CC BY
11
УДК 519.21
Representation of Preferences by Generalized Coherent Risk Measures
Tatyana A. Kustitskaya*
Institute of Space and Information Technology, Siberian Federal University, Kirensky, 26, Krasnoyarsk, 660074
Russia
Received 06.04.2012, received in revised form 06.07.2012, accepted 06.08.2012 In this paper the model of generalized coherent risk measures is considered. Within the bounds of the model the properties of acceptance set are examined. A notion of elliptic cone is introduced. It is shown that the elliptic cone can be used as an acceptance set. The properties of the elliptic acceptance cone, particularly the interrelation between the cone shape and the risk aversion value, are studied.
Keywords: preference relation, stochastic dominance, risk measure, risk aversion, generalized coherent risk measure, acceptance set, elliptic cone.
Introduction
We work in a probability space (Q, A, P), where Q is a reference set, A — a a-algebra specified on Q, P — a probability measure, specified on the sets of A.
Definition 0.1. A Risk X on (Q, A) is any measurable mapping from Q to R (a random variable).
The set of all risks on (Q, A) we denote by X.
1. Orders and Preferences on the Set of Risks Natural Orders
We can specify order relation < on the set X:
X < Y ^ P(w : X(w) < Y(w)) = 1.
Strict order < is an order relation determined by:
X < Y ^ P(w : X(w) < Y(w)) = 1.
Suppose |Q| = n. Then we can submit a a-algebra A in the form of A = Probability measures P on the measurable space can be represented as the elements of the standard simplex in Rn:
Sn = {P = (p\...,pn) e Rn : p1 > 0,... ,pn > 0, p1 +-----+ pn = 1}.
The set of all risks X is isomorphic to Rn. Renumbering the elements of Q in some arbitrary way: Q = {w1,..., wn} , we denote P(w*) = p®, X(w!) = Xi, i = 1,..., n. We identify random the variables X e X with the vectors X = (X1,..., Xn) e Rn.
* [email protected] © Siberian Federal University. All rights reserved
We assume that X < Y if X4 < Y4 for all i = 1,..., n.
Another way to specify an order on the set X is related to stochastic dominance. Denote by F the set of all distribution functions, by FX — the distribution function of a random variable X :
FX (x) = P(X < x).
Let Fk be a set of all distribution functions with finite values of k-th moments:
/œ
tk dF (t).
-œ
For a given F G F specify a sequence of functions F(k), k = 1,2,... :
/X
F(k)(t)dt, -to < x < to.
-œ
Suppose F, Q G Fk. We say that Q has k-order stochastic dominance over F (F <k Q), if
F(k)(x) > Q(k)(x), -to < x < to.
We can also introduce strict stochastic dominance. Suppose F, Q G Fk. We say that Q strictly dominates F with the order k (F <k Q), if
F <k Q and 3 x G R : F(k)(x) > Q(k)(x).
For the case of a finite reference set the orders ^ and ^ i are consistent - from X ^ Y it follows that X Y. This is a consequence of the fact that for all X, Y G X P(X= P(Yi = 1 , . . . , n.
By means of first-order stochastic dominance we can determine an order relation (<1) on X:
X Y (X <i Y) ^ fx Fy (fX <i Fy).
A Preference relation ^ on the set X is a complete transitive binary relation on X. Risks X and Y are called equivalent if X ^ Y and Y ^ X.
Suppose that a preference relation ^ reflects an individual attitude to risk of a certain investor. The relation X ^ Y means that in equal conditions the investor prefers a financial instrument with return Y to an instrument with return X (or both instruments are equally preferable if X - Y).
If to take into account that in equal conditions market participants seek to maximal profits, it is reasonable to require that a preference relation ^ on X should be conformed to the order < on X :
X < Y X ^ Y.
Such preference relations are called monotone. They are called strictly monotone if
X < Y X ^ Y.
One of the ways to describe preferences on the set of all risks is to represent them by a real-valued functional.
A preference relation is represented on X by a measure p : X ^ R if one of the following conditions holds:
p(X) < p(Y), if X ^ Y, X, Y GX (1)
p(X ) < p(Y ), if Y ^ X, X, Y GX (2)
The functional p is called a risk measure.
