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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
• 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
Arthur CHARPENTIER - Nonparametric quantile estimation.
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
3
Arthur CHARPENTIER - Nonparametric quantile estimation.
Risk measures and price of a risk
Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth centuryproposed to evaluate the “produit scalaire des probabilites et des gains”,
< p,x >=n∑i=1
pixi = EP(X),
based on the “regle des parties”.
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), “l’esperance mathematique est donc le juste prix des chances”(or the “fair price” mentioned in Feller (1953)).
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Arthur CHARPENTIER - Nonparametric quantile estimation.
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) =1α
log(EP(eαX)
).
5
Arthur CHARPENTIER - Nonparametric quantile estimation.
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) = TV aR(X,α) (also called expected
shortfall), Rg(X) = EP(X|X > V aR(X,α)).
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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 l’esperance mathématique
Fig. 1 – Expected value∫xdFX(x) =
∫P(X > x)dx.
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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 l’esperance d’utilité
Fig. 2 – Expected utility∫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 l’intégrale de Choquet
Fig. 3 – Distorted probabilities∫g(P(X > x))dx.
9
Arthur CHARPENTIER - Nonparametric quantile estimation.
Distorted risk measures and quantiles
Equivalently, note that E(X) =∫ 1
0F−1X (1− u)du, and
Rg(X) =∫ 1
0F−1X (1− u)dgu.
A very general class of risk measures can be defined as follows,
Rg(X) =∫ 1
0
F−1X (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 ofquantiles, where dg(1− ·) denotes the distribution of the weights.
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Arthur CHARPENTIER - Nonparametric quantile estimation.
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Distortion function, VaR (quantile) − cdf
1 − probability level
0.0 0.2 0.4 0.6 0.8 1.0
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0.2
0.4
0.6
0.8
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Distortion function, TVaR (expected shortfall) − cdf
1 − probability level
0.0 0.2 0.4 0.6 0.8 1.0
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0.2
0.4
0.6
0.8
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Distortion function, cdf
1 − probability level
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Distortion function, VaR (quantile) − pdf
1 − probability level
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01
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45
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Distortion function, TVaR (expected shortfall) − pdf
1 − probability level
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
Distortion function, pdf
1 − probability level
Fig. 4 – Distortion function, g and dg
11
Arthur CHARPENTIER - Nonparametric quantile estimation.
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
12
Arthur CHARPENTIER - Nonparametric quantile estimation.
Using a parametric approach
If FX ∈ F = {Fθ, θ ∈ Θ} (assumed to be continuous), qX(α) = F−1θ (α), and thus,
a natural estimator isqX(α) = F−1
θ(α), (1)
where θ is an estimator of θ (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 ∼ 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) parametricestimation of the α-quantile is
qn(α) = µ+ Φ−1(α)σ,
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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 − E(X))/σ might be closer to the Gaussiandistribution, and thus, consider the so-called Cornish-Fisher approximation, i.e.
Q(X,α) ∼ E(X) + zα√V (X), (2)
where
zα = Φ−1(α)+ζ16
[Φ−1(α)2−1]+ζ224
[Φ−1(α)3−3Φ−1(α)]− ζ21
36[2Φ−1(α)3−5Φ−1(α)],
(3)where ζ1 is the skewness of X, and ζ2 is the excess kurtosis, i.e. i.e.
ζ1 =E([X − E(X)]3)
E([X − E(X)]2)3/2and ζ1 =
E([X − E(X)]4)E([X − E(X)]2)2
− 3. (4)
15
Arthur CHARPENTIER - Nonparametric quantile estimation.
Using a parametric models
Definition 2. Given a n sample {X1, · · · , Xn}, the Cornish-Fisher estimation ofthe α-quantile is
qn(α) = µ+ zασ, where µ =1n
n∑i=1
Xi and σ =
√√√√ 1n− 1
n∑i=1
(Xi − µ)2,
and
zα = Φ−1(α)+ζ16
[Φ−1(α)2−1]+ζ224
[Φ−1(α)3−3Φ−1(α)]− ζ21
36[2Φ−1(α)3−5Φ−1(α)],
where ζ1 is the natural estimator for the skewness of X, and ζ2 is the natural
estimator of the excess kurtosis, i.e. ζ1 =
√n(n− 1)n− 2
√n∑ni=1(Xi − µ)3
(∑ni=1(Xi − µ)2)3/2
and
ζ2 = n−1(n−2)(n−3)
((n+ 1)ζ ′2 + 6
)where ζ ′2 = n
∑ni=1(Xi−µ)4
(∑ni=1(Xi−µ)2)2 − 3.
