1 Risk measures for portfolio vectors and

1 Risk measures for portfolio vectors and allocation of risks Ludger R¨ uschendorf University of Freiburg, Eckerstr. 1, 79104 Freiburg, Germany, [email protected]

Summary. In this paper we survey some recent developments on risk measures for portfolio vectors and on the allocation of risk problem. The main purpose to study risk measures for portfolio vectors X = (X1 , . . . , Xd ) is to measure not only the risk of the marginals separately but to measure the joint risk of X caused by the variation of the components and their possible dependence. Thus an important property of risk measures for portfolio vectors is consistency with respect to various classes of convex and dependence orderings. It turns out that axiomatically defined convex risk measures are consistent w.r.t. multivariate convex ordering. Two types of examples of risk measures for portfolio measures are introduced and their consistency properties are investigated w.r.t. various types of convex resp. dependence orderings. We introduce the general class of convex risk measures for portfolio vectors. These have a representation result based on penalized scenario measures. It turns out that maximal correlation risk measures play in the portfolio case the same role that average value at risk measures have in one dimensional case. The second part is concerned with applications of risk measures to the optimal risk allocation problem. The optimal risk allocation problem or, equivalently, the problem of risk sharing is the problem to allocate a risk in an optimal way to n traders endowed with risk measures %1 , . . . , %n . This problem has a long history in mathematical economics and insurance. We show that the optimal risk allocation problem is well defined only under an equilibrium condition. This condition can be characterized by the existence of a common scenario measure. A meaningful modification of the optimal risk allocation problem can be given also for markets without assuming the equilibrium condition. Optimal solutions are characterized by a suitable dual formulation. The basic idea of this extension is to restrict the class of admissible allocations in a proper way. We also discuss briefly some variants of the risk allocation problem as the capital allocation problem. Key words: risk measures, portfolio vector, allocation of risks AMS 2000 subject classification: 91B30, 62P05

2

Ludger R¨ uschendorf

1.1 Representation of convex risk measures for portfolio vectors Convex risk measures for real risk variables have been axiomatically introduced and studied in the mathematical finance literature by Artzner et al. (1998), Delbaen (2002), Föllmer and Schied (2004) and many others while there are independent and earlier studies of various aspects of risk measures and related premium principles in the economics and insurance literature. Various important subclasses of risk measures have been characterized. Law invariant, convex risk measures on L∞ (P ) (resp. Lr (P ), r > 1) have been characterized by a Kusuoka type representation of the form ´ ³Z %(X) = sup AV @Rλ (X)µ(dλ) − β(µ) (1.1) µ∈M1 ([0,1])

(0,1)

where %λ (X) = AV @Rλ (X) is the average value at risk R (also called expected shortfall or conditional value at risk), β(µ) = supX∈A% (0,1] AV @Rλ (X)µ(dλ) is the penalty function, and A% = {X ∈ L∞ (P ); %(X) ≤ 0} is the acceptance set of % (see Kusuoka (2001) and Föllmer and Schied (2004)). Thus in dimension d = 1 the average value at risk measures %λ are the basic building blocks of the class of law invariant convex risk measures. For some recent developments in the area of risk measures see [24]. For portfolio vectors X = (X1 , . . . , Xd ) ∈ L∞ d (P ) on (Ω, A, B) a risk 1 measure % : L∞ (P ) → IR is called convex risk measure if d M1) X ≥ Y ⇒ %(X) ≤ %(Y ) M2) %(X + mei ) = −m + %(X), m ∈ IR1 M3) %(αX + (1 − α)Y ) ≤ α%(X) + (1 − α)%(Y ) for all α ∈ (0, 1); thus % is a monotone translation invariant, convex risk functional (see [8, 19]). Like in d = 1 %(X) denotes the smallest amount m to be added to the portfolio vector X such that X +me1 is acceptable. ei denotes here the i-th unit vector. d A subset A ⊂ L∞ d (P ) with IR not contained in A is called (convex) acceptance set, if A1) A is closed (and convex) A2) Y ∈ A and Y ≤ X implies X ∈ A A3) X + mei ∈ A ⇔ X + mej ∈ A. With %A (X) := inf{m ∈ IR; X + me1 ∈ A} risk measures are identified with their acceptance sets: a) If A is a convex acceptance set, then %A is a convex risk measure b) If % is a convex risk measure, then A% = {X ∈ L∞ d ; %(X) ≤ 0} is a convex acceptance set.

