Risk Aggregation
with Dependence Uncertainty
Carole Bernard
(Grenoble Ecole de Management)
Hannover,
Current challenges in Actuarial Mathematics
November 2015
Carole Bernard
Risk Aggregation with Dependence Uncertainty
1
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Motivation on VaR aggregation with dependence uncertainty
Full information on marginal distributions:
Xj ∼ Fj
+
Full Information on dependence:
(known copula)
⇒
VaRq (X1 + X2 + ... + Xn ) can be computed!
Carole Bernard
Risk Aggregation with Dependence Uncertainty
2
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Motivation on VaR aggregation with dependence uncertainty
Full information on marginal distributions:
Xj ∼ Fj
+
Partial or no Information on dependence:
(incomplete information on copula)
⇒
VaRq (X1 + X2 + ... + Xn ) cannot be computed!
Only a range of possible values for VaRq (X1 + X2 + ... + Xn ).
Carole Bernard
Risk Aggregation with Dependence Uncertainty
3
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Model Risk
Pd
1
Goal: Assess the risk of a portfolio sum S =
2
Choose a risk measure ρ(·): variance, Value-at-Risk...
3
“Fit” a multivariate distribution for (X1 , X2 , ..., Xd ) and
compute ρ(S)
4
How about model risk? How wrong can we be?
Carole Bernard
i=1 Xi .
Risk Aggregation with Dependence Uncertainty
4
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Model Risk
Pd
1
Goal: Assess the risk of a portfolio sum S =
2
Choose a risk measure ρ(·): variance, Value-at-Risk...
3
“Fit” a multivariate distribution for (X1 , X2 , ..., Xd ) and
compute ρ(S)
4
How about model risk? How wrong can we be?
Assume ρ(S) = var (S),
!)
(
d
X
ρ+
Xi
,
F := sup var
i=1
(
ρ−
F := inf
var
i=1 Xi .
d
X
!)
Xi
i=1
where the bounds are taken over all other (joint distributions of)
random vectors (X1 , X2 , ..., Xd ) that “agree” with the available
information F
Carole Bernard
Risk Aggregation with Dependence Uncertainty
4
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Aggregation with dependence uncertainty:
Example - Credit Risk
I Marginals known:
I Dependence fully unknown
Consider a portfolio of 10,000 loans all having a default probability
p = 0.049. The default correlation is ρ = 0.0157 (for KMV).
q = 0.95
q = 0.995
KMV VaRq
10.1%
15.1%
Max VaRq
98%
100%
Min VaRq
0%
4.4%
Portfolio models are subject to significant model uncertainty
(defaults are rare and correlated events).
Using dependence information is crucial to try to get more
“reasonable” bounds.
Carole Bernard
Risk Aggregation with Dependence Uncertainty
5
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Objectives and Findings
ˆ Model uncertainty on the risk assessment of an aggregate
portfolio: the sum of d dependent risks.
I Given all information available in the market, what can we say
about the maximum and minimum possible values of a given
risk measure of a portfolio?
Carole Bernard
Risk Aggregation with Dependence Uncertainty
6
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Objectives and Findings
ˆ Model uncertainty on the risk assessment of an aggregate
portfolio: the sum of d dependent risks.
I Given all information available in the market, what can we say
about the maximum and minimum possible values of a given
risk measure of a portfolio?
ˆ Implications:
I Current VaR based regulation is subject to high model risk,
even if one knows the multivariate distribution “almost
completely”.
Carole Bernard
Risk Aggregation with Dependence Uncertainty
6
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Acknowledgement of Collaboration
with M. Denuit, X. Jiang, L. Rüschendorf, S. Vanduffel, J. Yao, R.
Wang including
• Bernard, C., Rüschendorf, L., Vanduffel, S. (2015).
Value-at-Risk bounds with variance constraints. Journal of
Risk and Insurance
• Bernard, C., Vanduffel, S. (2015). A new approach to
assessing model risk in high dimensions. Journal of Banking
and Finance
• Bernard, C. , Rüschendorf, L., Vanduffel, S., Yao, J. (2015).
How robust is the Value-at-Risk of credit risk portfolios?
