Positive homogeneity and multiplicativity of premium principles on positive risks

Insurance: Mathematics North-Holland

and Economics

8 (1989) 315-319

315

Positive homogeneity and multiplicativi of premium principles on Dositive risks Klaus

Received

We denote:

D. SCHMIDT

Fakuhiil fir Marhematik und Informarik, Universitiit heim, A.5. D-6800 Mannheim, West Germany 20 February

Mann-

Lx = the collection

of all risks, of all positive risks, Lz = the collection of all (positive) risks which are bounded away from 0, and 0: = the collection of all positive Bernoulli risks. LT = the collection

1989

In this note it is shown that the Swiss premium principle on positive risks is, under a mild regularity condition, positively homogeneous or multiplicative if and only if it is the net premium principle. A similar result is obtained for a generalized form of the Orlicz principle.

For a set B c L”, a functional B + R is said to be a premium principle on B. A premium principle H is multiplicative

independent Keywor& Swiss premium principle. Orlicz principle, tive homogeneity, Multiplicativity, Positive risks.

Posi-

positively

if H( XY) X, YE B,

homogeneous

all c~tR+

and

= H( X) H( Y)

for all

if H( cX) = cH( X)

for

XE B,

additive

if H( X + Y) = H(X) + H(Y) for all independent X, YE B, and translative if H( c + X) = c + H(X) for all c E Iw and XE B.

1. Introduction For a variety of premium principles, the consequences of homogeneity and multiplicativity have been studied by Goovaerts, De Vylder and Haezendonck (1984) and Reich (1984,1985). They proved that in many cases these requirements force the premium principle under consideration to reduce to the net premium principle. Some of their results, however, have only been obtained under the assumption of homogeneity or multiplicativity on arbitrary risks, whereas the risks encountered in practice are usually positive. In this note we shall study the impact of positive homogeneity and multiplicativity on positive risks for the Swiss premium principle and a generalized form of the Orlicz principle. Let us now fix some notation: A random variable X defined on this probability space with probability measure P is said to be a risk if there exist a, b E R satisfying P(a I XI b) = 1. A risk X is - positive if P(0 5 X) = 1, - bounded away from 0 if there exists some a E _ (0, CC) satisfying P(a I X) = 1, and - a Bernoulli risk if it assumes at most two distinct values with positive probability. 0167-6687/89/$3.50

Q 1989, Elsevier Science Publishers

The Swiss premium principle For /3 E [0, l] and a continuous strictly monotone function 9: Iw + 08, the functional H: Lx-R given by

E[+(X- PH( X))] = +((l - P)H( X)) is said to be the Swiss premium principle with respect to p and 9. Without loss of generality, we may and do assume that + is strictly increasing and satisfies ~(0) = 0. For /3 = 0, the Swiss premium principle reduces to the mean value principle with respect to 9, with H(X)

=+-‘(E[+(X)]);

for p = 1, it is equivalent with the principle of zero utility with respect to the (strictly increasing) function 1c,: R --, Iw given by q(z) := -+( - z), which is concave if and only if + is convex. Both of these premium principles include the exponential principle with respect to LYE Iw+, with

B.V. (North-Holland)

IE[Xl

if

cr=o

if

(Y> 0,

316

K. D. Schmdt

/ Premium

which, for any j3 E [0, 11, is obtained by letting +(-_):=zifa=Oand+(z):=(~-‘(ea’-l)ifa> 0. The exponential principle with respect to (Y= 0 is said to be the net premium principle.

pnnciples

on positice

risks

for all : E R, sufficiently small. The assertion now follows from ~(0) = 0 and o’(O) = 1. (B) If p=

1, then @(cz) = c@(z)

To prove (B), define, Lemma

2.1.

Zf p E [0, 1) and 4: R ---)R is an

arbitrary function satisfying E[G(X-PH(X))]

=+((I

for all X E Dt,

-P)H(X))

then there exist a, b E R satisfying

&:=

G(z) G(z)-$3(-l)

and

X, E 0:

for all z E R,.

for each z E R +,

by letting

P( x, = 0) := e,

+(z)=u+b$(z)

and

forallrER_.

P(X,=l+z):=l-6,. Lemma 2.1 can be proven in the same way as the corresponding result without the positivity condition; see Goovaerts, DeVylder and Haezendonck (1984. Theorem 2 of Section 2.10).

From (1) we obtain

H( X,) = 1. By (2), this yields

for all z E [w+, and hence Theorem 2.2.

