Random sets in economics, finance
and insurance
Part II
Ilya Molchanov
University of Bern, Switzerland
based on joint works with
I.Cascos (Madrid, Statistics), E.Lepinette (Paris, Finance),
F.Molinari (Cornell, Economics), M.Schmutz (Bern, Probability and Finance),
A.Haier (Swiss Financial Market Supervision)
University Austin TX, May 2015
1
Non-linearity
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The capacity functional
T (K ) = P{X \ K 6= ;}
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determines the distribution of random closed set X.
T is not additive — it is subadditive.
Further examples of such functionals appear as
payoffs of coalitional games.
Another source of such functionals is upper
envelopes of measures:
'(K ) = sup µ(K )
µ2M
2
Choquet integral
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Non-additive measure '(A) is a set function with
values in [0, 1] such that '(;) = 0.
' is called a capacity if it satisfies some continuity
conditions.
Definition
If f : Rd 7! R+ , then
Z
Z
fd' =
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1
0
'({x : f (x)
t})dt
If '(K ) = T (K ) = P{X \ K 6= ;}, then
Z
fdT = E sup f (X) = E sup f (⇠) = sup Ef (⇠).
⇠2X
⇠2X
3
Measuring risk
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Let ⇠ be the terminal financial position (gain if
positive, loss if negative) of an agent.
The regulator requires to set aside capital r (⇠) as a
security in case of possible eventual distress.
4
Univariate coherent Lp -risk measure
r : Lp (R) 7! ( 1, 1]
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cash-invariant
r (⇠ + a) = r (⇠)
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monotonone
⇠⌘
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a;
)
r (⇠)
r (⌘);
subadditive
r (⇠ + ⌘)  r (⇠) + r (⌘);
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homogeneous
r (c⇠) = cr (⇠),
c > 0.
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Examples
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Expectation
r (⇠) =
Essential infimum
r (⇠) =
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ess inf ⇠.
Expected Shortfall at level ↵
r (⇠) =
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E⇠
E(⇠|⇠  q↵ ),
where q↵ is the ↵-quantile of ⇠ (assume non-atomic
distribution of ⇠).
R
Choquet integrals r (⇠) = ⇠d' for an appropriate
capacity on ⌦.
6
Multivariate risks
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The theory of risk measures started with the
univariate case
and then moved to the dynamic setting.
The situation with multivariate risks only recently
received a proper attention.
7
Kabanov’s exchange cone model
X=X +K
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X is Lp -integrable random vector in Rd (gains on d
assets/currencies).
K is a random exchange cone (e.g. generated by
bid-ask exchange rates for currencies).
K is the family of portfolios available at price zero.
K describes transaction rules at the time when the
gain X is assessed.
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Exchange cone
X
X +K
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Aim: measure the risk of X taking K into account.
9
Set-valued portfolio
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Idea: consider X = X + K as a random set.
Consider also other random sets X not necessarily of
point-plus-cone type.
Portfolio X is a random convex closed set such that
X = X + Rd (lower set).
Risk ⇢(X) is an upper set.
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Portfolio and risk measure
X
⇢(X + K)
X +K
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Position is acceptable if the risk measure contains the
origin.
Larger set means lower risk.
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Acceptability
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A portfolio X is acceptable if there exists a random
vector
⇠2X
a.s.,
so that all components of ⇠ are individually
acceptable.
This means that there exists a terminal transfer that
makes all lines acceptable.
12
X
X +K
Although components of ⇠ are considered separately, the
⇠ itself appears as linear combination of components of X
and K, so that the results and not marginalised.
13
X +K
X
Although components of ⇠ are considered separately, the
⇠ itself appears as linear combination of components of X
and K, so that the results and not marginalised.
14
X
X +K
Although components of ⇠ are considered separately, the
⇠ itself appears as linear combination of components of X
and K, so that the results and not marginalised.
15
Example
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Let X = (X1 , X2 ) be bivariate normal with means
(0.5, 0.5), variances (1, 1) and correlation 0.6.
