2.3 The case of Comonotonic Risk Measures
2.3.3 Inf-Convolution of Choquet Integrals
Remark 2.3.2 The same proof shows that the result is true for risk measures defined onLp, provided that C(X) =R
X dc is finite, where c(A) :=ρ(1A).
In view of these results, the convolution of comonotonic risk measures reduces to the inf-convolution of Choquet integrals, the study of which will require some preliminary results that we enonciate and prove in section2.3.2. We will represent (2.3.8) the inf-convolution of two comonotonic and cash additive functionnals as a Choquet integral in section 2.3.3.
2.3.3 Inf-Convolution of Choquet Integrals
As shown in the following proposition, the Kusuoka-type quantile representation (2.3.5) is stable under inf-convolution. This extends a result in [67] to non necessarily law invariant measures.
Proposition 2.3.2 Letρ1 andρ2 be convex comonotonic risk measures satisfying the Fatou property and letX ∈ X. Assume that ρ1ρ2(0)>−∞and the functionsx→ρi(1{X>x}) inherit the constant intervals of FX.
If X = Lp, we also assume that ρi and its associated comonotonic risk measure is finite on Lp, i= 1,2.
Then there exist two distortion functionsgX1 and gX2 such that ρ1ρ2(X) =
Z
qX(u)(gX1 ∧g2X)0(u)du (2.3.6) where (g1X ∧g2X)(x) := min gX1 (x), gX2 (x)
.
Proof. Since ρ1 and ρ2 are convex and comonotonic, using either (2.3.1) or (2.3.2) we have the existence of two submodular capacitiesc1andc2such thatρ1,ρ2are the associated Choquet integrals.
Letc1,2(A) := min(c1(A), c2(A)), ∀A∈ F. The dual representation ofρ1 and ρ2 is given by:
ρi(X) = sup
Q∈M1,f
{EQ(X)−αi(Q)} (2.3.7)
withαi(Q) = 0 if Q∈ Ci and +∞ if not.
Ci is the core of ci defined by Ci ={Q ∈ M1,f|Q(A) ≤ci(A), ∀A ∈ F } and M1,f denotes the set of finitely additive set functions. Now theorem 3.6 in [5] says thatρ1ρ2 is a convex risk measure with a penalty function given by α(Q) =˜ α1(Q) +α2(Q), which equals 0 if Q∈C˜=C1∩ C2 ={Q∈ M1,f|Q(A)≤min(c1(A), c2(A)), ∀A∈ F }and+∞ if not. α(Q)˜ is the core ofc1,2. c1,2 is obviously a capacity, and using theorem 4.88 in [54],c1,2 is a submodular capacity (here the Assumption??of atomless probability space is important) and we have
ρ1ρ2(X) = Z
Xdc1,2 (2.3.8)
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Now since c1,2(X > x) inherits the constant intervals of FX(x) (since c1 and c2 do), we can use Lemma 2.3.1 to provide the existence of a distortion function g˜ such that R
Xdc1,2 coincides with the Choquet integral with respect to the capacityg(P(.))˜ at X. We have
˜
g(u) =c1,2(X > qX(u)) =c1(X > qX(u))∧c2(X > qX(u)) = min gX1 (u), gX2 (u) ,
with giX(u) =ci(X > qX(u)), i= 1,2. This implies, as in the demonstration of Proposition 2.3.1, thatρ1ρ2(X) =R
qX(u)(g1X ∧g2X)0(u)du(this is where the Fatou property is needed).
Remark 2.3.3 The formula (2.3.8) in the previous proof tells us that the inf-convolutionρ1ρ2(X) is again a Choquet integral with respect to the capacity
c1,2(A) = min(c1(A), c2(A)).
In the particular case where c1 and c2 are constructed with distortion functions ψ1 and ψ2 as in Example 2.2.1, then c1,2 is also a distorted probability with respect to the function ψ1,2(u) :=
min(ψ1(u), ψ2(u)).
