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Coherent allocation of risk capital

Michel Denault*

The allocation problem stems from the diversification effect observed in risk

measurements of financial portfolios: the sum of the risks of many portfolios is

larger than the risk of the sum of the portfolios. The allocation problem is to

apportion this diversification advantage to the portfolios in a fair manner, yield-

ing, for each portfolio, a risk appraisal that takes diversification into account.

Our approach is axiomatic, in the sense that we first argue for the necessary

properties of an allocation principle and then consider principles that fulfill the

properties. Important results from the area of game theory find a direct applica-

tion. Our main result is to provide conceptual justification for a risk allocationprinciple that is based on the gradient of the risk measure. A numerical example

based on the rules of the Securities Exchange Commission for margin compu-

tations is provided.

1. Introduction

As an insurance against the uncertainty of the future net worths of its consti-

tuents, a firm could well, and would often be regulated to, hold an amount of

riskless investments. We will call this buffer the risk capital of the firm. From thefirms perspective, holding an amount of money dormant, ie, in extremely low

risk, low return instruments, is a burden. It is therefore natural to look for a fair

allocation of that burden between the constituents of the firm, especially when

the allocation provides a basis for performance comparisons of the constituents

between themselves. We will refer to the constituents as portfolios, but they

could just as well be business units.

The problem of allocation is interesting and non-trivial because the sum of the

risk capitals of each constituent is usually larger than the risk capital of the firm

taken as a whole, something called the diversification effect. This decrease

of total costs, or rebate, needs to be shared fairly between the constituents.

We stress fairness, as all constituents are from the same firm and none should

receive preferential treatment for the purpose of this allocation exercise.

1

The author expresses special thanks to F. Delbaen, who provided both the initial inspiration

for this work and generous subsequent advice, and to Ph. Artzner for drawing his attention to

Aubins literature on fuzzy games. Discussions with P. Embrechts, H.-J. Lthi, D. Straumann,

and S. Bernegger have been most fruitful. The author would also like to thank the Editor and

an anonymous referee for their useful comments. Finally, he gratefully acknowledges the

financial support of the S.S.H.R.C. (Canada), and of RiskLab (Switzerland), where most of

this research was carried out.

*Assistant Professor, Quantitative Methods; HEC Montral, 3000 Chemin de la Cte-Sainte-

Catherine, Montral, Canada, H3T 2A7.

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2. Risk measure and risk capital

In this paper we follow Artzner et al. (1999) in relating the risk of a firm to the

uncertainty of its future worth. The danger, inherent to the idea of risk, is that the

firms worth reaches such a low net value at a point in the future that it must stop

its activities. Risk is then defined as a random variable Xrepresenting a firms

net worth at a specified point in the future.

A risk measure quantifies the level of risk. Specifically, it is a mapping froma set of random variables (risks) to the real numbers: (X) is the amount of anumraire (eg, cash dollars) which, added to the firms assets, ensures that its

future worth is acceptable to the regulator, the chief risk officer or others. (For a

discussion of acceptable worths see Artzner et al., 1999.) Clearly, the heftier the

required safety net, the riskier the firm is perceived to be. We call (X) the risk

capital of the firm. The risk capital allocation problem is to allocate the amountof risk(X) between the portfolios of the firm.

We will assume that all random variables are from the space L(, A, ) ofbounded random variables over a fixed probability space. The reader who wishes

to do so can generalize the results along the lines of Delbaen (1998).

In their papers, Artzner et al. (1999) and (1997) have suggested a set of prop-

erties that risk measures should satisfy, thus defining the concept of coherent

measures of risk:2

DEFINITION 1 A risk measure :L is coherent if it satisfies the following

properties:

Subadditivity For all bounded random variables X and Y

(X+ Y) (X) + (Y)

Monotonicity For all bounded random variables X, Y such that3X Y

(X) (Y)

Positive homogeneity For all 0 and bounded random variable X

(X) = (X)

Translation invariance For all and bounded random variable X

(X+ B) = (X)

where B is the price, at some point in the future, of a reference, riskless invest-

ment whose price is 1 today.

For example, value-at-risk (VAR) is in general not a coherent risk measure,

although it is so for normally distributed random variables; expected shortfall is

a coherent risk measure.


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The properties that define coherent risk measures are to be understood as

necessary conditions for a risk measure to be reasonable. Let us briefly justify

them. Subadditivity reflects the diversification of portfolios, or that a mergerdoes not create extra risk (Artzner et al., 1999, p. 209). Monotonicity says that

if a portfolio Y is always worth more than X, then Ycannot be riskier than X.

Homogeneity is a limit case of subadditivity, representing what happens when

there is precisely no diversification effect. Translation invariance is a natural

requirement, given the meaning of the risk measure given above and its relation

to the numraire.

In this paper we require risk measures to be coherent; as a consequence,

VAR-based allocation principles, such as component value-at-risk, do not

qualify. We will not be concerned with specific risk measures until our example

in section 7.

3. Coherence of the allocation principle

An allocation principle provides a generic solution to the risk capital allocation

problem. We suggest in this section a set of axioms, which we argue are

necessary properties of a reasonable allocation principle. We will call coherent

an allocation principle that satisfies the set of axioms. The following definitions

are used:

N= {1, 2,, n} is the set of all portfolios of the firm.

Xi, i N, is a bounded random variable representing the net worth at time Tof the i th portfolio of a firm. We assume that the nth portfolio is a risklessinstrument with net worth at time Tequal toXn = B, whereB is the time Tprice of a riskless instrument with price 1 today.

X, the bounded random variable representing the firms net worth at some

point in the future T, is defined asX= ni=1Xi.

A is the set of risk capital allocation problems: pairs (N, ) composed of a setofn portfolios and a coherent risk measure .

K= (X) is the risk capital of the firm.

We can now define:

DEFINITION 2 An allocation principle is a function :A n that maps eachallocation problem (N, ) into a unique allocation:

The condition ensures that the risk capital is fully allocated. The Ki-notation is

used when the arguments are clear from the context.

: ( , )

( , )

( , )

( , )

( )N

N

N

N

K

K

K

K X

n n

i

i N

aM M

1

2

1

2

=

=such that

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DEFINITION 3 An allocation principle is coherent if for every allocationproblem (N, ) the allocation (N, ) satisfies the three properties:

No undercut

Symmetry If by joining any subset MN\{i,j},4 portfolios i and j bothmake the same contribution to the risk capital, then Ki = Kj.

Riskless allocation

Kn = (B) =

Recall that the n thportfolio is a riskless instrument.

Furthermore, we call non-negative coherent allocation a coherent allocation

which satisfies Ki 0, i N.It is our proposition that the three axioms of Definition 3 are necessary con-

ditions of the fairness, and thus credibility, of allocation principles. In that sense,

coherence is a yardstick by which allocation principles can be evaluated.

