A notion of multivariate Value at Risk from a directional ...
Transcript of A notion of multivariate Value at Risk from a directional ...
A NOTION OF MULTIVARIATE VALUE AT RISK FROM A
DIRECTIONAL PERSPECTIVE
Raúl A. TORRES
Henry LANIADO
Rosa E. LILLO
EAFIT
Department of Statistics
Universidad Carlos III de Madrid
August 2015
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 1 / 44
OUTLINE
1 INTRODUCTION
2 DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVAR)
3 MARGINAL VAR VS. MVAR
4 COPULAS AND VaRuα(X)
5 NON-PARAMETRIC ESTIMATION
6 ROBUSTNESS
7 CONCLUSIONS
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 2 / 44
Introduction
OUTLINE
1 INTRODUCTION
2 DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVAR)
3 MARGINAL VAR VS. MVAR
4 COPULAS AND VaRuα(X)
5 NON-PARAMETRIC ESTIMATION
6 ROBUSTNESS
7 CONCLUSIONS
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 3 / 44
Introduction
VALUE AT RISK (VaR)
Let X be a random variable representing loss, F its distribution
function and 0 ≤ α ≤ 1. Then,
VaRα(X) := infx ∈ R | F(x) ≥ α.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 4 / 44
Introduction
VALUE AT RISK (VaR)
Let X be a random variable representing loss, F its distribution
function and 0 ≤ α ≤ 1. Then,
VaRα(X) := infx ∈ R | F(x) ≥ α.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 4 / 44
Introduction
VALUE AT RISK (VaR)
Let X be a random variable representing loss, F its distribution
function and 0 ≤ α ≤ 1. Then,
VaRα(X) := infx ∈ R | F(x) ≥ α.
b
VaRα(X)
α
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 4 / 44
Introduction
VALUE AT RISK (VaR)
The VaR has became in a benchmark for risk management.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 5 / 44
Introduction
VALUE AT RISK (VaR)
The VaR has became in a benchmark for risk management.
The VaR has been criticized by Artzner et al. (1999) since it
does not encourage diversification.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 5 / 44
Introduction
VALUE AT RISK (VaR)
The VaR has became in a benchmark for risk management.
The VaR has been criticized by Artzner et al. (1999) since it
does not encourage diversification.
But defended by Heyde et al. (2009) for its robustness and
recently by Daníelsson et al. (2013) for its tail subadditivity.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 5 / 44
Introduction
VALUE AT RISK (VaR)
But, what is one of the problems with this measure?
It is its extension to the multivariate setting
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 6 / 44
Introduction
VALUE AT RISK (VaR)
But, what is one of the problems with this measure?
It is its extension to the multivariate setting, where
There is not a unique definition of a multivariate quantile.
There are a lot of assets in a portfolio. (High Dimension)
There is dependence among them.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 6 / 44
Introduction
REVIEW ON MULTIVARIATE VALUE AT RISK
An initial idea to study risk measures related to portfolios
X = (X1, . . . ,Xn),
is to consider a function f : Rn −→ R and then:
The VaR of the joint portfolio is the univariate-one associated
to f (X).
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 7 / 44
Introduction
REVIEW ON MULTIVARIATE VALUE AT RISK
An initial idea to study risk measures related to portfolios
X = (X1, . . . ,Xn),
is to consider a function f : Rn −→ R and then:
The VaR of the joint portfolio is the univariate-one associated
to f (X).
In Burgert and Rüschendorf (2006),
f (X) =n
∑
i=1
Xi or f (X) = maxi≤n
Xi.
Output: A NUMBER
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 7 / 44
Introduction
REVIEW ON MULTIVARIATE VALUE AT RISK
Embrechts and Puccetti (2006) introduced a multivariate approach
of the Value at Risk,
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 8 / 44
Introduction
REVIEW ON MULTIVARIATE VALUE AT RISK
Embrechts and Puccetti (2006) introduced a multivariate approach
of the Value at Risk,
Multivariate lower-orthant Value at Risk
VaRα(X) := ∂x ∈ Rn | FX(x) ≥ α.