1 Risk measures for portfolio vectors and allocation of risks

3

Let bad (P ) denote the set of finite additive, normed, positive measures on L∞ d (P ). Convex risk measures on portfolio vectors allow a representation similar to d = 1. 1 Theorem 1 (see [8]). % : L∞ d (P ) → IR is a convex risk measure if and only if there exists some function α : bad (P ) → (−∞, ∞] such that

%(X) =

sup

(EQ (−X) − α(Q)).

(1.2)

Q∈bad (P )

α can be chosen as Legendre–Fenchel inverse α(Q) =

sup

(EQ (−X) − %(X))

X∈L∞ d (P )

= sup EQ (−X). x∈A%

P

For risk measures % which are Fatou-continuous, i.e. Xn → X, (Xn ) uniformly bounded implies %(X) ≤ lim%(X), bad (P ) can be replaced by the class Md1 (P ) of P -continuous, σ-additive normed measures which can be identified by the class of P -densities D = {(Y1 , . . . , Yd ); Yi ≥ 0, EP Yi = 1, 1 ≤ i ≤ d}. For coherent risk measures i.e. homogeneous, subadditive, monotone, translation invariant risk measures the representation in (1.2) simplifies to %(X) = sup EQ (−X),

(1.3)

Q∈P

where P ⊂ ba(P ), resp. P ⊂ Md1 (P ), if the Fatou property holds, can be interpreted as class of scenario measures. d e e it For law invariant convex risk measures i.e. X = X implies %(X) = %(X) has been found recently that maximal correlation risk measures play the role of basic building blocks as the average value at risk measures do in the Kusuoka representation result. Let for some density vector Y ∈ D, ΨY (X) = EX · Y denote the correlation of X and Y (up to normalization) and define d e= ΨbY (X) = sup{ΨY (Ye ); X X}

(1.4)

the maximal correlation risk measure (in direction Y ). Theorem 2 (see [29]). Let Ψ be a Fatou continuous convex risk measure on L∞ d (P ) with penalty function α. Then it holds: Ψ is law invariant ⇔ Ψ has a representation of the form Ψ (X) = sup (ΨbY (X) − α(Y )) Y ∈D0

with law invariant penalty function α and D0 = {Y ∈ D; α(Y ) < ∞}.

(1.5)

4


Remark 3 a) In particular, the law invariant coherent risk measures on L∞ d (P ) have a representation of the form Ψ (X) = sup ΨbY (X)

(1.6)

Y ∈A

for some subset A ⊂ D. Thus the maximal correlation risk measures ΨbY are the basic building blocks of all law invariant convex risk measures on portfolio vectors. b) For d = 1 the representation in (1.5) can be shown to be equivalent to the Kusuoka representation result in (1.1). For d ≥ 1 optimal couplings as in the definition of the maximal correlation risk measure ΨbY arise, have been characterized in R¨ uschendorf and Rachev (1990). There are some examples where ΨbY can be calculated in explicit form but in general one does not have explicit formulas. Therefore, it is useful to give more explicit constructions of risk measures for portfolio vectors which generalize the known classes of one dimensional risk measures. For some partial extensions of distortion type risk measures see [8, 29].

1.2 Consistency w.r.t. convex orderings and some classes of examples For some class of functions F ⊂ {f : IRd → IR1 } the ordering ≤F is defined for random vectors X, Y by X ≤F Y

if Ef (X) ≤ Ef (Y ),

∀f ∈ F,

(1.7)

such that the integrals exist. In particular for the class of nondecreasing functions this leads to the stochastic ordering ≤st , for the class Fcx of convex functions this leads to the convex ordering ≤cx . Interesting dependence orderings are by the classes Fdcx of directionally convex functions, Fsm the class of supermodular functions, F∆ the class of ∆-monotone functions. The corresponding orderings are denoted by ≤dcx , ≤sm , ≤∆ (see M¨ uller and Stoyan (2002) for details on these orderings). From Strassen’s well-known representation result it follows that any risk measure % on L∞ d (P ), which satisfies the monotonicity condition M1) is consistent w.r.t. stochastic ordering ≤st , i.e. X ≤st Y ⇒ %(Y ) ≤ %(X).

(1.8)

It is of particular interest to study consistency of risk measures w.r.t. the above mentioned various convexity and dependence orderings. Let ≤decx , ≤icx denote the ordering by decreasing resp. increasing convex functions. Then it turns out that all law invariant axiomatically defined convex risk measures are consistent w.r.t. decreasing convex ordering ≤decx (see [8]).


5

Theorem 4. Let % be a law invariant, Fatou continuous convex risk measure on L∞ d (P ). Then % is consistent w.r.t. ≤decx , i.e. X ≤decx Y ⇒ %(X) ≤ %(Y ).

