How Robust is the Value-at-Risk of Credit Risk Portfolios?

(joint work with L. Rüschendorf, S. Vanduel, J. Yao)

Carole Bernard, Grenoble Ecole de Management

14th Scientic Day, 30.04.2015

1

14th Scientic Day 30.04.2015

Agenda

1

Background

2

Unconstrained VaR Bounds

3

VaR Bounds with Dependence Information

4

Approximate VaR Bounds

5

Case Study: Credit Risk Portfolio

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14th Scientic Day 30.04.2015

Background

Agenda

1

Background Literature Observations

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14th Scientic Day 30.04.2015

Background,

Credit risk management

1

Management of credit risk is of utmost importance (Crisis 2008).

2

Portfolio models are subject to signicant

(defaults 3

are

model uncertainty rare and correlated events).

Recent studies (Embrechts et al. (2013,2014)) show that the impact of model uncertainty on Value-at-Risk (VaR) estimates is huge.

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14th Scientic Day 30.04.2015

Background,

Credit risk management: Notation

• n •

individual risks

(L1 , L2 , ..., Ln )

(risky loans)

S := L1 + ... + Ln • Value-at-Risk of S at level q ∈ (0, 1) A portfolio

VaRq

(S) = FS−1 (q) = inf {x ∈ R | FS (x) ≥ q}

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14th Scientic Day 30.04.2015

Background,

Motivation on VaR aggregation

Full

marginal distributions: −1 represent risks as L =F (U )

information on

Lj ∼ Fj and where Uj is U(0, 1).

j

j

j

+ Full

Information on

(U1 , U2 , ..., Un ) ∼ C

dependence: (C is called the copula)

⇒

VaRq (L1 + L2 + ... + Ln )

can be computed!

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14th Scientic Day 30.04.2015

Background,

Motivation on VaR aggregation

Full

information on

marginal distributions: −1 risks as L =F (U )

Lj ∼ Fj and represent where Uj is U(0, 1).

j

j

j

+

Partial

or

no

Information on

dependence:

(U1 , U2 , ..., Un ) ∼??? ⇒

VaRq (L1 + L2 + ... + Ln ) cannot be computed! Only a range of possible values for VaRq (L1 + L2 + ... + Ln ). 7

14th Scientic Day 30.04.2015

Background, Literature

Maximum VaR under Dependence Uncertainty Bounds on Value-at-Risk

M := sup VaRq [L1 +L2 +... + Ln ] , subject to Lj ∼ Fj , copula C = unknown • Explicit sharp bounds · n = 2 Makarov (1981), Rüschendorf (1982) · homogeneous portfolios: Rüschendorf & Uckelmann

(1991), Denuit,

Genest & Marceau (1999), Embrechts & Puccetti (2006), Wang & Wang (2011), Bernard, Jiang and Wang (2014)

· heterogeneous portfolios: Wang & Wang (2015) • Approximate sharp bounds · The Rearrangement Algorithm (Puccetti & Rüschendorf Embrechts, Puccetti & Rüschendorf (2013))

(2012), 8

14th Scientic Day 30.04.2015

Background, Observations

Observations

•

The

bound M may be too wide

to be practically useful:

a feature that can only be explained by the absence of dependence information.

• Our objective: incorporate dependence information

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14th Scientic Day 30.04.2015

Background, Observations

Bounds on Value-at-Risk

I

VaRq is

not

maximized for the comonotonic scenario:

S c = Lc1 + Lc2 + ... + Lcn where all

Lci

are comonotonic.

M ≥ VaRq [Lc1 + Lc2 + ... + Lcn ] = VaRq [L1 ] + VaRq [L2 ] + ... + VaRq [Ln ] c

c

c

where (L1 , L2 , ...Ln ) is a comonotonic copy of (L1 , L2 , ...Ln ), i.e.