Exchanges are free from transaction costs.
Initial exchange rate ⇡0 = 1.5, terminal rate ⇡ is
log-normal mean 1.5 and volatility 1.4, independent
of X .
Take Expected Shortfall at level 0.05 as the risk
measure.
16
Example
X
X
The required capital in the first currency
No transfers
2.0645
Terminal transfer to the 1st currency 1.801
Terminal transfer to the 2nd currency 1.784
“Cleverer” transfer
1.661
17
Example
X
X
This “cleverer” transfer is given by
✓
◆
⇡(X1 ⇡ + X2 ) X1 ⇡ + X2
⇠ = (⇠1 , ⇠2 ) =
,
1 + ⇡2
1 + ⇡2
18
Acceptable selections
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A random vector ⇠ 2 Rd is called a selection of X if
⇠ 2 X a.s.
Assume that X contains at least one Lp -integrable
selection, i.e. the set Lp (X) is not empty.
Consider d-tuple r = (r1 , . . . , rd ) of univariate
coherent law invariant Lp -risk measures.
Random set X is acceptable if it possesses at least
one acceptable selection ⇠, meaning that
r(⇠) = (r1 (⇠1 ), . . . , rd (⇠d ))  0 ,
i.e. all individual coordinates of ⇠ are acceptable.
19
Selection risk measure
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The selection risk measure ⇢s (X) is the topological
closure of the set
⇢s,0 (X) = {a 2 Rd : X + a is acceptable} .
20
Portfolio and risk measure
[r(⇠), 1)
⇢(X + K)
X
X +K
⇠
The risk measure is the closed union of quadrants
r(⇠) + Rd+ determined by all possible selections ⇠:
[
⇢s (X) = cl
[r(⇠), 1) .
⇠2Lp (X)
21
Main properties
Theorem
The selection risk measure takes values being upper
convex closed sets, is law invariant and is coherent, i.e.
satisfies the following conditions
1. ⇢s (X + a) = ⇢s (X) a for all a 2 Rd (cash invariance).
2. If X ⇢ Y a.s., then ⇢s (X) ⇢ ⇢s (Y) (monotonicity).
3. ⇢s (cX) = c⇢s (X) for all c > 0 (homogeneity).
4. ⇢s (X + Y) ⇢s (X) + ⇢s (Y) (superadditivity for inclusion
= subadditivity of risks).
22
Law invariance
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We assume that the probability space is non-atomic.
Two identically distributed random sets X and Y might
have rather different families of selections.
Let FX be the -algebra generated by X.
If ⇠ 2 Lp (X) and r(⇠)  0, then
r(E(⇠|FX ))  r(⇠)  0
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and ⌘ = E(⇠|FX ) is also a selection of X, since X is
a.s. convex.
Use the fact that the families of FX -measurable
selections of X and FY -measurable selections of Y
coincide.
23
Selection (Aumann) expectation
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The closure of the set of expectations of all integrable
selections
EX = cl{E⇠ : ⇠ 2 L1 (X)}
is called the selection (Aumann) expectation of X.
The closure is not needed if X is integrably bounded,
i.e. kXk = sup{kxk : x 2 X} is integrable.
If X , Y are integrably bounded,
E(X + Y) = EX + EY
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If
EX = { x : x 2 EX}, then
⇢s (X) =
EX
is a (rather trivial) selection risk measure.
24
Capital reserving
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Let X = X + K for a possibly random exchange cone
K.
The necessary capital should be allocated at time
zero, with the exchange rules determined by a
non-random exchange cone K0 .
Then the family of all possible initial capital
requirements is given by
A0 = ⇢s (X + K) + ( K0 ) .
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Optimal capital requirements are given by the
extremal points from A0 in the order generated by the
cone K0 .
If A0 is the whole space, then it is possible to make
an infinite capital gain (risk arbitrage).