European Journal of Finance
• Bernard, C., X. Jiang, R. Wang, (2013) Risk Aggregation with
Dependence Uncertainty, Insurance: Mathematics and
Economics
Carole Bernard
Risk Aggregation with Dependence Uncertainty
7
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Risk Aggregation and full dependence uncertainty
I Marginals known:
I Dependence fully unknown
I In two dimensions d = 2, assessing model risk on variance is
linked to the Fréchet-Hoeffding bounds
var (F1−1 (U)+F2−1 (1−U)) 6 var (X1 +X2 ) 6 var (F1−1 (U)+F2−1 (U))
I A challenging problem in d > 3 dimensions
ˆ Puccetti and Rüschendorf (2012): algorithm (RA) useful to
approximate the minimum variance.
ˆ Embrechts, Puccetti, Rüschendorf (2013): algorithm (RA) to
find bounds on VaR
I Issues
ˆ bounds are generally very wide
ˆ ignore all information on dependence.
Carole Bernard
Risk Aggregation with Dependence Uncertainty
8
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Risk Aggregation and full dependence uncertainty
I Marginals known:
I Dependence fully unknown
I In two dimensions d = 2, assessing model risk on variance is
linked to the Fréchet-Hoeffding bounds
var (F1−1 (U)+F2−1 (1−U)) 6 var (X1 +X2 ) 6 var (F1−1 (U)+F2−1 (U))
I A challenging problem in d > 3 dimensions
ˆ Puccetti and Rüschendorf (2012): algorithm (RA) useful to
approximate the minimum variance.
ˆ Embrechts, Puccetti, Rüschendorf (2013): algorithm (RA) to
find bounds on VaR
I Issues
ˆ bounds are generally very wide
ˆ ignore all information on dependence.
I Our answer:
ˆ incorporating in a natural way dependence information.
Carole Bernard
Risk Aggregation with Dependence Uncertainty
8
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
VaR Bounds with full dependence uncertainty
(Unconstrained VaR bounds)
Carole Bernard
Risk Aggregation with Dependence Uncertainty
9
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
“Riskiest” Dependence: maximum VaRq in 2 dims
If X1 and X2 are U(0,1) comonotonic, then
VaRq (S c ) = VaRq (X1 ) + VaRq (X2 ) = 2q.
q
q
Carole Bernard
Risk Aggregation with Dependence Uncertainty
10
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
“Riskiest” Dependence: maximum VaRq in 2 dims
If X1 and X2 are U(0,1) comonotonic, then
VaRq (S c ) = VaRq (X1 ) + VaRq (X2 ) = 2q.
q
q
Note that TVaRq )(S c ) =
Carole Bernard
R1
q
2pdp/(1 − q) = 1 + q.
Risk Aggregation with Dependence Uncertainty
11
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
“Riskiest” Dependence: maximum VaRq in 2 dims
If X1 and X2 are U(0,1) and antimonotonic in the tail, then
VaRq (S ∗ ) = 1 + q.
q
q
VaRq (S ∗ ) = 1 + q > VaRq (S c ) = 2q
⇒ to maximize VaRq , the idea is to change the comonotonic
dependence such that the sum is constant in the tail
Carole Bernard
Risk Aggregation with Dependence Uncertainty
12
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
VaR at level q of the comonotonic sum w.r.t. q
VaRq(Sc)
q
Carole Bernard
1
p
Risk Aggregation with Dependence Uncertainty
13
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
VaR at level q of the comonotonic sum w.r.t. q
TVaRq(Sc)
VaRq(Sc)
q
1
1
where TVaR (Expected shortfall):TVaRq (X ) =
1−q
Carole Bernard
p
Z
1
VaRu (X )du
q
Risk Aggregation with Dependence Uncertainty
14
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Riskiest Dependence Structure VaR at level q
S* => VaRq(S*) =TVaRq(Sc)?