(Lc1 , Lc2 , ...Lcn ) = (FL−11 (U), FL−21 (U), ..., FL−n1 (U)). 10

14th Scientic Day 30.04.2015

Unconstrained VaR Bounds

Agenda

2

Unconstrained VaR Bounds VaR Bounds with 2 risks VaR Bounds with

n

risks

Example

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14th Scientic Day 30.04.2015

Unconstrained VaR Bounds, VaR Bounds with 2 risks

Riskiest Dependence: maximum VaRq in 2 dims If

L1

and

L2

are U(0,1) comonotonic, then

VaRq (S c ) = VaRq (X1 ) + VaRq (X2 ) = 2q.

q

q

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14th Scientic Day 30.04.2015

Unconstrained VaR Bounds, VaR Bounds with 2 risks

Riskiest Dependence: maximum VaRq in 2 dims

L1 and L2 are U(0,1) VaRq (S ∗ ) = 1 + q .

If

and antimonotonic in the tail, then

q

q

VaRq (S ∗ ) = 1 + q > VaRq (S c ) = 2q 13

14th Scientic Day 30.04.2015

Unconstrained VaR Bounds, VaR Bounds with n risks

VaR at level

q

of the comonotonic sum w.r.t.

q

VaRq(Sc)

q

1

p 14

14th Scientic Day 30.04.2015

Unconstrained VaR Bounds, VaR Bounds with n risks

VaR at level

q

of the comonotonic sum w.r.t.

q

TVaRq(Sc) VaRq(Sc)

q

where TVaR (Expected shortfall):TVaRq (X )

1

=

1 1

−q

p

Z 1 q

du

VaRu (X )

q 15

14th Scientic Day 30.04.2015

Unconstrained VaR Bounds, VaR Bounds with n risks

Riskiest Dependence Structure VaR at level

q

S* => VaRq(S*) =TVaRq(Sc)? VaRq(Sc)

q

1

p 16

14th Scientic Day 30.04.2015

Unconstrained VaR Bounds, VaR Bounds with n risks

Analytic expressions

Analytical Unconstrained Bounds with Lj ∼ Fj

A = LTVaRq (S c ) ≤ VaRq [L1 + L2 + ... + Ln ] ≤ B = TVaRq (S c )

B:=TVaRq(Sc)

A:=LTVaRq(Sc)

q

1

p 17

14th Scientic Day 30.04.2015

Unconstrained VaR Bounds, VaR Bounds with n risks

Proof for

B

Upper bound for VaR with given marginals

VaRq [X1 + X2 + ... + Xn ] ≤ B := TVaRq [X1c + X2c + ... + Xnc ] c

c

c

Here (X1 , X2 , ...Xn ) is a comonotonic copy of (X1 , X2 , ...Xn ), i.e.

(X1c , X2c , ...Xnc ) = (FX−11 (U), FX−21 (U), ..., FX−n1 (U)). Proof:

VaRq [X1 + X2 + ... + Xn ]

≤ ≤

TVaRq [X1 + X2 + ... + Xn ] TVaRq [X1c + X2c + ... + Xnc ] 18

14th Scientic Day 30.04.2015

Unconstrained VaR Bounds, Example

Illustration for the maximum VaR (1/3)

q

1-q

8 10 11 12

0 1 7 8

3 4 7 9

Sum= 11 Sum= 15 Sum= 25 Sum= 29 19

14th Scientic Day 30.04.2015

Unconstrained VaR Bounds, Example

Illustration for the maximum VaR (2/3)

Rearrange within columns..to make the sums as constant as possible… B=(11+15+25+29)/4=20

q

1-q

8 10 11 12

0 1 7 8

3 4 7 9

Sum= 11 Sum= 15 Sum= 25 Sum= 29 20

14th Scientic Day 30.04.2015

Unconstrained VaR Bounds, Example

Illustration for the maximum VaR (3/3)

q

1-q

8 10 12 11

8 7 1 0

4 3 7 9

Sum= 20 Sum= 20

=B!

