The Rearrangement Algorithm - Peopleembrecht/ftp/Rearrangement...The rearrangement algorithm...
Transcript of The Rearrangement Algorithm - Peopleembrecht/ftp/Rearrangement...The rearrangement algorithm...
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Paul Embrechts
The Rearrangement Algorithma new tool for computing bounds on risk measures
joint work with Giovanni Puccetti (university of Firenze, Italy) and
Ludger Rüschendorf (university of Freiburg, Germany)
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1. QRM framework
2. The Rearrangement Algorithm
3. Applications
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1. QRM framework
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QRM framework under Basel 2,3,Solvency 2,...
L1 ⇠ F1, L2 ⇠ F2, . . . , Ld ⇠ Fd
one period risks with statistically estimated marginals.
⇢(L1 + · · · + Ld)
total loss exposure
amount of capital to be reserved
L+ = L1 + · · · + Ld
If a dependence model is not specified there exist infinitely many values for the risk measure which are consistent with
the choice of the marginals
⇢ = inf{ ⇢(L1 + · · · + Ld) : Lj ⇠ F j, 1 j d}⇢ = sup{ ⇢(L1 + · · · + Ld) : Lj ⇠ F j, 1 j d}
⇢⇢
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-
i.e.
Risk measures: definition
-
i.e. if is continuousES↵(L+) = E[L+|L+ > VaR↵(L+)] L+
ES↵(L+) =1
1 � ↵
Z 1
↵VaRq(L+) dq, ↵ 2 (0, 1)
VaR↵(L+) = F
�1L
+ (↵) = inf{x 2 R : F
L
+ (x) � ↵}, ↵ 2 (0, 1)
Value-at-Risk (VaR)
Expected Shortfall (ES)
P(L+ > VaR↵(L+)) 1 � ↵
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QRM framework
L1 ⇠ F1, L2 ⇠ F2, . . . , Ld ⇠ Fd
one period risks with statistically estimated marginals.
model uncertainty for VaR
model uncertainty for ES
ES↵(L+)ES↵(L+)
VaR↵(L+) VaR↵(L+)
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- Subadditivity of ES implies that
-
are known only in the case and for under special assumptions, e.g. for identically distributed risks having monotone densities; see Puccetti (2013) and Puccetti, G. and L. Rüschendorf (2013).
For general inhomogenous marginals, there does not exist an analytical tool to compute them.
Known bounds
ES↵(L+) =dX
j=1
ES↵(Lj) =dX
j=1
11 � ↵
Z 1
↵VaRq(Lj) dq
d = 2 d � 3
ES↵(L+),VaR↵(L
+),VaR↵(L+)
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2. The Rearrangement Algorithm
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1 1 22 4 63 3 64 2 65 5 10
X =
The Rearrangement Algorithm
Solution: Arrange the second column oppositely to the first
Question: Given the matrix X, rearrange the second column to obtain rowwise sums with minimal variance
1 5 62 4 63 3 64 2 65 1 6
Y =
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1 1 2 42 4 1 73 3 4 104 2 3 95 5 5 15
X =
1 24 13 42 35 5
3575
10
5 1 2 83 4 1 82 3 4 94 2 3 91 5 5 11
1 24 13 42 3
1 5 5
3575
10
1 24 1
2 3 42 3
1 5 5
3575
10
1 23 4 12 3 44 2 31 5 5
3575
10
5 1 23 4 12 3 44 2 31 5 5
3575
10
Strategy: rearrange the entries of column j oppositely to the sum of the other columns. Then iterate for all j.
The Rearrangement AlgorithmQuestion (more difficult): Given the matrix X, rearrange the entries within each column to obtain rowwise sums with minimal variance
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5 1 2 83 4 1 82 3 4 94 2 3 91 5 5 11
5 1 2 83 5 1 92 3 4 94 2 3 91 4 5 10
5 1 2 83 5 1 92 3 4 94 2 3 91 4 5 10
Rearrangement Algorithm:
Rearrange the entries in the columns of X until you find an ordered matrix Y,
i.e. a matrix in which
each column is oppositely ordered to the sum of the others.
Y =
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Let +(X) and +(Y) be the vectors having as components the componentwise sum of each row of X and, respectively, Y.
47
109
15
+(X) =
8999
10
+(Y) =
5 1 2 83 5 1 92 3 4 94 2 3 91 4 5 10
1 1 2 42 4 1 73 3 4 104 2 3 95 5 5 15
X = Y =
�cx
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The convex order is defined as
for all convex functions such that the expectations exist.