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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
−4 −2 0 2 4
0.0
0.1
0.2
0.3
Density, theoritical versus empirical
Theoritical StudentFitted lStudentFitted Gaussian
Fig. 5 – 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 ≤ Y2:n ≤ . . .≤ Yn:n denotes the associatedorder statistics.
If u large enough, Y − u given Y > u has a Generalized Pareto distribution withparameters 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 maximumlikelihood 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
((nk
(1− α))−ξk
− 1
), (5)
An alternative is to use Hill’s estimator if ξ > 0,
Q(Y, α) = Yn−k:n
(nk
(1− α))−ξk
, ξk =1k
k∑i=1
log Yn+1−i:n − log Yn−k:n. (6)
18
Arthur CHARPENTIER - Nonparametric quantile estimation.
On nonparametric estimation for quantiles
For continuous distribution q(α) = F−1X (α), thus, a natural idea would be to
consider q(α) = F−1X (α), for some nonparametric estimation of FX .
Definition 3. The empirical cumulative distribution function Fn, based on
sample {X1, . . . , Xn} is Fn(x) =1n
n∑i=1
1(Xi ≤ x).
Definition 4. The kernel based cumulative distribution function, based onsample {X1, . . . , Xn} is
Fn(x) =1nh
n∑i=1
∫ x
−∞k
(Xi − th
)dt =
1n
n∑i=1
K
(Xi − xh
)
where K(x) =∫ x
−∞k(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.
• consider a linear combinaison of order statistics,
The classical empirical quantile estimate is simply
Qn(p) = F−1n
(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. 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) = (1− γ)X[np]:n + γX[np]+1:n where γ = np− [np].
20
Arthur CHARPENTIER - Nonparametric quantile estimation.
0.0 0.2 0.4 0.6 0.8 1.0
24
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The quantile function in R
probability level
quan
tile
leve
l
●
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type=1type=3type=5type=7
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23
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The quantile function in R
probability levelqu
antil
e le
vel
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type=1type=3type=5type=7
Fig. 6 – 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) =n∑i=1
Wi,n,pXi:n =n∑i=1
Wi,n,pF−1n
(i
n
)=∫ 1
0
F−1n (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) =n∑i=1
[∫ in
(i−1)n
k
(t− ph
)dt
]X(i) =
n∑i=1
[IK
(in − ph
)− IK
(i−1n − ph
)]X(i)
(9)
where again IK (x) =∫ x
−∞k (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 – 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 – 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 – 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 – 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 – 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) =n∑i=1
[∫ in
(i−1)n
Γ(n+ 1)Γ((n+ 1)p)Γ((n+ 1)q)
y(n+1)p−1(1− y)(n+1)q−1
]Xi:n,
• 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) isqn(α) such that Fn ◦ qn(α) = α, obtained using e.g. Gauss-Newton algorithm.
28
Arthur CHARPENTIER - Nonparametric quantile estimation.
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
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
n∑i=1
k
(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
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 – 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,
– 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 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
n∑i=1
k
(Xi, 1 +
x
b, 1 +
1− xb
), x ∈ [0, 1],
where k(u, α, β) =uα−1(1− u)β−1
B(α, β), u ∈ [0, 1].
If {X1, · · · , Xn} are i.i.d. variables with density f0, if n→∞, b→ 0, thenBouzmarni & Scaillet (2005)
fb(x)→ f0(x), x ∈ [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 – Shape of Beta kernels, different x’s and b’s.
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 ,1−t
b(u) , if t ∈ [2b, 1− 2b]
kρb(t),1−t
b(u) , if t ∈ [0, 2b)
k tb ,ρb(1−t) (u) , if t ∈ (1− 2b, 1]
where ρb (t) = 2b2 + 2.5−√
4b4 + 6b2 + 2.25− t2 − t
b.
35
Arthur CHARPENTIER - Nonparametric quantile estimation.
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, theestimator becomes inconsistent, i.e.
fb(x) =1n
∑k
(Xi, 1 +
x
b, 1 +
1− xb
), x ∈ [0, 1]
=1n
∑Xi 6=0,1
k
(Xi, 1 +
x
b, 1 +
1− xb
), x ∈ [0, 1]
=n− n0 − n1
n
1n− n0 − n1
∑Xi 6=0,1
k
(Xi, 1 +
x
b, 1 +
1− xb
), 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 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)∫ 1
0fb(t)dt
, for all x ∈ [0, 1].
It is called macro-β since the correction is performed globally.