Hereinafter we deal with risk measures that represent preference relations like in (1).
2. Risk Aversion
Risk aversion is a disposition of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain, but possibly lower, expected payoff. A preference ^ on X is called risk averse if for all nonsingular risks A : EA = 0 and any a e R
a + A -< a. (3)
In terms of risk measure the property of risk aversion can be written as
p(a + A) < p(a).
For some preferences we can get a numerical characteristic of risk aversion.
By Wa denote a degenerate risk localized at a e R (P(X=a)=1). If a preference relation ^ on X is conformed with then Wa ^ Wb if a < b. Moreover it is logical to assume that the preference is strict Wa -< Wb.
A preference ^ on X is called regular if it is conformed with stochastic dominance <1, for all a, b e R : a < b Wa -< Wb and in every equivalence class K e X^ there is exactly one degenerate distribution.
It was shown in [1] that for regular preference (3) can be written as VA e X : EA = 0, a e R 3 c > 0
a + A ~ a - c (4)
If a regular preference on X is represented by a risk measure p (which is also called regular in this case) then (4) can be written as
p(a + A) = p(a — c).
Value c can be interpreted as a price for which a person agrees to accept uncertainty. It can be used as a quantitative assessment of risk aversion that was introduced in [1].
A functional p : X ^ R is called canonical if p(Wa) = a Va e R. Every regular risk measure has a canonical analog.
For the canonical risk measure p value of risk aversion ca,p(A) is a solution of
p(a + A) = p(a — c). (5)
3. Generalized Coherent Risk Measures
The term "generalized coherent risk measure" was introduced in [2] and presented a modification of classical coherent risk measures introduced in [3].
We consider another modification. It is also called generalized coherent risk measures because it defines a broader class of functionals then classic coherent measures.
The axiomatics of generalized coherent risk measures is bases on the axiomatics of classical coherent risk measures with some distinctions.
A risk is called acceptable if investor agrees to work with it without investing any capital.
The set of all acceptable to an investor risks is called an acceptance set and is denoted by A (A c X).
Suppose that |Q| = n. Then X = (x1,..., xn) is a vector from Rn.
Also we introduce a norm || • || in Rn. It can be for example || • ||p (1 < p < to), given by
/ n \ 1/p
||X||p = 1**1* , ||X|U = lim ||X||p = max{|x11,..., K|}
We postulate that any acceptance set A satisfies the following axioms: A1: C+ C A, C+ = {X e X : X > 0} A2: Ap|C_ = 0, C_ = {X eX : X < 0}
A3: A is a convex cone (if X e A, Y e A, then aiX + a2Y e A, ai, a2 ^ 0).
A generalized coherent risk measure /A, associated with A is determined by
/a(X) = /A,|M|(X) = ¿A(X) inf ||X - Y||,
Y GdA
MX )={'■ X e A', (6)
AV ; [-1, X e Ac
where dA is a boundary of A.
The functional /A(X) exhibits the following properties: M) monotonicity:
/a(x) < /a(y), VX, Y e X, X < Y; PH) positive homogeneity:
/a(ax)= a/a(x), VA > 0, X eX;
S) superadditivity:
/a(x + Y) > /a(x ) + /a(y ), VX, Y eX;
Sh) shortcut property:
: dA tha
/a(x + Au(X)) = /a(x) + A, -TO < A < aa(x),
VX e X 3 X'(X) e <9 A that ||X - X'(X)|| = ^if ||X - Y|| and
X — X' (X)
where AA(X) > 0, u(X) = JA(X), M ;
i|x - XA (X )||
It is obvious that the classical coherent risk measure is a particular case of generalized coherent risk measures corresponding to the norm || • || = || • ||TO.
In [4] there is given a representation theorem for a generalized coherent risk measure, associated with an acceptance set A.
By X* denote the dual space (the space of linear continuous functionals on X), the dual cone A* is defined as
A* = {g e X* : g(X) > 0, X e A}. (7)
Distinguish the subset of functionals with unit norm:
A* = {g e A* : ||g||* = 1}.
