value-at-Risk and related risk measures
Quantile estimation : classical techniques
β’ Parametric estimationβ’ Semiparametric estimation, extreme value theoryβ’ Nonparametric estimation
Quantile estimation : use of Beta kernels
β’ Beta kernel estimationβ’ Transforming observations
A simulation based study
2
Arthur CHARPENTIER - Nonparametric quantile estimation.
Agenda
β’ General introduction
Risk measures
β’ Distorted risk measuresβ’ Value-at-Risk and related risk measures
Quantile estimation : classical techniques
β’ Parametric estimationβ’ Semiparametric estimation, extreme value theoryβ’ Nonparametric estimation
Quantile estimation : use of Beta kernels
β’ Beta kernel estimationβ’ Transforming observations
A simulation based study
3
Arthur CHARPENTIER - Nonparametric quantile estimation.
Risk measures and price of a risk
Pascal, Fermat, Condorcet, Huygens, dβAlembert in the XVIIIth centuryproposed to evaluate the βproduit scalaire des probabilites et des gainsβ,
< p,x >=nβi=1
pixi = EP(X),
based on the βregle des partiesβ.
For Quetelet, the expected value was, in the context of insurance, the price thatguarantees a financial equilibrium.
From this idea, we consider in insurance the pure premium as EP(X). As inCournot (1843), βlβesperance mathematique est donc le juste prix des chancesβ(or the βfair priceβ mentioned in Feller (1953)).
4
Arthur CHARPENTIER - Nonparametric quantile estimation.
Risk measures : the expected utility approach
Ru(X) =β«u(x)dP =
β«P(u(X) > x))dx
where u : [0,β)β [0,β) is a utility function.
Example with an exponential utility, u(x) = [1β eβΞ±x]/Ξ±,
Ru(X) =1Ξ±
log(EP(eΞ±X)
).
5
Arthur CHARPENTIER - Nonparametric quantile estimation.
Risk measures : Yarriβs dual approach
Rg(X) =β«xdg β¦ P =
β«g(P(X > x))dx
where g : [0, 1]β [0, 1] is a distorted function.
Exampleβ if g(x) = I(X β₯ 1β Ξ±) Rg(X) = V aR(X,Ξ±),β if g(x) = min{x/(1β Ξ±), 1} Rg(X) = TV aR(X,Ξ±) (also called expected
shortfall), Rg(X) = EP(X|X > V aR(X,Ξ±)).
6
Arthur CHARPENTIER - Nonparametric quantile estimation.
Distortion of values versus distortion of probabilities
0 1 2 3 4 5 6
0.00.2
0.40.6
0.81.0
Calcul de lβesperance mathΓ©matique
Fig. 1 β Expected valueβ«xdFX(x) =
β«P(X > x)dx.
7
Arthur CHARPENTIER - Nonparametric quantile estimation.
Distortion of values versus distortion of probabilities
0 1 2 3 4 5 6
0.00.2
0.40.6
0.81.0
Calcul de lβesperance dβutilitΓ©
Fig. 2 β Expected utilityβ«u(x)dFX(x).
8
Arthur CHARPENTIER - Nonparametric quantile estimation.
Distortion of values versus distortion of probabilities
0 1 2 3 4 5 6
0.00.2
0.40.6
0.81.0
Calcul de lβintΓ©grale de Choquet
Fig. 3 β Distorted probabilitiesβ«g(P(X > x))dx.
9
Arthur CHARPENTIER - Nonparametric quantile estimation.
Distorted risk measures and quantiles
Equivalently, note that E(X) =β« 1
0Fβ1X (1β u)du, and
Rg(X) =β« 1
0Fβ1X (1β u)dgu.
A very general class of risk measures can be defined as follows,
Rg(X) =β« 1
0
Fβ1X (1β 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 ofquantiles, where dg(1β Β·) denotes the distribution of the weights.
10
Arthur CHARPENTIER - Nonparametric quantile estimation.
ββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Distortion function, VaR (quantile) β cdf
1 β probability level
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Distortion function, TVaR (expected shortfall) β cdf
1 β probability level
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Distortion function, cdf
1 β probability level
βββββββββββββββββββββββββββββββββββββββββββββ
β
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Distortion function, VaR (quantile) β pdf
1 β probability level
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Distortion function, TVaR (expected shortfall) β pdf
1 β probability level
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
Distortion function, pdf
1 β probability level
Fig. 4 β Distortion function, g and dg
11
Arthur CHARPENTIER - Nonparametric quantile estimation.
Agenda
β’ General introduction
Risk measures
β’ Distorted risk measuresβ’ Value-at-Risk and related risk measures
Quantile estimation : classical techniques
β’ Parametric estimationβ’ Semiparametric estimation, extreme value theoryβ’ Nonparametric estimation
Quantile estimation : use of Beta kernels
β’ Beta kernel estimationβ’ Transforming observations
A simulation based study
12
Arthur CHARPENTIER - Nonparametric quantile estimation.
Using a parametric approach
If FX β F = {FΞΈ, ΞΈ β Ξ} (assumed to be continuous), qX(Ξ±) = Fβ1ΞΈ (Ξ±), and thus,
a natural estimator isqX(Ξ±) = Fβ1
ΞΈ(Ξ±), (1)
where ΞΈ is an estimator of ΞΈ (maximum likelihood, moments estimator...).