The properties can be justified as follows. The no undercut property ensures

that no portfolio can undercut the proposed allocation: an undercut occurs when aportfolios allocation is higher than the amount of risk capital it would face as an

entity separate from the firm. Given subadditivity, the rationale is simple. Upon a

portfolio joining the firm (or any subset thereof), the total risk capital increases by

no more than the portfolios own risk capital: in all fairness, that portfolio cannot

justifiably be allocated more risk capital than it can possibly have brought to the

firm. The property also ensures that coalitions of portfolios cannot undercut, with

the same rationale. The symmetry property ensures that a portfolios allocation

depends only on its contribution to risk within the firm, and nothing else.

According to the riskless allocation axiom, a riskless portfolio should be allocated

exactly its risk measure, which, incidentally, will be negative. It also means that,

all other things being equal, a portfolio that increases its cash position should see

its allocated capital decrease by the same amount.

In the next section we introduce some game theory and see how it applies to

allocation problems.

4. Game theory and allocation to atomic players

Game theory is the study of situations where players adopt various strategies to

best attain their individual goals. Game theory definitions and theorems are

given in sections 4.1 and 4.2 and are then applied to risk capital allocation in

section 4.3.

M N K X i ii Mi M

,



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For now, players will be atomic, meaning that fractions of players are not

considered possible (players who are humans are naturally atomic, but when

players are portfolios fractions of players can be considered). Furthermore, wewill focus here on coalitional games.

DEFINITION 4 A coalitional game (N, c) consists of:

a finite set, N, of n players; and

a costfunction, c, that associates a real number, c(S), with each subset, S, of

N (called a coalition).

We denote by G the set of games with n players.

The goal of each player is to minimize the cost she incurs, and her strategies

consist of agreeing or not to take part in coalitions (including the coalition of all

players).

In the literature the cost function is usually assumed to be subadditive:

c(S T) c(S) + c(T) for all subsets S and TofNwith empty intersection; anassumption which we make as well.

One of the main questions tackled in coalitional games is the allocation of

the cost, c(N), between all players; this question is formalized by the concept of

value:

DEFINITION 5 A value is a function : G n that maps each game (N, c) intoa unique allocation:

Again, the Ki-notation can be used when the arguments are clear from the

context and when it is also clear whether we mean i(N, ) or i(N, c).

4.1 The core of a game

Given the subadditivity ofc, the players of a game have an incentive to form the

largest coalition N, since this brings an improvement of the total cost when

compared with the sum of their individual costs. They need only find a way to

allocate the cost, c(N), of the full coalition,N, between themselves; but in doing

so players still try to minimize their own share of the burden. Player i will even

threaten to leave the coalition Nif she is allocated a share, Ki , of the total cost

that is higher than her own individual cost c({i}). Similar threats may come from

coalitions S N: if i S Ki exceeds c(S), every player i in S could carry anallocated cost lower than his current K

iifS separated fromN.

: ( , )

( , )

( , )

( , )

( )N c

N c

N c

N c

K

K

K

K c N

n n

i

i N

aM M

1

2

1

2

=

=where

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The set of allocations that do not allow such threat from any player or

coalition is called the core:

DEFINITION 6 The core of a coalitional game, (N, c), is the set of allocations

Knfor which i S Ki c(S)for all coalitions S N.

A condition for the core to be non-empty is the BondarevaShapley theorem.

Let C be the set of all coalitions ofN, and let us denote by 1S n

the

characteristic vector of the coalition S:

A balanced collection of weights is a collection of|C| numbers S in [0,1] suchthat S C S1S = 1N. An interpretation is that each player has one time unit todistribute between the coalitions, and each coalition S is active during a fraction

S of a unit of time, meaning that each player in S devotes her time solely to Sduring that period. Balancedness is then a feasibility condition on the players

distribution of her time.

A game is balanced if S C S c(S) c(N) for all balanced collections ofweights. Then:

THEOREM 1 (BondarevaShapley (Bondareva, 1963; Shapley, 1967)) A coali-

tional game has a non-empty core if and only if it is balanced.

PROOF See, for example, Osborne and Rubinstein (1994).

4.2 The Shapley value

The Shapley value was introduced by Lloyd Shapley (1953) and ever since has

received a considerable amount of interest (see Roth, 1988).

We use the abbreviation i(S) = c(S i) c(S) for any set S N, i S. Twoplayers i and j are interchangeable in (N, c) if either one makes the same

contribution to any coalition S it may join that contains neither i norj : i(S) =

j(S) for each S Nand i,j S. A player i is a dummy if it brings the contri-bution c(i) to any coalition S that does not contain it already: i(S) =c(i). Weneed to define the three properties:

Symmetry If players i andj are interchangeable, (N, c)i = (N, c)j . Dummy player For a dummy player, (N, c)i = c(i). Additivity over games For two games (N, c1) and (N, c2), (N, c1 + c2) =

(N, c1) + (N, c2), where the game (N, c1 + c2) is defined by (c1 + c2)(S) =c1(S) + c2(S) for all S N.

The rationale of these properties will be discussed in the next section. The

axiomatic definition of the Shapley value is then:

11

0S i

i S( ) =

if

otherwise



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DEFINITION 7 (Shapley, 1953) The Shapley value is the only value that satisfies

the properties of symmetry, dummy player, and additivity over games.

Let us now bring together the core and the Shapley value: when does the

Shapley value yield allocations that are in the core of the game? To our knowl-

edge the only relevant results are those of Shapley (1971) and Aubin (1979). The

former involves the property of strong subadditivity:

DEFINITION 8A coalitional game is strongly subadditive if it is based on a strongly

subadditive5 cost function:

c(S) +c(T) c(S T) + c(S T)

for all coalitions S Nand TN.

THEOREM 2 (Shapley, 1971) If a game (N, c) is strongly subadditive, its core

contains the Shapley value.

The second condition that ensures that the Shapley value is in the core is:

THEOREM 3 (Aubin, 1979) If for all coalitions S, | S | 2,

then the core contains the Shapley value.

The implications of these two results are discussed in the next section.

Let us end this section with the algebraic definition of the Shapley value,

which provides both an interpretation (see Shapley, 1953, or Roth, 1988) and an

explicit computational approach.

DEFINITION 9 The Shapley value, KSh, for the game (N, c) is defined as:

(1)

where s = | S |, andCi represents all coalitions of N that contain i.

Note that this requires the evaluation ofc for each of the 2n

possible coalitions,

unless the problem has some specific structure. Depending on what c is, this task

may become impossibly long, even for moderate n.

Ks n s

nc S c S i i N i

S i

Sh =

( ) ( )! ( )!

!( ) ( \ { }) ,

1

C

( ) ( )

1 0S T

T Sc T

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4.3 Risk capital allocations modelled as coalitional games

Clearly, we intend to model risk capital allocation problems (N, ) as coalitionalgames (N, c). We can associate the portfolios of a firm with the players of a game

and the risk measure with the cost function c :

(2)

Allocation principles naturally become values.

Note that given (2), being coherent and thus subadditive in the sense(X+ Y) (X) + (Y) of Definition 1, implies that c is subadditive in the sensec(S T) c(S) +c(T) given above.