VaRq(Sc)
q
Carole Bernard
1
p
Risk Aggregation with Dependence Uncertainty
15
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Analytic expressions (not sharp)
Analytical Unconstrained Bounds with Xj ∼ Fj
A = LTVaRq (S c ) 6 VaRq [X1 + X2 + ... + Xn ] 6 B = TVaRq (S c )
B:=TVaRq(Sc)
A:=LTVaRq(Sc)
q
Carole Bernard
1
p
Risk Aggregation with Dependence Uncertainty
16
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
VaR Bounds with full dependence uncertainty
Approximate sharp bounds:
ˆ Puccetti and Rüschendorf (2012): algorithm (RA) useful to
approximate the minimum variance.
ˆ Embrechts, Puccetti, Rüschendorf (2013): algorithm (RA) to
find bounds on VaR
Carole Bernard
Risk Aggregation with Dependence Uncertainty
17
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Illustration for the maximum VaR (1/3)
q
1-q
Carole Bernard
8
10
11
12
0
1
7
8
3
4
7
9
Sum= 11
Sum= 15
Sum= 25
Sum= 29
Risk Aggregation with Dependence Uncertainty
18
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
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
Carole Bernard
8
10
11
12
0
1
7
8
3
4
7
9
Sum= 11
Sum= 15
Sum= 25
Sum= 29
Risk Aggregation with Dependence Uncertainty
19
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Illustration for the maximum VaR (3/3)
q
1-q
Carole Bernard
8
10
12
11
8
7
1
0
4
3
7
9
Sum= 20
Sum= 20
=B!
Sum= 20
Sum= 20
Risk Aggregation with Dependence Uncertainty
20
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
VaR Bounds with partial dependence uncertainty
VaR Bounds with Dependence Information...
Carole Bernard
Risk Aggregation with Dependence Uncertainty
21
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Adding dependence information
Finding minimum and maximum
possible values for VaR of the
Pn
credit portfolio loss, S = i=1 Xi , given that
ˆ known marginal distributions of the risks Xi .
ˆ some dependence information.
Example 1: variance constraint - with Rüschendorf and Vanduffel
M := sup VaRq [X1 + X2 + ... + Xn ] ,
subject to Xj ∼ Fj , var(X1 + X2 + ... + Xn ) 6 s 2
Example 2: Moments constraint - with Denuit, Rüschendorf,
Vanduffel, Yao
M := sup VaRq [X1 + X2 + ... + Xn ] ,
subject to Xj ∼ Fj , E(X1 + X2 + ... + Xn )k 6 ck
Carole Bernard
Risk Aggregation with Dependence Uncertainty
22
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Adding dependence information
Example 3: VaR bounds when the joint distribution of
(X1 , X2 , ..., Xn ) is known on a subset of the sample space: with
Vanduffel.
Example 4: with Rüschendorf, Vanduffel and Wang
M := sup VaRq [X1 + X2 + ... + Xn ] ,
subject to (Xj , Z ) ∼ Hj ,
where Z is a factor.
Carole Bernard
Risk Aggregation with Dependence Uncertainty
23
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Examples 1 and 2
Example 1: variance constraint
M := sup VaRq [X1 + X2 + ... + Xn ] ,
subject to Xj ∼ Fj , var(X1 + X2 + ... + Xn ) 6 s 2
Example 2: Moments constraint
M := sup VaRq [X1 + X2 + ... + Xn ] ,
subject to Xj ∼ Fj , E(X1 + X2 + ... + Xn )k 6 ck
for all k in 2,...,K
Carole Bernard
Risk Aggregation with Dependence Uncertainty
24
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
VaR bounds with moment constraints
I Without moment constraints, VaR bounds are attained if
there exists a dependence among risks Xi such that
A
probability q
S=
a.s.
B probability 1 − q
ˆ If the “distance” between A and B is too wide then improved
bounds are obtained with
a
with probability q
∗
S =
b with probability 1 − q
such that
ak q + b k (1 − q) 6 ck
aq + b(1 − q) = E [S]
in which a and b are “as distant as possible while satisfying all
constraints”(for all k)
Carole Bernard
Risk Aggregation with Dependence Uncertainty
25
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
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 Average
Default probability
0.0001
0.15
0.0119
EAD
0
750.2
116.7
LGD
0
0.90
0.41
Carole Bernard
Risk Aggregation with Dependence Uncertainty
26
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Comparison of Industry Models
VaRs of a corporate portfolio
q=
Comon.