(1.9)

Since X ≤decx Y is equivalent to Y ≤icv X, ≤icv the ordering by increasing concave functions, (1.9) is equivalent for d = 1 with consistency w.r.t. the second order stochastic dominance. The proof of Theorem 4 is based essentially on the following important property: For all X, Y ∈ L∞ d (P ) holds %(X) ≥ %(E(X | Y )),

(1.10)

i.e. smoothing by conditional expectation reduces the risk (for d = 1 see Schied (2004) or Föllmer and Schied (2004)). In insurance mathematics the monotonicity axiom M1) of a risk measure has to be changed to monotonicity in the usual componentwise ordering. We shall use the notation Ψ (X) for risk measures satisfying this kind of monotonicity. The relation Ψ (X) = %(−X) gives a one to one relation risk measures % in the financial context and risk measures Ψ in the insurance context. A natural idea to construct risk measures for portfolio vectors X is to measure the risk of some real aggregation of the risk vector like the joint portfolio or the maximal risk, i.e. to consider Ψ (X) = Ψ1

d ³X

´ Xi

or

i=1

(1.11)

Ψ (X) = Ψ1 (max Xi ), i

where Ψ1 is a suitable one dimensional risk measure like expected shortfall or some distortion type risk measure. More generally for some class of real aggregation functions F0 = {fα ; α ∈ A} the following classes of risk measures have been introduced in Burgert and R¨ uschendorf (2006). Define ΨA (X) = sup Ψ1 (fα (X)), α∈A Z ΨM (X) = sup Ψ1 (fα (X))dµ(α),

(1.12) (1.13)

µ∈M

where M ⊂ Mσ (A) is a class of weighting measures on A. ΨA (X) is the maximal risk of some class of aggregation functions, while ΨM (X) considers the maximum risk over some weighted average. If for example A = ∆ = {α ∈ Pd IRd+ ;R i=1 αi = 1}, then one gets in thisRway risk measures like supα∈∆ Ψ1 (α · X), Ψ1 (α·X)dµ(α), Ψ1 (maxi αi Xi ) or ∆ Ψ1 (max αi Xi )dµ(α) measuring the risk in all positive directions α. It is important to assume that Ψ1 is consistent with respect to ≤icx – the increasingR convex ordering. This is e.g. the case for distortion risk measures ∞ Ψ1 (X) = 0 g(F X (t))dt where g is a concave distortion function and F X (t) =

6


1 − FX (t) is the survival function. Then the following consistency results hold true (see [8]): a)

If F0 ⊂ Ficx , then ΨA , ΨM are consistent w.r.t. ≤icx .

(1.14)

b)

If F0 ⊂ Fism , (Fidcx ), then ΨA , ΨM are consistent w.r.t. ≤ism (≤idcx ). (1.15)

As consequence of a), b) one gets that more positive dependent risk vectors have higher risks. This extends some classical results on comparison of risk vectors. Let Fi−1 denote the generalized inverse of the distribution function Fi of Xi , then d X

Xi ≤icx

i=1

d X

Fi−1 (U ),

(1.16)

i=1

where U is uniformly distributed on [0, 1] (see Meilijson and Nadas (1979), R¨ uschendorf (1983)). Further with the comonotonic vector X c := (F1−1 (U ), . . . , Fd−1 (U )) holds the following basic comparison result wich extends (1.16) X ≤sm X c

and

X ≤∆ X c

(1.17)

(see Tchen (1980) and R¨ uschendorf (1980)). Thus as consequence of (1.15) and (1.17) we conclude under the conditions of (1.14), (1.15) ΨM (X) ≤ ΨM (X c ),

ΨA (X) ≤ ΨA (X c );

(1.18)

the comonotonic risk vector leads to the highest possible risk under all risk measures of type ΨM , ΨA . Extensions of (1.17) to compare risks also of two risk vectors X, Y are given in [11, 27]. For a review of this type of comparison results for risk vectors see the survey paper [28].

1.3 Risk allocation and equilibrium The classical risk sharing problem is to consider a market, described by some probability space (Ω, A, P ), and n traders in the market supplied with risk ∞ measures %1 , . . . , %n . The problem Pnis to allocate a risk X ∈ L (P ) in an optimal way to the traders X = i=1 Xi , such that the risk vector (%i (Xi )) is PnPareto optimal in the class of all allocations or such that the total risk i=1 %i (Xi ) is minimal under all allocations. This problem goes back to early work in the economics and insurance literature (see the early contributions of Borch (1960a,b, 1962), B¨ uhlmann and Jewell (1979), Chevallier and M¨ uller (1994), and many others). It was later on extended to risk allocations in financial context (see e.g. Barrieu and El Karoui (2005) and references therein.