The core

Allocations satisfying the no undercut property lie in the core of the game,

and, if none does, the core is empty. There is only an interpretational distinction

between the two concepts: while a real player can threaten to leave the full

coalitionN, a portfolio cannot walk away from a bank. However, if the allocation

is to be fair, undercutting should be avoided. Again, this holds also for coalitions

of individual players/portfolios.

The non-emptiness of the core is therefore crucial to the existence of coherent

allocation principles. From Theorem 1, we have:

THEOREM 4 If a risk capital allocation problem is modelled as a coalitional game

whose cost function c is defined with a coherent risk measure through (2), thenits core is non-empty.

PROOF Let 0 S 1 for S C, and S C S1S = 1N. Then

By Theorem 1 the core of the game is non-empty. s

S

S S

S i

i S

S i

i SS

S i

S S ii N

c S X

X

X

c N

=

=

=

C C

C

C

( )

( )

,

c S X S N ii S

( ) =

for



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The Shapley value

With the allocation problem modelled as a game, the Shapley value yields a risk

capital allocation principle. Much more, it is a coherent allocation principle, butfor the no undercut axiom. Symmetry is satisfied by definition. The riskless

allocation axiom of Definition 3 is implied by the dummy player axiom: from

our definitions in section 3, the reference, riskless instrument (cash and equiva-

lents) is a dummy player.

Note that additivity over games is a property that the Shapley value possesses

but which is not required of a coherent allocation principle. As discussed in

section 5.3, additivity conflicts with the coherence of the risk measures.

The Shapley value as a coherent allocation principle

From the above, the Shapley value provides us with a coherent allocation principle

ifit maps games to elements of the core. This is the case when the conditions of

either Theorem 2 or 3 are satisfied. The case of Theorem 2 is perhaps disappoin-

ting as the strong subadditivity ofc implies an overly stringent condition on :

THEOREM 5 Let be a positively homogeneous risk measure such that(0) = 0. Let c be defined over the set of subsets of random variables in L

through

c(S) = (i SXi). Then, if c is strongly subadditive, is linear.

PROOF Consider any random variablesX, Y,ZinL

. The strong subadditivity of

c implies

(X+Z) + (Y+Z) (X+ Y+Z) + (Z)

but also

(X+Z) + (Y+Z) = (X+ (Y+Z) Y) + (Y+Z)

(X+ (Y+Z)) + ((Y+Z) Y)

= (X+ Y+Z) + (Z)

so that

(X+Z) + (Y+Z) = (X+ Y+Z) + (Z)

By takingZ= 0, we obtain the additivity of. Then, combining

(X) = (XX) (X) = (X)

with the positive homogeneity of, we obtain that is homogeneous, and thuslinear. s

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That risks be plainly additive is difficult to accept since it eliminates all possi-

bility of diversification effects.

Unfortunately, the condition of Theorem 3 is also a strong one, at least in noway implied by the coherence of the risk measure . We thus fall short of aconvincing proof of the existence of coherent allocations. However, we consider

next another type of coalitional game where a slightly different definition of

coherence yields much stronger existence results.

5. Allocation to fractional players

In the previous section portfolios were modelled as players of a game, each of

them indivisible. This indivisibility assumption is not a natural one as we could

consider fractions of portfolios as well as coalitions involving fractions of port-

folios. The purpose of this section is to examine a variant of the allocation gamewhich allows divisible players.

This time around, we dispense with the initial separation of risk capital allo-

cation problems and games and introduce the two simultaneously. As before,

players and cost functions are used to model, respectively, portfolios and risk

measures, and values give us allocation principles.

5.1 Games with fractional players

The theory of coalitional games has been extended to continuous players who

need neither be in nor out of a coalition but who have a scalable presence.

This point of view seems much less incongruous if the players in questionrepresent portfolios: a coalition could consist of 60% of portfolio A, and 50% of

portfolio B. Of course, this meansx% ofeach instrument in the portfolio.

Aumann and Shapleys book Values of Non-Atomic Games (1974) was the

seminal work on the game concepts discussed in this section. There, the interval

[0, 1] represents the set of all players, and coalitions are measurable sub-

intervals (in fact, elements of a -algebra). Any sub-interval contains one ofsmaller measure, so there are no atoms ie, smallest entities that could be called

players; hence the name non-atomic games.

Some of the non-atomic game theory was later recast in a more intuitive set-

ting: an n-dimensional vector

n

+represents the level of presence of each

of the n players in a coalition. The original papers on the topic are Aubin (1979)

and (1981), Billera and Raanan(1981), Billera and Heath (1982) and Mirman

and Tauman (1982). Aubin called such games fuzzy; we call them (coalitional)

games with fractional players:

DEFINITION 10 A coalitional game with fractional players (N, , r) consists of

a finite set N of players, with |N| = n; a positive vector n+ , each component representing for one of the n

players his full involvement; and

a real-valued cost function r:

n

, r: a

r() such that r(0) = 0.



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Players are portfolios; the vector represents, for each portfolio, the size ofthe portfolio, in a reference unit. (The could also represent the business

volumes of the business units). The ratio ii then denotes a presence oractivity level, for player/portfolio i , so that a vector n+ can be used torepresent a coalition of parts of players. We still denote by Xi the random

variable of the net worth of portfolio i at a future time T, and Xn keeps its

riskless instrument definition of section 3, with time Tnet worth equal to B,with some constant. Then the cost function r can be identified with a riskmeasure through

so that r() = (N). By extension, we also call r() a risk measure. The expres-sionXii is theper-unitfuture net worth of portfolio i.

The definition of coherent risk measure (Definition 1) is adapted to the frac-

tional players context as:

DEFINITION 11 A risk measure, r, is coherent if it satisfies the four properties:

Subadditivity6 For all * and** in n,

r(* + **) r(*) + r(**)

Monotonicity For all * and** in n,

where the left-hand side inequality is again understood as in footnote 3.

Degree-one homogeneity For all n and for all +,

r( ) = r()

Translation invariance For all n,

One can check that ris coherent if and only if is.

r r

n

n

n

( )

=

1

2

1

0

M

i

ii

i N

i

ii

i N

X X r r

* *** **( ) ( )

r Xi

ii

i N

( )

=

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5.2 Coherent cost allocation to fractional players

The portfolio sizes given by allow us to treat allocations on a per-unitbasis.We thus introduce a vector kn, each component of which represents theper-unit allocation of risk capital to each portfolio. The capital allocated to each

portfolio is obtained by a simple Hadamard (ie, component-wise) product

(3) .* k = K

Let us also define, in a manner equivalent to the concepts of section 4:

DEFINITION 12 A fuzzy value7 is a mapping assigning to each coalitional game

with fractional players (N, , r) a unique per-unit allocation vector

with

tk = r() (4)

Again, we use the k-notation when the arguments are clear from the context.Clearly, a fuzzy value provides us with an allocation principle if we generalize

the latter to the context of divisible portfolios.