13
Arthur CHARPENTIER - Nonparametric quantile estimation.
Using the Gaussian distribution
A natural idea (that can be found in classical financial models) is to assumeGaussian distributions : if X βΌ N (Β΅, Ο), then the Ξ±-quantile is simply
q(Ξ±) = Β΅+ Ξ¦β1(Ξ±)Ο,
where Ξ¦β1(Ξ±) is obtained in statistical tables (or any statistical software), e.g.u = β1.64 if Ξ± = 90%, or u = β1.96 if Ξ± = 95%.
Definition 1. Given a n sample {X1, Β· Β· Β· , Xn}, the (Gaussian) parametricestimation of the Ξ±-quantile is
qn(Ξ±) = Β΅+ Ξ¦β1(Ξ±)Ο,
14
Arthur CHARPENTIER - Nonparametric quantile estimation.
Using a parametric models
Actually, is the Gaussian model does not fit very well, it is still possible to useGaussian approximation
If the variance is finite, (X β E(X))/Ο might be closer to the Gaussiandistribution, and thus, consider the so-called Cornish-Fisher approximation, i.e.
Q(X,Ξ±) βΌ E(X) + zΞ±βV (X), (2)
where
zΞ± = Ξ¦β1(Ξ±)+ΞΆ16
[Ξ¦β1(Ξ±)2β1]+ΞΆ224
[Ξ¦β1(Ξ±)3β3Ξ¦β1(Ξ±)]β ΞΆ21
36[2Ξ¦β1(Ξ±)3β5Ξ¦β1(Ξ±)],
(3)where ΞΆ1 is the skewness of X, and ΞΆ2 is the excess kurtosis, i.e. i.e.
ΞΆ1 =E([X β E(X)]3)
E([X β E(X)]2)3/2and ΞΆ1 =
E([X β E(X)]4)E([X β E(X)]2)2
β 3. (4)
15
Arthur CHARPENTIER - Nonparametric quantile estimation.
Using a parametric models
Definition 2. Given a n sample {X1, Β· Β· Β· , Xn}, the Cornish-Fisher estimation ofthe Ξ±-quantile is
qn(Ξ±) = Β΅+ zΞ±Ο, where Β΅ =1n
nβi=1
Xi and Ο =
ββββ 1nβ 1
nβi=1
(Xi β Β΅)2,
and
zΞ± = Ξ¦β1(Ξ±)+ΞΆ16
[Ξ¦β1(Ξ±)2β1]+ΞΆ224
[Ξ¦β1(Ξ±)3β3Ξ¦β1(Ξ±)]β ΞΆ21
36[2Ξ¦β1(Ξ±)3β5Ξ¦β1(Ξ±)],
where ΞΆ1 is the natural estimator for the skewness of X, and ΞΆ2 is the natural
estimator of the excess kurtosis, i.e. ΞΆ1 =
βn(nβ 1)nβ 2
βnβni=1(Xi β Β΅)3
(βni=1(Xi β Β΅)2)3/2
and
ΞΆ2 = nβ1(nβ2)(nβ3)
((n+ 1)ΞΆ β²2 + 6
)where ΞΆ β²2 = n
βni=1(XiβΒ΅)4
(βni=1(XiβΒ΅)2)2 β 3.
16
Arthur CHARPENTIER - Nonparametric quantile estimation.
Parametrics estimator and error model
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
Density, theoritical versus empirical
Theoritical lognormalFitted lognormalFitted gamma
β4 β2 0 2 4
0.0
0.1
0.2
0.3
Density, theoritical versus empirical
Theoritical StudentFitted lStudentFitted Gaussian
Fig. 5 β Estimation of Value-at-Risk, model error.
17
Arthur CHARPENTIER - Nonparametric quantile estimation.
Using a semiparametric models
Given a n-sample {Y1, . . . , Yn}, let Y1:n β€ Y2:n β€ . . .β€ Yn:n denotes the associatedorder statistics.
If u large enough, Y β u given Y > u has a Generalized Pareto distribution withparameters ΞΎ and Ξ² ( Pickands-Balkema-de Haan theorem)
If u = Ynβk:n for k large enough, and if ΞΎ>0, denote by Ξ²k and ΞΎk maximumlikelihood estimators of the Genralized Pareto distribution of sample{Ynβk+1:n β Ynβk:n, ..., Yn:n β Ynβk:n},
Q(Y, Ξ±) = Ynβk:n +Ξ²k
ΞΎk
((nk
(1β Ξ±))βΞΎk
β 1
), (5)
An alternative is to use Hillβs estimator if ΞΎ > 0,
Q(Y, Ξ±) = Ynβk:n
(nk
(1β Ξ±))βΞΎk
, ΞΎk =1k
kβi=1
log Yn+1βi:n β log Ynβk:n. (6)
18
Arthur CHARPENTIER - Nonparametric quantile estimation.
On nonparametric estimation for quantiles
For continuous distribution q(Ξ±) = Fβ1X (Ξ±), thus, a natural idea would be to
consider q(Ξ±) = Fβ1X (Ξ±), for some nonparametric estimation of FX .
Definition 3. The empirical cumulative distribution function Fn, based on
sample {X1, . . . , Xn} is Fn(x) =1n
nβi=1
1(Xi β€ x).