25
Risk arbitrage: step by step
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Two currencies: exchange rate ⇡ at time one
lognormally distributed. No transaction costs.
Position at time one X = (0, 0).
Position ( a, ⇡a) is reachable from X = (0, 0) at price
zero.
Its risk is (a, ar (⇡)).
So we need a of the first currency and a⇢(⇡) in the
second (note that r (⇡) < 0).
If the exchange rate at tme zero is ⇡0 , this costs
⇡0 a + ar (⇡) = a(⇡0 + r (⇡))
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If ⇡0 is small enough, then ⇡0 + r (⇡) < 0 and we let a
grow to release infinite capital.
The model does not admit financial arbitrage (there
exists a martingale measure).
26
Bounds
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The family of all selections of X is immense.
Exact computation of ⇢s (X) requires multicriterial
optimisation algorithms.
Aim: provide bounds for the selection risk measure.
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Upper bound
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Consider any selections ⇠1 , . . . , ⇠N of X.
Determine r(⇠1 ), . . . , r(⇠N ).
Take the convex hull of the union of the
corresponding upper quadrants.
This corresponds to a higher risk (subset of ⇢s (X)).
r(⇠ 0 )
r(⇠ 00 )
r(⇠ 000 )
28
Lower bound: univariate example
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A univariate coherent Lp -risk measure r can be
represented as
E( ⇣⇠)
r (⇠) = sup
,
E⇣
⇣2Z
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where Z ⇢ Lq (R).
An alternative expression would be
\ E([ ⇠, 1)⇣)
\ E(X̌⇣)
r (⇠) =
=
,
E⇣
E⇣
⇣2Z
⇣2Z
where X = ( 1, ⇠].
29
Lower bound
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Intersection of half-spaces:
⇢(X + K)
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Lower bound
⇢s (X) ⇢
\
Z 2Z,u2Rd+
{x : Ehx, uZ i
EhX (uZ )} ,
where
I Z is a family of Z = (⇣1 , . . . , ⇣d ) 2 Lq (Rd ) such that ⇣i
belongs to a set Zi providing the dual representation
for ri , i = 1, . . . , d.
I hX (·) is the support function of X.
31
Normal distribution X = X + K
−4
1.0
−2
1.5
0
2.0
2
2.5
4
6
3.0
X is standard normal;
K has ( 5, 1) and (1, 5) on its boundary
−4
−2
0
2
4
6
1.0
1.5
2.0
2.5
3.0
32
Dual representation: univariate
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r is a coherent risk measure that is lower
semicontinuous, i.e. r (⇠)  lim inf r (⇠n ) if ⇠n ! ⇠ in Lp ,
p 2 [1, 1).
For p = 1 one requires the weak star convergence
⇠n ! ⇠, i.e. ⇠n ! ⇠ in probability and |⇠n |  1 a.s. for
all n.
Then
E( XZ )
r (⇠) = sup
EZ
Z 2Z
where Z is a family of random variables in Lq .
33
Dual representation
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Replace random set X with its support function
hX (u) = sup{hu, xi : x 2 X} .
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Sums of sets turn into sums of support functions.
However, hX (u) may be infinite (even with probability
one for all u), e.g. if X is a half-space.
Idea: consider hX (Y ), where Y is a selection of the
random set (on the unit sphere) being the efficient
domain of the support function of X.
34
Support functions
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G = {u : hX (u) < 1} is a random cone.
G1 = {u 2 G : kuk = 1} is a random subset of the
unit sphere.
Consider random sets such that hX is p-integrable
and a.s. Lipschitz on G1 with the p-integrable
Lipschitz constant kXkLip .
Then we say that X (or hX ) belongs to the space
Lipp (G1 ).
Theorem
This is the case if and only if ⇣ = supu2G1 |hX (u)| is
p-integrable.
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For X = X + K the condition amounts to the
p-integrability of X and G is the dual cone to K.