We can now define the coherence of fuzzy values:

DEFINITION 13 Let r be a coherent risk measure. A fuzzy value

: (N, , r) a kn

is coherent if it satisfies the properties defined below and if k is an element of the

fuzzy core:

Aggregation invariance Suppose the risk measures r and r satisfy r() =r()for some m n matrix and all such that0 , then

(5)(N, , r) = t(N, , r)

Continuity The mapping is continuous over the normed vector space Mn ofcontinuously differentiable functions r:

n+ that vanish at the origin.

Non-negativity underr non-decreasing8 If r is non-decreasing, in the sense

that r() r(*) whenever0 * , then

(6)(N, , r) 0

: ( , , )

( , , )

( , , )

( , , )

N r

N r

N r

N r

k

k

kn n

aM M

1

2

1

2

=



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Dummy player allocation If i is a dummy player in the sense that

whenever0 and* = except in the ith component, then

(7)

Fuzzy core The allocation (N, , r) belongs to the fuzzy core of the game(N, , r) if for all such that0

(8)t(N, , r) r()

as well as t(N, , r) r().

The properties required of a coherent fuzzy value can be justified essentially

in the same manner as was done in Definition 3. Aggregation invariance is akin

to the symmetry property: equivalent risks should receive equivalent alloca-

tions. Continuity is desirable to ensure that similar risk measures yield similar

allocations. Non-negativity under non-decreasing risk measures is a natural

requirement to enforce that more risk implies more allocation". The dummy

player property is the equivalent of the riskless allocation of Definition 3 and is

necessary to give risk capital the sense we gave it in section 2: an amount of

riskless instrument necessary to make a portfolio acceptable, riskwise. Finally,

note that the fuzzy core is a simple extension of the concept of core: allocationsobtained from the fuzzy core through (3) allow no undercut from any player,

coalition of players or coalition with fractional players. Such allocations are

fair in the same sense that core elements were considered fair in section 4.3.

Much less is known about this allocation problem than is known about the

similar problem described in section 4. On the other hand, one solution concept

has been well investigated: the AumannShapley pricing principle.

5.3 The AumannShapley value

Aumann and Shapley extended the concept of Shapley value to non-atomic

games in their original book (1974). The result was called the AumannShapley value, and later it was recast in the context of fractional-player games,

where it is defined as:

(9)

for player i ofN. The per-unit cost kiAS is thus an average of the marginal costs of

the ith portfolio, as the level of activity or volume increases uniformly for all port-

folios from 0 to . The value has a simpler expression, given our assumed coher-ence of the risk measure r; indeed, consider the result from standard calculus:

i ii

N r k rAS AS

d( , , ) ( ) = =

0

1

kX

ii

i

= ( )

r r

X

i i

i

i( ) ( )

( )* *

= ( )

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LEMMA 1 If f is a k-homogeneous function, ie, f(x) = kf(x), then f(x) xi is(k 1)-homogeneous.

As a result, since ris 1-homogeneous,

(10)

and the per-unit allocation vector is the gradient of the mapping revaluated at

the full-presence level :

(11)AS(N, , r) = kAS = r()

We call this gradient AumannShapley per-unit allocation, or simply Aumann

Shapley prices. The amount of risk capital allocated to each portfolio is thengiven by the components of the vector

(12)KAS = kAS .*

Note that the gradient (11) is called marginal VAR in the value-at-risk literature.

However, marginal VAR is based on VAR, a not-coherent risk measure, and the

allocation principle derived from it (called component VAR) is not a coherent

one.

Axiomatic characterizations of the AumannShapley value

As in the Shapley value case, a characterization consists of a set of properties

that uniquely define the AumannShapley value. Many characterizations exist

(see Tauman, 1988); we concentrate here on those of Aubin (1979) and (1981)

and Billera and Heath (1982). Both characterizations are for values of games

with fractional players as defined above, only their assumptions on rdiffer from

ours: their cost functions are taken to vanish at zero and to be continuously

differentiable but are not assumed coherent. Aubin also implicitly assumes rto

be homogeneous of degree one. Let us define:

A fuzzy value is linear if for any two games (N, , r1) and (N, , r2) andscalars 1 and 2 it is additive and 1-homogeneous in the risk measure:

(N, , 1r1 + 2 r2) = 1(N, , r1) + 2 (N, , r2)

Then, the following properties of a fuzzy value are sufficient to uniquely define

the AumannShapley value (9):

Aubins Billera and Heaths

linearity linearity

aggregation invariance aggregation invariance

continuity non-negativity under rnon-decreasing

i i i

N r k rAS AS

( , , )( )

= =



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In fact, both Aubin, and Billera and Heath prove that the AumannShapley value

satisfies all four properties in the table above.

So, is the AumannShapley value a coherent fuzzy value when ris a coherentrisk measure? Note first that the coherence ofrimplies its homogeneity, as well

as r(0) = 0. Being continuously differentiable is not automatic however; let us

assume for now that r does have continuous derivatives. The eventual non-

differentiability will be discussed later.

Clearly, two properties are missing from the set above for to qualify ascoherent: the dummy player property and the fuzzy core property. The former

causes no problem: given (10), the very meaning of a dummy player in Defini-

tion 13 implies:

LEMMA 2 When the allocation process is based on a coherent risk measure r, the

AumannShapley prices (10) satisfy the dummy player property.

Concerning the fuzzy core property, one very interesting result of Aubin is the

following:

THEOREM 6 (Aubin, 1979) The fuzzy core (8) of a fuzzy game (N, r, ) with posi-tively homogeneous r is equal to the sub-differential r() of r at .

As Aubin noted, the theorem has two very important consequences:

THEOREM

7 (Aubin, 1979) If the cost function r is convex (as well as positivelyhomogeneous), the fuzzy core is non-empty, convex and compact.

If furthermore r is differentiable at, the core consists of a single vector, thegradientr().

The direct consequence of this is that the AumannShapley value is indeed a

coherent fuzzy value, given that it exists:

COROLLARY 1 If(N, r, ) is a game with fractional players, with r a coherent cost function that is differentiable at , then the AumannShapley value (11) is acoherent fuzzy value.

PROOF The corollary follows directly from (10), Theorem 7, and the fact that

under positive homogeneity ris subadditive if and only if it is convex.

This corollary is our most useful result from a practical point of view. It says that

if we use a coherent but also differentiable risk measure, and if we deem impor-

tant the properties of allocation given in Definition 13, the allocation r() .* is a right way to go.

As a final note, let us mention that the condition of non-increasing marginal

costs, given in Billera and Heath (1982) for the the membership ofAS(N, , r) inthe fuzzy core, in fact implies that ris linear whenever it is homogeneous.