If C#I is differentiable

in a neighbour-

hood of 0 and satisfies ~‘(0) = 1, then the following are equivalent: (a)

H is the net premium

principle

(b)

H is multiplicative on LT.

(c)

H is positively homogeneous

for all z E R + sufficiently small. now follows from +‘(O) = 1.

on LT.

on LT.

(C) ff /3 E (0, 11, then G(C)

N(X))]

= #-

P>H(X))

assertion

= &I(-_) for all : c

I--1,01.

Proof. It is sufficient to prove that (c) implies (a). To this end, let us assume that (c) holds. Then we have E[+(X-

The

To prove (C), define,

for each : E [ - 1, 0).

(1)

and, by hypothesis, E[+(c(X-PH(X)))]

=+@(I

-PMX))

(2)

for all XE 0: and c E R +. In order to prove that (a) holds, we first establish some auxiliary results. Consider c E Iw+. (A) Zf j? E [0, l),

P(X,=1+z):=6Jz and P

x:=1+1 i

To prove (A), define a function 4 : R +R letting 4(z) := +(cz). By (2), we have =+((l

by

-P)H(X))

for all XE Di, and it now follows from Lemma 2.1 that there exist Q, b E !R satisfying +(cz)=u+b~(z)

for all z E WT, and hence c+‘( cz) = b+‘(z)

by letting

then C$(CZ) = CC#B(.Z) for all t E

[w+-

E[#(X-@H(X))]

Xz E 03

and

From yields

:=1-e,. P i

(1) we obtain

H( X2) = l//3.

By (2), this

9(c4[0($)-0(+-l)] ++#o-1)-o(l)] =O(c(~-l))[~(f)-9(z)].

317

K.D. Schmidt / Premium principles on posltice risks

Since l/p E R + and l/p - 1 E R +, the assertion now follows from (A) and (B). We are now in a position to prove that (a) holds. In the case p = 0, assertion (A) yields

The following lemma relates each Orlicz principle to a Swiss premium principle:

+(x> = 41)

e(y)

for all x E R +, and hence G’(x)

Then,

small.

that

H is the

(A) and

(B)

(P(x) =x9(1)

@ : 02 --) R by let-

- 1.

for each X E Lz

P

(C) yields

and p E (0. x).

the follow-

=(-x)+(--l) Since C$is differentiable

9(l)

-1).

at 0,

=x that

- P) ln P>.

Since each Swiss premium principle is defined on Lx. Lemma 3.1 yields L~c B(P. p). 3.2.

If /3 E [0, 1) and p is differentiable are equivalent:

(a)

H is the net premium

(b)

H is multiplicative.

(c)

H is positive& homogeneous.

H is the

p(e’)

KY):= Then with

3. The Orlicz principle For /? E [0, l] and a continuous ing function p : R + + R +, H : B(j?, p) --) (0, 00) given by

P>] = +((I

principle

on LC.

Proof. It is sufficient to prove that (c) implies (a). To this end, assume that (c) holds and define a function Q: DB+ Iw by letting

It now follows from 6’(O) = 1 that

holds for all x E R, which implies net premium principle. Cl

=P(P'+).

with p’ > 0, then the following

for all xElR\lR+. we obtain = G’(O) = -+(

$

(b) E[@(lnX-P ln

Theorem

for all x E R +, and assertion

EP ii H(X)’)I

:= p(e!)

(0) E

holds for all x E R +, which implies net premium principle on LT. In the case fi E (0, 11, assertions yield

X

a function

[r)I

It now follows

(P(x) =x

6)

Define

ing are equivalent:

=9(l)

for all x E R + sufficiently from $‘(O) = 1 that

e(x)

Lemma 3.1. ting

strictly increasthe functional

=p(H(X)‘-‘)

is said to be the (general) Orlicz principle with respect to p and p, where B( p, p) is defined to be the collection of all positive risks for which the above equation has a (unique) solution. Without loss of generality, we may and do assume that p satisfies p(O) = 0 and p(l) = 1. For j3 = 0, the Orlicz principle reduces to the mean value principle on positive risks; for p = 1, we obtain the special case which is usually studied in the literature. -

.

+ is strictly

@‘(VI=

p’(e’)

in particular, well as P(Z)

- 1

p’(l)

=

increasing

and

differentiable

e” , o.

p’(l)

’

we have ~(0) = 0 and

0 t p’(l)+(ln

z) + 1

if if

O’(0) = 1, as

z=O z > 0.