Sum= 20 Sum= 20 21

14th Scientic Day 30.04.2015

VaR Bounds with Dependence Information, Literature

Agenda

3

VaR Bounds with Dependence Information Literature Problem

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14th Scientic Day 30.04.2015

VaR Bounds with Dependence Information, Literature

Constrained Problem Finding minimum and maximum possible values for VaR of the credit portfolio loss,

L=

Pn

i=1 Li , given that

we know the marginal distributions of the risks we have some

Li .

dependence information.

Example 1: variance constraint - Bernard, Rüchendorf and Vanduel (2015)

M := sup VaRq [L1 + L2 + ... + Ln ] , 2 subject to Lj ∼ Fj , var(L1 + L2 + ... + Ln ) ≤ s Example 2: VaR bounds when the joint distribution of

(L1 , L2 , ..., Ln )

is known on a subset of the sample space: Bernard and Vanduel (2015).

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14th Scientic Day 30.04.2015

VaR Bounds with Dependence Information, Problem

Description It appears that adding dependence information can sharpen the bounds considerably. Here,

I

VaR bounds with higher order moments on the portfolio sum I

Portfolio loss

L=

n X i=1

I

Li where Li ∼ vi B(pi ) (vi ≥ 0)

Hence, Li is a scaled Bernoulli rv. We are interested in the problem: M:= sup VaRq [L] subject to Li ∼vi B(p i ) and E [Lk ] ≤ c k (k = 2, 3, ..., K ).

I I

Extended version of the RA Assess model risk of industry credit risk models for VaR

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14th Scientic Day 30.04.2015

VaR Bounds with Dependence Information, Problem

VaR bounds with moment constraints

I

Without moment constraints, VaR bounds are attained if there exists a dependence among risks

L=



A B

If the distance between

Li

such that

probability probability 1

A

and

B

q −q

a.s.

is too wide then improved

bounds are obtained with

∗

L = such that

in which

a

constraint

and

b



a b



ak q + b k (1 − q) ≤ ck aq + b(1 − q) = E [L]

q 1−q

with probability with probability

are as distant as possible while satisfying the 25

14th Scientic Day 30.04.2015

VaR Bounds with Dependence Information, Problem

Dealing with moment constraints

To nd a (A ≤ B)

and

and obtain

b,

solve for each

 K −1

pairs

k = 2, 3, .., K

the system of equations

Aq + B(1 − q) = E (L) Ak q + B k (1 − q) = ck {Aj , Bj }.

b = a =

Then, take

min {B j |j

= 2, 3, ..., K }

E [L] − b(1 − q) . q

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14th Scientic Day 30.04.2015

Approximate VaR Bounds, Rearrangement Algorithm

Agenda

4

Approximate VaR Bounds Rearrangement Algorithm Standard Rearrangement Algorithm Extended Rearrangement Algorithm

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14th Scientic Day 30.04.2015

Approximate VaR Bounds, Rearrangement Algorithm

Approximating Sharp Bounds

The bounds a and b are sharp if one can construct dependence among the risks Li such that quantile function of their sum L becomes at on [0, q] and on [q, 1]. This holds true under certain conditions (see eg Wang and Wang, 2014). To approximate sharp VaR bounds: Extended Rearrangement Algorithm (RA). Standard RA

(Puccetti and Rüschendorf, 2012):

I

Put the margins in a matrix

I

Rearrange each column (adapt the dependence) such that

L

(row-sums) approximates a constant (E [L]) 28

14th Scientic Day 30.04.2015

July 14, 2014

Approximate VaR Bounds, Standard Rearrangement Algorithm

Example

N=4

observations of

d =3

variables:



1  0 M=  4 6

L1 , L2 , L3

1 6 0 3

iance sum Each column:

marginal

 2 3   0  4

SN



 = 

distribution

Interaction among columns:

dependence

among the risks

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14th Scientic Day 30.04.2015

Approximate VaR Bounds, Standard Rearrangement Algorithm

Standard RA: Sum with Minimum Variance

minimum variance with d = 2 risks L1 and L2 Antimonotonicity:

var (La1 + L2 ) ≤ var (L1 + L2 )

Aggregate Risk with Minimum Variance I

Columns of

M

are rearranged such that they become

anti-monotonic with the sum of all other columns.