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On linear values and uniqueness

From the results given above, the AumannShapley value is the only linear

coherent allocation principle when the cost function is adequately differen-tiable. However, linearity over risk measures is not required of a coherent

allocation principle: while 1-homogeneity is quite acceptable, the additivity

part (N, , r1 + r2) = (N, , r1) + (N, , r2) causes the following problem.Because of the riskless condition, a coherent risk measure cannot be the sum of

two other coherent risk measures as it leads to the contradiction (written in the

less cluttered but equivalent notation)

(X) = (X+ B) = 1(X+ B) + 2(X+ B)

= 1(X) +

2(X) 2

= (X) 2

Therefore, the very definition of additivity would imply that we consider non-

coherent risk measures. On the other hand, convex combinations of coherent risk

measures are coherent (see Delbaen, 2000), so we could make the following

condition part of the definition of coherent allocations.

DEFINITION 14 A fuzzy value satisfies the convex combination property if forany two games (N, , r1) and(N, , r2) and any scalar [0,1]

(N, , r1 + (1 )r2) = (N, , r1) + (1 ) (N, , r2)

That condition implies linearity when combined with the 1-homogeneity with

respect to r of AS(N, , r) (which holds given the aggregation invarianceproperty). This would make the AumannShapley value the unique coherent

allocation principle. However, we see no compelling, intuitive reason to include

linearity (under one form or another) in the definition of coherent fuzzy

allocation, allowing for the existence of non-linear coherent fuzzy allocation

principles, a topic that is left for further investigation.

The same remarks on uniqueness and linearity apply to the Shapley value andallocation in the non-divisible players context. Note that the debate on the perti-

nence of linearity is far from new: as early as 1957 Luce and Raiffa observed

that [additivity] strikes us as a flaw in the concept of [Shapley] value.

On the differentiability requirement

Concerning the differentiability of the risk measures/cost functions, recent

results are encouraging. Tasche (1999) and Scaillet (2000) give conditions under

which a coherent risk measure, the expected shortfall, is differentiable. The

conditions are relatively mild, especially in comparison with the temerarious

assumptions common in the area of risk management. Explicit first derivatives



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are provided, which have the following interpretation: they are expectations of

the risk factors, conditioned on the portfolio value being below a certain quantile

of its distribution. This is very interesting: it shows that when the AumannShapley value is used with a shortfall risk measure, the resulting (coherent)

allocation is again of a shortfall type:

where q is a quantile of the distribution ofiXi.Even when r is not differentiable something can often be saved. Indeed,

suppose that r is not differentiable at but is the supremum of a set of para-meterized functions that are themselves convex, positively homogeneous and

differentiable at :

(13)r() = supp P

w(,p)

where P is a compact set of parameters of the functions w, and w(,p) is uppersemi-continuous inp. Then Aubin (1979) proved:

The fuzzy core is the closed convex hull of all the values AS(N, , w(,p)) of thefunctions w that are active at, ie, that are equal to r().

Thus, should (13) arise (which is not unlikely think of Lagrangian relaxation

when ris defined by an optimization problem), the above result provides a setofcoherent values to choose from.

Alternative paths to the AumannShapley value

It is very interesting that the recent report by Tasche (1999) comes, fundamen-

tally, to the same result that is obtained in this section namely, that given some

differentiability conditions on the risk measure , the correct way of allocatingrisk capital is through the AumannShapley prices (10). Tasches justification of

this contention is, however, completely different; he defines as suitable capital

allocations such that if the risk-adjusted return of a portfolio is above average,

then, at least locally, increasing the share of this portfolio improves the overallreturn of the firm. Note that the work of Schmock and Straumann (1999) points,

again, to the same conclusion. In the approach of Tasche (1999) and Schmock

and Straumann (1999) the AumannShapley prices are in fact the unique satis-

factory allocation principle.

Other important results on the topic include those of Artzner and Ostroy (1983),

who, working in a non-atomic measure setting, provide alternative characteriza-

tions of differentiability and sub-differentiability, with the goal of establishing

the existence of allocations through, basically, Eulers theorem. See also the

forthcoming Delbaen (2000).

On Eulers theorem,9 note also that condition (4) of the allocation vector

K E X X qi i ii

=

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follows directly from it, and that out of consideration for this some authors have

called the allocation principle (10) the Euler principle. See, for example, the

attachment to the report of Patrik, Bernegger and Regg (1999), which providessome properties of this principle.

We shall end this section by drawing the attention of the reader to the

importance of the coherence of the risk measure (and the rderived from it) forthe allocation.

The subadditivity of the risk measure is a necessary condition for the exis-

tence of an allocation with no undercut in both the atomic and the fractional

players contexts.

The homogeneity of the risk measure ensures the simple form (10) of the

AumannShapley prices.

Both subadditivity and homogeneity are used to prove that the core is non-empty (Theorem 4) in the atomic game setting. In the fuzzy game setting, the

two properties are used to show that the AumannShapley value is in the

fuzzy core (under differentiability). They are also used in the non-negativity

proof in Appendix A.

The riskless property is central to the definition of the riskless allocation

(dummy player) property.

6. The non-negativity of the allocation

Given our definition of risk measure, a portfolio may well have a negative riskmeasure, with the interpretation that the portfolio is then safer than deemed

necessary.

Similarly, the allocation of a negative amount to a portfolio does not pose a

conceptual problem; the interpretation is that this portfolio is a hedge within the

firms global portfolio, ie, a constituent that reduces the level of risk.

Unfortunately, in some applications we may consider, negativity poses a

problem. If the amount is to be used in a RAPM-type quotientReturnAllocated

capital, negativity has a rather nasty drawback in that a portfolio with an allo-

cated capital slightly below zero ends up with a negative risk-adjusted measure

of large magnitude, whose interpretation is less than obvious. A negative alloca-

tion is therefore not so much a concern with the allocation itself as with the use

we would like to make of it.

A crossed-fingers, and perhaps most pragmatic approach, is to assume that

the coherent allocation is inherently non-negative. In fact, one could reasonably

expect non-negative allocations to be the norm in real-life situations. For

example, provided that no portfolio of the firm ever reduces the risk measure

when added to any subset of portfolios of the firm:

c (S {i}) c (S) S N, i N\ S

then the Shapley value is necessarily non-negative. The equivalent condition for



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the AumannShapley prices is the property of non-negativity under non-decreas-

ing r(Equation (6)): if the antecedent always holds, the per-unit allocations are

non-negative.Another approach would be to enforce non-negativity by requiring more of

the risk measure. For example, the core and the non-negativity of the Ki s form

a set of linear inequalities (and one linear equality), so that the existence of a

non-negative core solution is equivalent to the existence of a solution to a linear

system. Specifically, a hyperplane separation argument proves that such a

solution will exist if the following condition on holds:

(14)

The proof is given in Appendix A. The condition could be interpreted as follows.

First, assume that (iNXi) > 0, which is reasonable if we are indeed to allo-cate some risk capital. Then (14) says that there is no positive linear combination

of (each and every) portfolios that runs no risk. In other words, a perfectly

hedged portfolio cannot be attained by simply reweighting the portfolios if all

portfolios are to have a positive weight. However, unless one is willing to

impose such a condition on the risk measure, the fact is that the issue of the non-

negativity remains unsatisfactorily resolved for the moment.