Definition 4. The kernel based cumulative distribution function, based onsample {X1, . . . , Xn} is
Fn(x) =1nh
nβi=1
β« x
ββk
(Xi β th
)dt =
1n
nβi=1
K
(Xi β xh
)
where K(x) =β« x
ββk(t)dt, k being a kernel and h the bandwidth.
19
Arthur CHARPENTIER - Nonparametric quantile estimation.
Smoothing nonparametric estimators
Two techniques have been considered to smooth estimation of quantiles, eitherimplicit, or explicit.
β’ consider a linear combinaison of order statistics,
The classical empirical quantile estimate is simply
Qn(p) = Fβ1n
(i
n
)= Xi:n = X[np]:n where [Β·] denotes the integer part. (7)
The estimator is simple to obtain, but depends only on one observation. Anatural extention will be to use - at least - two observations, if np is not aninteger. The weighted empirical quantile estimate is then defined as
Qn(p) = (1β Ξ³)X[np]:n + Ξ³X[np]+1:n where Ξ³ = npβ [np].
20
Arthur CHARPENTIER - Nonparametric quantile estimation.
0.0 0.2 0.4 0.6 0.8 1.0
24
68
The quantile function in R
probability level
quan
tile
leve
l
β
β
β
β
β
β
β
β
β
β
type=1type=3type=5type=7
0.0 0.2 0.4 0.6 0.8 1.0
23
45
67
The quantile function in R
probability levelqu
antil
e le
vel
ββ
β
βββ
βββ
βββ
ββ
ββ
ββββββββ
ββββββββββββββ
ββββ
βββ
ββ
β
ββ
ββ
β
βββ
βββ
βββ
ββ
ββ
ββββββββ
ββββββββββββββ
ββββ
βββ
ββ
β
ββ
type=1type=3type=5type=7
Fig. 6 β Several quantile estimators in R.
21
Arthur CHARPENTIER - Nonparametric quantile estimation.
Smoothing nonparametric estimators
In order to increase efficiency, L-statistics can be considered i.e.
Qn (p) =nβi=1
Wi,n,pXi:n =nβi=1
Wi,n,pFβ1n
(i
n
)=β« 1
0
Fβ1n (t) k (p, h, t) dt (8)
where Fn is the empirical distribution function of FX , where k is a kernel and h abandwidth. This expression can be written equivalently
Qn (p) =nβi=1
[β« in
(iβ1)n
k
(tβ ph
)dt
]X(i) =
nβi=1
[IK
(in β ph
)β IK
(iβ1n β ph
)]X(i)
(9)
where again IK (x) =β« x
ββk (t) dt. The idea is to give more weight to order
statistics X(i) such that i is closed to pn.
22
Arthur CHARPENTIER - Nonparametric quantile estimation.
0.0 0.2 0.4 0.6 0.8 1.0
01
23
quantile (probability) level
Fig. 7 β Quantile estimator as wieghted sum of order statistics.
23
Arthur CHARPENTIER - Nonparametric quantile estimation.
0.0 0.2 0.4 0.6 0.8 1.0
01
23
quantile (probability) level
Fig. 8 β Quantile estimator as wieghted sum of order statistics.
24
Arthur CHARPENTIER - Nonparametric quantile estimation.
0.0 0.2 0.4 0.6 0.8 1.0
01
23
quantile (probability) level
Fig. 9 β Quantile estimator as wieghted sum of order statistics.
25
Arthur CHARPENTIER - Nonparametric quantile estimation.
0.0 0.2 0.4 0.6 0.8 1.0
01
23
quantile (probability) level
Fig. 10 β Quantile estimator as wieghted sum of order statistics.
26
Arthur CHARPENTIER - Nonparametric quantile estimation.
0.0 0.2 0.4 0.6 0.8 1.0
01
23
quantile (probability) level
Fig. 11 β Quantile estimator as wieghted sum of order statistics.
27
Arthur CHARPENTIER - Nonparametric quantile estimation.
Smoothing nonparametric estimators
E.g. the so-called Harrell-Davis estimator is defined as
Qn(p) =nβi=1
[β« in
(iβ1)n
Ξ(n+ 1)Ξ((n+ 1)p)Ξ((n+ 1)q)
y(n+1)pβ1(1β y)(n+1)qβ1
]Xi:n,
β’ find a smooth estimator for FX , and then find (numerically) the inverse,
The Ξ±-quantile is defined as the solution of FX β¦ qX(Ξ±) = Ξ±.
If Fn denotes a continuous estimate of F , then a natural estimate for qX(Ξ±) isqn(Ξ±) such that Fn β¦ qn(Ξ±) = Ξ±, obtained using e.g. Gauss-Newton algorithm.
28
Arthur CHARPENTIER - Nonparametric quantile estimation.
Agenda
β’ General introduction
Risk measures
β’ Distorted risk measuresβ’ Value-at-Risk and related risk measures
Quantile estimation : classical techniques
β’ Parametric estimationβ’ Semiparametric estimation, extreme value theoryβ’ Nonparametric estimation
Quantile estimation : use of Beta kernels
β’ Beta kernel estimationβ’ Transforming observations
A simulation based study
29
Arthur CHARPENTIER - Nonparametric quantile estimation.