If at a given pointu∗ in [0,1], we haveψ1(u∗)< ψ2(u∗), then we can say that at the given quantile levelu∗, agent1 is less risk averse than agent2, since he uses a less severe distortion of the a-priori probability P at this point u∗. Now the form of ψ1,2 suggests that the inf-convolution splits the total risk X in many parts, which are asigned to the agent being the less risk averse on the corresponding quantile intervals (see Theorem 2.3.3 and Section2.4 for more details).
Using this result, we are able to solve explicitely the inf-convolution of convex Choquet integrals.
Recall that the core Ci of the capacity ci is defined by
Ci={Q∈ M1,f|Q(A)≤ci(A), ∀A∈ F } whereM1,f denotes the set of finitely additive set functions onΩ.
In order to correctly formulate our next theorem, we need the following definitions.
Definition 2.3.3 We will say that a subset B of [0,1] is densely ordered if for any x, y in B with x < y, there existsz inB such that x < z < y.
For instance the setQ∩[0,1]is numerable and densely ordered.
Definition 2.3.4 Letf be a real valued function defined on[0,1]. We define the changing sign set of f by thoseu in(0,1)such that for every sequence(un) increasing towardsu and every sequence(vn) decreasing towards u, there is an integerN, such that ∀n≥N,sgn[f(un)] =−sgn[f(vn)], where as usual, sgn(x) =−1 if x <0 andsgn(x) = 1 if x >0.
The two previous definitions are used to formulate the following assumption on two distortion functions.
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Assumption 2.3.2 Letψ1 andψ2 be two distortion functions and denotef :=ψ1−ψ2. The set of changing sign points of f contains no densely ordered subset.
The economic interpretation of the previous assumption is as follows: it means thatψ1−ψ2 does not change sign too often. Recall from the previous remark that these functions represent uncertainty weighting, ifψ1(u)≥ψ2(u)for a certain pointu, an economic agent usingψ1 as a distortion function asigns more uncertainty to the point u than an agent 2 using ψ2. Now when the sign of ψ1−ψ2 changes, it means that the order of uncertainty weighting between agent 1 and agent 2 changes. If it changes very often in the sense that Assumption 2.3.2 is not satisfied, this means that the agents relative agreement on the probability weighting changes very often.
A sufficient condition for Assumption2.3.2 to hold is that the distortionsψ1 andψ2 are continuous and have a finite number of crossing points. This will be the case in all the examples we consider.
Theorem 2.3.3 Let X be in L∞. Let ρ1 and ρ2 be two comonotonic risk measures satisfying the assumptions of Proposition 2.3.2. Suppose also that the associated local distortions g1X and g2X are continuous and satisfy Assumption 2.3.2.
Then there exists a possibly infinite increasing sequence {kn, n≤N} of real numbers corresponding to quantile values of X, such that
ρ1ρ2(X) =ρ1(X−Y∗) +ρ2(Y∗) where Y∗ is given by:
Y∗ =
N
X
p=0
(X−k2p)+−(X−k2p+1)+.
Remark 2.3.4 If the sequence {kn, n≥0} only takes two non zero values k1 andk2, then the first agent (the insurer in that case) takes in charge X∧k1+ (X−k2)+ and agent two (the reinsurer) is exposed to (X−k1)+−(X−k2)+. This is exactly the definition of the Excess-of-Loss contract, denoted "A-XS-B", that we find in a large majority of catastrophe reinsurance market exchanges (here B =k1 andA=k2−k1).
Remark 2.3.5 This result means that the inf-convolution of law invariant and comonotonic risk measures is given by a generalization of the Excess-of-Loss contract, with more treshold values. The domain R+ of attainable losses is divided in "ranges", and each range is alternatively at the charge of one of the two agents.
The case where N is infinite is of course impossible in practice, but it gives an additional conceptual insight to the optimal structure of the risk transfer.
Proof.