7. Allocation with a SEC-like risk measure

In this section we provide some examples of applications of the Shapley and

AumannShapley concepts to a problem of margin (ie, risk capital) allocation.

The risk measure we use is derived from the Securities and Exchange

Commission (SEC) rules for margin requirements (Regulation T), as described

in the National Association of Securities Dealers (NASD) document (1996).

These rules are used by stock exchanges to establish the margins required of

their members as a guarantee against the risk that the members portfolios

involve (the Chicago Board of Options Exchange is one such exchange). The

rules themselves are not constructive in that they do not specify how the margin

should be computed; this computation is left to each member of the exchange,

who must find the smallest margin complying with the rules. Rudd andSchroeder (1982) proved that a linear optimization problem (LP) modelled the

rules adequately and was sufficient to establish the minimum margin of a port-

folio that is, to evaluate its risk measure. It is worth mentioning that given this

LP-based risk measure, the corresponding coalitional game has been called the

linear production game by Owen (1975); see also Billera and Raanan (1981).

For the purpose of the article we restrict the risk measure to simplistic

portfolios of calls on the same underlying stock and with the same expiration

date. This restriction of the SEC rules is taken from Artzner et al. (1999), who

use it as an example of a non-coherent risk measure. In the case of a portfolio of

calls the margin is calculated through a representation of the calls by a set of

{ }

+

n ii N

i Ni i i

i N

X X, min

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spread options, each of which carries a fixed margin. To obtain a coherent

measure of risk, we prove later that it is sufficient to represent the calls by a set

of spreads andbutterfly options. Note that such a change to the margin rules wasproposed by the NASD and recently accepted by the SEC (NASD, 2000).

7.1 Coherent, SEC-like margin calculation

We consider a portfolio consisting ofCP calls at strike price P, where P belongs

to a set of strike prices P= {Pmin, Pmin + 10,, Pmax 10, Pmax}. This assump-

tion about the format of the strike prices set P, including the intervals of 10,

makes the notation more palatable without loss of generality. We also make the

simplifying assumption that there are as many long calls as short calls in the

portfolio, ie, PP CP = 0. Both assumptions remain valid throughout section 7.

We will denote by CP the vector of the CP parameters, P P. While CP fullydescribes the portfolio, it certainly does not describe the future value of the port-

folio, which depends on the price of the underlying stock at a future date.

Although risk measures were defined as a mapping on random variables, we

nevertheless write (CP) since the considered here can be defined by usingonly CP. On the other hand, there is a simple linear relationship between CP and

the future worths (under an appropriate discretization of the stock price space),

so that an expression such as (C*P + CP**) is also justified. Only later, in the caseof the proof of monotonicity of, need we treat with more care the distinctionbetween number of calls and future worth.

We can now define our SEC-like margin requirement. To evaluate the margin(or risk measure) of the portfolio CP, we first replicate its calls with spreadsand butterflies, defined as in Table 1. The variables shall represent the number of

each specific instrument. AllHand Kare understood to be in P for the spreads,

withH K; allHare in P \{Pmin, Pmax} for the butterflies.As in the SEC rules, fixed margins are attributed to the instruments used for the

replicating portfolio, ie, spreads and butterflies in our case. Spreads carry a mar-

gin of (H K)+ = max(0,H K), which effectively means that bull spreads carry

no margin, while bear spreads do. Short butterflies are given a margin of 10, while

long butterflies require no margin. In simple language, each instrument requires a

margin equal to the worst potential loss, or negative payoff, it could yield.



21

TABLE 1 Call equivalents of spreads and butterflies

Variable Instrument Calls equivalent

SH,K Spread, long in H, short in K

BH

long Long butterfly, centred at H

BH

short Short butterfly, centred atH

One long call at price H, one short call at

strike K

One long call at H 10, two short

calls at H, one long call at H + 10

One short call at H 10, two long calls at

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By definition, the margin of a portfolio of spreads and butterflies is the sum of

the margins of the instruments taken separately.

It is easy to extend the results of Rudd and Schroeder (1982) to take intoaccount our introduction of fixed margins on butterflies; the margin (CP) of theportfolio can be evaluated with the linear optimization problem (SECLP):

minimize ftY

(SECLP)

subject to AY= CPY 0

The column of variables Ystands for

where S is a column vector of all spread variables considered (appropriately

ordered: bull spreads, then bear spreads) and Blong and Bshort are appropriately

ordered column vectors of butterfly variables. The objective function ftY is

shorthand notation for

andA is

The matrix is read as follows. Its first column indicates that the first component

of Y, the variable associated with the bull spread SPmin, Pmin+10, contributes, per

spread, one long call with strike Pmin and one short call with strike Pmin + 10 to

the portfolio of calls CP. Similarly, from the first column of the third part ofA,

the component B longPmin+10 of Ycontributes, per butterfly, one long call with each

strike Pmin

and Pmin

+ 20 and two short calls with strike Pmin

+ 10.

A =

1 1 0 1 1 0 1 0 0 1 0 0

1 0 0 1 0 0 2 1 0 2 1 0

0 1 0 0 1 0 1 2 0 1 2 0

0 0 0 0 0 0 0 1 0 0 1 0

0 0 0 0 0 0 0 0 1 0 0 1

0 0 1 0 0 1 0 0

L L L L

L L L L

L L L L

L L L L

M M M M M M M M M M M M

L L L L

L L L

2 0 0 2

0 0 1 0 0 1 0 0 1 0 0 1

L

L L L L

f H K S Bt

H K

H K

H

H P P

Y= ++

( ) ,

, \{ , }min maxP P

10short

Y

S

B

B

=

long

short

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The objective function represents the margin; the equality constraints ensure

that the portfolio is exactly replicated. The risk measure thus defined is coherent;

the proof is given in Appendix B.

7.2 Computation of the allocations

Given this risk measure as a linear optimization problem, the Shapley value is

easy to compute when the firm is divided into a small number of portfolios (the

firm is defined as the sum of the portfolios). First, the margin of every possible

coalition of portfolios is calculated. Then the margin allocated to each portfolio

is computed using the formula of the Shapley value given in Definition 9.

The computation of the AumannShapley value is even simpler. Note that by

working in the fractional players framework we implicitly assume that fractions

of portfolios are sensible instruments. We choose the vector of full presence ofall players, , to be the vector of ones denoted by e. Recall that the AumannShapleyper-unitmargin allocated to the ith portfolio is

(15)

where r() is the margin required of the sum of all the portfolios i, each scaledby a scalar i, so that r(e) =(CP). In vector notation, k

AS = r(e).Let us first define the linear operator L which maps the level of presence of

the portfolios to numbers of calls in the firm. If there are |P | different calls(equivalently here, |P | strike prices) and n portfolios, thenL is a |P | n matrix,such thatLe = CP. Examples ofL

tare the three-by-five matrices at the top of the

tables given in section 7.3.

