Capital Requirements, Risk Measures and Optimal Capital ... · Capital Requirements, Risk Measures...

CapitalRequirements,Risk Measuresand Optimal

CapitalInjections

Speaker:Walter Farkas

Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation

Capitalrequirementswith a singleeligible asset

Effectiveness

Robustness

Efficiency

Capitalrequirementswith severaleligible assets

Acceptabilityarbitrage

Portfoliosbounded awayfrom zero

Examples

References

Capital Requirements, Risk Measures andOptimal Capital Injections

Walter FarkasUniversity of Zurich and ETH Zurich

Pablo Koch-MedinaSwiss Reinsurance Company

Cosimo-Andrea MunariETH Zurich

Birmingham, 25. January 2012


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

1 Preliminaries

2 IntroductionAcceptance setsEligible assetsRequired capital and risk measuresMotivation

3 Capital requirements: the case of a single eligible assetWhen is required capital well defined for all financial positions?When is required capital a continuous function of financial positions?Does an optimal eligible asset exist?

4 Capital requirements: the case of several eligible assetsAbsence of acceptability arbitragePortfolios bounded away from zeroAbsence of acceptability arbitrage for particular acceptance sets

5 References


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Financial institutions and capital adequacy

• Liability holders of a financial institution are credit sensitive: they, andregulators on their behalf, are concerned that the institution may fail tohonor its future obligations.

• This will be the case if the institution’s financial position, i.e. the value ofits assets less the value of its liabilities, becomes negative in some futurestate of the world.

• To address this concern financial institutions hold risk capital, which ismeant to absorb unexpected losses thereby reducing the likelihood thatthey may become insolvent.

• A key question is how much capital a financial institution should berequired to hold to be deemed adequately capitalized by the regulator.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Capital adequacy and acceptance sets

• The previous question can be usefully framed from a mathematical point ofview using the concept of an acceptance set.

• Let Ω be a given sample space representing the set of all possible futurestates of the world.

• Let X be the set of all possible net financial positions X : Ω→ R of a givenfinancial institution, i.e. X = A− L, where the random variables A : Ω→ Rand L : Ω→ R represent the value of the assets and liabilities respectively.

• An acceptance set A is a subset of X whose elements represent allfinancial positions that are deemed to provide a reasonable security to theliability holders.

• Hence, testing whether a financial institution is adequately capitalized ornot reduces to establishing whether its financial position belongs to A ornot.

• Coherent acceptance sets were introduced in 1999 in the seminal paper byArtzner, Delbaen, Eber and Heath for finite sample spaces and in Delbaen(2000) for general probability spaces.

• This setting was extended to convex acceptance sets in Follmer and Schied(2002) and Frittelli and Rosazza Gianin (2002).


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Risk measures and capital requirements

• A fundamental question to ask is: can certain actions by the managementof a financial institution turn an unacceptable financial position into anacceptable one and at which cost?

• The standard theory of risk measures was designed to answer this questionfor a very particular case: can a badly capitalized financial institution bere-capitalized to acceptable levels by raising additional capital and investingit in a pre-specified risk-free traded asset?

• More generally, an investment in a class of fixed tradable assets leads to arisk measure with respect to these assets: it represents the minimumamount of capital, the so-called required capital, that needs to be injectedin order to make a position acceptable.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Effectiveness, robustness, efficiency

In our papers we discuss the dependency of required capital on the choice of theacceptance set and the class of tradable assets, and we study the main propertiesof the corresponding risk measures.

Our three driving questions are therefore the following:

1 is required capital a well-defined number for any financial position?

This is related to the effectiveness of the choice of the eligible assets, i.e.to the ability to make any unacceptable position acceptable.

2 is required capital a continuous function of financial positions?

This is related to the robustness of the required capital figure, i.e. to thereliability of required capital in a world of estimates.

3 can the eligible assets be chosen in such a way that for every financialposition the corresponding required capital is lower than if any other classhad been chosen?

This is related to the efficiency of the choice of the eligible assets, i.e. tothe ability to reach acceptance with the least possible amount of capital.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

References

The main references can be downloaded at these links:

• http://ssrn.com/abstract=1966645

• http://ssrn.com/abstract=1989077


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

1 Preliminaries




5 References


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Financial positionsWe will set our discussion in the following framework. We consider:

• a single-period economy with dates t = 0 and t = 1;• a sample space Ω (not necessarily finite) endowed with a σ-algebra F ,

representing uncertainty at time t = 1;• the Banach space X of bounded random variables of the form X : Ω→ R

equipped with the supremum norm

‖X‖ := supω∈Ω|X (ω)| ;

• the partial order ≤ on X such that X ≤ Y whenever X (ω) ≤ Y (ω) for allω ∈ Ω;

• the corresponding positive cone X + := X ∈ X ; X ≥ 0, whose interioris non-empty and consists of those positions X that are bounded away fromzero, i.e. such that for some ε > 0 we have X ≥ ε1Ω, with 1Ω(ω) = 1 forany ω ∈ Ω.