Y cx
X
Y cx
X i↵ E[ f (Y)] E[ f (X)]
f
Y cx
X
E(Y) = E(X) and var(Y) var(X)
implies
ES↵(Y) ES↵(X), ↵ 2 (0, 1)
and is equivalent to
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Associate to a (N × d ) matrix X the N-discrete d-variate distribution giving probability mass 1/N to each one of its N row vectors.
Theorem (see Puccetti, 2013)
Let Y be the matrix obtained by applying the RA to X. Then, the distribution associated to Y has the same univariate marginals of the distribution associated to X.
and ES↵(Y1 + · · · + Yd) ES↵(X1 + · · · + Xd), ↵ 2 (0, 1)
Y1 + · · · + Y
d
cx
X1 + · · · + X
d
,
(X1, . . . , Xd) ⇠ X and (Y1, . . . ,Yd) ⇠ YMoreover, if , then
5 1 2 83 5 1 92 3 4 94 2 3 91 4 5 10
1 1 2 42 4 1 73 3 4 104 2 3 95 5 5 15
X = Y =
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The RA finds a finite sequence of matrices with a decreasing expected shortfall for the the sum of the components of the random
vectors having the associated distributions.
Permutation����������� ������������������ ����������� ������������������ matrices
ordered
optimal
X1 X2 X3
5 1 2 83 5 1 92 3 4 94 2 3 91 4 5 10
ordered
5 3 1 93 4 2 91 5 3 94 1 4 92 2 5 9
optimal
It may fail in general to minimize ES
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3. Applications
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General distributions1) Approximate the support of each marginal from above and below:
2) Iteratively rearrange the column of each matrix and find two matrices X* and Y* with each column oppositely ordered to the sum of the other columns.
F j
F j � F j � F j
4) Run the algorithm with N large enough.
and create two matrices X and Y with N rows and d columns.
3) If and , then
Pareto (4)
(X1, . . . , Xd) ⇠ X⇤ (Y1, . . . ,Yd) ⇠ Y⇤
' ES↵(L+) ES↵(X1 + · · · + Xd) ES↵(Y1 + · · · + Yd)
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Pareto(4) marginals and ORDERED MATRIX↵ = 0.90
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2.13717
2.13720
2.13717
2.13716
ordered
X1
X2
X3
X4
With , we obtain the first three decimals of in 2 mins.
ES0.9(L+) = 2.1377N = 2 ⇥ 106
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VaR0.99(L1)
VaR1(L1)
takes only values all having the same probability . (1 � ↵)/N
N↵ 2 (0, 1) F�1
j |[↵, 1]Fix and assume that each
VaR↵(L1
+ · · · + Ld) � min(rowSums(X))
Analogous procedure for VaR↵(L+)
Pareto (2)
max
˜X2P(X)
min(rowSums(
˜X))VaR↵ =VaR↵(L+)
(proof in Puccetti and Rüschendorf (2012b))
P
0BBBBBB@
3X
j=1
Lj � min(rowSums(X))
1CCCCCCA � 1 � ↵
d
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ORDERED MATRIXWORST-ES SCENARIOPareto(2) marginals and ↵ = 0.99
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45.98906
45.98908
45.98907
45.98911
ordered
X1
X2
X3
X4
With N=10^5, we obtain the first three decimals of in 0.2 sec.
VaR0.99(L+) = 45.9898
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Model uncertainty for id risks
ES↵(L+)
VaR↵(L+) VaR↵(L+)
d=56, Pareto(2),
52.57 3453.99
Li ⇠ ↵ = 99.9%
3485.75
1714.88
ES↵(L+)=Pd
j=1 ES↵(Lj)
472.30
superadditive VaR
VaR+↵(L+) =dX
j=1
VaR↵(Lj)
Several inhomogeneous examples are given in Embrechts, P., Puccetti, G. and L. Rüschendorf (2013).
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For a risk vector , we define the superadditivity ratio
limd!1�↵(d) =
ES↵(L1)VaR↵(L1)
(L1, . . . , Ld)
Assume that the random variables are positive, identically distributed like , an unbounded continuous distribution having an ultimately decreasing density and finite mean. Then
LjF
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For a risk vector , we define the superadditivity ratio (L1, . . . , Ld)
limd!1�↵(d) = 1
Assume that the random variables are positive, identically distributed like , an unbounded continuous distribution having an ultimately decreasing density and infinite mean. Then
LjF
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Model uncertainty for id risks
ES↵(L+)
VaR↵(L+) VaR↵(L+)
d=56, Pareto(2),
52.57 3453.99
Li ⇠ ↵ = 99.9%
3485.75
1714.88
ES↵(L+)=Pd
j=1 ES↵(Lj)
472.30
VaR+↵(L+) =dX
j=1
VaR↵(Lj)
Several inhomogeneous examples are given in Embrechts, P., Puccetti, G. and L. Rüschendorf (2013).