Now, the optimal dual solution, *, of the linear program (SECLP), obtainedautomatically when computing the margin of the firm, provides the rates of change

of the margin when the presence ofeach specific call varies. However, using the

complementarity condition satisfied at the optimal solution pair (Y*, *), we canwrite

(16)

(17)

so that the components ofLt* give the marginal rates of change of the objective

value of (SECLP), as a function of portfolio presence, evaluated at the point of

full presence of all portfolios.

Put in one sentence, the most interesting result of this section is that the

AumannShapley allocation is only a matrix product away from the lone evalu-

ation of the margin for the firm.

Finally, concerning the uniqueness of the allocation and the differentiability

of the risk measure, we can only say that they depend directly on the uniqueness

f

L e

t t

t

Y CP* =

= ( )

( )

( )

*

*

kr e

ii

AS =

( )



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of the optimal solution of the dual problem of (SECLP). Although there is no

special reason for multiple optimal dual solutions to occur here, it can well

happen in which case we have obtained one of many acceptable allocations, asnoted in the differentiality discussion of section 5.3.

7.3 Numerical examples of coherent allocation

We can obtain a somewhat more practical feeling of Shapley and Aumann

Shapley allocations by looking at examples.

For all allocation examples below, the reference firm is the same; its values

ofCP, P {10, 20, 30, 40, 50}, are

meaning one short call at strike 10, two short calls at strike 20, eight long calls

at strike 30, etc. The linear programming problem (SECLP) is solved with any

optimization software (we used MATLAB), and yields a margin (CP) = 40,obtained with the solution

and all other variables equal to 0.10 The vector of dual variables, also obtained

directly from the software, is (*)t= [ 20 10 0 0 0 ].In each of the tables below, we give the division of the firm in three portfolios

such that CP1+ CP2

+ CP3= CP, followed by the Shapley allocation and the

AumannShapley allocation whose computations were explained in the previous

section. The tables are followed by some observations. Consider first the division:

In this example, coalitions of portfolios incur margins as follows:

With these margins, the Shapley allocation can be computed using the formula (1)

C C C C C C

C C C

P P P P P P

P P P

1 2 1 3 2 3

1 2 3

+( ) = +( ) = +( ) =

( ) = ( ) = ( ) =40 20 30

20 20 10

C C C C C 10 20 30 40 50

1 0 6 6 1

0 2 2 0 0

0 0 0 1 1

1 2 8 7 2

Shapley AumannShapley

15 20

20 20

5 0

Total 40 40

C

C

C

P

P

P

1

2

3

S S B B

S S

S

20 10 30 40 30 40 50 20 30 40

30 10 30 50

30 20

0 125 0 625 2 5 1 375

0 875 0 500

0 750

, , , , , ,

, ,

,

. . . .

. .

.

= = = == ==

short long

C C C C C 10 20 30 40 50

1 2 8 7 2Total

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of section 4.2. The AumannShapley allocation is given by Lt*, as explained

in the previous section:

Consider a second example:

Here, coalitions of portfolios incur the margins

Finally, the third example is

where the coalitions of portfolios incur

On these examples, we note that:

The Shapley and AumannShapley allocations do not agree. An equal alloca-

tion in this setting would have been fortuitous.

Null allocations do occur. Null (or negative) allocations cannot be ruled out

and do happen here, both for the Shapley and AumannShapley principles.

For use in a risk-adjusted return calculation (return divided by allocated

capital), these would indeed be problematic.

C C C C C C

C C C

P P P P P P

P P P

1 2 1 3 2 3

1 2 3

+( ) = +( ) = +( ) =( ) = ( ) = ( ) =

40 30 10

30 10 20

C C C C C 10 20 30 40 50

1 1 4 2 0

0 1 4 3 0

0 0 0 2 2

1 2 8 7 2


26.66 30

6.66 10

6.66 0

Total 40 40

C

C

C

P

P

P

1

2

3

C C C C C C

C C C

P P P P P P

P P P

1 2 1 3 2 3

1 2 3

+( ) = +( ) = +( ) =

( ) = ( ) = ( ) =30 50 20

20 10 30

C C C C C 10 20 30 40 50

1 0 2 2 1

0 1 6 5 0

0 1 0 0 1

1 2 8 7 2


20 20

0 10

20 10

Total 40 40

CC

C

P

P

P

1

2

3

=

=

Lt*1 0 6 6 1

0 2 2 0 0

0 0 0 1 1

20

10

0

0

0

20

20

0



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The AumannShapley allocation is a marginal rate of risk capital. It can be

checked that, for example in the first case, the risk capital of the whole firm

does not change when the presence of the third portfolio varies slightlyaround unity.

The Shapley allocation may not be in the core. Again, this was to be expected.

The Shapley allocation of the third example is not part of the core, as can be

checked manually.

As a final note, let us mention that we could have verified whether the Shapley

allocation is or is not part of the fuzzy core (whenever it is part of the core). One

way of doing this is through a bi-level linear program, which, unfortunately,

is known to be computationally demanding for large instances; see Marcotte

(1997) or Marcotte and Savard (2000). Further details on this issue are reserved

for later.

8. Conclusion

In this article we have discussed the allocation of risk capital from an axiomatic

perspective, defining in the process what we call coherent allocation principles.

Our original goal was to establish a framework within which financial risk

allocation principles could be compared as meaningfully as possible. Our stand

is that this can be achieved by binding the concept of coherent risk measures to

the existing game theory results on allocation.

We suggest two sets of desirable properties of allocation principles, each set

defining the coherence of risk capital allocation in a specific setting: either theconstituents of the firm are considered indivisible entities (in the coalitional

game setting) or, to the contrary, they are considered to be divisible (in the context

of games with fractional players). In the former case, we find that the Shapley

value is a coherent allocation principle, though only under rather restrictive con-

ditions on the risk measure used.

In the latter, fractional players setting, the AumannShapley value is a coher-

ent allocation principle under a much laxer differentiability condition on the risk

measure; under linearity it is also the unique coherent principle. In fact, given

that the allocation process starts with a coherent risk measure, this coherent allo-

cation simply corresponds to the gradient of the risk measure with respect to thepresence level of the constituents of the firm.

Our main message is that the gradient-of-the-risk-measure, AumannShapley

principle has beyond the computational advantage of being as easy to evaluate

as the risk measure itself a strong conceptual justification.

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Appendix A: Proof of the non-negativity condition

The following result from section 6 is proved here.