By Lemma 3.1, the Orlicz principle with respect to j3 and p corresponds to the Swiss premium principle with respect to p and +, and an easy computation shows that the Orlicz principle is positively homogeneous on Lg if and only if the corresponding Swiss premium principle is translative (on L”). Therefore, we have either G(Y) =Y

K. D. Schmidt / Premium princtples on positice risks

318

with respect to /? and case we have either

or

q(y) =

i(eUi-1)

9(4’)

for some (YE (0. x): see Goovaerts, DeVylder and Haezendonck (1984; Theorem 4 of Section 3.3). In the case +(y) =y, we obtain P(L) =

1

0 P’(l)

if if

ln z+l

p(z)

=

P’(l);(z” i

-l)+l

hence p’(l) = (Y,by continuity, p(z)

and in this

=>

or +(_y) = i(eay-

1)

for some a E (0, 30); for a proof of this result, see e.g. Mammitzsch (1986). The assertion now follows as in the proof of Theorem 3.2. Cl

== 0 : > 0,

which contradicts the continuity of p. In the case 9(y) = a-‘(eaJ - 1). we obtain 0

Q is additive,

if

r=O

if

z>O,

We remark that in the hypothesis of Theorem 3.3 the inequality involving p” is fulfilled whenever p is convex.

4. Remarks

and thus

=zU.

For (Y# 1, the derivative of p at z = 0 is either equal to 0 or does not exist, which contradicts the assumption on p. For CY= 1, we obtain P(Z) = z,

In Theorems 2.2, 3.2, and 3.3, each of the equivalent properties may be replaced by the corresponding one for positive Bernoulli risks; this is immediate from the proofs. For (uER,, the functional H: L” ---)R given by

which implies that H is the net premium on Ly. 0

principle H(X)

:=

E[ Xe,*] E[eaX]

For j3 = 1, the Orlicz principle is always positively homogeneous (which corresponds to the fact that in this case the Swiss premium principle is always translative), and we have the following result on multiplicativity: Theorem 3.3. differentiable

If /3 = 1 and p is twice continuously with

p’ >

0

and such

that p’(z) +

:p”( I) r 0 holds for all z E R +, then the following are equivalent : (a) (6)

H is the net premium H is multiplicative.

principle

is said to be the Esscher principle with respect to 0~. Formally, the Esscher principle can be obtained from the defining equation of the Swiss premium principle, with j3 = 1 and +: W + R given by e(z) := zea’, but for cy # 0 the function 0 is not monotone. For the Esscher principle, we have the following analogue of Theorem 2.2:

on Lz.

Proof. It is sufficient to prove that (b) implies (a). To this end, assume that (b) holds and define a function Q: R ---,!R as in the proof of Theorem 3.2. Then 9 is strictly increasing and twice continuously differentiable with +’ > 0 and +” 2 0; in particular, we have r+(O) and o’(O) = 1. As in the proof of Theorem 3.2, we conclude that the Orlicz principle with respect to j? and 9 is multiplicative on Lz if and only if the Swiss premium principle -

Theorem 4.1.

The following

are equivalent:

(a)

H is the net premium

(b)

H is multiplicative

principle.

(c)

H is positively homogeneous

on LT. on Lz.

The proof of Theorem 4.1 is straightforward. The general form of the Orlicz principle considered in Section 3 seems to be new; it is the obvious multiplicative analogue of the Swiss premium principle. As we have shown in Lemma 3.1 and in the proofs of Theorems 3.2 and 3.3, the function 0: R + R given by G(V) := p(eY) - 1 or +(y) := (p(e!) - 1)/p’(l) provides a useful tool to deduce properties of the Orlicz principle with re-

K. D. Schmrdt / Premwm

spect to p and p from corresponding a Swiss premium principle.

properties

of

Acknowledgement I would like to thank Professor A. Reich for his comments on an earlier version of this paper.

References Goovaerts, M.J., F. DeVylder and J. Haezendonck (1984). Insurance Premiums. North-Holland, Amsterdam-New York.

principles

on positice ruks

319

Mammitzsch. V. (1986). A rigorous proof of a property of the premium principle of zero utility in the case of additivity. In: Insurance and Risk Theo7 (Maratea 1985). Reidel. Dordrecht-Boston. MA. 189-194. Reich, A. (1984). Homogeneous premium calculation principies. ASTIN Bulletin 14. 123-133. Reich, A. (1985). Eine Charakterisierung des Standardabweichungsprinzips. Bltitrer DGVM 17, 93-102.