Mathematical and Statistical Methods for Actuarial Sciences and Finance

Distortion risk measures,XLreinsurance and reinstatements 57

where the distortion functiongis defined as a non-decreasing functiong:[0,1]→
[0,1] such thatg( 0 )=0andg( 1 )=1. As is well known, thequantile risk measure
and theTail Value-at-Riskare examples of risk measures belonging to this class. In
the particular case of a powergfunction, i.e.,g(x)=x^1 /ρ,ρ≥1, the corresponding
risk measure is thePH-transform risk measure, which is the choice made by Walhin
and Paris [6].
Distortion risk measures satisfy the following properties (see Wang [7] and Dhaene
et al. [2]):

P1. Additivity for comonotonic risks

Wg(Sc)=

∑n

i= 1

Wg(Xi) (10)

whereScis the sum of the components of the random vectorXcwith the same
marginal distributions ofXand with the comonotonic dependence structure.
P2. Positive homogeneity

Wg(aX)=aWg(X) for any non-negative constanta; (11)

P3. Translation invariance

Wg(X+b)=Wg(X)+b for any constantb; (12)

P4. Monotonicity
Wg(X)≤Wg(Y) (13)
for any two random variablesXandYwhereX≤Ywith probability 1.

In the particular case of a concave distortion measure, the related distortion risk
measure satisfying propertiesP1-P4is also sub-additive and it preserves stop-loss
order. It is well known that examples of concave distortion risk measures are theTa i l
Value-at-Riskand thePH-transform risk measure, whereasquantile risk measureis
not a concave risk measure.

4 Risk-adjusted premiums

In equation (8) the expected total premium income is set equal to the expected aggre-
gate claims payments: in order to refer to a class of premium principles that is more
general than the pure premium principle, we consider a new expected value condition
with reference to the class of distortion risk measures.
We impose that the distorted expected value of the total premium incomeδ(P)
equals the distorted expected value of the aggregate claimsS, given two distortion
functionsg 1 andg 2. Note that in our proposal it is possible to consider distortion
functions that are not necessarily the same.
The equilibrium condition may be studied as an equation on the initial premium
P: if it admits a solution which is unique, then we callinitial risk-adjusted premium
the corresponding premiumP. This is formalised in the following definition.