∀k ∈ {1, 2, ..., d}, Lak antimonotonic with I

After each step, where

Lak



var Lak +

is antimonotonic

P





j6=k Lj ≤ var Lk + P with j6=k Lj

X

Lj

j6=k

P

j6=k Lj

 30

et...

X1 +X2 +X3 6 6Standard 4 Rearrangement Algorithm 16 Approximate VaR Bounds,  3 3minimum   9  Aggregate risk4 with   SN variance  =  3  Step 1: First column 1 1 2 0 0 0 0 

 ↓ 6 4  1 0

6 3 1 0



14th Scientic Day 30.04.2015

4 2 1 0

 X 2 + X3 10  5   2 0



0 becomes  1 4 6

6 3 1 0

 4 2  1 0 31

14th Scientic Day 30.04.2015

Approximate VaR Bounds, Standard Rearrangement Algorithm

Aggregate risk with minimum variance

 ↓ 6  4   1 0 

0  1   4 6 

0  1   4 6

6 3 1 0 ↓ 6 3 1 0 3 6 1 0

 4 2   1  0

 4 2   1  0

↓ 4 2 1 0

   

X 2 + X3 10 5 2 0 X 1 + X3 4 3 5 6 X 1 + X2 3 7 5 6



0 becomes   1  4 6

6 3 1 0

0 becomes   1  4 6

3 6 1 0

0 becomes   1  4 6

3 6 1 0





umns are antimonotonic with the sum of the others:

 4 2   1  0  4 2   1  0

 4 0   2  1

32

e

 4 6

1  0 0

1 0

 4

5

1

↓  X 1 + X2   6 3 Day 6 0 314th4 Scientic 0 3 4  1 630.04.2015    2 7 becomes  1 6 0   Approximate VaR Bounds, Algorithm  4 with antimonotonic sum of 1 Standard 1  the 5Rearrangement 4the 1 2 others: 6 with 0 0 minimum 6 6 0 1 Aggregate risk variance

+ X3 ↓ X 1 + X3  All columns are antimonotonic with the sumof the others: 7 Each column is antimonotonic 0 3 with 4 the sum of4the others: ↓ X +X ↓ 6  ↓ ,  X2 + X3 1 6 0     1 3 1  0 3 4 , 0 3 4 7  0 3 4  4 3  4 1 2  6  1 6 0 6 1 6 0 1 1 6 0 , ,    1 4 1 2 3 6 04 1 12  6 7  4 1 2  6 0 1

1

6 0 1

7

6 0 1

2  1 (5)



X 1 + X2   3  7 5 6

0 1 4 6

nce sum

Minimum variance sum



0  1   4 6

3 6 1 0



40 01  4 2 6 1



3 4



 6 0   1 2  0 1

X2 X1X +X12 ++ X 3 7 7 7  7    SS N = = N 7  7 7 7

+  X3   

(6)

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14th Scientic Day 30.04.2015

Approximate VaR Bounds, Extended Rearrangement Algorithm

Illustration

Extended RA q

… … … …

… … … …

… … … …

-a

8 10

8 7

4 3

-b -b

12

1

7

-b

11

0

9

-b

-a -a

-a

1-q

Rearrange now within all columns such that all sums becomes close to zero

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14th Scientic Day 30.04.2015

Approximate VaR Bounds, Extended Rearrangement Algorithm

Extended RA

ERA: Apply RA on the new matrix and check:  If all constraints are satised, then

L∗

readily generates the

approximate solutions to the problem  If not, decrease expectation of

L

b

by

ε,

and compute

a

such as the

is satised. Apply the extended RA again.

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14th Scientic Day 30.04.2015

Case Study: Credit Risk Portfolio

Agenda

5

Case Study: Credit Risk Portfolio

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14th Scientic Day 30.04.2015

Case Study: Credit Risk Portfolio

Corporate portfolio

I

a corporate portfolio of a major European Bank.