Kernel based estimation for bounded supports
Classical symmetric kernel work well when estimating densities withnon-bounded support,
fh(x) =1nh
nβi=1
k
(xβXi
h
),
where k is a kernel function (e.g. k(Ο) = I(|Ο| β€ 1)/2).
If K is a symmetric kernel, note that
E(fh(0) =12f(0) +O(h)
30
Arthur CHARPENTIER - Nonparametric quantile estimation.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Kernel based estimation of the uniform density on [0,1]
Dens
ity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Kernel based estimation of the uniform density on [0,1]
De
nsity
Fig. 12 β Density estimation of an uniform density on [0, 1].
31
Arthur CHARPENTIER - Nonparametric quantile estimation.
Kernel based estimation for bounded supports
Several techniques have been introduce to get a better estimation on the border,
β boundary kernel (Muller (1991))β mirror image modification (Deheuvels & Hominal (1989), Schuster
(1985))β transformed kernel (Devroye & Gyrfi (1981), Wand, Marron &
Ruppert (1991))β Beta kernel (Brown & Chen (1999), Chen (1999, 2000)),
see Charpentier, Fermanian & Scaillet (2006) for a survey withapplication on copulas.
32
Arthur CHARPENTIER - Nonparametric quantile estimation.
Beta kernel estimators
A Beta kernel estimator of the density (see Chen (1999)) - on [0, 1] is
fb(x) =1n
nβi=1
k
(Xi, 1 +
x
b, 1 +
1β xb
), x β [0, 1],
where k(u, Ξ±, Ξ²) =uΞ±β1(1β u)Ξ²β1
B(Ξ±, Ξ²), u β [0, 1].
If {X1, Β· Β· Β· , Xn} are i.i.d. variables with density f0, if nββ, bβ 0, thenBouzmarni & Scaillet (2005)
fb(x)β f0(x), x β [0, 1].
This is the Beta 1 estimator.
33
Arthur CHARPENTIER - Nonparametric quantile estimation.
0.0 0.2 0.4 0.6 0.8 1.0
05
1015
Beta kernel, x=0.05
0.0 0.2 0.4 0.6 0.8 1.0
02
46
810
12
Beta kernel, x=0.10
0.0 0.2 0.4 0.6 0.8 1.0
02
46
810
Beta kernel, x=0.20
0.0 0.2 0.4 0.6 0.8 1.00
24
68
Beta kernel, x=0.45
Fig. 13 β Shape of Beta kernels, different xβs and bβs.
34
Arthur CHARPENTIER - Nonparametric quantile estimation.
Improving Beta kernel estimators
Problem : the convergence is not uniform, and there is large second order biason borders, i.e. 0 and 1.
Chen (1999) proposed a modified Beta 2 kernel estimator, based on
k2 (u; b; t) =
k t
b ,1βt
b(u) , if t β [2b, 1β 2b]
kΟb(t),1βt
b(u) , if t β [0, 2b)
k tb ,Οb(1βt) (u) , if t β (1β 2b, 1]
where Οb (t) = 2b2 + 2.5ββ
4b4 + 6b2 + 2.25β t2 β t
b.
35
Arthur CHARPENTIER - Nonparametric quantile estimation.
Non-consistency of Beta kernel estimators
Problem : k(0, Ξ±, Ξ²) = k(1, Ξ±, Ξ²) = 0. So if there are point mass at 0 or 1, theestimator becomes inconsistent, i.e.
fb(x) =1n
βk
(Xi, 1 +
x
b, 1 +
1β xb
), x β [0, 1]
=1n
βXi 6=0,1
k
(Xi, 1 +
x
b, 1 +
1β xb
), x β [0, 1]
=nβ n0 β n1
n
1nβ n0 β n1
βXi 6=0,1
k
(Xi, 1 +
x
b, 1 +
1β xb
), x β [0, 1]
β (1β P(X = 0)β P(X = 1)) Β· f0(x), x β [0, 1]
and therefore Fb(x) β (1β P(X = 0)β P(X = 1)) Β· F0(x), and we may haveproblem finding a 95% or 99% quantile since the total mass will be lower.
36
Arthur CHARPENTIER - Nonparametric quantile estimation.
Non-consistency of Beta kernel estimators
Gourieroux & Monfort (2007) proposed
f(1)b (x) =
fb(x)β« 1
0fb(t)dt
, for all x β [0, 1].
It is called macro-Ξ² since the correction is performed globally.
Gourieroux & Monfort (2007) proposed
f(2)b (x) =
1n
nβi=1
kΞ²(Xi; b;x)β« 1
0kΞ²(Xi; b; t)dt
, for all x β [0, 1].
It is called micro-Ξ² since the correction is performed locally.
37
Arthur CHARPENTIER - Nonparametric quantile estimation.
Transforming observations ?
In the context of density estimation, Devroye and Gyβorfi (1985) suggestedto use a so-called transformed kernel estimate
Given a random variable Y , if H is a strictly increasing function, then thep-quantile of H(Y ) is equal to H(q(Y ; p)).
An idea is to transform initial observations {X1, Β· Β· Β· , Xn} into a sample{Y1, Β· Β· Β· , Yn} where Yi = H(Xi), and then to use a beta-kernel based estimator, ifH : Rβ [0, 1]. Then qn(X; p) = Hβ1(qn(Y ; p)).