Multivariate upper-orthant Value at Risk
VaRα(X) := ∂x ∈ Rn | FX(x) ≤ 1 − α.
Output: A SURFACE ON Rn
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 8 / 44
Introduction
REVIEW ON MULTIVARIATE VALUE AT RISK
Cousin and Di Bernardino (2013) introduced a multivariate risk
measure related to the measure introduced by Embrechts and Puc-
cetti (2006).
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 9 / 44
Introduction
REVIEW ON MULTIVARIATE VALUE AT RISK
Cousin and Di Bernardino (2013) introduced a multivariate risk
measure related to the measure introduced by Embrechts and Puc-
cetti (2006).
Multivariate lower-orthant Value at Risk
VaRα(X) := E [X|FX(x) = α] .
Multivariate upper-orthant Value at Risk
VaRα(X) := E [X|FX(x) = 1 − α] .
Output: A POINT IN Rn
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 9 / 44
Introduction
DRAWBACKS IN THE MULTIVARIATE SETTING
The lack of a total order in high dimensions.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 10 / 44
Introduction
DRAWBACKS IN THE MULTIVARIATE SETTING
The lack of a total order in high dimensions.
The dependence among the variables.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 10 / 44
Introduction
DRAWBACKS IN THE MULTIVARIATE SETTING
The lack of a total order in high dimensions.
The dependence among the variables.
There are many interesting directions to analyze the data.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 10 / 44
Introduction
DRAWBACKS IN THE MULTIVARIATE SETTING
The lack of a total order in high dimensions.
The dependence among the variables.
There are many interesting directions to analyze the data.
The computation in high dimensions.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 10 / 44
Introduction
OBJECTIVES
Introduce a directional multivariate value at risk
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 11 / 44
Introduction
OBJECTIVES
Introduce a directional multivariate value at risk
1 Consider the dependence among the variables.
2 Give the possibility of analyzing the losses considering the
manager preferences.
3 Improve the interpretation of the risk measure.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 11 / 44
Introduction
OBJECTIVES
Introduce a directional multivariate value at risk
1 Consider the dependence among the variables.
2 Give the possibility of analyzing the losses considering the
manager preferences.
3 Improve the interpretation of the risk measure.
4 Provide a non-parametric estimation to compute the risk mea-
sure in high dimensions.
5 Provide analytic expressions of the risk measure with copulas.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 11 / 44
Directional MVaR
OUTLINE
1 INTRODUCTION
2 DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVAR)
3 MARGINAL VAR VS. MVAR
4 COPULAS AND VaRuα(X)
5 NON-PARAMETRIC ESTIMATION
6 ROBUSTNESS
7 CONCLUSIONS
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 12 / 44
Directional MVaR
DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVaR)
DIRECTIONAL MVAR
Let X be a random vector satisfying "the regularity conditions", then the
Value at Risk of X in direction u and confidence parameter α is defined
as
VaRuα(X) =
(
QX(α,u)⋂
λu + E[X])
,
where λ ∈ R and 0 ≤ α ≤ 1.
Output: A POINT IN Rn
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 13 / 44
Directional MVaR
QX(α, u) ≡ Directional Multivariate Quantile
(Laniado et al. (2012)).
DEFINITION
Given u ∈ Rn, ||u|| = 1 and a random vector X with distribution proba-
bility P, the α-quantile curve in direction u is defined as:
QX(α,u) := ∂x ∈ Rn : P [Cu
x ] ≤ α,
where ∂ mans the boundary and 0 ≤ α ≤ 1
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 14 / 44
Directional MVaR
Cux ≡ Oriented Orthant.