35
Lipschitz space built from support functions
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Consider the space Lipp (G1 ) of functions
hX (⌘) : (L0 (G1 ), k · k1 ) 7! Lp (R)
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These maps are Lipschitz, since
E|hX (⌘)
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hX (⌘ 0 )|p  EkX kpLip k⌘
⌘ 0 k1 .
Linear functionals on Lipp (G1 ) are given by
X 7! E
n
X
hX (Zi )
i=1
where Z1 , . . . , Zn 2 Lq (G).
36
Linear functionals
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While the weak duals to Lipschitz spaces are not yet
known, the weak-star convergence hXn to hX is well
understood (Johnson, 1970):
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pointwise weak-star convergence hXn (⌘) ! hX (⌘) in
Lp (R);
uniform boundedness of hXn .
Theorem
Xn weak-star converges to X if and only if the Hausdorff
distance ⇢H (Xn , X) ! 0 in Lp if p 2 [1, 1) and in
probability together with Xn ⇢ M + G0 for a deterministic
set M and all n for p = 1.
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In case Xn = Xn + K for a deterministic exchange
cone K corresponds to the usual weak-star
convergence Xn ! X .
37
Dual representation
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Fatou property:
If Xn weak-star converges to X, then
lim sup ⇢s,0 (Xn ) ⇢ ⇢s,0 (X) .
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Equivalent to the weak-star closedness of the family
{X 2 Lipp (G1 ) : ⇢s,0 (X) 3 0}.
Theorem
If ⇢s has the Fatou property, then
\
⇢s (X) =
{x : Ehx, uZ i
EhX (uZ )}
Z 2Z,u2Rd+
for a certain family Z ⇢ Lq (Rd ).
38
Classical Fatou lemma for selection
expectation
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Define EI X = {E⇠ : ⇠ 2 L1 (X)} (the Aumann integral,
i.e. the expectation without closure).
Aumann (1965), Schmeidler (1970), etc.
If supn kXn k is integrable (random sets in Rd are
uniformly integrably bounded), then
lim sup EI (Xn ) ⇢ EI lim sup Xn .
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Implies the closedness of EI X if we take Xn = X.
Generalisations for unbounded random sets are
complicated and restrictive (Balder and Hess, 1995).
39
Fatou property for selection risk measures
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Fatou property:
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If Xn weak-star converges to X, then
lim sup ⇢s,0 (Xn ) ⇢ ⇢s,0 (X) .
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holds
in the Lp -case with p 2 [1, 1);
in the L1 -case if X is quasi-bounded;
in the L1 -case if X = X + K is for deterministic cone
K.
In all these cases ⇢s,0 (X) is closed.
40
Financial networks
Holding
Subsidiary I
Subsidiary II
Reinsurance
Life Non−life
Investment Junk
I
IntraGroup Transfers (in order to help distressed
members of the group)
41
Allianz Group
Allianz Group Structure
H6 – Insurance USA
H7 – Insurance
German Speaking
Countries
H8 – Insurance
Growth Markets
H9 – Asset
Management
Worldwide
H10 – Global
Insurance Lines
& Anglo Markets
H11 – Insurance
Iberia &
Latin America
Allianz Life
Insurance Company
of North America
United States,
Minneapolis (MN)
Allianz
Versicherungs-AG
Germany, Munich
Allianz
Hungária
Biztosító Zrt.
Hungary, Budapest
Allianz
Global Investors
Europe GmbH
Germany,
Frankfurt am Main
Allianz
Insurance plc
United Kingdom,
Guildford
Allianz compañía
de Seguros y
Reaseguros S.A.
Spain, Barcelona
Allianz
Lebensversicherungs-AG
Germany. Stuttgart
AllianzSlovenská
poist'ovna a.s.
Slovakia, Bratislava
PIMCO
Deutschland GmbH
Germany, Munich
Allianz p.l.c.
Ireland, Dublin
Allianz
Popular S.L.
Spain, Madrid
Allianz
Private Krankenversicherungs-AG
Germany, Munich
Allianz
pojistovna a.s.
