V ALUE AT R ISK ( VaR ) Let X be a random variable representing loss, - - PowerPoint PPT Presentation

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 VaRu α(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 VaRu α(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) := inf{x ∈ 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) := inf{x ∈ 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) := inf{x ∈ 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) = max

i≤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

• f 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

• f 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 VaRu α(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 VaRu

α(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

Cu

x

≡ Oriented Orthant.

DEFINITION Given x, u ∈ Rn and ||u|| = 1, the orthant with vertex x and direction u is: Cu

x = {z ∈ Rn|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)

• 5
• 4
• 3
• 2
• 1

• 5
• 4
• 3
• 2
• 1

(A) Bivariate Uniform (C) Bivariate Exponential (B) Bivariate Normal VaR−e

0.7(X)

Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 18 / 44

Directional MVaR

DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVaR)

• 5
• 4
• 3
• 2
• 1

• 5
• 4
• 3
• 2
• 1

(A) Bivariate Uniform (C) Bivariate Exponential (B) Bivariate Normal VaRe

0.3(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 VaRu

α(X),

where the order is given by PREORDER (LANIADO ET AL. (2010)) x is said to be less than y if: x u y ≡ Cu

x ⊇ Cu y

≡ Rux ≤ Ruy.

Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 19 / 44

Directional MVaR Properties

MVAR PROPERTIES

Quasi-Odd Measure: VaRu

α(−X) = −VaR−u α (X).

Positive Homogeneity and Translation Invariance: Given c ∈ R+ and b ∈ Rn, then VaRu

α(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, VaRw

α(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: VaRu

α(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 [Cu

z] ≤ 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), VaRu

α(X) u R′ u sup ω∈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, VaRu

α(X + Y) u VaRu α(X) + VaRu α(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, VaRu

α(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 VaRu

α(X),

Upper Multivariate VaR in the direction u as 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

47 48 49 50 51 52 53 47 48 49 50 51 52 53

Lower Multivariate VaR = VaRe

0.3(X) and

Upper Multivariate VaR = VaR−e

0.7(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 53 47 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 VaRu α(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) ≥ [VaRe

α(X)]i ,

for all i = 1, ..., n. Moreover, if its distribution function F is quasi-concave, then, for all α ∈ (0, 1),

• VaR−e

1−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) ≥ [RuVaRu

α(X)]i ,

for all i = 1, ..., n. And if RuX has a quasi-concavity cumulative distribution, we have that

• RuVaR−u

1−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 VaRu

α(X)

OUTLINE

1 INTRODUCTION 2 DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVAR) 3 MARGINAL VAR VS. MVAR 4 COPULAS AND VaRu α(X) 5 NON-PARAMETRIC ESTIMATION 6 ROBUSTNESS 7 CONCLUSIONS

Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 27 / 44

Copulas and VaRu

α(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 VaRu

α(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)− 1

2 (sin(θ)−cos(θ)

cos(θ)

, if cos(θ) = 0 and x1 ∈ [0, 1],

1 2,

if cos(θ) = 0 and x1 ∈ [0, 1].

Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 28 / 44

Copulas and VaRu

α(X)

BIVARIATE COPULAS

Example of Dθ(x1) for θ ∈ (π

4 , π 2 )

Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 28 / 44

Copulas and VaRu

α(X)

BIVARIATE COPULAS

Results of VaRu

α(X) with the Frank’s copula, for different values on the

dependence parameter β:

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

α - level VaRα

u(X) β= -10 β= -3 β= 1 β= 3 β= 10 0.2 0.4 0.6 0.8 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

α - level VaRα

u(X) β= -10 β= -3 β= 1 β= 3 β= 10

a) Direction u = −e b) Direction u = − 3

√ 5 5 [1 3, 2 3]′

Behavior of the first component of VaRu

α(X)

Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 28 / 44

Copulas and VaRu

α(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−e

1−α(X)

• i = φ−1

1 − φ(α) n

• .

If X has a survival copula given by an Archimedean copula generated by ¯ φ(·), then: [VaRe

α(X)]i = 1 − ¯

φ−1 ¯ φ(α) n

• .

Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 29 / 44

Copulas and VaRu

α(X)

n-DIMENSIONAL ARCHIMEDEAN COPULAS

Then, we compare VaR−e

α (X) (Our) with VaRα(X) (Cousin and Di

Bernardino (2013)) and VaRe

1−α(1 − X) with VaRα(1 − X), using the

Clayton’s family of copulas.

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

α - level VaRα

• e (X)

Beta= -1 Beta= 0 Beta= 1 Beta= 8 Beta= ∞

a) Lower Case b) Upper Case Dashed line ≡ Cousin and Di Bernardino. Solid line ≡ VaRu

α(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 VaRu α(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: ˆ QXm(α, u) :=

• xi : PXm
• Cu

xi

• = α
• ,

where PXm[Cu

xi] = 1

m

m

• j=1

xi}. Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 31 / 44

Non-Parametric Estimation

NON-PARAMETRIC ESTIMATION

However, it is possible that ˆ QXm(α, u) = ∅. This can be solved allowing a slack h: ˆ Qh

Xm(α, u) :=

• xj : |PXm
• Cu

xj

• − α| ≤ h
• ,

where ˆ QXm(α, u) ⊂ ˆ Qh

Xm(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

• Cu

xi

• ,

If |Pi − α| ≤ h xi ∈ ˆ Qh

Xm(α, u),

end for xj ∈ ˆ Qh

Xm(α, u)

dj = dist(xj, {µXm + λu}), end end VaRu

α(Xm) = {xk|dk = min{dj}}.

Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 31 / 44

Non-Parametric Estimation

EXECUTION TIME Time in Seconds

Dim\ 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 VaRu α(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 X1

d

= 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 = 0,

∆Σ = 0.

2 4 6 8 10 0.01 0.02 0.03 0.04 0.05 0.06 0.07

ω

VaR0.1

e (X)

CB VaR 2 4 6 8 10 0.01 0.02 0.03 0.04 0.05 0.06

ω

VaR0.1

e (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 6.5

• ,

2 4 6 8 10 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

ω

VaR0.1

e (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 = 0,

∆Σ = 4.5 0.2 0.3 6.5

• ,

2 4 6 8 10 1 2 3 4 5 6 7 x 10

• 3

ω

VaR0.1

e (X)

CB VaR 2 4 6 8 10 0.5 1 1.5 2 2.5 3 3.5 4 x 10

• 3

ω

VaR0.1

e (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 VaRu α(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

Torres Díaz, Raúl A. Multivariate VaR: Directional perspective August 2015 44 / 44