Theorem 3.1. Let / be a generalized coherent risk measure, defined by an acceptance set A. Then the following representation is valid:
/a(x )= inf g(X), X eX. (8)
gGAi
Value of Risk Aversion for Generalized Coherent Risk Measures
For a classical coherent risk measure p we can easily find the value of risk aversion. Since this measure of risk is canonical and possesses the property of translation invariance [3], we have:
cp (A) = —p(A).
Consider a generalized coherent risk measure f (x). As it is not canonical, first we find its canonical analog fK:
f(X)
X e A, X 4 A.
fK <X) = \ f) f (—i)'
Solving equation (5) for the measure fK we find:
fA(a/ + A)
f (1) , al + A e A,
^(A)=< f(+A) (9)
a + : , n , al + A e A
We obtain that in the model of general coherent risk measures risk aversion depends on a. In particular,
f (A) (10)
co,A = f(—1) (10)
4. The Properties of Acceptance Sets
Consider two probability spaces (Q, A, P) and (Q, A, Q). The sets of all risks, defined on them, we denote by XP and Xq , the acceptance sets - by AP and Aq . Suppose XP = (X1,..., Xn) e XP and XQ = (X1,..., Xn) e Xq.
We assume AP = Aq if VXP e AP ^ Xq e Aq and vice versa, VXq e Aq ^ XP e AP
Theorem 4.1. Let the preference relation ^ on XP and Xq be consistent with the stochastic dominance If P = Q and 3 XP such that XP <1 Xq (or Xq <1 XP), then AP = Aq.
Theorem 4.2. Let the preference relation ^ be risk averse. Then for all X e A, X = 0 is true that EX > 0. t.
Theorem 4.3. Let the preference relation ^ be consistent with stochastic dominance and P = (n, n,..., n). Then for every X = (X1, X2,..., Xn) e A, the vector Y which components are obtained by interchanging the components of X also lies in the cone A (the cone is symmetric about the coordinate axes).
Proof. Consider X = (X 1,X2,.. .,Xn) e AP and a vector Y = (X ** ,X *2,.. .,X ), obtained by interchanging the components of X.
Since P(Xj) = P(Yj) = 1/n, it follows that (x) = FY (x) Vx e R, then X - Y and fAp(X) = fAp(Y). Therefore, X and Y belong or do not belong to the cone AP contemporaneously. □
Theorem 4.4. Let the preference relation ^ be consistent with stochastic dominance. Then for the acceptance cone AP (P = (p1,... ,pk-1, 0,pk+1,... ,pn)) it is true that
X = (X1,...,xk-1,Xk,xfc+1,...,xn) e AP ^ y = (X1,..., Xk-1, y, Xfc+1,..., Xn) e AP Vy e R.
tTheorems 4.1 and 4.2 were proved in [5]
Proof. Suppose X e AP.
FX (x) = FY (x) Vx e R
Fx <1 Fy ^ f (X) < f (Y);
Fy <1 Fx ^ f (Y) < f (X).
Therefore, f (Y) = f (X), thus we have X e Ap ^ Y e Ap. □
Theorem 4.5. Let the preference relation ^ be consistent with stochastic dominance. Then the acceptance cone AP, corresponding to P = (p1,...,pn), where pk = 1; pj =0,« = k can be defined by the inequality:
Xk > 0 (11)
Proof. Assume that there exists such vector
X = (X 1,...,Xk-1,Xk,xfc+1,...,X") e Ap,
that Xk < 0, but then by Theorem 4.4 the vector
Z = (-1,..., —1,Xk,-1,...,-1) e AP,
but this is impossible since Z e C_, and C_ n AP = 0 by the axiom A2. So, if X : Xk < 0, then X e AP.
Suppose Xk > 0. Then by Axiom A1 X = (0,..., 0,Xk, 0,..., 0) e AP, hence, by Theorem 4.4
Y = (Y 1,...,Yk-1,Xk,Yfc+1 , ...,Y") e AP VY 1,...,Yk-1,Yfc+1,...,Y" e R.