Gourieroux & Monfort (2007) proposed
f(2)b (x) =
1n
n∑i=1
kβ(Xi; b;x)∫ 1
0kβ(Xi; b; t)dt
, for all x ∈ [0, 1].
It is called micro-β since the correction is performed locally.
37
Arthur CHARPENTIER - Nonparametric quantile estimation.
Transforming observations ?
In the context of density estimation, Devroye and Gy’orfi (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, · · · , Xn} into a sample{Y1, · · · , Yn} where Yi = H(Xi), and then to use a beta-kernel based estimator, ifH : 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 inDevroye and Gyorfi (1985) (p 245), “for a transformed histogram histogramestimate, the optimal H gives a uniform [0, 1] density and should therefore beequal to H(x) = F (x), for all x”.
38
Arthur CHARPENTIER - Nonparametric quantile estimation.
Transforming observations ? a monte carlo study
Assume that sample {X1, · · · , Xn} have been generated from Fθ0 (from a famillyF = (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)
39
Arthur CHARPENTIER - Nonparametric quantile estimation.
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0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
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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
Tra
nsfo
rmed
obs
erva
tions
Fig. 14 – F (Xi) versus Fθ(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
●
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 – Nonparametric estimation of the density of the Fθ(Xi)’s.
41
Arthur CHARPENTIER - Nonparametric quantile estimation.
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●
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●●
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●
●
●
●
●
●
●
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
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●
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●●
●●
●
●
●
●
●
●
●
0.80 0.85 0.90 0.95 1.00
12
34
5
Estimated optimal transformation
Probability level
Qua
ntile
Fig. 16 – Nonparametric estimation of the quantile function, F−1
θ(q).
42
Arthur CHARPENTIER - Nonparametric quantile estimation.
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0.0 0.2 0.4 0.6 0.8 1.0
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0.2
0.4
0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
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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
Tra
nsfo
rmed
obs
erva
tions
Fig. 17 – F (Xi) versus Fθ0(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
●
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 – Nonparametric estimation of the density of the Fθ0(Xi)’s.
44
Arthur CHARPENTIER - Nonparametric quantile estimation.
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●
●●
●●
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●
●
●
●
●
●
●
●
0.80 0.85 0.90 0.95 1.00
12
34
Estimated optimal transformation
Probability level
Qua
ntile
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●
●●
●●
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●
●
●
●
●
●
●
●
0.80 0.85 0.90 0.95 1.00
12
34
Estimated optimal transformation
Probability level
Qua
ntile
Fig. 19 – Nonparametric estimation of the quantile function, F−1θ0
(q).
45
Arthur CHARPENTIER - Nonparametric quantile estimation.
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0.0 0.2 0.4 0.6 0.8 1.0
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0.2
0.4
0.6
0.8
1.0
Tra
nsfo
rmed
obs
erva
tions
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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
Tra
nsfo
rmed
obs
erva
tions
Fig. 20 – F (Xi) versus Fθ(Xi), i.e. PP plot, θ < θ0 (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
●
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 – Estimation of the density of the Fθ(Xi)’s, θ < θ0 (heavier tails).
47
Arthur CHARPENTIER - Nonparametric quantile estimation.
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●
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●●
●
●
●
●
●
●
●
0.80 0.85 0.90 0.95 1.00
24
68
1012
Estimated optimal transformation
Probability level
Qua
ntile
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●
●●
●●
●
●
●
●
●
●
●
0.80 0.85 0.90 0.95 1.00
24
68
1012
Estimated optimal transformation
Probability level
Qua
ntile
Fig. 22 – Estimation of quantile function, F−1θ (q), θ < θ0 (heavier tails).
48
Arthur CHARPENTIER - Nonparametric quantile estimation.
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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
Tra
nsfo
rmed
obs
erva
tions
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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
Tra
nsfo
rmed
obs
erva
tions
Fig. 23 – F (Xi) versus Fθ(Xi), i.e. PP plot, θ > θ0 (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
●
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 – Estimation of density of Fθ(Xi)’s, θ > θ0 (lighter tails).
50
Arthur CHARPENTIER - Nonparametric quantile estimation.
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●
●●
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●
●
●
●
●
●
●
●
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
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●
●●
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●
●
●
●
●
●
●
●
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 – Estimation of quantile function, F−1θ (q), θ > θ0 (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,
Fα,M,c (y) =(y + c)α − cα
(y + c)α + (M + c)α − 2cαwhere α > 0, c ≥ 0 and M > 0.
The associated density is then
fα,M,c (y) =α (y + c)α−1 ((M + c)α − cα)
((y + c)α + (M + c)α − 2cα)2.