Assumption

• The net financial position of a financial institution at time t = 1 will berepresented as a random variable in the space X .

• We will require a probabilistic model on (Ω,F) only for some specificexamples. In this sense, most of the next results will be “model free”.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Acceptable financial positions

DefinitionA set A ⊂ X is called an acceptance set whenever the following two conditionsare satisfied:

(A1) A is a non-empty, proper subset of X (non-triviality);

(A2) if X ∈ A and Y ≥ X then Y ∈ A (monotonicity).

RemarkAn acceptance set encapsulates the minimum requirements we would place on aconcept of non-trivial capital adequacy test:

• some – but not all – positions should be acceptable;

• any financial position dominating an already accepted position should alsobe acceptable.

DefinitionA financial institution with net financial position X ∈ X will be said to beadequately capitalized with respect to A if X ∈ A .


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Coherent acceptance sets

Introduced by Artzner, Delbaen, Eber and Heath (1999), coherent acceptancesets A ⊂ X are convex cones. As such for any X , Y ∈ A and any λ ∈ [0, 1]

λX + (1− λ)Y ∈ A (convexity) ,

and for any X ∈ A and any λ ≥ 0

λX ∈ A (cone) .

The most known and used example of coherent acceptance set is the acceptanceset based on Expected Shortfall at level α ∈ (0, 1)

A α :=

X ∈ X ; ESα(X ) :=

1

α

∫ α

0VaRβ(X )dβ ≤ 0

where VaRα(X ) is the Value-at-Risk at level α of the position X ∈ X defined by

VaRα(X ) := infm ∈ R ; P(X + m < 0) ≤ α .

Here P is a given probability measure on the measurable space (Ω,F).


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Convex acceptance sets

Convex acceptance sets were introduced by Follmer and Schied (2002) andFrittelli and Rosazza Gianin (2002). Being convex, any coherent acceptance set isalso a convex acceptance set, while the converse is not true.

Genuine convex acceptance sets are for example the ones associated with anon-empty collection Q of finitely additive set functions

Q : F → [0, 1]

with Q(Ω) = 1 and with a floor

γ : Q → R

such that supQ∈Q γ(Q) <∞, and having the form

A (Q, γ) :=⋂

Q∈Q

X ∈ X ; EQ[X ] ≥ γ(Q) .

Note that the previous condition supQ∈Q γ(Q) <∞ guarantees that the setA (Q, γ) is not empty.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Value-at-Risk based acceptance sets

The most widely used acceptance set is the one based on Value-at-Risk at levelα ∈ (0, 1) and it is defined by

Aα := X ∈ X ; VaRα(X ) ≤ 0= X ∈ X ; P(X < 0) ≤ α ,

where P is a given probability measure on the measurable space (Ω,F) and asabove

VaRα(X ) := infm ∈ R ; P(X + m < 0) ≤ α .

It is well known that Aα is not convex, henceforth not coherent.

Remark

• In the sequel we will assume neither coherence nor convexity, making theresults as general as possible.

• In particular all the following results will apply to Value-at-Risk acceptance.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

The class of eligible assets

We allow to modify the acceptability of financial positions by investing in a classof assets S made of N fixed tradable eligible assets S1, · · · ,SN with

• initial price S j0 > 0,

• final payoff S j1 ∈ X such that S j

1(ω) > 0 for all ω ∈ Ω.

The corresponding marketed space, i.e. the set of all possible portfolios made ofsuch assets, is given by

M (S ) := span( S11 , · · · , SN

1 ) =

N∑

j=1

λj Sj1 ; λ1, · · · , λN ∈ R

.

Assumption

• There exists a linear pricing functional π : M (S )→ R and we set

Mm(S ) := Z ∈M (S ) ; π(Z) = m ;

• Absence of arbitrage: M0(S ) ∩ (X + \ 0) = ∅.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

From unacceptable to acceptable

DefinitionWe define for an arbitrary set A ⊂ X the function ρA ,S : X → [−∞,∞] bysetting for any X ∈ X

ρA ,S (X ) := inf π(Z) ∈ R ; Z ∈M (S ) : X + Z ∈ A .

In case S = S we will simply write ρA ,S and we have

ρA ,S (X ) = inf

m ∈ R ; X +

m

S0S1 ∈ A

.

Remark

• The quantity ρA ,S (X ) represents, if finite, the least amount of capital tobe invested in the class S that is able to guarantee the acceptability of X .

• If positive, the quantity ρA ,S (X ) can thus be regarded as a capitalrequirement: it defines the capital required to turn the position X into anacceptable position, or the cost of making X acceptable.