We start by writing the representation (2.3.6) for the inf-convolution and divide it in two parts.
For X∈ X, letA:={x∈[0,1] :g1X(x)≤g2X(x)} (we will write g1 and g2 instead of gX1 and g2X for
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convenience),
ρ1ρ2(X) = Z 1
0
qX(u)(g1∧g2)0(u)du
= Z 1
0
qX(u)g10(u)1A(u)du+ Z 1
0
qX(u)g02(u)1Ac(u)du.
Sinceg1 andg2 satisfy Assumption2.3.2 and are continuous, the following non increasing sequence {un, n≥1}is well defined: it corresponds to the points of equality of the functionsg1 andg2 defined as follows :
u0= 1; un+1= sup{x < un such that g1(x) =g2(x) and g1(x−)6=g2(x−)}
For each nwe definekn:=qX(un).
Then we can write the set A as the numerable union of disjoint intervals A = S
n≥0[un+1, un], where u0 = 1, limn→+∞un = 0 and possibly there are integers k ∈ N such that g1(x) = g2(x) for anyx∈[uk+1, uk]. The non increasing sequence{un, n≥1}corresponds to points of equality of the functionsg1 and g2. We can rewrite the inf-convolution
ρ1ρ2(X) =X
p≥1
Z 1
0
qX(u)1(u2p+1≤u≤u2p)g10(u)du
+X
p≥1
Z 1 0
qX(u)1(u2p+2≤u≤u2p+1)g10(u)du. (2.3.9) Then we remark that
qX(u)1(ui+1≤u≤ui) = (qX(u)−qX(ui))+−(qX(u)−qX(ui+1))+
−qX(ui+1)1{u≤ui+1}+qX(ui)1{u≤ui},
We define the increasing sequence (ki)by ki :=qX(ui), and the random variablesZ := (X−ki)+− (X−ki+1)+ and Zi =1{X>ki}, then we have
qX(u)1(ui+1≤u≤ui)=qZ(u)−ki+1qZi+1(u) +kiqZi(u).
We can plug this equality into (2.3.9) to obtain ρ1ρ2(X) =ρ1
X
p≥1
(X−k2p)+−(X−k2p+1)+
+ρ2
X
p≥1
(X−k2p+1)+−(X−k2p+2)+
−X
p≥1
k2p+1ρ1(X > k2p+1)−k2pρ1(X > k2p)
−X
p≥1
k2p+2ρ2(X > k2p+2)−k2p+1ρ2(X > k2p+1)
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Where we denoted ρ(X > k) :=ρ(1{X>k}).
Finally, by noticing that for any i, ρ1(1(X>ki)) = g1(ui) = g2(ui) = ρ2(1(X>ki)), we obtain the
desired claim.
The last proof shows in particular that the pointskidefining the risk transfer are such thatρ1(X >
ki) =ρ2(X > ki). This is the counterpart onRof the equality at the pointsui belonging to[0,1]of the local distortion functions g1X and gX2 .
In Figure2.1below, we represented two possible functionsg1X andgX2 crossing twice on(0,1), and thus generating a layer as the optimal structure for the risk sharing. Assume that an insurer uses the local distortion atXcorresponding to the red curve, and that a reinsurer uses the one corresponding to the green curve.
In the interval(w,1), the reinsurer uses a more severe probability distortion, and thus all the risk is assumed by the insurer in that region, since (w,1)corresponds to losses X belonging to (0, k1), with k1 =qX(w). The same thing happens for the interval (0, v) corresponding to losses belonging to (k2,+∞), with of course k2 =qX(v). Finally, the reinsurer assumes all the losses belonging to (k1, k2), since he has a less severe distortion on the interval(v, w).
With the notations of the last proof,k1 and k2 are the only two finite points such that we have the
"agreement" ρ1(X > ki) =ρ2(X > ki).
Figure 2.1: Example of configuration leading to a layer as optimal risk transfer