I

4495 loans mainly to medium sized and large corporate clients

I

total exposure (EAD) is 18642.7 (million Euros), and the top 10% of the portfolio (in terms of EAD) accounts for 70.1% of it.

I

portfolio exhibits some heterogeneity.

Summary statistics of a corporate portfolio Minimum

Maximum

Default probability

0.0001

0.15

EAD

0

750.2

LGD

0

0.90

Average 0.0119 116.7 0.41 37

14th Scientic Day 30.04.2015

Case Study: Credit Risk Portfolio

Comparison of Industry Models

VaRs of a corporate portfolio under dierent industry models

ρ = 0.10

q= 95% 95% 99% 99.5%

+

Comon.

KMV

Credit Risk

Beta

393.5

281.3

281.8

282.5

393.5

340.6

346.2

347.4

2374.1

539.4

513.4

520.2

5088.5

631.5

582.9

593.5

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14th Scientic Day 30.04.2015

Case Study: Credit Risk Portfolio

VaR bounds

With

ρ = 0.1,

VaR assessment of a corporate portfolio

q=

KMV

Comon.

Unconstrained

95% 99% 99.5% 99.9%

340.6 539.4 631.5 862.4

393.3 2374.1 5088.5 12905.1

(34.0 ; 2083.3) (56.5 ; 6973.1) (89.4 ; 10119.9) (111.8 ; 14784.9)

K (97.3 (111.8 (114.9 (119.2

=2 ; 614.8) ; 1245.0) ; 1709.4) ; 3692.3)

K =3 (100.9 ; 562.8) (115.0 ; 941.2) (117.6 ; 1177.8) (120.8 ; 1995.9)

K =4 (100.9 ; 560.6) (115.9 ; 834.7) (118.5 ; 989.5) (121.2 ; 1472.7)

Obs 1: Comparison with analytical bounds Obs 2: Signicant bounds reduction with moments information

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14th Scientic Day 30.04.2015

Conclusions

1

We propose simple bounds for VaR of a portfolio when there is information on the higher order moments of the portfolio sum.

2

We propose a new algorithm to approximate sharp VaR bounds.

3

Considering additional moment constraints can strengthen the unconstrained VaR bounds signicantly.

4

Illustration with credit risk models

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14th Scientic Day 30.04.2015

References I

Bernard, C., X. Jiang, and R. Wang (2014): Risk Aggregation with Dependence Uncertainty, .

I

Bernard, C., McLeish D. (2014): Algorithms for Finding Copulas Minimizing the Variance of Sums, .

I

Bernard, C., L. Rüschendorf, and S. Vanduel (2014): VaR Bounds with a Variance Constraint, .

I

Bernard, C., Vanduel S. (2014): A new approach to assessing model risk in high dimensions,

I

Embrechts, P., G. Puccetti, and L. Rüschendorf (2013): Model uncertainty and VaR aggregation, .

I

Embrechts, P., G. Puccetti, L. Rüschendorf, R. Wang, and A. Beleraj (2014): An academic response to Basel 3.5, .

I

Haus U.-U. (2014): Bounding Stochastic Dependence, Complete Mixability of Matrices, and Multidimensional Bottleneck Assignment Problems, .

I

Puccetti, G., and L. Rüschendorf (2012a): Computation of sharp bounds on the distribution of a function of dependent risks, .

I

Puccetti, G., Rüschendorf L. (2012b). Bounds for joint portfolios of dependent risks. .

I

Wang, B., and R. Wang (2011): The complete mixability and convex minimization problems with monotone marginal densities, .

I

Wang, B., and R. Wang (2015): Joint Mixability,

Insurance: Mathematics and Economics Working Paper

Working Paper

Journal of Banking and Finance. Journal of Banking & Finance Risks

Working Paper

Journal of Computational and Applied Mathematics

Risk Modeling

Statistics and

Journal of Multivariate Analysis Working paper.

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