7

An interesting point is that for translation invariant risk measures %i , 1 ≤ i ≤ n, the principle of Pareto optimal risk allocations is equivalent to minimizing the total risk. This follows from the separating hyperplane theorem and some simple arguments involving translation invariance. In particular solutions are not unique and several additional (game theoretic) postulates like fairness have been introduced to single out specific solutions of the risk sharing problem. For example Chevallier and M¨ uller (1994) single out conditions which yield as possible solutions only portfolio insurance, tactical asset allocation, and collar strategies. Classical results are the derivation of linear quota sharing rules and of stop loss contracts as optimal sharing rules. We discuss in the following some developments on the risk allocation problem in the case where %i are coherent risk measures with representation %i (X) = supQ∈Pi EQ (−X) and scenario measures Pi . The more general case of convex risk measures is discussed in [7, 9]. There is a naturally associated equilibrium condition coming from similar equilibria conditions in game theory saying that in a balance of supply and demand it is not possible to lower some risks without increasing others. In formal terms this condition is formulated as: n X (E) If Xi ∈ L∞ (P ) satisfy Xi = 0 and %i (Xi ) ≤ 0, ∀i, then %i (Xi ) = 0, i=1

∀i. To investigate this equilibrium condition we introduce two naturally associated risk measures to the risk allocation problem. The first one is Ψ (X) = inf{m : X + m ∈ A},

(1.19)

with A the closed Sn cone generated by the union of the acceptance sets A%i of %i , A = cone( i=1 A%i ). W.r.t. Ψ every risk is acceptable, which is acceptable to any one of the traders in the market. Thus Ψ corresponds to some kind of optimistic view towards risk. The second related risk measure is the infimal convolution %ˆ = %1 ∧ · · · ∧ %n %ˆ(X) = inf

n nX

%i (Xi );

i=1

n X

o Xi = X ,

(1.20)

i=1

which describes the optimal reachable total risk of an allocation. Both risk measures have been considered in the literature (see [12, 2]). It turns out (see [7]) that %ˆ is a coherent risk measure ⇔ %ˆ(0) = 0

(1.21)

⇔ The equilibrium condition (E) holds true ⇔ Ψ is a coherent risk measure and in this case %ˆ = Ψ and the scenario set P ∼ %ˆ satisfies

8


P = P%ˆ = PΨ =

n \

Pi .

(1.22)

i=1

As consequence one obtains an interesting result of Heath and Ku (2004) (derived there for finite spaces Ω) saying: The equilibrium condition (E) is equivalent to n \ Pi 6= ∅, (1.23) i=1

i.e. to the existence of a common scenario measure of all traders. In particular (1.21) implies that the optimal risk allocation problem makes sense only under the equilibrium condition (E). Without (E) it is not possible to determine Pareto optimal allocation rules or allocation rules which minimize the total risk and a natural question is what to do in case the equilibrium condition does not hold true. To consider a useful version of the optimal risk allocation problem we define for X ∈ L∞ (P ) n n o X A(X) = (Xi ); X = Xi , (Xi ) admissible ,

(1.24)

i=1

where (Xi ) is called an admissible allocation of X if X(ω) ≥ 0 ⇒ Xi (ω) ≥ 0 X(ω) ≤ 0 ⇒ Xi (ω) ≤ 0.

(1.25)

The idea of introducing restrictions as above on the class of decompositions is similar to portfolio optimization theory, where restrictions on the trading strategies are introduced in order to prevent doubling strategies and thus to prevent the possibility of arbitrage. In the risk sharing problem we want to prevent risk arbitrage by restricting the class of admissible allocations. We define the admissible infimal convolution %∗ by %∗ (X) = inf

n nX

o %i (Xi ); (Xi ) ∈ A(X) .

(1.26)

i=1

Considering the connection with multiple decision problems and using a nonconvex version of the minimax theorem we get the following dual representation of %∗ , which essentially simplifies the calculation (see Burgert and R¨ uschendorf (2005)). Let X− , X+ denote the negative (positive) parts of W V X and Pj , Pj denote the lattice supremum resp.Winfimum Vof Pj . Thus in the case of P -continuous probability measure Pj , Pj and Pj are the probability measures with the max resp. inf of the P -densities as their density with respect to P . Theorem 5. For coherent risk measures %i = %Pi holds


nZ

Z

^ _ a) %∗ (X) = sup X− d Pj − X+ d Pj ; Pj ∈ Pj , 1 ≤ j ≤ n Z Z n o ^ _ ∞ b) A%∗ = X ∈ L (P ); X+ d Pj ≤ X+ d Pj , ∀Pj ∈ Pj .