THEOREM 8 A sufficient condition for a non-negative, no undercut allocation

to exist is

PROOF Let us recall that we denoted by 1S n

the characteristic vector of the

coalition S:

A non-negative, no undercut allocation Kexists when

(18)

Using Farkas lemma, this is equivalent to

(19)

which in turn is equivalent to

(20)

y y y S N i N

X y X y

N S

S i

S

i

i SS N

S i

i N

N

, , ,0 and

1 1 0

0

N N S S

S N

N S

i

i N

N i

i SS N

S

y y y y S N

X y X y

+

+

+

, , ,

=

K

K X S N

K X

K

n

tS i

i S

tN i

i N

such that

1

1

0

11

0S i

i S( ) =

if

otherwise

{ }

+

n ii N

i Ni i i

i N

X X, min



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Now, using the homogeneity and the subadditivity of,

Therefore, a sufficient condition for (18) (or (19) or (20)) to hold is

Finally, using the definition i = SiyS, we can write the sufficient condition

for (18)

Note that in the last step, we also used (i NXi) 0, a necessary conditionfor the existence of a non-negative, no undercut allocation; checking yN= 1,

yS = 0 in (19) shows this point. s

Appendix B: Proof of the coherence of the measure

We prove here that the risk measure obtained through (SECLP) is coherentinthe sense of Definition 1. We prove each of the four properties in turn below.

Subadditivity

For any two portfolios C*P and CP**

(C*P + CP**) (C*P) + (CP**)

PROOF If solving (SECLP) with C*P as the right-hand side of the equality

constraints yields a solution Y* and solving with CP** yields a solution Y**, then

Y* + Y** is a feasible solution for the (SECLP) with C*P + CP** as the right-hand

side. Subadditivity follows directly, given the linearity of the objective function.

S N

( ) i i i

i N

i

i N i Nii N X X 0, , min

y y y S N i N

X y X y

N S

S i

S

i S

S ii N

i

i N

N

, , , ,

( )

0

X y y X

y X

X y

ii SS N

S S ii SS N

S i

i SS N

i S

S ii N

=

=

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Degree-one homogeneity

For any 0 and any portfolio CP,

(CP) = (CP)

PROOF This is again a direct consequence of the linear optimization nature ofas S is a solution of the (SECLP) with CP as the right-hand side of theconstraints when S is a solution of the (SECLP) with CP as the right-hand side.

Of course, the very definition of homogeneity implies that we allow fractions of

calls to be sold and bought.

Translation invariance11

Adding to any portfolio of calls CP an amount of riskless instrument worth today reduces the margin ofCP by .

PROOF There is little to prove here; rather we need to define the behaviour of

in the presence of a riskless instrument, and naturally we choose the transla-tion invariance property to do so. This property simply anchors the meaning of

margin.

Monotonicity

For any two portfolios C*

P and CP**

such that the future worth ofC*

P is always lessthan or equal to that ofCP**,

(C*P) (CP**)

Before proving monotonicity, let us first introduce the values VP, for

P {Pmin + 10, , Pmax, Pmax + 10}, which represent the future payoffs, or worths,of the portfolio for the future prices, P, of the underlying.12 Again, we write VPto denote the vector of all VPs. The components ofVP are completely determined

by the number of calls in the portfolio:

which is alternatively written VP =MCP, with the square, invertible matrixM:

M=

10 0 0

20 10 0

30 20 10

L

L

L

M M M O

V C P p P P P P PP pp P

P= + + +{ }

=

( ) , , , ,min

min min max max

1010 20 10



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The antecedent of the monotonicity property is, of course, the componentwise

V*P VP**.

We will also use the following lemma.

LEMMA 3 Under subadditivity, two equivalent formulations of monotonicity are,

for any three portfolios CP, C*

P and CP**,

and

PROOF The upper condition is sufficient, as it implies

and (0) = 0 from the very structure of (SECLP). The upper condition isnecessary as

s

PROOF OF MONOTONICITY As a consequence of the above lemma, it is sufficient

to prove that if a portfolio of calls always has non-negative future payoff, its

associated margin is non-positive.

A look at (SECLP) shows that the margin assigned to the portfolio will be

non-positive (in fact, zero) if and only if a non-negative, feasible solution of(SECLP) exists in which all spread variables SH,K with H> K and all short

butterfly variablesBshort have value zero. This means that there exists a solution

to the linear system

(21)A~Y = CP

(22)Y 0

where A~

is made of the first and third parts of the A that was defined for

(SECLP):

M M

M M

M

( ) = + ( )( )( )

( ) + ( )( )

( )

1 1

1 1

1

V V V V

V V V

V

P**

P*

P**

P*

P*

P**

P*

P*

( )0 ( )M 1 VP

0 01 ( ) V VP P M

V V V V P*

P**

P*

P** ( ) ( ) M M1 1

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and Y is the appropriate vector of spread and butterfly variables. We obtain a

new, equivalent system of equations MA~

Y=MCP

= VP

by pre-multiplying by

the invertible matrix M introduced above. Recall now that we have made the

assumption that the portfolio contains as many short calls as long calls, ie,

PP CP = 0. Thus, we need only prove that there exists a non-negative solution

to the system

MA~

Y= VP whenever VP 0 and e tM1VP = 0

(e t is a row vector of 1s). A simple observation ofMA~

shows that its columns

span the same subspace as the set of columns

Observing furthermore that

so that any VP satisfying etM1VP = 0 has identical last two components, the

right-hand side ofMA~

Y= VP can always be expressed as a non-negative linear

combination of the columns ofMA~. s

e Mt

t

=

1

00

0

1

1

M

1

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

1

1

M M M M

, , , ,

A =

1 1 0 1 0 0

1 0 0 2 1 0

0 1 0 1 2 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 1 0 0 2

0 0 1 0 0 1

L L

L L

L L

L L

M M M M M M

L L

L L

L L



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1 At a more general level the interested reader may consult a game theory reference such as

the book by Osborne and Rubinstein (1994), the book edited by Roth (1988) (which

includes the survey by Tauman, 1988), or the survey article by Young (1994), which con-

tain legions of references on the subject.

2 On this topic see also Artzner (1999) and Delbaen (1998) and (2000).

3 The relationX Ybetween two random variables is taken to mean X() Y() for almostall in the probability space (, A, ).

4 Here \ denotes the set relative complement:A\B contains all the elements ofA less those of

the intersection ofA andB.

5 By definition, a strongly subadditive set function is subadditive. We follow Shapley (1971)

in our terminology; note that he calls convex a function satisfying the reverse relation of

strong subadditivity.

6 Note that under degree-one homogeneity, subadditivity is equivalent to convexity:r( * + (1 ) **) r(*) + (1 ) r(**)

7 Like fuzzy game, the term seems to have been coined by Aubin (1979).

8 Called monotonicity by some authors.

9 Which states that ifFis a real, n-variable, homogeneous function of degree k,

10 Note that our extension of Rudd and Schroeders result given in section 7.1 to account

for the butterflies destroys the transportation problem structure that ensured integer

solutions. Consequently, non-integer solutions (eg, 1.125 contracts of a certain spread)

are to be expected, as is the case in our example. For portfolios with a large number of

instruments, rounding the solution would be an adequate approach. Alternatively, integ-

rality constraints could be imposed in the optimization.

11 Note that, in the interest of a tidier notation, we departed from our previous usage and did

not use the last component ofCP to denote the riskless instrument.

12 Obviously, the latter set of prices may be too coarse a representation of possible future

prices and is used to keep the notation compact; starting with a finer P relieves this

problem; similarly, Pmin and Pmax can be set as low and high as desired.

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