We obtain that if X is such that Xk > 0, then X e AP. Hence, (11) actually determines the acceptance cone for the given risk measure. □
5. Elliptic Acceptance Set
Consider a set
V (Xj — (P,X)npj)2 < (P,X)2, X = (X 1,...,x") e R" (12)
r2(pi)
and suppose that it satisfies the following conditions:
1. (P,X) > 0;
2. r(p) ^ — + n3p2
n . 3 2 (13)
p2
Theorem 5.1. The set A, determined by inequality (12), is an acceptance set for some preference.
Proof. For A to be an acceptance set it is sufficient to satisfy Axioms A1—A3.
1. First we prove that A satisfies A2: A n C_ = 0. Consider an arbitrary X e C_ Xj < 0 Vi = 1,2,..., n ^ (P, X) < 0, therefore, X doesn't satisfy (13).
2. Then we prove A3: A is a convex set. Let X e A. Then (P, X) = a > 0 and
" (X j — anPj)2 2 (,As
> -2—N-< a (14)
r2(pj)
The set {Y : (P, Y) = a} also satisfies (14) and it forms a n-dimensional ellipsoid Ea, which is a convex set.
Suppose that X' = AX, A > 0. Then (P, X') = Aa. X' also belongs to A (It can be verified by substituting in(12)).
Moreover, it belongs to the ellipsoid
(X1 - Aanpi)2 r2(pi)
+ ••• +
(Xn - Aanpn)2
r2(p„)
< Aa
2
(15)
like all other vectors Y' = AY, Y e Ea. Hence, A is a cone and all its hyperplane sections (P, X) = a, a > 0 are ellipsoids. Therefore, A is a convex set. 3. At last we prove A1: C+ c A.
Consider the basis e = {e*, i = 1,..., n : e* = 1, ej =0}.
Any vector X e C+ can be represented as a convex linear combination of the elements of the basis e:
X = X 1ei + X2e2 + • • • + Xnen, Xi > 0, i = 1,..., n
Since Axiom A3 is satisfied, we can assert that C+ c A, if e* e A Vi = 1,..., n.
Then we prove that e1 e A (for the rest e* the proof is similar). Substitute coordinates of e1 in (12):
(1 - npi2)2 (npip2)2
r2(pi)
r2(p„ )
,2„4
+
1 - 2np2 + n2pi 2 2 pi + n3P2
(nPiPn)2 r2(p„)
p2
P2 + n3p2
1 - 2npi2 + n2pi 2 2I
-+ n2pi
2 p2
+ ••• +
r2(pi)
pn N
P2 + n3pn ,
r2 (P2 )
+
, n — 1
P2
^ Pi , 2 2 '
<--+ n2pi - 3
n n3
'(Pn),
= Pi = (ei,P )2
Hence, ej e A j = 1,..., n, thus, C+ c A. □
Theorem 5.2. An elliptic cone A defines a reference consistent with stochastic dominance.
Proof. A preference relation determined by is consisted with stochastic dominance if VX, Y : Fx FY it is true that pa(X) < pA(Y). In our case X and Y are discrete:
X = (X 1,X2, ...,Xn), Y = (Y1, Y2,..., Yn)
It means that FX (x) > FY (x) Vx is true iff Xi < Y* i = 1,..., n, that is X < Y.
By Theorem 5.1 A is an acceptance cone and hence the risk measure is coherent. Respectively p(X) < p(Y) if X < Y. It means that if FX FY it is true that p(X) < p(Y).
□
6. Some Special Cases of the Elliptic Acceptance Set
Case 1. Consider an elliptic cone AP such that P = (p1,... ,pk-1,0,pfc+1,... ,pn). From Theorem 4.4 it follows that since X = 0 e AP we have Y = (0,..., 0, y, 0,..., 0) e AP. Substituting P and Y in (12) we get
y2
< 0 Vy e R
r2(0)
Therefore, we have
r(0) = to. (16)
One can easily see that there are no summands corresponding to i1,... ,im coordinates in
(12) if Pj! = ••• = pjm =0.