DEFINITION
Given x, u ∈ Rn and ||u|| = 1, the orthant with vertex x and direction u
is:
Cux = z ∈ R
n|Ru(z − x) ≥ 0,
where e = 1√n(1, ..., 1)′ and Ru is a matrix such that Ruu = e.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 15 / 44
Directional MVaR
EXAMPLES OF ORIENTED ORTHANTS
(A) Orthant in direction u = (0, 1) (B) Orthant in direction u = −e
Examples of oriented orthants in R2
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 16 / 44
Directional MVaR
DIRECTIONAL MULTIVARIATE QUANTILES
u ∈ U =
(
− 1√2,− 1√
2
)
,
(
− 1√2,
1√2
)
,
(
1√2,− 1√
2
)
,
(
1√2,
1√2
)
(A) Bivariate Uniform (B) Bivariate Exponential (C) Bivariate Normal
CLASSICAL DIRECTIONS
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 17 / 44
Directional MVaR
DIRECTIONAL MULTIVARIATE QUANTILES
u ∈ U = (1, 0) , (0, 1) , (−1, 0) , (0,−1)
(A) Bivariate Uniform (B) Bivariate Exponential (C) Bivariate Normal
CANONICAL DIRECTIONS
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 17 / 44
Directional MVaR
DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVaR)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
(A) Bivariate Uniform (C) Bivariate Exponential (B) Bivariate Normal
VaR−e0.7(X)
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 18 / 44
Directional MVaR
DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVaR)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
(A) Bivariate Uniform (C) Bivariate Exponential (B) Bivariate Normal
VaRe0.3(X)
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 18 / 44
Directional MVaR Properties
MVAR PROPERTIES
Non-Negative Loading: If λ > 0,
E[X] u VaRuα(X),
where the order is given by
PREORDER (LANIADO ET AL. (2010))
x is said to be less than y if:
x u y ≡ Cux ⊇ C
uy ≡ Rux ≤ Ruy.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 19 / 44
Directional MVaR Properties
MVAR PROPERTIES
Quasi-Odd Measure: VaRuα(−X) = −VaR−u
α (X).
Positive Homogeneity and Translation Invariance: Given c ∈R+ and b ∈ R
n, then
VaRuα(cX + b) = cVaRu
α(X) + b.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 20 / 44
Directional MVaR Properties
MVAR PROPERTIES
Orthogonal Quasi-Invariance: Let w and Q be an unit vector and
a particular orthogonal matrix obtained by a QR decomposition
such that Qu = w. Then,
VaRwα(QX) = QVaRu
α(X).
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 21 / 44
Directional MVaR Properties
MVAR PROPERTIES
Consistency: Let X and Y be random vectors such that E[Y] =cu + E[X], for c > 0 and X ≤Eu Y. Then:
VaRuα(X) u VaRu
α(Y),
where the stochastic order is defined by
STOCHASTIC EXTREMALITY ORDER (LANIADO ET AL. (2012))
Let X and Y be two random vectors in Rn,
X ≤Eu Y ≡ P [Ru(X − z) ≥ 0] ≤ P [Ru(Y − z) ≥ 0] ≡ PX [Cuz ] ≤ PY [Cu
z ] ,
for all z in Rn.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 22 / 44
Directional MVaR Properties
MVAR PROPERTIES
Non-Excessive Loading: For all α ∈ (0, 1) and u ∈ B(0),
VaRuα(X) u R′
u supω∈Ω
RuX(ω).
Subadditivity in the Tail Region: Let X and Y be random vectors,
with the same mean µ and let (RuX,RuY) be a regularly varying
random vector. Then,
VaRuα(X + Y) u VaRu
α(X) + VaRuα(Y).
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 23 / 44
Directional MVaR
LOWER AND UPPER VERSIONS OF DIRECTIONAL MVaR
RESULT
Let X be a random vector and u a direction. Then for all 0 ≤ α ≤ 1,
VaRuα(X) u VaR−u
1−α(X).