52
Arthur CHARPENTIER - Nonparametric quantile estimation.
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
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 value−at−risk
de
nsity o
f e
stim
ato
rs
Benchmark (R estimator)HD (Harrell−Davis)PRK (Park)B1 (Beta 1)B2 (Beta 2)
●
●
Box−plot for the 11 quantile estimators
5 10 15 20
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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 – Distribution of the 95% quantile of the mixture distribution, n = 200,and associated box-plots.
54
Arthur CHARPENTIER - Nonparametric quantile estimation.
●
●
MSE ratio, normal distribution, HD (Harrell−Davis)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●● ● ● ● ● ● ●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, normal distribution, B1 (Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
●
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●
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●
n= 50n=100n=200n=500
●
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MSE ratio, normal distribution, MACB1 (MACRO−Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●
●
●● ●
●
●●●
●
n= 50n=100n=200n=500
●
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MSE ratio, normal distribution, PRK (Park)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●
●
●
●
●
●
●
●●
●
n= 50n=100n=200n=500
●
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MSE ratio, normal distribution, B1 (Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
●
●●
●
●●●
●
n= 50n=100n=200n=500
●
●
MSE ratio, normal distribution, MACB1 (MACRO−Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●
●
●● ●
●
●●●
●
n= 50n=100n=200n=500
Fig. 27 – Comparing MSE for 6 estimators, the normal distribution case.
55
Arthur CHARPENTIER - Nonparametric quantile estimation.
●
●
MSE ratio, Weibull distribution, HD (Harrell−Davis)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
● ● ● ●●
●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, Weibull distribution, MACB1 (MACRO−Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
●●
●
●
●
●●
●
n= 50n=100n=200n=500
●
●
MSE ratio, Weibull distribution, MICB1 (MICRO−Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
● ●●
●●
●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, Weibull distribution, PRK (Park)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0 ●
●
●
●
●
●●
●
n= 50n=100n=200n=500
●
●
MSE ratio, Weibull distribution, MACB1 (MACRO−Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
●●
●
●
●
●●
●
n= 50n=100n=200n=500
●
●
MSE ratio, Weibull distribution, MICB1 (MICRO−Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
● ●●
●●
●
●
●
n= 50n=100n=200n=500
Fig. 28 – Comparing MSE for 6 estimators, the Weibull distribution case.
56
Arthur CHARPENTIER - Nonparametric quantile estimation.
●
●
MSE ratio, lognormal distribution, HD (Harrell−Davis)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
● ● ● ● ● ● ●
●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, lognormal distribution, MACB1 (MACRO−Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.00.
00.
51.
01.
52.
0
● ● ●●
● ●●
●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, lognormal distribution, MICB1 (MICRO−Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●
●
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●
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●
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●
n= 50n=100n=200n=500
●
●
MSE ratio, lognormal distribution, B1 (Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
● ● ●●
●●
●
●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, lognormal distribution, PRK (Park)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●
●
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●
n= 50n=100n=200n=500
●
●
MSE ratio, lognormal distribution, MACB2 (MACRO−Beta2)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●● ● ●
● ●● ●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, lognormal distribution, MICB2 (MICRO−Beta2)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.00.
00.
51.
01.
52.
0
●
●● ● ● ●
●
●●
●
n= 50n=100n=200n=500
●
●
MSE ratio, lognormal distribution, B2 (Beta2)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●● ●
● ● ●●
●
●
●
n= 50n=100n=200n=500
Fig. 29 – Comparing MSE for 9 estimators, the lognormal distribution case.
57
Arthur CHARPENTIER - Nonparametric quantile estimation.
●
●
MSE ratio, mixture distribution, HD (Harrell−Davis)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
● ● ● ● ●
●●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, mixture distribution, MACB1 (MACRO−Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.00.
00.
51.
01.
52.
0
●●
●
●
●
●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, mixture distribution, MICB1 (MICRO−Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
●
● ●
● ●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, mixture distribution, B1 (Beta1)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
● ●●
●
●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, mixture distribution, PRK (Park)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
● ●
●
●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, mixture distribution, MACB2 (MACRO−Beta2)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
● ● ● ●
● ●
●●
●
n= 50n=100n=200n=500
●
●
MSE ratio, mixture distribution, MICB2 (MICRO−Beta2)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.00.
00.
51.
01.
52.