Case 2. Suppose AP is an elliptic cone corresponding to P = (p1,... ,pn), pk = 1, p» = 0,i = k .
From Theorem 4.5 it follows that AP is specified by Xk > 0. Substitute P in (12) and (13). We obtain
(Xk )2 (1 — n)2 , 2 . V 7 V 7 < (Xk)2, Xk > 0
r2(1)
-(1) > n - 1
(17)
The first inequality in (17) is satisfied automatically if the last inequality in (17) is satisfied. Hence, we get one more condition on the function r(p):
r(1) ^ n — 1.
Case 3 Consider P = (",..., "). Let the norm in X be Euclidean (|| • || = || • ||2). Substituting P (12), we get the following inequality for AP:
(X1 — (P, X))2 + • • • + (Xn — (P, X))2 < (P, X)2r2(1/n).
By I we denote the unit vector (1,..., 1), and by r0 the following form r0 = r (1/n). Combining this with (19), we obtain
||X — (P, X)1|| < (P, X)ro
(18)
(19)
(I, X)
^ ——-ro, (I,X) > 0.
(20)
All sections of the cone by hyperplanes (I, X) = a are n-dimensional spheres of radius (1,X )ro/n.
An acceptance cone is called a spherical cone if it is defined by (20).
Risk Aversion for a Spherical Cone
Lemma 6.1. Let X be an Euclidean space, P = (", ",..., "), and let the acceptance cone A be
determined by the inequality Then УД : ЕД = 0, ||Д|| = 1
х-(IX i
. (I,X)
< -ro.
inf ||X - Д|| = ||Z - Д||,
X eöA
(21)
where Z = (1-ZI + iM)l0. A, (I, Z) > 0.
nn
Proof. We claim that (A,1) = 0. Indeed,
EA = ^A1 + • • • + ^An = 1(A, I)
n n n
1. At first we prove that Z e dA:
(Д, I ) = 0
Z - (I,Z) I =
n
(I,Z) I + (I,Z)ro Д - MI
(I, Z)ro
Д
(1, Z)r0 цд| = (1, Z)ro
n
n
n
0
n
n
n
therefore, Z e dA.
2. Then we prove that ||Z - Ay = ^mf^ ||X - Ay. Suppose 3 Y e dA, then
IIY - Ay < ||Z - Ay,
(22)
Z — A and Y — A belong to the same plane. Hence, the triangle, constructed on these vectors, belongs to this plane plane. Therefore, the condition (22) holds iff cos ft ^ 0, where ft = (Y — Z HZ — A). Y can be written as
Y =(M) I a+(Y — Z).
Solve the optimization problem
||Y - A|2 ^ mm
Y EdA
(23)
(1,Z)r o n
- 1) A2+
2
yY — A|2 = ||Y — Z + tig* + (^ — l) A|| = yY — Z||2 + ^I2 + ( +2(Y — Z, HZ-I) +2 (Y — Z, (^ — l) A) = ||Y — Z||2 + (^ — l)%
+2 (Y — Z, I + ^ A — A = || Y — Z||2 + ^ + (— l)2 + 2(Y — Z, Z — A) The optimization problem (23) is equivalent to the problem
b = ||Y — Z||2 + 2(Y — Z, Z — A) ^ min
b = ||Y — Z ||2 + 2||Y — Z || • ||Z — A|| cos ft > 0 b = 0 if Y = Z Finally, we obtain Xf ||X — A|| = ||Z — A||.
Theorem 6.1. Let X be an Euclidean space. The cone A is determined by
(I,X) -r0.
X - ^ J
n
Then VA : EA = 0, ||A|| = 1 is true that
CQ,A =
Proof. Since Z±(A — Z), by Pythagorean theorem we obtain
||Z ||2 + ||Z — A||2 = ||A||2
<
(24)
| Z — A| 2 = l — | Z| 2
(I Z) (I Z) On the other hand, —-— I±(Z---— I), therefore, we get
||Z||2
(J,Z)
+
Z - ^ J
n
J||2||Z||2 cos2a
(J,Z )2
1 + ^
n
+
(J,Z)
22
■r.