Czech Republic,
Prague
Allianz
Australia Limited
Australia, Sydney
Companhia de
Seguros Allianz
Portugal S.A.
Portugal, Lisbon
Allianz
Hayat ve
Emeklilik A.S.
Turkey, Istanbul
Allianz
Beratungsund Vertriebs-AG
Germany, Munich
TU Allianz
Polska S.A.
Poland, Warsaw
Allianz
Global Corporate
& Specialty SE
Germany, Munich
Allianz
Latin America
(Brazil, Argentina
Colombia, Mexico)
Allianz
Sigorta A.S.
Turkey, Istanbul
Allianz
Elementar
Versicherungs-AG
Austria, Vienna
OJSC Insurance
Company Allianz
Russia, Moscow
Yapi Kredi
Sigorta A.S.
Turkey, Istanbul
Allianz
Elementar Lebensversicherungs- AG
Austria, Vienna
Allianz
Other CEE
(Croatia, Bulgaria,
Romania)
Allianz
Yasam ve
Emeklilik A.S.
Turkey, Istanbul
Allianz Suisse
VersicherungsGesellschaft AG
Switzerland, Zurich
Allianz
Life Insurance
Co. Ltd.
Korea, Seoul
Allianz Suisse
LebensversicherungsGesellschaft AG
Switzerland, Zurich
Allianz
Taiwan Life
Insurance Co. Ltd.
Taiwan, Taipei
Oldenburgische
Landesbank AG
Germany, Oldenburg
Allianz Malaysia
Berhad p.l.c.
Malaysia,
Kuala Lumpur
H5 – Insurance Western Europe &
Southern Europe
H4 – Operations
Allianz SE
Allianz
Global Assistance
S.A.S.1
France, Paris
Allianz
S.p.A.
Italy, Trieste
Allianz
Vie S.A.
France, Paris
Allianz
Worldwide Care Ltd.1
Ireland, Dublin
Allianz
Belgium S.A.
Belgium, Brussels
Allianz
IARD S.A.
France, Paris
Allianz
Nederland Groep N.V.
Netherlands,
Rotterdam
Allianz
Hellas Insurance
Company S.A.
Greece, Athens
Fireman‘s Fund
Insurance Company
Corp.
United States,
Novato (CA)
The functional divisions
H1 – Chairman of the Board
H2 – Finance, Controlling, Risk
Allianz
MENA
(Egypt, Lebanon)
H3 – Investments
do not have major operating entities in their
area of responsibilities and are therefore not
shown in the overview.
Allianz Other
Asia-Pacific
(Indonesia, Thailand)
Allianz
Global Investors
France S.A.
France, Paris
Allianz
Global Investors
Asia Pacific Group
Pacific Investment
Management
Company LLC
United States,
Dover (DE)
Allianz Asset Management of America
United States,
Dover (DE)
Allianz Global Risks
US Insurance
Company Corp.
United States,
Burbank (CA)
Euler Hermes S.A.
France, Paris
© Allianz SE 2014
This overview is simplified. It focuses on major operating entities and does not contain all entities
of Allianz Group. It does not show whether a shareholding is direct or indirect. Indications show
status as of December 31, 2013.
1) Starting January 1, 2014: AWP Group, including Allianz Global Assistance and Allianz Worldwide Care
42
Stand-alone regulation
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Consider each agent separetely and assess his
liabilities and gains (balance sheets).
Suggest a probability model for his terminal position
Ci .
Calculate the risk r (Ci ) using the chosen risk
measure.
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In the EU mostly used Value-at-Risk: it is not
subadditive and is not sensitive for high losses.
In Switzerland: Expected Shortfall (more conservative
and secure).
Request each agent to reserve capital r (Ci ) (or allow
to release capital if r (Ci ) is negative).
43
Systemic risk
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In a network possible default of an agent increases
the pressure on the whole network.
Possible influences of losses between the agents
should be taken into account.
This leads to an increase of the required capital
reserves.