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 24 / 44
Directional MVaR
LOWER AND UPPER VERSIONS OF DIRECTIONAL MVaR
Then, analogously as Embrechts and Puccetti (2006) and Cousin
and Di Bernardino (2013), we can define:
Lower Multivariate VaR in the direction u as
VaRuα(X),
Upper Multivariate VaR in the direction u as
VaR−u1−α(X).
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 24 / 44
Directional MVaR
LOWER AND UPPER VERSIONS OF DIRECTIONAL MVaR
47 48 49 50 51 52 5347
48
49
50
51
52
53
Lower Multivariate VaR = VaRe0.3(X) and
Upper Multivariate VaR = VaR−e0.7(X)
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 24 / 44
Directional MVaR
LOWER AND UPPER VERSIONS OF DIRECTIONAL MVaR
47 48 49 50 51 52 5347
48
49
50
51
52
53
Lower Multivariate VaR = VaR( 1√
5, 2√
5)
0.3 (X) and
Upper Multivariate VaR = VaR−( 1√
5, 2√
5)
0.7 (X)
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 24 / 44
Marginal VaR vs. MVaR
OUTLINE
1 INTRODUCTION
2 DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVAR)
3 MARGINAL VAR VS. MVAR
4 COPULAS AND VaRuα(X)
5 NON-PARAMETRIC ESTIMATION
6 ROBUSTNESS
7 CONCLUSIONS
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 25 / 44
Marginal VaR vs. MVaR
RELATION BETWEEN THE MARGINAL VaR AND THE
MVaR
RESULT
Let X be a random vector with survival function F quasi-concave. Then,
for all α ∈ (0, 1):
VaR1−α(Xi) ≥ [VaReα(X)]i , for all i = 1, ..., n.
Moreover, if its distribution function F is quasi-concave, then, for all α ∈(0, 1),
[
VaR−e1−α(X)
]
i≥ VaR1−α(Xi), for all i = 1, ..., n.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 26 / 44
Marginal VaR vs. MVaR
RELATION BETWEEN THE MARGINAL VaR AND THE
MVaR
RESULT
Let X be a random vector and u a direction. If the survival function of
RuX is quasi-concave. Then, for all 0 ≤ α ≤ 1,
VaR1−α([RuX]i) ≥ [RuVaRuα(X)]i , for all i = 1, ..., n.
And if RuX has a quasi-concavity cumulative distribution, we have that
[
RuVaR−u1−α(X)
]
i≥ VaR1−α([RuX]i), for all i = 1, ..., n.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 26 / 44
Copulas and VaRuα(X)
OUTLINE
1 INTRODUCTION
2 DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVAR)
3 MARGINAL VAR VS. MVAR
4 COPULAS AND VaRuα(X)
5 NON-PARAMETRIC ESTIMATION
6 ROBUSTNESS
7 CONCLUSIONS
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 27 / 44
Copulas and VaRuα(X)
BIVARIATE COPULAS
In R2 ⇒ u = (cos θ, sin θ).
Let X be a bivariate vector with density given by a copula density
c(·, ·). Then, the first component of VaRuα(X) can be obtained by
solving the equation on the domain,
∫ ∫
Dθ(x1)c(s, t)dtds = α,
where Dθ(x1) = Cu(x1,lθ(x1))
⋂
[0, 1]2 and
lθ(x1) :=
x1 sin(θ)− 12(sin(θ)−cos(θ)
cos(θ) , if cos(θ) 6= 0 and x1 ∈ [0, 1],12, if cos(θ) = 0 and x1 ∈ [0, 1].
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 28 / 44
Copulas and VaRuα(X)
BIVARIATE COPULAS
Example of Dθ(x1) for θ ∈ (π4, π
2)
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 28 / 44
Copulas and VaRuα(X)
BIVARIATE COPULAS
Results of VaRuα(X) with the Frank’s copula, for different values on the
dependence parameter β:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
α - level
Va
Rαu(X
)
β= -10β= -3β= 1β= 3β= 10
0 0.2 0.4 0.6 0.80.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
α - level
Va
Rαu(X
)
β= -10β= -3β= 1β= 3β= 10
a) Direction u = −e b) Direction u = − 3√
55[1
3, 2
3]′
Behavior of the first component of VaRuα(X)
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 28 / 44
Copulas and VaRuα(X)
n-DIMENSIONAL ARCHIMEDEAN COPULAS
Let X be a n-dimensional random vector with [0, 1]-uniform marginals.