Q
2
n2 n
7A
(J,Z )V r2 + 1
n
where a = ZAJ
(25)
|Z||2 = ||J||2|Z||2 cos2 a rQi nn
1
2
Q
2
2
n
n
2
2
cos2 a
1 + ГЛ n + r2
By Lemma 6.1 vector Z belongs to same two-dimensional plane as the vectors I and Д.
n
Since /_LД, we have y = ZЛД =--a.
' 2
||ZII2 = ||Д||2 cos2 в =1 - cos2 a = —
n + r2
Substituting this result in (25), we get
n
II Z - Д|| = —2 УД: ЕД = 0, ||Д|| = 1.
n + rg
2 _ /2(Д) _ n 1
f 2(-I) (n + r2 )n n + r2
It now follows that (24) holds.
From Theorem 6.1 we get the following condition on r(p):
r - =
1 - nc0,A
n со,д
(26)
n
n
1
7. Some Classes of Axial Functions
It follows from (12) that the function r(p) is a parameter of a cone which determines an individual attitude to risk. It also determines how the attitude to risk changes depending on the changes of the probability measure.
We have shown that the function satisfies the conditions (16), (18), (26). Is is also reasonable to assume that r(p) is monotonically decreasing on [0,1].
One of the ways to define an individual preference by an elliptic acceptance cone is to take r(p) from some class of one-parameter functions (that satisfy (16), (18), (26)) and evaluate the parameter according to the previous decisions of the individual.
8. An Example of an Axial Function
\/n + n3
Assume that rm(p) = -, m > 1 is an axial function. It obviously satisfies (16),
vn + n3 \/n + n3
(18), (26). Consider two representatives of this class: ri(p) = -, r2(p) = -2-• As
p p2
r1(p) < r2(p) Vp e [0,1] we get that an individuum whose preferences are determined by the function r1(p) is more cautious than an individuum whose preferences are determined by r2(p).
On Figure 1 we see acceptance cones for the case of two-dimensional space and P = (1,1). Ai is an elliptic acceptance cone corresponded to r1, and A2 corresponds to r2. Note that A1 c A2.
Conclusion
The generalized coherent risk measures afford the opportunity to value risk according to individual preferences. The elliptic acceptance cone introduced in the paper is an instrument for constructing such measures. As the parameter of the elliptic cone determines an individual attitude to risk it should be studied more detail.
Figure 1. Axial functions r1 (p) and r2(p) and acceptance cones corresponded to them
Acknowledgements
The author is grateful to professor A.A.Novosyolov and professor O.Yu.Vorobyev for useful discussions of the topic.
References
[1] A.A.Novosyolov, Risk aversion: qualitative approach and quantitative assessments, Automatics and telemechanics, 7(2003), 165-177 (Russian).
[2] R.A.Jarrow, A.K.Purnanandam, Generalized Coherent Risk Measures: The Firm's perspective, Finance Research Letters, 2(2005), 23-29.
[3] Ph.Artzner, F.Delbaen, J.-M.Eber, D.Heath, Coherent measures of risk, Mathematical Finance, 9(1999), 203-228.
[4] A.A.Novosyolov, Generalized coherent risk measures in decision-making under risk, Proceedings of the International Scientific School "Modelling and Analysis of Safety and Risk in Complex Systems", 2005, 145-150.
[5] T.A.Kustitskaya, Acceptance cones for some preferences, Proceedings of the X International FAMET'2011 Conference / Oleg Vorobyov ed. - Krasnoyarsk, KSTEI, SFU, 2011, 192-196.
Описание предпочтений на множестве рисков с помощью обобщенных когерентных мер риска
Татьяна А. Кустицкая
В 'работе рассматривается модель обобщенных когерентных мер риска. В рамках этой модели изучаются свойства множеств приемлемых рисков. Вводится понятие эллиптического конуса приемлемых рисков. Рассматриваются его свойства, в частности взаимосвязь между формой конуса и величиной неприятия риска.
Ключевые слова: отношение предпочтения, стохастическое доминирование, мера риска, неприятие риска, обобщенные когерентные меры риска, множество приемлемых рисков, эллиптический конус.