In the context of density estimation fX(x) = fY (H(x))H β²(x). As mentioned inDevroye and Gyorfi (1985) (p 245), βfor a transformed histogram histogramestimate, the optimal H gives a uniform [0, 1] density and should therefore beequal to H(x) = F (x), for all xβ.
38
Arthur CHARPENTIER - Nonparametric quantile estimation.
Transforming observations ? a monte carlo study
Assume that sample {X1, Β· Β· Β· , Xn} have been generated from FΞΈ0 (from a famillyF = (FΞΈ, ΞΈ β Ξ). 4 transformations will be consideredβ H = FΞΈ (based on a maximum likelihood procedure)β H = FΞΈ0 (theoritical optimal transformation)β H = FΞΈ with ΞΈ < ΞΈ0 (heavier tails)β H = FΞΈ with ΞΈ > ΞΈ0 (lower tails)
39
Arthur CHARPENTIER - Nonparametric quantile estimation.
ββββββββββββββ
βββββββββββββββββββββ
ββββββββββββββ
ββββββββββββββββββββββββββ
ββββββββββββββββββ
ββββββββββββββββββ
βββββββββββββββββββ
βββββββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββββββ
ββββββββββββββββ
βββββββββββββββββββββββ
βββββββββββββββββββββ
ββββββββββββββ
ββββββββββββββββββββββββ
ββββββββββββββββ
βββββββββββββββββββ
βββββββββββββββ
ββββββββββ
ββββββββββββββββββββ
βββββββββββββββββ
βββββββββββββββ
ββββββββββββββ
βββββββββββββββββββ
ββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββ
ββββββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
ββββββββββββββββββββ
βββββββββββββββββ
ββββββββββββββββββββββββββββββ
βββββββββββββββββββββ
ββββββββββββββββββ
ββββββββββββββββββββββ
ββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββββββββββββββ
βββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββββββββββββ
ββββββββββββββββββββββ
ββββββββββββββββββββ
ββββββββββββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββββββββ
ββββββββββββββββββββββββββ
ββββββββββββββ
ββββββββββββββββββββββββ
ββββββββββββββ
βββββββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
Fig. 14 β F (Xi) versus FΞΈ(Xi), i.e. PP plot.
40
Arthur CHARPENTIER - Nonparametric quantile estimation.
Est
imat
ed d
ensi
ty
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
β
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
Est
imat
ed d
ensi
ty
Fig. 15 β Nonparametric estimation of the density of the FΞΈ(Xi)βs.
41
Arthur CHARPENTIER - Nonparametric quantile estimation.
β β β β β β β β β β β β β β β β β β β β β β ββ
ββ
ββ
ββ
ββ
ββ
β
β
β
β
β
β
β
0.80 0.85 0.90 0.95 1.00
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Estimated optimal transformation
Probability level
Qua
ntile
β β β β β β β β β β β β β β β β β β β β β β β β β β ββ
ββ
ββ
ββ
β
β
β
β
β
β
β
0.80 0.85 0.90 0.95 1.00
12
34
5
Estimated optimal transformation
Probability level
Qua
ntile
Fig. 16 β Nonparametric estimation of the quantile function, Fβ1
ΞΈ(q).
42
Arthur CHARPENTIER - Nonparametric quantile estimation.
βββββββββββββββ
βββββββββββββββββββββββ
βββββββββββ
ββββββββββββββββββ
βββββββββββββββ
βββββββββββββββββββββββ
βββββββββββββββββββ
ββββββββββββββββββββ
βββββββββββββ
βββββββββββββββββββ
βββββββββββββββ
ββββββββββββββββββ
ββββββββββββββ
ββββββββββββββββββββ
βββββββββββββββββββββββββββββ
βββββββββββββββββββββββ
βββββββββββββββββ
ββββββββββββββββ
ββββββββββββββββββββββ
βββββββββββββββ
ββββββββββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββββ
ββββββββββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
ββββββββββββββββββββ
βββββββββββββββββ
ββββββββββββββββββββββββββββββ
βββββββββββββββββββββ
ββββββββββββββββββ
ββββββββββββββββββββββ
ββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββββββββββββββ
βββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββββββββββββ
ββββββββββββββββββββββ
ββββββββββββββββββββ
ββββββββββββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββββββββ
ββββββββββββββββββββββββββ
ββββββββββββββ
ββββββββββββββββββββββββ
ββββββββββββββ
βββββββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
Fig. 17 β F (Xi) versus FΞΈ0(Xi), i.e. PP plot.
43
Arthur CHARPENTIER - Nonparametric quantile estimation.
Est
imat
ed d
ensi
ty
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
β
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
Est
imat
ed d
ensi
ty
Fig. 18 β Nonparametric estimation of the density of the FΞΈ0(Xi)βs.
44
Arthur CHARPENTIER - Nonparametric quantile estimation.
β β β β β β β β β β β β β β β β β β β β β β β ββ
ββ
ββ
ββ
ββ
β
β
β
β
β
β
β
β
0.80 0.85 0.90 0.95 1.00
12
34
Estimated optimal transformation
Probability level
Qua
ntile
β β β β β β β β β β β β β β β β β β β β β β β ββ
ββ
ββ
ββ
ββ
β
β
β
β
β
β
β
β
0.80 0.85 0.90 0.95 1.00
12
34
Estimated optimal transformation
Probability level
Qua
ntile
Fig. 19 β Nonparametric estimation of the quantile function, Fβ1ΞΈ0
(q).