95%
393.5
95%
393.5
ρ = 0.10 99%
2374.1
99.5%
5088.5
Carole Bernard
under different industry
KMV Credit Risk+
281.3
281.8
340.6
346.2
539.4
513.4
631.5
582.9
models
Beta
282.5
347.4
520.2
593.5
Risk Aggregation with Dependence Uncertainty
27
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
VaR bounds
With ρ = 0.1,
q=
95%
99%
99.5%
KMV
340.6
539.4
631.5
VaR assessment of a corporate portfolio
Comon. Unconstrained
K =2
393.3
(34.0 ; 2083.3) (97.3 ; 614.8)
2374.1 (56.5 ; 6973.1) (111.8 ; 1245)
5088.5
(89.4 ; 10120) (114.9 ; 1709)
K =3
(100.9 ; 562.8)
(115.0 ; 941.2)
(117.6 ; 1177.8)
ˆ Obs 1: Comparison with analytical bounds
ˆ Obs 2: Significant bounds reduction with moments
information
ˆ Obs 3: Significant model risk
Carole Bernard
Risk Aggregation with Dependence Uncertainty
28
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Example 3
Example 3: VaR bounds when the joint distribution of
(X1 , X2 , ..., Xn ) is known on a subset of the sample space.
Carole Bernard
Risk Aggregation with Dependence Uncertainty
29
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Bounds on variance
Analytical Bounds on Standard Deviation
Consider d risks Xi with standard deviation σi
0 6 std(X1 + X2 + ... + Xd ) 6 σ1 + σ2 + ... + σd
Carole Bernard
Risk Aggregation with Dependence Uncertainty
30
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Bounds on variance
Analytical Bounds on Standard Deviation
Consider d risks Xi with standard deviation σi
0 6 std(X1 + X2 + ... + Xd ) 6 σ1 + σ2 + ... + σd
Example with 20 standard normal N(0,1)
0 6 std(X1 + X2 + ... + X20 ) 6 20
and in this case, both bounds are sharp but too wide for practical
use!
Our idea: Incorporate information on dependence.
Carole Bernard
Risk Aggregation with Dependence Uncertainty
30
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Illustration with 2 risks with marginals N(0,1)
3
2
X2
1
0
−1
−2
−3
−3
−2
−1
0
X
1
2
3
1
Carole Bernard
Risk Aggregation with Dependence Uncertainty
31
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Illustration with 2 risks with marginals N(0,1)
3
2
X2
1
0
−1
−2
−3
−3
−2
−1
0
X1
1
Assumption: Independence on F =
2
2
\
3
{qβ 6 Xk 6 q1−β }
k=1
Carole Bernard
Risk Aggregation with Dependence Uncertainty
32
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
3
3
2
2
1
1
X2
X
2
Illustration with marginals N(0,1)
0
0
−1
−1
−2
−2
−3
−3
−3
−2
−1
0
X
1
Carole Bernard
1
2
3
−3
−2
−1
0
X
1
2
3
1
Risk Aggregation with Dependence Uncertainty
33
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Illustration with marginals N(0,1)
3
2
X2
1
0
−1
−2
−3
−3
−2
−1
0
X
1
2
3
1
2
\
F1 =
{qβ 6 Xk 6 q1−β }
k=1
Carole Bernard
Risk Aggregation with Dependence Uncertainty
34
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Illustration with marginals N(0,1)
2
\
F1 =
k=1
Carole Bernard
{qβ 6 Xk 6 q1−β }
F=
2
[
{Xk > qp }
[
F1
k=1
Risk Aggregation with Dependence Uncertainty
35
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Illustration with marginals N(0,1)
F1 =contour of MVN at β
F=
2
[
{Xk > qp }
[
F1
k=1
Carole Bernard
Risk Aggregation with Dependence Uncertainty
36
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Our assumptions on the cdf of (X1 , X2 , ..., Xd )
F ⊂ Rd (“trusted” or “fixed” area)
U =Rd \F (“untrusted”).
We assume that we know:
(i) the marginal distribution Fi of Xi on R for i = 1, 2, ..., d,
(ii) the distribution of (X1 , X2 , ..., Xd ) | {(X1 , X2 , ..., Xd ) ∈ F}.