• If negative, ρA ,S (X ) represents the amount of capital that can beextracted from a well-capitalized financial institution without compromisingthe acceptability of its financial position X .


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Capital requirements and risk measures

The capital requirement ρA ,S satisfies the two defining properties of a riskmeasure.

PropositionFix A ⊂ X . Then

(i) ρA ,S is translation invariant with respect to S , i.e. for all X ∈ X andZ ∈M (S )

ρA ,S (X + Z) = ρA ,S (X )− π(Z) .

In particular if S = S then ρA ,S is translation invariant with respect toS, i.e. for all X ∈ X and m ∈ R

ρA ,S (X + mS1) = ρA ,S (X )−mS0 .

(ii) If A satisfies the monotonicity axiom (A2) then ρA ,S is monotone, i.e. forany X , Y ∈ X such that X ≤ Y

ρA ,S (X ) ≥ ρA ,S (Y ) .


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Comparison with existing literature (1)

Our context is more general than the standard setting for risk measures as it ispresented in

- Artzner, Delbaen, Eber and Heath (1999),- Follmer and Schied (2002),- Frittelli and Rosazza Gianin (2002).

• We work with general acceptance sets. We require neither coherence norconvexity.

• We allow for investments in a class of assets rather than one single asset.

• Even in case of a single eligible asset S , we don’t require that it is risk-free,i.e. S1 = 1Ω.

• As a result we deal with a more general form of translation invariance thansimple cash-additivity.

• The function ρA ,S may consequently take both the value ∞ and −∞.

• Even if ρA ,S is finitely valued, its continuity is not ensured any more, asin Follmer and Schied (2011).


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References


Frittelli and Scandolo (2006) introduce capital requirements with respect to aclass of assets (see Definition 3.1 and 3.4) and extend the treatment to amulti-period economy. Yet for their main results they always assume the standardone-asset cash-additive framework (from Proposition 4.5 on).

Recent papers study risk measures defined on more general spaces than the spaceX of bounded F-measurable random variables of the form X : Ω→ R, but theystill deal with standard cash-additive risk measures which are assumed to befinitely valued, or to take at most the value ∞.

• Lp spaces: Kaina and Ruschendorf (2009)

• Morse spaces: Cheridito and Li (2009)

• Orlicz spaces: Biagini and Frittelli (2008), Arai (2010)

• Frechet lattices: Biagini and Frittelli (2009)


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References


Some very recent papers deal indeed with risk measures that are invariant withrespect to a single pre-specified asset (whose payoff is often called numeraire).Even if they don’t require the numeraire to be the risk-free asset, they explicitlyor implicitly assume more than just asking that it is everywhere positive.

• Filipovic and Kupper (2007) deal with ordered normed spaces and areinterested in inf-convolutions of convex risk measures. In Lemma 3.5 theyassume a condition that in our setting is equivalent to boundedness awayfrom zero of S1.

• Konstantinides and Kountzakis (2011) work on ordered reflexive spaceswith non-empty cone interior, and assume that the numeraire belongs tothe interior of the cone generating the partial order: in our setting, this isequivalent to the boundedness away from zero of S1.

• Kountzakis (2011) extend the previous results on non-reflexive orderednormed spaces with non-empty cone interior, but again asks the numeraireto be bounded away from zero or more generally to be an everywherepositive quasi-interior point.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Motivation

This work has been mainly motivated by the following issues.

• The intent of re-emphasizing (as done in Artzner, Delbaen, Eber and Heath(1999)) the primacy of acceptance sets with respect to risk measures in acapital adequacy environment: the raison d’etre of a risk measure is tomeasure how far a position is from acceptability using a particular eligibleasset as a sort of yardstick, which is meaningful only if we already knowwhat acceptability is supposed to mean.

• The intent of dealing with general (non-convex) acceptance sets, exploitingmonotonicity as the only essential property of acceptability.

• The intent of allowing for general forms of investments in order to establishcapital requirements.

• The crucial question about the existence of optimal investments leading tolower capital requirements than any other investment.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

1 Preliminaries




5 References


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Capital requirements with respect to a single

eligible asset

We recall the definition of a capital requirement with respect to a single eligibleasset and we list its main properties.

PropositionFix an acceptance set A ⊂ X and an eligible asset S with price S0 > 0 andpayoff S1 ∈ X such that S1(ω) > 0 for any ω ∈ Ω. We set

ρA ,S (X ) = inf

m ∈ R ; X +

m

S0S1 ∈ A

.

The basic properties of ρA ,S : X → [−∞,∞] are the following:

(i) ρA ,S cannot be identically equal to ∞ or −∞ ;

(ii) ρA ,S is translation invariant with respect to S, i.e. for all X ∈ X andm ∈ R we have ρA ,S (X + mS1) = ρA ,S (X )−mS0 ;

(iii) ρA ,S is monotone, i.e. for any X , Y ∈ X such that X ≤ Y we haveρA ,S (X ) ≥ ρA ,S (Y ) .