45
Arthur CHARPENTIER - Nonparametric quantile estimation.
βββββββββββββββββββββββββββββββββ
βββββββββββββββ
βββββββββββββββββ
βββββββββββ
ββββββββββββββββββ
βββββββββββββββββββββ
βββββββββββββββββ
βββββββββββββββ
βββββββββββββββββββββββββββ
ββββββββββββββββ
ββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββ
βββββββββββββββ
ββββββββββββββββββ
ββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββ
ββββββββββββββββ
βββββββββββββββββββββββββββββ
βββββββββββββββ
ββββββββββββββββββββ
ββββββββββββββββββββ
βββββββββββββββββββββββ
βββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
ββββββββββββββββββββ
βββββββββββββββββ
ββββββββββββββββββββββββββββββ
βββββββββββββββββββββ
ββββββββββββββββββ
ββββββββββββββββββββββ
ββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββββββββββββββ
βββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββββββββββββ
ββββββββββββββββββββββ
ββββββββββββββββββββ
ββββββββββββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββββββββ
ββββββββββββββββββββββββββ
ββββββββββββββ
ββββββββββββββββββββββββ
ββββββββββββββ
βββββββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
Fig. 20 β F (Xi) versus FΞΈ(Xi), i.e. PP plot, ΞΈ < ΞΈ0 (heavier tails).
46
Arthur CHARPENTIER - Nonparametric quantile estimation.
Est
imat
ed d
ensi
ty
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
β
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
Est
imat
ed d
ensi
ty
Fig. 21 β Estimation of the density of the FΞΈ(Xi)βs, ΞΈ < ΞΈ0 (heavier tails).
47
Arthur CHARPENTIER - Nonparametric quantile estimation.
β β β β β β β β β β β β β β β β β β β β β β β β β β β β ββ
ββ
ββ
β
β
β
β
β
β
β
0.80 0.85 0.90 0.95 1.00
24
68
1012
Estimated optimal transformation
Probability level
Qua
ntile
β β β β β β β β β β β β β β β β β β β β β β β β β β β β ββ
ββ
ββ
β
β
β
β
β
β
β
0.80 0.85 0.90 0.95 1.00
24
68
1012
Estimated optimal transformation
Probability level
Qua
ntile
Fig. 22 β Estimation of quantile function, Fβ1ΞΈ (q), ΞΈ < ΞΈ0 (heavier tails).
48
Arthur CHARPENTIER - Nonparametric quantile estimation.
ββββββββββββββββββββββββ
βββββββββββββββββββββββββ
βββββββββββββββββ
βββββββββββββββ
ββββββββββββββββββββββββ
βββββββββββββββββββββββββ
βββββββββββββββββββ
βββββββββββββββββββββββββ
ββββββββββββββββββββ
βββββββββββββββββββ
ββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββ
βββββββββββββ
ββββββββββββββββ
ββββββββββββββββββββββ
βββββββββββββββββββββββββββββββ
ββββββββββββββ
βββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
βββββββββββββ
βββββββββββββββββββββββ
βββββββββββββββ
ββββββββββββββββββββββββ
ββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
ββββββββββββββββββββ
βββββββββββββββββ
ββββββββββββββββββββββββββββββ
βββββββββββββββββββββ
ββββββββββββββββββ
ββββββββββββββββββββββ
ββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββββββββββββββ
βββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββ
ββββββββββββββ
βββββββββββββββββββββββββββββ
ββββββββββββββββββββββ
ββββββββββββββββββββ
ββββββββββββββββββββββββ
βββββββββββββββββββββββ
ββββββββββββββββββββ
ββββββββββββββββββββββββββ
ββββββββββββββ
ββββββββββββββββββββββββ
ββββββββββββββ
βββββββββββββ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
Fig. 23 β F (Xi) versus FΞΈ(Xi), i.e. PP plot, ΞΈ > ΞΈ0 (lighter tails).
49
Arthur CHARPENTIER - Nonparametric quantile estimation.
Est
imat
ed d
ensi
ty
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
β
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
Est
imat
ed d
ensi
ty
Fig. 24 β Estimation of density of FΞΈ(Xi)βs, ΞΈ > ΞΈ0 (lighter tails).
50
Arthur CHARPENTIER - Nonparametric quantile estimation.
β β β β β β β β β β β β β β β β β β β β β ββ
ββ
ββ
ββ
ββ
ββ
β
β
β
β
β
β
β
β
0.80 0.85 0.90 0.95 1.00
1.0
1.5
2.0
2.5
3.0
3.5
Estimated optimal transformation
Probability level
Qua
ntile
β β β β β β β β β β β β β β β β β β β β β ββ
ββ
ββ
ββ
ββ
ββ
β
β
β
β
β
β
β
β
0.80 0.85 0.90 0.95 1.00
1.0
1.5
2.0
2.5
3.0
3.5
Estimated optimal transformation
Probability level
Qua
ntile
Fig. 25 β Estimation of quantile function, Fβ1ΞΈ (q), ΞΈ > ΞΈ0 (lighter tails).
51
Arthur CHARPENTIER - Nonparametric quantile estimation.