(iii) P ((X1 , X2 , ..., Xd ) ∈ F)
Carole Bernard
Risk Aggregation with Dependence Uncertainty
37
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Our assumptions on the cdf of (X1 , X2 , ..., Xd )
F ⊂ Rd (“trusted” or “fixed” area)
U =Rd \F (“untrusted”).
We assume that we know:
(i) the marginal distribution Fi of Xi on R for i = 1, 2, ..., d,
(ii) the distribution of (X1 , X2 , ..., Xd ) | {(X1 , X2 , ..., Xd ) ∈ F}.
(iii) P ((X1 , X2 , ..., Xd ) ∈ F)
I When only marginals are known: U = Rd and F = ∅.
I Our Goal: Find bounds on var (S) := var (X1 + ... + Xd )
when (X1 , ..., Xd ) satisfy (i), (ii) and (iii).
Carole Bernard
Risk Aggregation with Dependence Uncertainty
37
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Example d = 20 risks N(0,1)
I (X1 , ..., X20 ) correlated N(0,1) on
F := [qβ , q1−β ]d ⊂ Rd
pf = P ((X1 , ..., X20 ) ∈ F)
(for some β 6 50%) where qγ : γ-quantile of N(0,1)
I β = 0%: no uncertainty (20 correlated N(0,1))
I β = 50%: full uncertainty
d
F = [qβ , q1−β ]
ρ=0
U =∅
β = 0%
4.47
pf ≈ 98%
β = 0.05%
(4.4 , 5.65)
pf ≈ 82%
β = 0.5%
(3.89 , 10.6)
U = Rd
β = 50%
(0 , 20)
Model risk on the volatility of a portfolio is reduced a lot by
incorporating information on dependence!
Carole Bernard
Risk Aggregation with Dependence Uncertainty
38
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Example d = 20 risks N(0,1)
I (X1 , ..., X20 ) correlated N(0,1) on
F := [qβ , q1−β ]d ⊂ Rd
pf = P ((X1 , ..., X20 ) ∈ F)
(for some β 6 50%) where qγ : γ-quantile of N(0,1)
I β = 0%: no uncertainty (20 correlated N(0,1))
I β = 50%: full uncertainty
d
F = [qβ , q1−β ]
ρ=0
U =∅
β = 0%
4.47
pf ≈ 98%
β = 0.05%
(4.4 , 5.65)
pf ≈ 82%
β = 0.5%
(3.89 , 10.6)
U = Rd
β = 50%
(0 , 20)
Model risk on the volatility of a portfolio is reduced a lot by
incorporating information on dependence!
Carole Bernard
Risk Aggregation with Dependence Uncertainty
39
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Numerical Results for VaR, 20 risks N(0, 1)
When marginal distributions are given,
ˆ What is the maximum Value-at-Risk?
ˆ What is the minimum Value-at-Risk?
ˆ A portfolio of 20 risks normally distributed N(0,1). Bounds on
VaRq (by the rearrangement algorithm applied on each tail)
q=95%
q=99.95%
( -2.17 , 41.3 )
( -0.035 , 71.1 )
I More examples in Embrechts, Puccetti, and Rüschendorf
(2013): “Model uncertainty and VaR aggregation,” Journal of
Banking and Finance
I Very wide bounds
I All dependence information ignored
Idea: add information on dependence from a fitted model where
data is available...
Carole Bernard
Risk Aggregation with Dependence Uncertainty
40
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Numerical Results, 20 correlated N(0, 1) on F = [qβ , q1−β ]d
F
q=95%
q=99.95%
U =∅
β = 0%
12.5
25.1
pf ≈ 98%
β = 0.05%
( 12.2 , 13.3 )
pf ≈ 82%
β = 0.5%
( 10.7 , 27.7 )
U = Rd
β = 50%
( -2.17 , 41.3 )
( -0.035 , 71.1 )
ˆ U = ∅ : 20 correlated standard normal variables (ρ = 0.1).
VaR95% = 12.5
VaR99.95% = 25.1
ff The risk for an underestimation of VaR is increasing in
the probability level used to assess the VaR.
ff For VaR at high probability levels (q = 99.95%), despite all
the added information on dependence, the bounds are
still wide!