9

o

The choice of restrictions in the definition of admissibility is justified by the following theorem which is based on Theorem 5. Theorem 6 (see [7]). Define the coherent admissible infimal convolution %ˆ∗ (X) = inf{m ∈ IR; X + m ∈ A%∗ } = inf{m ∈ IR; %∗ (X + m) ≤ 0}. a) Under the equilibrium condition (E) holds %ˆ∗ = %ˆ = Ψ . b) %ˆ∗ is the largest coherent risk measure % ≤ mini %i . Part b) says that our chosen restrictions on decompositions are not too restrictive since as a result of them we get the largest possible coherent risk measure below %i . Several related classes of restrictions can be given which lead to the same coherent risk measure. In particular we get a new useful coherent risk measure describing the value of the total risk of the optimal modified risk allocation problem. A different new type of restrictions on the allocation problem has been introduced in a recent paper by Filipovic and Kupper (2006) who consider Pn for a given risk allocation X = C i=1 i as admissible risk transfers only allocations of the form X=

n X

Xi with Xi = Ci + xi · Z,

(1.27)

i=1

where Z = (Z1 , . . . , Zd ) is a finite vector of d fixed random instruments in the Pn market, xi ∈ IRd are admissible allocation vectors such that i=1 xi · Z ≤ 0. Thus the optimal restricted risk allocation problem n X i=1

%i (Ci + xi · Z) =

inf

xi admissible

(1.28)

leads to an optimization problem with vector valued variables x1 , . . . , xn ∈ IRd and methods from game theory can be applied to characterize optimal solutions. Problem (1.28) can be seen as a variant of the classical portfolio optimization problem, i.e.P to minimize the risk %(x·Z) over all portfolio vectors n x = (x1 , . . . , xd ), xi ≥ 0, i=1 xi = 1. There is an alternative related form of the risk allocation problem which may be called the capital allocation problem (see [12, Chapter 9]). For a firm with N trading units there are expected future wealthPX1 , . . . , XN ∈ N L∞ (P ). If risk is measured by a risk measured %, then k = %( i=1 Xi ) is the necessary capital the firm needs to cover the total risk. The problem is to find

10


a fair allocation of the risk capital k = k1 + · · · + kN to the N trading units. Alternatively for subadditive risk measures Pn % one can see Pnthis as the problem to distribute the gain of diversification i=1 %(Xi ) − %( i=1 Xi ) ≥ 0 over the different business units of a financial institution. An allocation k1 , . . . , kN of the diversification gain is called fair if N X

ki = %

N ³X

i=1

´ Xi

(1.29)

´ Xj .

(1.30)

i=1

and for all J ⊂ {1, . . . , N } holds X

kj ≤ %

³X

j∈J

j∈J

The existence of fair allocations (Bondarava–Shapley theorem for risk measures) is proved in Delbaen (2000) [12, Theorem 22] for coherent risk measures. Assuming continuity of % from below (see [16, p. 167]) we get a simple proof of this existence result and more information on the fair allocation. Let P (see [16, p. 165]) denote the maximal representation set of scenario measures in the representation of %. Theorem 7. Let % be a coherent risk measure continuous from below and let PN X1 , . . . , XN be N wealth variables with k = %( i=1 Xi ). Then there exists ∗ with ki∗ := EQ∗ (−Xi ) is some scenario measure Q∗ ∈ P such that k1∗ , . . . , kN a fair allocation of the risk capital k. Proof. By the representation of % we have k=%

N ³X i=1

N ´ ³ X ´ Xi = sup EQ − Xi . Q∈P

(1.31)

i=1

Using that % is continuous from below Corollary 4.35 of Föllmer and Schied (2004) implies the existence of some Q∗ ∈ P such that the supremum in (1.31) is attained in Q∗ and with ki∗ = EQ∗ (−Xi ) holds N N ³ X ´ X k = EQ∗ − Xi = ki∗ . i=1

i=1

Further for any J ⊂ {1, . . . , N } holds ³X ³ X ´ X % Xj ) ≥ EQ∗ − Xj = kj∗ . j∈J

j∈J

j∈J

∗ Thus k1∗ , . . . , kN is a fair allocation of the risk capital.

(1.32)


11

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