A universal distribution for losses
BNGB considered the Champernowne generalized distribution to modelinsurance claims, i.e. positive variables,
FΞ±,M,c (y) =(y + c)Ξ± β cΞ±
(y + c)Ξ± + (M + c)Ξ± β 2cΞ±where Ξ± > 0, c β₯ 0 and M > 0.
The associated density is then
fΞ±,M,c (y) =Ξ± (y + c)Ξ±β1 ((M + c)Ξ± β cΞ±)
((y + c)Ξ± + (M + c)Ξ± β 2cΞ±)2.
52
Arthur CHARPENTIER - Nonparametric quantile estimation.
A Monte Carlo study to compare those nonparametric
estimators
As in ....., 4 distributions were consideredβ normal distribution,β Weibull distribution,β log-normal distribution,β mixture of Pareto and log-normal distributions,
53
Arthur CHARPENTIER - Nonparametric quantile estimation.
5 10 15 20
0.0
00
.0
50
.1
00
.1
50
.2
00
.2
50
.3
0
Density of quantile estimators (mixture longnormal/pareto)
Estimated valueβatβrisk
de
nsity o
f e
stim
ato
rs
Benchmark (R estimator)HD (HarrellβDavis)PRK (Park)B1 (Beta 1)B2 (Beta 2)
β
β
Boxβplot for the 11 quantile estimators
5 10 15 20
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ βββββ ββββββ ββββββ ββ β β
ββ ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ ββ β β β ββ β β
β βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ ββββββββββββββββββββββ βββββββββ ββββββββββββββββββ β βββ β β βββ βββ β β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ ββββββ ββ
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MICRO Beta2
MACRO Beta2
Beta2
MICRO Beta1
MACRO Beta1
Beta1
PRK Park
PDG Padgett
HD Harrell Davis
E Epanechnikov
R benchmark
Fig. 26 β Distribution of the 95% quantile of the mixture distribution, n = 200,and associated box-plots.
54
Arthur CHARPENTIER - Nonparametric quantile estimation.
β
β
MSE ratio, normal distribution, HD (HarrellβDavis)
Probability, confidence levels (p)
MS
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atio
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n= 50n=100n=200n=500
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Probability, confidence levels (p)
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atio
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0.0 0.2 0.4 0.6 0.8 1.0
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Probability, confidence levels (p)
MS
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atio
0.0 0.2 0.4 0.6 0.8 1.0
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β
β
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β
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β
n= 50n=100n=200n=500
Fig. 27 β Comparing MSE for 6 estimators, the normal distribution case.
55
Arthur CHARPENTIER - Nonparametric quantile estimation.
β
β
MSE ratio, Weibull distribution, HD (HarrellβDavis)
Probability, confidence levels (p)
MS
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atio
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atio
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Fig. 28 β Comparing MSE for 6 estimators, the Weibull distribution case.
56
Arthur CHARPENTIER - Nonparametric quantile estimation.
β
β
MSE ratio, lognormal distribution, HD (HarrellβDavis)
Probability, confidence levels (p)
MS
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atio
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Fig. 29 β Comparing MSE for 9 estimators, the lognormal distribution case.
57
Arthur CHARPENTIER - Nonparametric quantile estimation.
β
β
MSE ratio, mixture distribution, HD (HarrellβDavis)
Probability, confidence levels (p)
MS
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Fig. 30 β Comparing MSE for 9 estimators, the mixture distribution case.
58
Arthur CHARPENTIER - Nonparametric quantile estimation.
Portfolio optimal allocation
Classical problem as formulated in Markowitz (1952), Journal of Finance, Οβ β argmin{Οβ²Ξ£Ο}u.c. Οβ²Β΅ β₯ Ξ· and Οβ²1 = 1
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59
Arthur CHARPENTIER - Nonparametric quantile estimation.
Empirical data, Eurostocks
DAX (1)
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FTSE (4)
Fig. 31 β Scatterplot of log-returns.
60
Arthur CHARPENTIER - Nonparametric quantile estimation.
Alloca
tion (
3rd as
set)
Allocation (4th asset)
Quantile
ValueβatβRisk (75%) on the grid ValueβatβRisk (75%) on the grid
Allocation (3rd asset)
Alloc
ation
(4th
asse
t)
β3 β2 β1 0 1 2
β3β2
β10
12
Alloca
tion (
3rd as
set)
Allocation (4th asset)
Quantile
ValueβatβRisk (97.5%) on the grid ValueβatβRisk (97,5%) on the grid
Allocation (3rd asset)
Alloc
ation
(4th
asse
t)
β3 β2 β1 0 1 2β3
β2β1
01
2
Fig. 32 β Value-at-Risk for all possible allocations on the grid G (surface and levelcurves), with Ξ± = 75% on the left and Ξ± = 97.5% on the right.
61
Arthur CHARPENTIER - Nonparametric quantile estimation.