Carole Bernard
Risk Aggregation with Dependence Uncertainty
41
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Numerical Results, 20 correlated N(0, 1) on F = [qβ , q1−β ]d
F
q=95%
q=99.95%
U =∅
β = 0%
12.5
25.1
pf ≈ 98%
β = 0.05%
( 12.2 , 13.3 )
( 24.2 , 71.1 )
pf ≈ 82%
β = 0.5%
( 10.7 , 27.7 )
( 21.5 , 71.1 )
U = Rd
β = 50%
( -2.17 , 41.3 )
( -0.035 , 71.1 )
ˆ U = ∅ : 20 correlated standard normal variables (ρ = 0.1).
VaR95% = 12.5
VaR99.95% = 25.1
I The risk for an underestimation of VaR is increasing in
the probability level used to assess the VaR.
I For VaR at high probability levels (q = 99.95%), despite
all the added information on dependence, the bounds
are still wide!
Carole Bernard
Risk Aggregation with Dependence Uncertainty
42
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Conclusions
We have shown that
ˆ Maximum Value-at-Risk is not caused by the comonotonic
scenario.
ˆ Maximum Value-at-Risk is achieved when the variance is
minimum in the tail. The RA is then used in the tails only.
ˆ Bounds on Value-at-Risk at high confidence level stay wide
even if the multivariate dependence is known in 98% of the
space!
I Assess model risk with partial information and given marginals
I Design algorithms for bounds on variance, TVaR and VaR and
many more risk measures.
I Challenges:
ˆ How to choose the trusted area F optimally?
ˆ Re-discretizing using the fitted marginal fˆi to increase N
ˆ Incorporate uncertainty on marginals
Carole Bernard
Risk Aggregation with Dependence Uncertainty
43
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Regulation challenge
The Basel Committee (2013) insists that a desired objective of a
Solvency framework concerns comparability:
“Two banks with portfolios having identical risk profiles apply the
framework’s rules and arrive at the same amount of risk-weighted
assets, and two banks with different risk profiles should
produce risk numbers that are different proportionally
to the differences in risk”
Carole Bernard
Risk Aggregation with Dependence Uncertainty
44
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
Acknowledgments
ˆ BNP Paribas Fortis Chair in Banking.
ˆ Research project on “Risk Aggregation and Diversification”
with Steven Vanduffel for the Canadian Institute of
Actuaries.
ˆ Humboldt Research Foundation.
ˆ Project on “Systemic Risk” funded by the Global Risk
Institute in Financial Services.
ˆ Natural Sciences and Engineering Research Council of
Canada
ˆ Society of Actuaries Center of Actuarial Excellence
Research Grant
Carole Bernard
Risk Aggregation with Dependence Uncertainty
45
Model Risk
Unconstrained VaR bounds
Dependence Information
Examples 1 and 2
Example 3
Conclusions
References
I Bernard, C., Vanduffel S. (2015): “A new approach to assessing model
risk in high dimensions”, Journal of Banking and Finance.
I Bernard, C., M. Denuit, and S. Vanduffel (2014): “Measuring Portfolio
Risk under Partial Dependence Information,” Working Paper.
I Bernard, C., X. Jiang, and R. Wang (2014): “Risk Aggregation with
Dependence Uncertainty,” Insurance: Mathematics and Economics.
I Bernard, C., L. Rüschendorf, and S. Vanduffel (2014): “VaR Bounds with
a Variance Constraint,”Journal of Risk and Insurance.
I Embrechts, P., G. Puccetti, and L. Rüschendorf (2013): “Model
uncertainty and VaR aggregation,” Journal of Banking & Finance.
I Puccetti, G., and L. Rüschendorf (2012): “Computation of sharp bounds
on the distribution of a function of dependent risks,” Journal of
Computational and Applied Mathematics, 236(7), 1833–1840.
I Wang, B., and R. Wang (2011): “The complete mixability and convex
minimization problems with monotone marginal densities,” Journal of
Multivariate Analysis, 102(10), 1344–1360.
I Wang, B., and R. Wang (2015): “Joint Mixability,” Mathematics
Operational Research.
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