44
Bulding groups
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If the agents form a group, some may voluntarily (or
may be forced by the holding) compensate for losses
of other group members.
This may and should lead to a decrease of capital
reserves.
Holding
Subsidiary I
Subsidiary II
Reinsurance
Life Non−life
Investment Junk
45
Example: Financial network, two agents
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Two agents A and B are assessed by a regulator.
The regulator evaluates their individual exposure to
risks and requests that they set aside (and freeze)
necessary capital reserves.
The agents want to minimise these reserves and
conclude an agreement that at the terminal time (and
under certain conditions) one of them would help to
offset the deficit of another one.
The family of allowed transactions is a random closed
set in the plane.
Despite the network is not geometric, it is described
by a geometric object.
46
Assumptions
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d agents operate with the same currency.
Their terminal positions are C1 , . . . , Cd (assume
d = 2).
All risks are assessed using the same coherent risk
measure r (say Expected Shortfall).
47
Unrestricted transfers
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The regulator allows all transfers.
Moving from C = (C1 , . . . , Cd ) to
C + ⌘ = (C1 + ⌘1 , . . . , Cd + ⌘d )
with
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⌘1 + · · · + ⌘d  0.
The resulting attainable positions build a half-space.
C = (C1 , C2 )
X
C+⌘
48
Theorem: Jouini, Schachermayer, Touzi,
Filipovic, Kupper
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The total required capital for all agents is
r (C1 + · · · + Cd )
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It is less than the capital in the stand-alone approach
X
r (C1 + · · · + Cd ) 
r (Ci )
There exist “best” transfers:
r (C1 + · · · + Cd ) =
⌘:
inf
P
⌘i 0
X
r (Ci + ⌘i ),
so the infimum is attained at ⌘ = ⌘ ⇤ , the transfers do
not worsen the risk assessment of each of agent:
r (Ci + ⌘i⇤ )  r (Ci ),
P
and Ci + ⌘i = f ( Ci ) are all monotonic functions of
the total position.
49
Unrestricted transfers
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The result can be alternatively derived using the
closedness of the risk measure of the random set X.
C = (C1 , C2 )
X
50
Imposing restrictions
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The optimal transfers may lead to a bankruptcy of
some agents.
The capital available for transfers normally is not
available as cash: the agents have to take a loan
using their assets as a deposit.
The interest on such loan depends on the balance
sheet of the agent (fungibility).
51
Example: no transfers that lead to bankruptcy
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The terminal capital is (C1 , C2 ).
Transfers from a solvent company to another one are
allowed up to the available positive capital.
No fungibility difficulties.
Disposal of assets is allowed (e.g. as dividends to
shareholders).
(C1 , C2 )
X
(C1 , C2 )
X
52
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X = X(C) is a random set of attainable positions for
group members.
Determine
X
inf{
ai : 0 2 ⇢s (X(C + a))}
This is the minimal total capital that should be
reserved in order to make X(C + a) acceptable as a
random set.
Theorem
The infimum is achieved.
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How to ensure that none of the agents needs to
reserve more than it would do in case of a
stand-alone approach?
In which case the optimal transfers are realised as a
monotonic function of X?
53
Summary
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In many applications one deals with non-stationary
random sets.
Compact random sets or possibly unbounded
random sets or sometimes even non-convex
(indivisible assets), possibly high-dimensional.
Selections and support functions form the central tool
to solve the corresponding problems.
Models, inference tools, and efficient computation
algorithms are needed.
54
References
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A. Beresteanu, I. Molchanov, and F. Molinari. Sharp
identification regions in models with convex moment
predictions. Econometrica, 79:1785–1821, 2011.
I. Molchanov and I. Cascos. Multivariate risk
measures: a constructive approach based on
selections. Math. Finance, 2015.
I. Molchanov and M. Schmutz. Multivariate
extensions of put-call symmetry. SIAM J. Financial
Math., 1:396–426, 2010.
Works in progress ...
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