If X has an Archimedean copula distribution generated by φ(·),then:
[
VaR−e1−α(X)
]
i= φ−1
(
1 − φ(α)
n
)
.
If X has a survival copula given by an Archimedean copula
generated by φ(·), then:
[VaReα(X)]i = 1 − φ−1
(
φ(α)
n
)
.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 29 / 44
Copulas and VaRuα(X)
n-DIMENSIONAL ARCHIMEDEAN COPULAS
Then, we compare VaR−eα (X) (Our) with VaRα(X) (Cousin and Di
Bernardino (2013)) and VaRe1−α(1 − X) with VaRα(1 − X), using the
Clayton’s family of copulas.
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
α - level
VaR
α-e(X
)
Beta= -1Beta= 0Beta= 1Beta= 8Beta= ∞
a) Lower Case b) Upper Case
Dashed line ≡ Cousin and Di Bernardino. Solid line ≡ VaRuα(X)
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 29 / 44
Non-Parametric Estimation
OUTLINE
1 INTRODUCTION
2 DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVAR)
3 MARGINAL VAR VS. MVAR
4 COPULAS AND VaRuα(X)
5 NON-PARAMETRIC ESTIMATION
6 ROBUSTNESS
7 CONCLUSIONS
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 30 / 44
Non-Parametric Estimation
NON-PARAMETRIC ESTIMATION
Given the sample Xm := x1, · · · , xm of the random loss X, the direction
u and the value of α. We find the directional quantile curve as:
QXm(α,u) :=
xi : PXm
[
Cuxi
]
= α
,
where
PXm [Cuxi] =
1
m
m∑
j=1
1xj∈Cuxi.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 31 / 44
Non-Parametric Estimation
NON-PARAMETRIC ESTIMATION
However, it is possible that QXm(α,u) = ∅. This can be solved allowing
a slack h:
QhXm
(α,u) :=
xj : |PXm
[
Cuxj
]
− α| ≤ h
,
where QXm(α,u) ⊂ QhXm
(α,u), for all h.
Once the directional α-quantile curve is obtained, we cross it with the
line µXm+ λu where
µXm= E[Xm].
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 31 / 44
Non-Parametric Estimation
NON-PARAMETRIC ESTIMATION
Input: u, α, h and the multivariate sample Xm.
for i = 1 to m
Pi = PXm
[
Cuxi
]
,
If |Pi − α| ≤ h
xi ∈ QhXm
(α,u),
end
for xj ∈ QhXm
(α,u)
dj = dist(xj, µXm+ λu),
end
end
VaRuα(Xm) = xk|dk = mindj.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 31 / 44
Non-Parametric Estimation
EXECUTION TIME
Time in SecondsDim\ Size 1000 5000 10000 50000
5 2 49 199 4903
10 2 53 208 5191
50 4 82 325 7656
100 6 139 561 12487
In an Intel core i7 (3,4 GH) computer with 32 Gb RAM.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 32 / 44
Robustness
OUTLINE
1 INTRODUCTION
2 DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVAR)
3 MARGINAL VAR VS. MVAR
4 COPULAS AND VaRuα(X)
5 NON-PARAMETRIC ESTIMATION
6 ROBUSTNESS
7 CONCLUSIONS
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 33 / 44
Robustness
ROBUSTNESS
We analyze the behavior of the MVaR when a sample is contami-
nated with different types of outliers.
We use as a benchmark the measurement given by the multivariate
VaR in Cousin and Di Bernardino (2013).