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Risk measures and capital requirements

We say that a function ρ : X → R is a risk measure with respect to a giveneligible asset S if it is translation invariant with respect to S , i.e. if for anyX ∈ X and m ∈ R

ρ(X + mS1) = ρ(X )−mS0 ,

and if it is monotone, i.e. if for any X , Y ∈ X such that X ≤ Y

ρ(X ) ≥ ρ(Y ) .

PropositionLet ρ be a risk measure with respect to a given eligible asset S. Then

(i) Aρ := X ∈ X ; ρ(X ) ≤ 0 is an acceptance set;

(ii) ρ = ρAρ,S .

RemarkAny risk measure of this kind can be therefore expressed as a capital requirementρA ,S for a suitable choice of the acceptance set A .


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Convex and coherent acceptance sets

If we require that the acceptance set A is convex or coherent we derive peculiarproperties for ρA ,S .

Proposition

(i) If A is convex and ρA ,S : X → (−∞,∞] then ρA ,S is convex, i.e. forX , Y ∈ X and λ ∈ [0, 1] we have

ρA ,S (λX + (1− λ)Y ) ≤ λρA ,S (X ) + (1− λ)ρA ,S (Y ) .

(ii) If A is a cone (in particular if A is coherent) and ρA ,S (0) = 0 then ρA ,S

is positively homogeneous, i.e. for all X ∈ X and λ ≥ 0

ρA ,S (λX ) = λρA ,S (X ) .

(iii) If A is coherent, ρA ,S : X → (−∞,∞] and ρA ,S (0) = 0, then ρA ,S issubadditive, i.e. for X , Y ∈ X we have

ρA ,S (X + Y ) ≤ ρA ,S (X ) + ρA ,S (Y ) .


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Eligible asset and numeraire asset

The eligible asset is by definition the only asset which is eligible to modify theacceptability properties of a financial position.

It is yet standard procedure to choose the eligible asset S as the numeraire, i.e.as the unit of account.

• In this case one does not consider financial positions X ∈ X but rathertheir discounted versions X

S1.

In this setting the translation invariance formula takes the simpler form

ρA ,S (X + m) = ρA ,S (X )−m .


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References


Even though we might find it occasionally convenient to use the eligible asset asthe numeraire, we will refrain from adopting this convention for three mainreasons:

1 because converting the question of capital adequacy into a problem aboutdiscounted values may lead to lose sight on the role of the eligible asset;

2 because it hides the critical dependence on the chosen eligible asset andinvesting in the eligible asset is just one of the means to make anunacceptable position acceptable. It is useful to remain aware at all timesthat the eligible asset is indeed a choice;

3 because a theory based on discounted assets on the space of boundedF-measurable functions implicitly assumes that the payoff S1 of the eligibleasset is bounded away from zero, and this assumption critically limits thechoice of the eligible asset itself.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

When is required capital well defined for all

financial positions?

Requiring that ρA ,S is real valued is economically reasonable. Indeed fix aposition X ∈ X .

• If ρA ,S (X ) =∞ then X cannot be made acceptable by investing anyamount of capital in the eligible asset, suggesting that we should choose adifferent eligible asset.

• If ρA ,S (X ) = −∞, we can extract arbitrary amounts of capital retainingthe acceptability of X , suggesting that the particular acceptance set mightbe too large and therefore might also accept positions leading to hugelosses.

DefinitionWe will say that a given eligible asset S is well behaved with respect to a fixedacceptance set A if the required capital ρA ,S (X ) is finite for any positionX ∈ X .


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Effectiveness

PropositionThe function ρA ,S assigns a finite value to a position X ∈ X if and only if thereexists m0 ∈ R such that

X +m

S0S1 /∈ A if m < m0 and X +

m

S0S1 ∈ A if m > m0 .

If such an m0 exists, then m0 = ρA ,S (X ).

In general we can find examples of A and S that allow for non-finite requiredcapital.

ExampleLet Ω be infinite and assume that the payoff S1 is not bounded away from zero.

1 If we set A := X + then ρA ,S (−1Ω) =∞.

2 If we set A := X ∈ X ; ∃ω0 ∈ Ω : X (ω0) > 0 then ρA ,S (1Ω) = −∞.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

A sufficient condition for effectiveness

We present a sufficient condition on the payoff S1 of the eligible asset so that Sturns out to be well behaved with respect to any acceptance set.

PropositionLet A ⊂ X be an arbitrary acceptance set and assume that the payoff S1 of theeligible asset is bounded away from zero. Then ρA ,S (X ) is finite for any positionX ∈ X .

Remark

• Note that the payoff S1 := 1Ω of the risk-free asset is bounded away fromzero. Hence, in this case the function ρA ,S is real valued for anyacceptance set A .

