Mathematical and Statistical Methods for Actuarial Sciences and Finance

58 A. Campana and P. Ferretti

Definition 1.Let g 1 and g 2 be distortion functions. The initial risk-adjusted premium
P is the unique initial premium, if it does exist, for which the distorted expected value
of the total premium incomeδ(P)equals the distorted expected value of the aggregate
claims S, that is
Wg 1 (δ(P))=Wg 2 (S). (14)
Equation (14) may be studied from several different perspectives, mostly con-
cerned with the existence and uniqueness of the solutions. The next result presents a
set of conditions ensuring a positive answer to both these questions: the choice of an
excess of loss reinsurance for the layermxsdwith no aggregate deductibleDand
Kreinstatements plays the leading role.

Proposition 1.Given an XL reinsurance with K reinstatements and no aggregate
deductible and given two distortion functions g 1 and g 2 , the initial risk-adjusted
premium P results to be a function of the percentages of reinstatement c 1 ,c 2 ,...,cK.
Moreover, it satisfies the following properties:

i) P is a decreasing function of each percentage of reinstatement ci(i= 1 ,...,K);
ii) P is a convex, supermodular, quasiconcave and quasiconvex function of the per-
centages of reinstatement c 1 ,c 2 ,...,cK.

Proof.Given the equilibrium condition between the distorted expected premium in-
come and the distorted expected claim payments (14), the initial risk-adjusted pre-
miumPis well defined: in fact equation (14) admits a solution which is unique.
Since the layersLX(im,(i+ 1 )m),i= 1 , 2 ,...,K+1, are comonotonic risks
from (7) we find

Wg 2 (S)=

∑K

i= 0

Wg 2 (LX(im,(i+ 1 )m)). (15)

From (5), by assuming the absence of an aggregate deductible (i.e. ,D=0), we have

Wg 1 (δ(P))=P

(

1 +

1

m

K∑− 1

i= 0

ci+ 1 Wg 1 (LX(im,(i+ 1 )m))

)

. (16)

Therefore, the initial premiumPis well defined and it is given by

P=

∑K

i= 0 Wg 2 (LX(im,(i+^1 )m))

1 +m^1

∑K− 1

i= 0 ci+^1 Wg 1 (LX(im,(i+^1 )m))

. (17)

The initial risk-adjusted premiumPmay be considered a function of the percen-
tages of reinstatementc 1 ,c 2 ,...,cK.LetP=f(c 1 ,c 2 ,···,cK).
Clearly the functionfis a decreasing function of any percentage of reinstatement
ci(wherei= 1 ,...,K).

Moreover, if we set

A=

∑K

i= 0

Wg 2 (LX(im,(i+ 1 )m)),