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 34 / 44
Robustness
ROBUSTNESS
We simulate 5000 observations of the following random vector:
Xω d=
X1 with probability p = 1 − ω,
X2 with probability p = ω,
where X1d= N1(µ1,Σ1), X2
d= N2(µ1 + ∆µ,Σ1 + ∆Σ) and 0 ≤ ω ≤ 1.
Specifically:
µ1 = [50, 50]′, Σ1 =
(
0.5 0.3
0.3 0.5
)
.
Contaminating
1. Varying only the mean.
2. Varying only the variances.
3. Varying all the parameters.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 35 / 44
Robustness
ROBUSTNESS
To evaluate the impact of the contamination, we use:
PVω =||Measure(Xω)− Measure(X0)||2
||Measure(X0)||2,
where Measure(X0) is the sample with ω = 0% and Measure(Xω) is
the sample with level of contamination ω%, (ω = 1% → 10%).
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 36 / 44
Robustness
ROBUSTNESS
1. Varying only the mean, ∆µ 6= 0, ∆Σ = 0.
0 2 4 6 8 100
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ω
VaR
0.1e (X)
CB VaR
0 2 4 6 8 100
0.01
0.02
0.03
0.04
0.05
0.06
ω
VaR
0.1e (X)
CB VaR
(A) ∆µ = (20, 20)′ (B) ∆µ = (0, 50)′
Mean of PVω
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 37 / 44
Robustness
ROBUSTNESS
2. Varying only the variances, ∆µ = 0, ∆Σ =
[
4.5 0
0 6.5
]
,
0 2 4 6 8 100
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
ω
VaR
0.1e (X)
CB VaR
Mean of PVω
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 38 / 44
Robustness
ROBUSTNESS
3. Varying all the parameters, ∆µ 6= 0, ∆Σ =
[
4.5 0.2
0.3 6.5
]
,
0 2 4 6 8 100
1
2
3
4
5
6
7x 10
-3
ω
VaR
0.1e (X)
CB VaR
0 2 4 6 8 100
0.5
1
1.5
2
2.5
3
3.5
4x 10
-3
ω
VaR
0.1e (X)
CB VaR
(A) ∆µ = (20, 20)′ (B) ∆µ = (0, 50)′
Mean of PVω
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 39 / 44
Conclusions
OUTLINE
1 INTRODUCTION
2 DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVAR)
3 MARGINAL VAR VS. MVAR
4 COPULAS AND VaRuα(X)
5 NON-PARAMETRIC ESTIMATION
6 ROBUSTNESS
7 CONCLUSIONS
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 40 / 44
Conclusions
CONCLUSIONS
We introduce a directional multivariate value at risk and a non-
parametric estimation for this risk measure.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 41 / 44
Conclusions
CONCLUSIONS
We introduce a directional multivariate value at risk and a non-
parametric estimation for this risk measure.
The directional approach allows to consider external informa-
tion or management preferences in the analysis of the data.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 41 / 44
Conclusions
CONCLUSIONS
We introduce a directional multivariate value at risk and a non-
parametric estimation for this risk measure.
The directional approach allows to consider external informa-
tion or management preferences in the analysis of the data.
We provide good properties for this risk measure, including
the tail subadditivity property.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 41 / 44
Conclusions
CONCLUSIONS
We introduce a directional multivariate value at risk and a non-
parametric estimation for this risk measure.
The directional approach allows to consider external informa-
tion or management preferences in the analysis of the data.
We provide good properties for this risk measure, including
the tail subadditivity property.
We obtain analytic expressions with copulas.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 41 / 44
Conclusions
CONCLUSIONS
We introduce a directional multivariate value at risk and a non-
parametric estimation for this risk measure.
The directional approach allows to consider external informa-
tion or management preferences in the analysis of the data.
We provide good properties for this risk measure, including
the tail subadditivity property.
We obtain analytic expressions with copulas.
The simulation study of robustness shows good behavior of
the measure.
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 41 / 44
Thanks
Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 42 / 44
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Thanks
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