• Note that in case Ω is finite, S1 is bounded away from zero since S1(ω) > 0for all ω ∈ Ω. It follows that for finite Ω the function ρA ,S is real valuedfor any choice of A and S. The same happens if, more generally, theσ-algebra F is finite.


CapitalInjections


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Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

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Efficiency




Examples

References

Effectiveness: VaR and ES case

In general the previous condition is not necessary for the finiteness of ρA ,S .

ExampleFix α ∈ (0, 1) and a probability P on (Ω,F).

1 Any eligible asset S is well behaved with respect to the Value-at-Risk basedacceptance set

Aα = X ∈ X ; P(X < 0) ≤ α .

2 Any eligible asset S is well behaved with respect to the Expected Shortfallbased acceptance set

A α = X ∈ X ; ESα(X ) =1

α

∫ α

0VaRβ(X )dβ ≤ 0 .

Yet sometimes the only way to have finiteness for ρA ,S is that S1 is boundedaway from zero.

ExampleAn eligible asset S is well behaved with respect to the acceptance set X + if andonly if its payoff S1 is bounded away from zero.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

When is required capital a continuous function

of financial positions?

Differently from the familiar case of cash-additive risk measures, the capitalrequirement ρA ,S may be non continuous even if it is finite for any position in X .

ExampleLet Ω be infinite and fix a probability P on (Ω,F). Consider an eligible asset withpayoff S1 such that P(S1 < ε) > 0 for all ε > 0. Then we can find α ∈ (0, 1) sothat the Value-at-Risk based required capital ρAα,S : X → R is not continuous.

Indeed ρAα,S fails to satisfy the conditions given by the following theorem.

TheoremAssume that the function ρA ,S is real valued. Then ρA ,S is continuous if andonly if any of the following statements is true:

(i) int(A ) = X ∈ X ; ρA ,S (X ) < 0 and A = X ∈ X ; ρA ,S (X ) ≤ 0.

(ii) ρint(A ),S = ρA ,S = ρA ,S .

(iii) ρA ,S is continuous from above and from below.


CapitalInjections


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Introduction

Acceptance sets

Eligible assets

Required capital

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Effectiveness

Robustness

Efficiency




Examples

References

Lipschitz continuity

PropositionLet A ⊂ X be an arbitrary acceptance set and assume that the payoff S1 of theeligible asset is bounded away from zero. Then ρA ,S is everywhere finite andLipschitz continuous, i.e. there exists L > 0 such that for all X , Y ∈ X∣∣ρA ,S (X )− ρA ,S (Y )

∣∣ ≤ L ‖X − Y ‖ .

RemarkAgain the proposition applies in case S is the risk-free asset, Ω is finite or theσ-algebra F is finite.

The previous condition is sometimes but not always necessary.

Examples

1 Fix A := X +. Then ρA ,S is everywhere finite and continuous if and onlyif S1 is bounded away from zero.

2 Fix λ ∈ R and a probability P on (Ω,F), and consider the acceptance setA := X ∈ X ; EP[X ] ≥ λ. Then ρA ,S is everywhere finite andcontinuous for any choice of the eligible asset S.


CapitalInjections


Preliminaries

Introduction

Acceptance sets

Eligible assets

Required capital

Motivation


Effectiveness

Robustness

Efficiency




Examples

References

Continuity in case of convex acceptance sets

The class of convex acceptance sets is special: if A is a convex acceptance setthen any effective required capital ρA ,S is also robust.

PropositionLet A ⊂ X be a convex acceptance set and S an arbitrary eligible asset. Setdom(ρA ,S ) := X ∈ X ; ρA ,S (X ) ∈ R.

(i) ρA ,S is Lipschitz continuous on a convenient neighborhood of each positionX ∈ int(dom(ρA ,S )).

(ii) If in particular ρA ,S is real valued, then ρA ,S is locally Lipschitzcontinuous, i.e. it is Lipschitz continuous on some neighborhood of eachposition X ∈ X .

Note that if A is a convex acceptance set and S is a generic eligible asset, thenρA ,S may be not effective, i.e. may be not everywhere finite, as already shown.

ExampleConsider the convex (indeed coherent) acceptance set A := X +. If the payoff S1

is not bounded away from zero, then ρA ,S (−1Ω) =∞.


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Continuity in case of coherent acceptance sets

In case of coherent acceptance sets we have a stronger result about robustness.

PropositionIf A is coherent and ρA ,S is real valued, then ρA ,S is Lipschitz continuous.

As a remarkable corollary, the capital requirement based on Expected Shortfall isa Lipschitz-continuous function of financial positions for any choice of the eligibleasset.

CorollaryConsider the coherent acceptance set based on Expected Shortfall at levelα ∈ (0, 1)

A α = X ∈ X ; ESα(X ) =1

α

∫ α

0VaRβ(X )dβ ≤ 0 .