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For any portfolio , of course we have that
Theorem (see Puccetti and Rüschendorf, 2013pp)
(L1, . . . , Ld)
Conjecture: the same result holds also for non id rvs
Assume that the random variables are positive, identically distributed like , an unbounded continuous distribution having an ultimately decreasing density and finite mean. Then
LjF
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Application to inhomogeneous data
- marginal losses are distributed like a Generalized Pareto Distribution (GPD), that is
- Moscadelli (2004) contains an analysis of the Basel II data on Operational Risk coming out of the second Quantitative Impact Study (QIS)
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Model uncertainty for non id risks
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- we show that additional positive dependence information added on top of the marginal distributions does not improve the VaR bounds substantially;
- we show that additional information on higher dimensional sub-vectors of marginals leads to possibly much narrower VaR bounds;
- many examples
Table 7: Estimates for VaRα(L+) for the Moscadelli example under different dependence assumptions, i.e. (from left to right) best-case dependence, best-case under additional information, comonotonicity, independence, worst-case under additional information (risks are two-by-two independent), worst-case dependence.
Adding additional information
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SummaryThe rearrangement algorithm computes numerically sharp bounds on the ES/VaR of a sum of dependent random variables.
- it can be used with any set of inhomogeneous marginals, with dimensions d up in the several hundreds and for any quantile level
- using the connection to convex order, it can be used also to compute moment bounds on supermodular functions (-> asset pricing)
- accuracy/speed can be increased by introducing a randomized starting condition and a termination condition based on the required accuracy.
The main message coming from our papers is that currently a whole toolkit of analytical and numerical techniques is available to better understand the aggregation and diversification properties of non-coherent risk measures such as Value-at-Risk.
↵
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4. Further mathematical links
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WORST VAR SCENARIOWORST-ES SCENARIOWORST ES SCENARIO
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The worst-VaR scenario (and the best-ES scenario) yields a dependence in which:
- either the rvs are very close to each other and sum up to something very close to the worst-VaR estimate (complete mixability )
- or one of the components is large and the others are small (mutual exclusivity)
These scenarios exhibit
negative dependence!
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Complete mixabilityDefinition
A distribution F is called d-completely mixable if there exist d random variables identically distributed as F such that
Examples
- F is continuous with a monotone density on a bounded support and satisfies a moderate mean condition; see Wang and Wang (2011).
- F is continuous with a concave density on a bounded support; see Puccetti, Wang and Wang (2012).
Applications
Plays the role of the lower Frèchet bound in multidimensional optimization problems
X1, . . . , Xd
P(X1
+ · · · + Xd = constant) = 1
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rearrangement = dependence
For N large enough, it is possible to approximate any dependence between N-discrete marginals by a proper rearrangement of the columns of X ; see Rüschendorf (1983) and Durante, F. and J.F.
Sánchez (2012)
comonotonicity complete����������� ������������������ mixability
1 1 1 32 2 2 63 3 3 94 4 4 125 5 5 15
5 3 1 93 4 2 91 5 3 94 1 4 92 2 5 9
X = X’ =
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Overall conclusions
- We are able to compute reliable bounds on the VaR/ES of a sum.
- Rearrangements provide an effective way to handle dependence (alternative/complementary to copulas).
- The concept of complete mixability enters many important optimization problems as an extension of the lower Fréchet bound in dimensions . d � 3
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• Durante, F. and J.F. Sánchez (2012). On the approximation of copulas via shuffles of Min. Stat. Probab. Letters, 82, 1761-1767.
• Embrechts, P., Puccetti, G. and L. Rüschendorf (2013). Model uncertainty and VaR aggregation. J. Bank. Financ., to appear
• Moscadelli, M. (2004). The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee. Temi di discussione, Banca d’Italia.
• Puccetti, G. (2013). Sharp bounds on the expected shortfall for a sum of dependent random variables. Stat. Probabil. Lett., 83(4), 1227-1232
• Puccetti, G. and L. Rüschendorf (2013pp). Asymptotic equivalence of conservative VaR- and ES-based capital charges. preprint
• Puccetti, G. and L. Rüschendorf (2013). Sharp bounds for sums of dependent risks. J. Appl. Probab., 50(1), 42-53
• Puccetti, G., Wang, B., and R. Wang (2012). Advances in complete mixability. J. Appl. Probab., 49(2), 430–440
• Rüschendorf, L. (1983). Solution of a statistical optimization problem by rearrangement methods. Metrika, 30, 55–61.
• Wang, B. and R. Wang (2011). The complete mixability and convex minimization problems with monotone marginal densities. J. Multivariate Anal., 102, 1344–1360.
visit my web-page: https://sites.google.com/site/giovannipuccetti
References: