Optimal risk sharing for law invariant monetary utility
functions
E. Jouini
∗
W. Schachermayer
†
N. Touzi
‡
April 19, 2007
Abstract
We consider the problem of optimal risk sharing of some given total risk be-
tween two economic agents characterized by law-invariant monetary utility functions
or equivalently, law-invariant risk measures. We first prove existence of an optimal risk
sharing allocation which is in addition increasing in terms of the total risk. We next
provide an explicit characterization in the case where both agents’ utility functions are
comonotone. The general form of the optimal contracts turns out to be given by a sum
of options (stop-loss contracts, in the language of insurance) on the total risk. In order
to show the robustness of this type of contracts to more general utility functions, we
introduce a new notion of strict risk aversion conditionally on lower tail events, which
is typically satisfied by the semi-deviation and the entropic risk measures. Then, in
the context of an AV@R-agent facing an agent with strict monotone preferences and
exhibiting strict risk aversion conditional on lower tail events, we prove that optimal
contracts again are European options on the total risk.
MSC 1991 subject classifications: Primary 91B06, 46A20; secondary 91B70.
Key words : Monetary utility functions, comonotonicity, Pareto optimal allocations.
∗
Universit´e Paris Dauphine and CEREMADE, Place du Mar´echal de Lattre de Tassigny, F-75775 Paris
Cedex 16, France.
†
Vienna University of Technology, Wiedner Hauptstrasse 8-10/105, A-1040 Wien, Austria and Univer-
sit´e Paris Dauphine, Place du Mar´echal de Lattre de Tassigny, F-75775 Paris Cedex 16, France. Financial
support from the Austrian Science Fund (FWF) under the grant P15889 and from Vienna Science and
Technology Fund (WWTF) under Grant MA13 is gratefully acknowledged.
‡
CREST, Laboratoire de Finance et Assurance, 15 Bd Gabriel P´e ri, F-92245 Malakoff Cedex, France
and Universit´e Paris Dauphine, Place du Mar´echal de L attre de Tassigny, F-75775 Paris Cedex 16, France.
1
1 Introduction
The problem of optimal sharing of risk between two agents has been considered by many
authors, starting from Arrow [1], Borch [9], B¨uhlmann [7], B¨uhlmann and Jewell [8], see
also Gerber [26]. The main motivation was the application to insurance problems. For an
extensive list of references, we refer to the recent paper by Dana and Scarsini [15] and the
second edition of the b ook by F¨ollmer and Schied [25]. The general setting of the problem
is the following. The initial risk endowments of the agents are defined by the bounded
random variables X
0
and X
1
, so that the aggregate risk is given by X := X
0
+ X
1
. An
allocation (ξ
0
, ξ
1
), with ξ
0
+ξ
1
= X, is c alled an optimal risk sharing if it is Pareto optimal
and satisfies the individual rationality constraint, i.e. none of the agents exp e riences a loss
of utility by passing from X
i
to ξ
i
.
The purpose of this paper is to obtain an explicit characterization of the optimal risk
sharing in the context of monetary utility functions, see Definition 2.1 below. Up to the
sign, monetary utility functions are identical to convex risk measures [24, 21], a popular
notion in particular since the Basel II acc ord. Indeed, U is a monetary utility function
if and only if ρ = −U is a convex risk measure in the sense of F¨ollmer and Schied [24],
Frittelli and Rosazza-Gianin [21]. We shall rather use the language of monetary utility
functions in order to embed our analysis in the setting of classical utility theory. Our
framework is mainly motivated by Barrieu and El Karoui [3, 4, 5, 6] who showed that,
when the agents’ utilities are given by U
i
(ξ) := γ
−1
i
U(γ
i
ξ) for some γ
i
> 0 and some
monetary utility function U, the optimal risk sharing rule is proportional to the aggregate
risk. We also refer to Heath and Ku [27] for analyzing Pareto optimal risk between banks
defined by coherent risk measures in a finite probability space.
In comparison to the ge neral utility theory, the setting of monetary utility functions
induces a remarkable simplification, as it induces a clear separation between the Pareto
optimality and the individual rationality constraints which define an optimal risk sharing
rule. Pareto optimal allocations are defined up to a constant, and their characterization
reduces to the calculation of the sup-convolution of the utility functions, as observed in
[3]. As a second independent step, the choice of the constant, or the premium, inside
the interval of reservation prices of the agents then characterizes all optimal risk s haring
allocations.
In this paper, we specialize further the class of monetary utility functions by assuming
the law-invariance property. Under this condition, we prove that the set of optimal risk
sharing allocations is not empty. We next derive an explicit characterization of optimal
risk sharing allocations in two concrete settings :
(i) When the preferences of both agents are defined by comonotone law-invariant utility
functions (a typical example being U = −AV @R
α
to be discussed below), a precise de-
2
scription of the optimal risk sharing rule is provided. The key-ingredient is the quantile
representation of Kusuoka [32] which is further extended in [22], [25] and [29]. We prove
that Pareto optimal allocations are given by sums of European options (“stop-loss con-
tracts” or simply “deductibles” in the insurance terminology) written on the aggregate
risk X. We observe that this setting is intimately connected to the context of convex
distorsions of probability as studied in Carlier and Dana [12].
(ii) As a s ec ond example, we fix some α ∈ (0, 1), and we assume that the utility of Agent
0 is defined by U
0
= −AV @R
α
, where AV @R
α
is the so-called average value at risk or
expected shortfall risk measure. This is the prime example of a comonotone monetary
utility function. Agent 1 is defined by a law-invariant monetary utility function which is
strictly monotone. We further assume that Agent 1 is strictly risk averse conditionally on
lower tail events. This notion is defined in Section 3. Loosely s peaking, it states that the
agent has a strict preference for averaging lower tail events of risk. The class of monetary
utility functions which satisfy these two conditions include the so-called entropic utility
and the semi-deviation utility (see e.g. [23]). We remark that U
0
= −AV @R
α
is neither
strictly monotone, nor strictly risk averse conditionally on lower tail events. In the above
setting, we show that Pareto optimal allocations are defined by class ical options : Agent 0
offers a stop-loss contract defined by a threshold κ, and leaves Agent 1 with the aggregate
risk capped at the level κ, i.e. (ξ
0
, ξ
1
) = (−(X − κ)
−
, κ ∨ X). In particular this shows that
Agent 0 takes the extremal risks, and that the AV@R measure of risk is not so prudent.
We believe that the above stated results provide an additional justification for the ex-
istence of options in financial markets, and stop-loss contracts, deductibles and layers in
insurance markets.
We c onclude this introduction by pointing out the importance of the concavity of the
utility function. It was stressed in [2] that the V@R criterion for risk measuring leads
to incoherent results because of the lack of sub-additivity. The present context of risk
sharing provides another result in this direction. Indeed, let Agent 1 be defined by
any monetary utility function. Assuming that Agent 0 is defined by the utility function
U
0
= −V @R
α
(X), it follows clearly that the level of the random variable on the event set
{X < V@R
α
(X)} is not relevant for this agent. Then, an optimal risk sharing allocation
consists in endorsing her any large amount of risk on the event set {X < V@R
α
(X)} ! In
other words, the V @R
α
−agent endorses an infinite amount of risk with positive proba-
bility. Hence the very question of optimal risk sharing leads to silly results if one of the
agents uses the V@R criterion as a measure of risk.
The paper is organized as follows. Sec tion 2 collects some notions on monetary utility
functions. Our main results are state d in Section 3. Section 4 is devoted to the charac-
terization of the optimal sharing allocations when both agents are defined by comonotone
law-invariant monetary utility functions. In Section 5, we focus on the notion of strict
3
risk aversion, conditional on lower tail events, together with its implications in terms of
risk sharing. Finally, we report the existence of optimal risk sharing allocations under
law-invariant monetary utility functions in Section 6.
2 Preliminaries : utility functions and related notions
Throughout this paper, we work on a standard probability space (Ω, F, P), i.e. we suppose
that (Ω, F, P) has no atoms and that L
2
(Ω, F, P) is separable. For every p ∈ [0, ∞], we
shall denote by L
p
(P) the collection of all real-valued random variables with finite L
p
norm
under P. Here, L
0
:= L
0
(P) c onsists of all real-valued F−measurable random variables,
and L
∞
:= L
∞
(P) is the subset of all essentially bounded elements in L
0
.
For every random variable ξ with values in R, we denote its cumulative distribution
function by F
ξ
(x) := P[ξ ≤ x], and its generalized inverse (called quantile function) by
q
ξ
(α) = inf {x : F
ξ
(x) ≥ α} for α ∈ [0, 1].
Given two random variables ξ and ζ, we shall write ξ =
d
ζ to indicate equality in
distribution.
For a bounded finitely additive measure µ ∈ (L
∞
)
∗
, we denote by kµk
ba
its total mass.
Given a sub-σ-algebra G of F, we define the conditional expectation (L
∞
)
∗
3 µ 7−→
E[µ|G] ∈ (L
∞
)
∗
as the transpose of the G−conditional expectation operator on L
∞
, i.e.
hE[µ|G], ξi = hµ, E[ξ|G]i for all µ ∈ (L
∞
)
∗
and ξ ∈ L
∞
. If the singular part of µ is zero,
i.e. µ is absolutely continuous with respect to P with density dµ/dP = Z, then it is
immediately checked that this definition coincides with the classical notion of condition
expectation in the sense that E[µ|G] = E[Z|G] · P.
Let E be a Banach space with dual E
∗
. Given a function f : E −→ R, its sub-gradient
and super-gradients are respectively denoted by
∂
−
f(x) := {x
∗
∈ E
∗
: f(y) ≥ f(x) + hx
∗
, y − xi for every y ∈ E} ,
∂
+
f(x) := {x
∗
∈ E
∗
: f(y) ≤ f(x) + hx
∗
, y − xi for every y ∈ E} .
When f is convex (resp. concave), we shall simply denote ∂f := ∂
−
f (resp. ∂f := ∂
+
f).
2.1 Monetary utility functions
We say that a function f : L
∞
−→ [−∞, ∞] is proper if dom(U ) := {ξ ∈ L
∞
: U(ξ) ∈ R}
is non-empty.
Definition 2.1. A proper function U : L
∞
−→ R is called a mon etary utility function if it
is concave, monotone with respect to the order of L
∞
, satisfies the normalization condition
4
U(0) = 0 and has the cash-invariance property
U(ξ + c) = U(ξ) + c for every ξ ∈ L
∞
and c ∈ R .
Observe that the cash invariance and the monotonicity of U imply that U is finite and
Lipschitz-continuous on L
∞
. In particular, the normalization U(0) = 0 does not restrict
the generality as it may be obtained by adding a constant to U.
Monetary utility functions can be identified with convex risk measures by the formula
ρ(ξ) = −U(ξ). In particular, positively homogeneous monetary utility functions can be
identified with coherent risk measures, introduced in Artzner, Delbaen, Eber and Heath [2],
while the general notion of monetary utility functions corresponds to convex risk measures
as introduced independently by F¨ollmer and Schied [24], and Frittelli and Rossaza-Gianin
[21]. We shall frequently use the conjugate function
V (µ) := sup
ξ∈L
∞
(U(ξ) − hµ, ξi) ∈ R ∪ {∞} , µ ∈ (L
∞
)
∗
,
which is convex and lower semi-continuous with respect to σ ((L
∞
)
∗
, L
∞
). From classical
convex duality theory, see e.g. [10], and the L
∞
−continuity of U, it follows that U and V
are hL
∞
, (L
∞
)
∗
i−conjugate, i.e.
U(ξ) = inf
µ∈(L
∞
)
∗
(V (µ) + hµ, ξi) for every ξ ∈ L
∞
.
We shall denote by V
1
the restriction of V to L
1
(P), and remark that its domain is included
in the set of densities of P-absolutely continuous probability measures, i.e.
dom(V
1
) :=
Z ∈ L
1
(P) : V (Z) < ∞
⊂ Z :=
Z ∈ L
1
+
(P) : E[Z] = 1
.(2.1)
Theorem 2.1 ([19, 24]). For a monetary utility function U , the following statements are
equivalent :
(i) U and V
1
are hL
∞
, L
1
(P)i−conjugate, i.e. U(ξ) = inf
Z∈L
1
(P)
V
1
(Z) + E[Zξ]
for
ξ ∈ L
∞
,
(ii) U is σ
L
∞
, L
1
(P)
−upper semicontinuous,
(iii) U has the Fatou property, i.e. for every L
∞
−bounded sequence (ξ
n
)
n≥1
converging in
probability to some ξ ∈ L
∞
, we have U(ξ) ≥ lim sup
n
U(ξ
n
).
Remark 2.1. Assuming the Fatou property, suppose that a monetary utility function U
is positively homogeneous, i.e. U(ξ) = −ρ(ξ) for some coherent risk measure ρ. Then, the
conjugate function V is zero on its domain, and U (X) = inf
Z∈dom(V
1
)
E[ZX].
Given two monetary utility functions U
0
and U
1
, we denote by
U
0
U
1
(ξ) := sup
ξ
0
∈L
∞
U
0
(ξ
0
) + U
1
(ξ − ξ
0
) , ξ ∈ L
∞
,
5