Let S be an arbitrary eligible asset. Then ρAα,S is everywhere finite, hence it isLipschitz continuous.


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Lower semicontinuity

We now focus on a weakened form of continuity, i.e. lower semicontinuity, whichwill be important while investigating the existence of an optimal eligible asset.

We recall that a function f : X → [−∞,∞] is said to be lower semicontinuous ifthe set X ∈ X ; f (X ) ≤ λ is closed for any λ ∈ R.

PropositionLet A ⊂ X be an acceptance set and S an eligible asset. The followingstatements are equivalent.

(i) ρA ,S is lower semicontinuous.

(ii) A = X ∈ X ; ρA ,S (X ) ≤ 0.(iii) ρA ,S = ρA ,S .

(iv) ρA ,S is continuous from above, i.e. if Xn ↓ X then ρ(Xn) ↑ ρ(X ).

ExampleConsider Aα = X ∈ X ; VaRα(X ) ≤ 0 and A α = X ∈ X ; ESα(X ) ≤ 0.Then both ρAα,S and ρAα,S are lower semicontinuous for any choice of theeligible asset S . This follows from the well-known continuity of VaRα and ESα.


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Equality between risk measures

We assume that:

1 A and B are two acceptance sets;

2 S and R are two eligible assets with same price S0 = R0 > 0;

3 S and R are well behaved with respect to A and B, respectively;

4 ρA ,S and ρB,R are both lower semicontinuous.

TheoremLet S = S,R and m = ρA ,S (0).

(i) ρA ,S = ρB,R if and only if A = B and A + M0(S ) = A hold.

(ii) Assume that Mm(S ) ∩A =

mS0

S1

. Then ρA ,S = ρB,R if and only if

A = B and S1 = R1 hold.

RemarkWe note that in case ρA ,S (0) = 0, the condition M0(S ) ∩A = 0 implies thenon-acceptability of fully-leveraged positions in S and R, i.e. portfolios in S andR with zero initial value.


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Does an optimal eligible asset exist?

Given a fixed acceptance set A and an eligible asset S, we ask if it is possiblethat for any position X ∈ X

ρA ,S (X ) ≤ ρA ,R (X ) for any eligible asset R different from S ,

and for at least one position Y ∈ X we indeed have

ρA ,S (Y ) < ρA ,R (Y ) .

We will assume here that:

1 S and R are two eligible assets with same price S0 = R0 > 0;

2 S and R are both well behaved with respect to a fixed acceptance setA ⊂ X ;

3 ρA ,S and ρA ,R are both lower semicontinuous.

ExampleAs mentioned above, if A is the acceptance set based on Value-at-Risk or onExpected Shortfall such assumptions are satisfied by any eligible assets S and Rwith same initial price.


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There is no optimal eligible asset

TheoremUnder the above assumptions, the following statements hold.

(i) If ρA ,S (X ) ≤ ρA ,R (X ) for every X ∈ X then

ρA ,S = ρA ,R .

(ii) Assume that S = S ,R and MρA ,S (0)(S ) ∩A =ρA ,S (0)

S0S1

.

If ρA ,S (X ) ≤ ρA ,R (X ) for every X ∈ X then

S1 = R1 .

Remark

• Given an eligible asset, we cannot find any other eligible asset that will bestrictly more efficient.

• Fix an eligible asset R. If fully-leveraged positions are not acceptable, theredoes not exist any eligible asset different from R which is just more efficientthan R.


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In Filipovic (2008) a different sort of optimality is addressed.The author investigates the impact a change in numeraire has on a fixed convexrisk measure ρ : X → R.A key consideration in this treatment is that a numeraire U induces anacceptance set

A U :=

X ∈ X ; ρ

(X

U

)≤ 0

.

The central result is that no optimal numeraire exists: there does not exist anumeraire U leading to lower capital requirements than those associated to anyother numeraire, i.e. such that A V ⊂ A U for any other numeraire V .

With respect to our treatment, we point out the following remarks.

• First, note again that dealing with discounted positions forces thenumeraire to be bounded away from zero in order to guarantee thatdiscounted positions still belong to X .

• Second, it is unclear how to interpret this non-optimality result from aneconomic perspective: if acceptance is meant to reflect the adequatecapitalization of a financial institution, it should not depend on theparticular numeraire we have chosen to account in.


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1 Preliminaries




5 References


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Capital requirements with respect to several

eligible assets

We focus back on the setting where required capital is determined in the contextof more than one single eligible asset.

We recall that the required capital ρA ,S : X → [−∞,∞] is defined by

ρA ,S (X ) := infπ(Z) ∈ R ; Z ∈M (S ) : X + Z ∈ A = infm ∈ R ; Z ∈Mm(S ) : X + Z ∈ A

where

• A ⊂ X is a fixed acceptance set,

• S = S1, · · · , SN is a given class of eligible assets made of tradable

assets S j with initial price S j0 > 0 and payoff S j

1 ∈ X such that S j1(ω) > 0

for any ω ∈ Ω,

• M (S ) = span( S11 , · · · ,SN

1 ) is the corresponding arbitrage-freemarketed space,

• π : M (S )→ R is the linear pricing functional on M (S ),

• Mm(S ) := Z ∈M (S ) ; π(Z) = m.


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From several eligible assets to a single one

We start by showing that, surprisingly enough, the study of capital requirementswith respect to a class of eligible assets can be reduced to the single-asset case.

TheoremLet A be an arbitrary subset of X . Assume T is a non-empty subset of S . Then

ρA ,S = ρA +M0(S ),T .

In particular for any S ∈ S we have

ρA ,S = ρA +M0(S ),S .

Remark

• By properly enlarging the set A , we can always express the capitalrequirement ρA ,S with respect to the class S in terms of a capitalrequirement with respect to any particular asset S belonging to the class Sitself.

• If A is an acceptance set and A + M0(S ) 6= X , then A + M0(S ) is alsoan acceptance set.


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Absence of acceptability arbitrage

The condition A + M0(S ) 6= X has a clear economic interpretation.

• If it does not hold then any financial position can be made acceptable atzero cost, by hedging with portfolios long and short of eligible assets, asituation that would cast doubts on the effectiveness of the choice of theacceptance set A .

We are therefore lead to set the following assumption.

AssumptionWe will say that the assumption NAA(A ,S ) of absence of acceptability arbitrageholds if A + M0(S ) 6= X .

The following result shows that absence of acceptability arbitrage is a soundassumption: under NAA(A ,S ) there does not exist a price m ∈ R, in particulara negative price m < 0, such that we can make every financial position acceptableat cost m.

PropositionIf A + Mm0 (S ) 6= X for some m0 ∈ R, then A + Mm(S ) 6= X for all m ∈ R.


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Sufficient conditions for NAA(A ,S )

We start by noting that given an acceptance set A ⊂ X we obviously haveA ⊆ A + M0(S ) ⊆ X .

PropositionIf A = A + M0(S ) then NAA(A ,S ) holds. Moreover the following statementsare equivalent:

(i) ρA ,S is everywhere finite;

(ii) there exists S ∈ S such that ρA ,S is everywhere finite.

PropositionAssume that one of the following equivalent statements is fulfilled:

(i) int(A ) ∩Mm(S ) = ∅ for some m ∈ R;

(ii) int(A ) ∩Mm(S ) = ∅ for some m ≤ 0.

Then NAA(A ,S ) holds true.

The second sufficient condition becomes also necessary in case one additionalassumption is made.


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The case of a portfolio bounded away from zero

AssumptionWe will assume that the class S of eligible assets allows for a portfolioZ∗ ∈M (S ) which is bounded away from zero. This happens in particular if oneof the assets belonging to S is itself bounded away from zero, for example it isthe risk-free asset.

RemarkThis assumption is meaningful from an economic point of view.

• Adding one more asset to a given class of eligible assets allows for a lowerrequired capital for each financial position.

• The point is to guarantee that the new required capital is not too low, i.e.equal to −∞.


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Characterization of NAA(A ,S )

The following proposition displays sufficient and necessary conditions so that theassumption NAA(A ,S ) is fulfilled.

PropositionAssume there exists a portfolio Z∗ ∈M (S ) which is bounded away from zero.Then ρA ,S (X ) <∞ for any X ∈ X .

Moreover NAA(A ,S ) is equivalent to any of the following statements:

(i) ρA ,S (X ) > −∞ for all X ∈ X ;

(ii) ρA ,S (X ) > −∞ for some X ∈ X ;

(iii) int(A ) ∩Mm(S ) = ∅ for some m ∈ R ;

(iv) int(A ) ∩Mm(S ) = ∅ for some m ≤ 0 .


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Acceptance sets that are cones

PropositionConsider an acceptance set A ⊂ X that is a cone. Assume that M (S ) containsa portfolio that is bounded away from zero and that all the eligible assets haveprice 1. Then the following conditions are equivalent:

(i) NAA(A ,S ) holds;

(ii) int(A ) ∩M0(S ) = ∅.

CorollaryConsider an acceptance set A ⊂ X that is a cone. Assume that the payoff S1

1 ofthe first eligible asset is bounded away from zero. Then the following statementsare equivalent:


(ii) for any λ2, · · · , λN ∈ R we have

ρA ,S1

N∑j=2

λj (S j1 − S1

1 )

≥ 0 .


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Value-at-Risk and Expected Shortfall

Consider the Value-at-Risk based acceptance set at level α ∈ (0, 1)

Aα = X ∈ X ; VaRα(X ) ≤ 0 .

Assume that S11 = 1Ω and that all the eligible assets have price 1. Then the

following statements are equivalent:

(i) NAA(Aα,S ) holds;

(ii) for any λ2, · · · , λN ∈ R we have

VaRα

N∑j=2

λj (S j1 − 1Ω)

≥ 0 .

If S is made of just two assets, namely the risk-free asset and another asset S ,the previous equivalence becomes:

(i) NAA(Aα,S ) holds;

(ii) VaRα(S1 − 1Ω) ≥ 0 and VaRα(1Ω − S1) ≥ 0.

The result remains true with ESα instead of VaRα.


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Existence of acceptability arbitrage:

the VaR and ES case

ExampleTake S consisting of the risk-free asset and of an eligible asset S such that for aconvenient ε > 1

P(S1 ≥ ε) = 1 .

SinceP(S1 − 1Ω < ε− 1) = 0 ,

for any α ∈ (0, 1) we have

VaRα(S1 − 1Ω) < 0 .

Hence NAA(Aα,S ) does not hold for any α ∈ (0, 1).

Recall that for X ∈ X and α ∈ (0, 1) we have set

ESα(X ) =1

α

∫ α

0VaRβ(X )dβ .

As a simple consequence we obtain that NAA(A α,S ) does not hold for anyα ∈ (0, 1) as well.


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Further examples

Simple-scenario acceptance sets

Fix ω0 ∈ Ω and λ ∈ R, and consider the acceptance set

A := X ∈ X ; X (ω0) ≥ λ .

The following statements are equivalent:


(ii) S11 (ω0) = · · · = SN

1 (ω0).

Expected-value based acceptance sets

Fix a probability P on (Ω,F) and λ ∈ R. Consider the acceptance set

A := X ∈ X ; EP(X ) ≥ λ .

The following statements are equivalent:


(ii) EP

(S1

1

S10

)= · · · = EP

(SN

1

SN0

);

(iii) A + M0(S ) = A .


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Further examples

No-arbitrage and acceptability arbitrage

There is a class of acceptance sets A for which NAA(A ,S ) holds regardless ofthe choice of the eligible assets.

Fix a position U ∈ X and consider an acceptance set A ⊂ X such that

A ⊆ X ∈ X ; X ≥ U .

Then NAA(A ,S ) holds for any choice of the class S (satisfying no-arbitrage).Indeed set X := U − 1Ω. Then X /∈ A + M0(S ), since otherwise there wouldexist Z ∈M0(S ) such that Z ≥ 1Ω, contradicting no-arbitrage.


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1 Preliminaries




5 References


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References

T. Arai, Convex risk measures on Orlicz spaces: inf-convolution andshortfall, Mathematics and Financial Economics, 3: 73-88, 2010.

Ph. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures ofrisk, Mathematical Finance, 9: 203-228, 1999.

S. Biagini and M. Frittelli, A unified framework for utility maximizationproblems: an Orlicz space approach, Annals of Applied Probability, 18:929-966, 2008.

S. Biagini and M. Frittelli, On the extension of the Namioka-Klee theoremand on the Fatou property for risk measures, in Optimality and risk: moderntrends in Mathematical Finance, Springer, 1-28, 2009.

P. Cheridito and T. Li, Risk measures on Orlicz hearts, MathematicalFinance, 19: 184-214, 2009.

D. Filipovic and M. Kupper, Monotone and cash-invariant convex functionsand hulls, Insurance: Mathematics and Economics, 41: 1-16, 2007.

D. Filipovic, Optimal numeraires for risk measures, Mathematical Finance,18: 333-336, 2008.

H. Follmer and A. Schied, Convex measures of risk and trading constraints,Finance and Stochastics, 6: 429-447, 2002.


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H. Follmer and A. Schied, Stochastic finance: an introduction in discretetime, De Gruyter, 3rd edition, 2011.

M. Frittelli and E. Rosazza Gianin, Putting order in risk measures, Journalof Banking & Finance, 26: 1473-1486, 2002.

M. Frittelli and G. Scandolo, Risk measures and capital requirements forprocesses, Mathematical Finance, 16: 589-612, 2006.

M. Kaina and L. Ruschendorf, On convex risk measures on Lp-spaces,Mathematical Methods of Operations Research, 69: 475-495, 2009.

D.G. Konstantinides and C.E. Kountzakis, Risk measures in ordered normedlinear spaces with non-empty cone-interior, Insurance: Mathematics andEconomics, 48: 111-122, 2011.

C.E. Kountzakis, Risk measures on ordered non-reflexive Banach spaces,Journal of Mathematical Analysis and Applications, 373: 548-562, 2011.


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Contacts

For any further question and comment please refer to:

• Walter [email protected]

• Pablo Koch-MedinaPablo [email protected]

• Cosimo-Andrea [email protected]

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