0
● ● ● ●
●
●●
●
●
n= 50n=100n=200n=500
●
●
MSE ratio, mixture distribution, B2 (Beta2)
Probability, confidence levels (p)
MS
E r
atio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
●●
●
● ●
●●
●
●
n= 50n=100n=200n=500
Fig. 30 – 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, ω∗ ∈ argmin{ω′Σω}u.c. ω′µ ≥ η and ω′1 = 1
convex⇔
ω∗ ∈ argmax{ω′µ}u.c. ω′Σω ≤ η′ and ω′1 = 1
Roy (1952), Econometrica,“the optimal bundle of assets (investment) forinvestors who employ the safety first principle is the portfolio that minimizes theprobability 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
Arthur CHARPENTIER - Nonparametric quantile estimation.
Empirical data, Eurostocks
DAX (1)
−0.05 0.05
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FTSE (4)
Fig. 31 – Scatterplot of log-returns.
60
Arthur CHARPENTIER - Nonparametric quantile estimation.
Alloca
tion (
3rd as
set)
Allocation (4th asset)
Quantile
Value−at−Risk (75%) on the grid Value−at−Risk (75%) on the grid
Allocation (3rd asset)
Alloc
ation
(4th
asse
t)
−3 −2 −1 0 1 2
−3−2
−10
12
Alloca
tion (
3rd as
set)
Allocation (4th asset)
Quantile
Value−at−Risk (97.5%) on the grid Value−at−Risk (97,5%) on the grid
Allocation (3rd asset)
Alloc
ation
(4th
asse
t)
−3 −2 −1 0 1 2−3
−2−1
01
2
Fig. 32 – Value-at-Risk for all possible allocations on the grid G (surface and levelcurves), with α = 75% on the left and α = 97.5% on the right.
61
Arthur CHARPENTIER - Nonparametric quantile estimation.
●
●
−1.0
0.00.5
1.01.5
2.0
Optimal allocation (asset 1)
Probability level (97.5%−75%)
weigh
t of a
lloca
tion
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
97.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
95%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
92.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
90%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
87.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
85%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
82.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
80%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
77.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
75%
●
●
●
−1.0
0.00.5
1.01.5
2.0
Optimal allocation (asset 2)
Probability level (97.5%−75%)
weigh
t of a
lloca
tion
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
97.5%
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
95%
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
92.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
90%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
87.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
85%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
82.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
80%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
77.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
75%
●
●
●
−1.0
0.00.5
1.01.5
2.0
Optimal allocation (asset 3)
Probability level (97.5%−75%)
weigh
t of a
lloca
tion
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●97.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
95%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
92.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
90%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
87.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
85%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
82.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
80%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
77.5%
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
75%
●●
●
−1.0
0.00.5
1.01.5
2.0
Optimal allocation (asset 4)
Probability level (97.5%−75%)
weigh
t of a
lloca
tion
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
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) 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 −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)
−0.532(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 – Mean and standard deviation of estimated optimal allocation, for differentquantile levels.
1. raw estimator Q(Y, α) = Y[α·n]:n
2. mixture estimator Q(Y, α) =n∑i=1
λi(α)Yi:n, which is the standard quantile
estimate in R (see [?]),
3. Gaussian estimator Q(Y, α) = Y + z1−αsd(Y , where sd denotes the empiricalstandard deviation,
63
Arthur CHARPENTIER - Nonparametric quantile estimation.
4. Hill’s estimator, with k = [n/5], Q(Y, α) = Yn−k:n
(nk
(1− α))−ξk
, where
ξk =1k
k∑i=1
logYn+1−i:n
Yn−k:n(assuming that ξ > 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, α) =n∑i=1
λi(α)F−1(i/n).
64
Arthur CHARPENTIER - Nonparametric quantile estimation.
●
●−3
−2−1
01
2
Optimal allocation (asset 1)
Quantile estimator
weigh
t of a
llocatio
n ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 1raw
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 2mixture
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●
●
●
●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 3Gaussian
● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 4Hill
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 5Kernel
●
●
●
−3−2
−10
12
Optimal allocation (asset 2)
Quantile estimator
weigh
t of a
llocatio
n ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 1raw
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 2mixture
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 3Gaussian
● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 4Hill
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 5Kernel
●
●
●
−3−2
−10
12
Optimal allocation (asset 3)
Quantile estimator
weigh
t of a
llocatio
n
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 1raw
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 2mixture
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 3Gaussian
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 4Hill
● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 5Kernel
●
●
●
−3−2
−10
12
Optimal allocation (asset 4)
Quantile estimator
weigh
t of a
llocatio
n ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 1raw
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 2mixture
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 3Gaussian
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 4Hill
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
Est. 5Kernel
●
Fig. 34 – 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.
66