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

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

Introduction

OUTLINE

1 INTRODUCTION

6 ROBUSTNESS

7 CONCLUSIONS

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.

Introduction

VALUE AT RISK (VaR)

Introduction

VALUE AT RISK (VaR)

VaRα(X)

Introduction

VALUE AT RISK (VaR)

The VaR has became in a benchmark for risk management.

Introduction

VALUE AT RISK (VaR)

The VaR has been criticized by Artzner et al. (1999) since it

does not encourage diversification.

Introduction

VALUE AT RISK (VaR)

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.

Introduction

VALUE AT RISK (VaR)

But, what is one of the problems with this measure?

It is its extension to the multivariate setting

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.

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).

Introduction

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

Xi or f (X) = maxi≤n

Output: A NUMBER

Introduction

Embrechts and Puccetti (2006) introduced a multivariate approach

of the Value at Risk,

Introduction

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

Introduction

Cousin and Di Bernardino (2013) introduced a multivariate risk

measure related to the measure introduced by Embrechts and Puc-

cetti (2006).

Introduction

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

Introduction

DRAWBACKS IN THE MULTIVARIATE SETTING

The lack of a total order in high dimensions.

Introduction

The dependence among the variables.

Introduction

There are many interesting directions to analyze the data.

Introduction

There are many interesting directions to analyze the data.

The computation in high dimensions.

Introduction

OBJECTIVES

Introduce a directional multivariate value at risk

Introduction

OBJECTIVES

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.

Introduction

OBJECTIVES

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.

Directional MVaR

OUTLINE

1 INTRODUCTION

6 ROBUSTNESS

7 CONCLUSIONS

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

VaRuα(X) =

QX(α,u)⋂

λu + E[X])

where λ ∈ R and 0 ≤ α ≤ 1.

Output: A POINT IN Rn

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

Directional MVaR

Cux ≡ Oriented Orthant.

DEFINITION

Given x, u ∈ Rn and ||u|| = 1, the orthant with vertex x and direction u

Cux = z ∈ R

n|Ru(z − x) ≥ 0,

where e = 1√n(1, ..., 1)′ and Ru is a matrix such that Ruu = e.

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

Directional MVaR

DIRECTIONAL MULTIVARIATE QUANTILES

u ∈ U =

− 1√2,− 1√

− 1√2,

1√2,− 1√

1√2,

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

CLASSICAL DIRECTIONS

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

Directional MVaR

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0 1 2 3 4 5 6 7 8 9 100

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

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

VaR−e0.7(X)

Directional MVaR

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0 1 2 3 4 5 6 7 8 9 100

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

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

VaRe0.3(X)

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.

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.

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).

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

for all z in Rn.

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).

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).

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).

Directional MVaR

47 48 49 50 51 52 5347

Lower Multivariate VaR = VaRe0.3(X) and

Upper Multivariate VaR = VaR−e0.7(X)

Directional MVaR

47 48 49 50 51 52 5347

Lower Multivariate VaR = VaR( 1√

5, 2√

0.3 (X) and

Upper Multivariate VaR = VaR−( 1√

5, 2√

0.7 (X)

Marginal VaR vs. MVaR

OUTLINE

1 INTRODUCTION

6 ROBUSTNESS

7 CONCLUSIONS

RELATION BETWEEN THE MARGINAL VaR AND THE

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.

RELATION BETWEEN THE MARGINAL VaR AND THE

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.

Copulas and VaRuα(X)

OUTLINE

1 INTRODUCTION

6 ROBUSTNESS

7 CONCLUSIONS

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].

BIVARIATE COPULAS

Example of Dθ(x1) for θ ∈ (π4, π

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

α - level

Rαu(X

β= -10β= -3β= 1β= 3β= 10

0 0.2 0.4 0.6 0.80.25

α - level

Rαu(X

β= -10β= -3β= 1β= 3β= 10

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

Behavior of the first component of 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)

If X has a survival copula given by an Archimedean copula

generated by φ(·), then:

[VaReα(X)]i = 1 − φ−1

φ(α)

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

α - level

α-e(X

Beta= -1Beta= 0Beta= 1Beta= 8Beta= ∞

a) Lower Case b) Upper Case

Dashed line ≡ Cousin and Di Bernardino. Solid line ≡ VaRuα(X)

Non-Parametric Estimation

OUTLINE

1 INTRODUCTION

6 ROBUSTNESS

7 CONCLUSIONS

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

PXm [Cuxi] =

1xj∈Cuxi.

However, it is possible that QXm(α,u) = ∅. This can be solved allowing

a slack h:

(α,u) :=

xj : |PXm

− 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].

Input: u, α, h and the multivariate sample Xm.

for i = 1 to m

Pi = PXm

If |Pi − α| ≤ h

xi ∈ QhXm

(α,u),

for xj ∈ QhXm

(α,u)

dj = dist(xj, µXm+ λu),

VaRuα(Xm) = xk|dk = mindj.

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.

Robustness

OUTLINE

1 INTRODUCTION

6 ROBUSTNESS

7 CONCLUSIONS

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).

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.

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%).

Robustness

ROBUSTNESS

1. Varying only the mean, ∆µ 6= 0, ∆Σ = 0.

0 2 4 6 8 100

0.1e (X)

CB VaR

0 2 4 6 8 100

0.1e (X)

CB VaR

(A) ∆µ = (20, 20)′ (B) ∆µ = (0, 50)′

Mean of PVω

Robustness

ROBUSTNESS

2. Varying only the variances, ∆µ = 0, ∆Σ =

0 2 4 6 8 100

0.1e (X)

CB VaR

Mean of PVω

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

0.1e (X)

CB VaR

0 2 4 6 8 100

0.1e (X)

CB VaR

(A) ∆µ = (20, 20)′ (B) ∆µ = (0, 50)′

Mean of PVω

Conclusions

OUTLINE

1 INTRODUCTION

6 ROBUSTNESS

7 CONCLUSIONS

Conclusions

CONCLUSIONS

We introduce a directional multivariate value at risk and a non-

parametric estimation for this risk measure.

Conclusions

CONCLUSIONS

The directional approach allows to consider external informa-

tion or management preferences in the analysis of the data.

Conclusions

CONCLUSIONS

We provide good properties for this risk measure, including

the tail subadditivity property.

Conclusions

CONCLUSIONS

We obtain analytic expressions with copulas.

Conclusions

CONCLUSIONS

We obtain analytic expressions with copulas.

The simulation study of robustness shows good behavior of

the measure.

Thanks

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BIBLIOGRAPHY

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BIBLIOGRAPHY

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Thanks

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