β
β
β1.0
0.00.5
1.01.5
2.0
Optimal allocation (asset 1)
Probability level (97.5%β75%)
weigh
t of a
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97.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
95%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
92.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
90%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
87.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
85%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
82.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
80%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
77.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
75%
β
β
β
β1.0
0.00.5
1.01.5
2.0
Optimal allocation (asset 2)
Probability level (97.5%β75%)
weigh
t of a
lloca
tion
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
97.5%
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
95%
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
92.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
90%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
87.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
85%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
82.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
80%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
77.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
75%
β
β
β
β1.0
0.00.5
1.01.5
2.0
Optimal allocation (asset 3)
Probability level (97.5%β75%)
weigh
t of a
lloca
tion
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ97.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
95%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
92.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
90%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
87.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
85%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
82.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
80%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
77.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
75%
ββ
β
β1.0
0.00.5
1.01.5
2.0
Optimal allocation (asset 4)
Probability level (97.5%β75%)
weigh
t of a
lloca
tion
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
97.5%
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
95%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
92.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
90%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
87.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
85%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
82.5%
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
80%
β
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
77.5%
β
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β
β
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
75%
β
Fig. 33 β Optimal allocations for different probability levels (Ξ± =75%, 77.5%, 80%, ..., 95%, 97.5%), with allocation for the first asset (top left) upto the fourth asset (bottom right). 62
Arthur CHARPENTIER - Nonparametric quantile estimation.
mean 75% 77.5% 80% 82.5% 85% 87.5% 90% 92.5% 95% 97.5%
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)
0.339(0.268)
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)
0.467(0.151)
asset 3 β0.2516 β0.062(0.161)
β0.083(0.176)
β0.106(0.184)
β0.139(0.187)
β0.163(0.215)
β0.196(0.203)
β0.228(0.163)
β0.253(0.141)
β0.310(0.184)
β0.532(0.219)
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)
0.726(0.200)
Tab. 1 β Mean and standard deviation of estimated optimal allocation, for differentquantile levels.
1. raw estimator Q(Y, Ξ±) = Y[Ξ±Β·n]:n
2. mixture estimator Q(Y, Ξ±) =nβi=1
Ξ»i(Ξ±)Yi:n, which is the standard quantile
estimate in R (see [?]),
3. Gaussian estimator Q(Y, Ξ±) = Y + z1βΞ±sd(Y , where sd denotes the empiricalstandard deviation,
63
Arthur CHARPENTIER - Nonparametric quantile estimation.
4. Hillβs estimator, with k = [n/5], Q(Y, Ξ±) = Ynβk:n
(nk
(1β Ξ±))βΞΎk
, where
ΞΎk =1k
kβi=1
logYn+1βi:n
Ynβk:n(assuming that ΞΎ > 0),
5. kernel based estimator is obtained as a mixture of smoothed quantiles,derived as inverse values of a kernel based estimator of the cumulative
distribution function, i.e. Q(Y, Ξ±) =nβi=1
Ξ»i(Ξ±)Fβ1(i/n).
64
Arthur CHARPENTIER - Nonparametric quantile estimation.
β
ββ3
β2β1
01
2
Optimal allocation (asset 1)
Quantile estimator
weigh
t of a
llocatio
n βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 1raw
β
βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 2mixture
ββββββββββββββββββββββββββββββββββββββ
βββ
β
β
β
β
ββ
βββββββββββββββββββββββββββββββββββ
Est. 3Gaussian
β ββββββββββββββββββββββββββββββββββββββββ
β
ββββββββββββββββββββββββββββββββββββββββ
Est. 4Hill
β
βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 5Kernel
β
β
β
β3β2
β10
12
Optimal allocation (asset 2)
Quantile estimator
weigh
t of a
llocatio
n βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 1raw
ββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 2mixture
β
βββββββββββββββββββββββββββββββββββ
ββββ
ββ
β
βββββββββββββββββββββββββββββββββββββββ
Est. 3Gaussian
β βββββββββββββββββββββββββββββββββββββββββ
β
βββββββββββββββββββββββββββββββββββββββ
Est. 4Hill
β
βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 5Kernel
β
β
β
β3β2
β10
12
Optimal allocation (asset 3)
Quantile estimator
weigh
t of a
llocatio
n
βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 1raw
β
βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 2mixture
β
βββββββββββββββββββββββββββββββββββββββ
β
β
β
β
ββββββββββββββββββββββββββββββββββββββ
Est. 3Gaussian
β
βββββββββββββββββββββββββββββββββββββββ
β
β
β
βββββββββββββββββββββββββββββββββββββββ
Est. 4Hill
β βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 5Kernel
β
β
β
β3β2
β10
12
Optimal allocation (asset 4)
Quantile estimator
weigh
t of a
llocatio
n βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 1raw
β
βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 2mixture
β
βββββββββββββββββββββββββββββββββββββββ
β
β
β
ββββββββββββββββββββββββββββββββββββββ
Est. 3Gaussian
ββββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 4Hill
β
βββββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββ
Est. 5Kernel
β
Fig. 34 β Optimal allocations for different 95% quantile estimators, with allocationfor the first asset (top left) up to the fourth asset (bottom right).
65
Arthur CHARPENTIER - Nonparametric quantile estimation.
Some references
Charpentier, A. & Oulidi, A. (2007). Beta Kernel estimation forValue-At-Risk of heavy-tailed distributions. in revision Journal of ComputationalStatistics and Data Analysis.
Charpentier, A. & Oulidi, A. (2007). Estimating allocations forValue-at-Risk portfolio optimzation. to appear in Mathematical Methods inOperations 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 BetaKernel Estimator for Recovery Rate Distribution. CREST-DP 2006-32.
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