On the characterization of premium principle with respect to pointwise comonotonicity
A premium principle is an economic decision rule used by the insurer in order to determine the amount of the net premium for each risk in his portfolio. In this paper we investigate the problem how to determine the premium principle to be used. In Goovaerts & Dhaene (1997), DTEW Research Report...
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irk-123456789-44552009-11-12T12:00:20Z On the characterization of premium principle with respect to pointwise comonotonicity Dhaene, J. Kukush, A. Pupashenko, M. A premium principle is an economic decision rule used by the insurer in order to determine the amount of the net premium for each risk in his portfolio. In this paper we investigate the problem how to determine the premium principle to be used. In Goovaerts & Dhaene (1997), DTEW Research Report 9740, K.U.Leuven, we can see some desirable properties of a premium principle. We consider a premium principle for risks of any sign, and prove a representation of premium principle without some property which involves the distribution of a risk. Later we introduce this property as a corollary. 2006 Article On the characterization of premium principle with respect to pointwise comonotonicity / J. Dhaene, A. Kukush, M. Pupashenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 26–42. — Бібліогр.: 4 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4455 en Інститут математики НАН України |
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A premium principle is an economic decision rule used by the insurer in order to determine the amount of the net premium for each risk in his portfolio. In this paper we investigate the problem how to determine the
premium principle to be used. In Goovaerts & Dhaene (1997), DTEW Research Report 9740, K.U.Leuven, we can see some desirable properties of a premium principle. We consider a premium principle for risks of
any sign, and prove a representation of premium principle without some property which involves the distribution of a risk. Later we introduce this property as a corollary. |
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Dhaene, J. Kukush, A. Pupashenko, M. |
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Dhaene, J. Kukush, A. Pupashenko, M. On the characterization of premium principle with respect to pointwise comonotonicity |
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Dhaene, J. Kukush, A. Pupashenko, M. |
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Dhaene, J. |
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On the characterization of premium principle with respect to pointwise comonotonicity |
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On the characterization of premium principle with respect to pointwise comonotonicity |
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On the characterization of premium principle with respect to pointwise comonotonicity |
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On the characterization of premium principle with respect to pointwise comonotonicity |
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On the characterization of premium principle with respect to pointwise comonotonicity |
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on the characterization of premium principle with respect to pointwise comonotonicity |
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On the characterization of premium principle with respect to pointwise comonotonicity / J. Dhaene, A. Kukush, M. Pupashenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 26–42. — Бібліогр.: 4 назв.— англ. |
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AT dhaenej onthecharacterizationofpremiumprinciplewithrespecttopointwisecomonotonicity AT kukusha onthecharacterizationofpremiumprinciplewithrespecttopointwisecomonotonicity AT pupashenkom onthecharacterizationofpremiumprinciplewithrespecttopointwisecomonotonicity |
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Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 26–42
JAN DHAENE, ALEXANDER KUKUSH, AND MYHAILO PUPASHENKO
ON THE CHARACTERIZATION OF PREMIUM
PRINCIPLE WITH RESPECT TO POINTWISE
COMONOTONICITY1
A premium principle is an economic decision rule used by the insurer in
order to determine the amount of the net premium for each risk in his
portfolio. In this paper we investigate the problem how to determine the
premium principle to be used. In Goovaerts & Dhaene (1997), DTEW
Research Report 9740, K.U.Leuven, we can see some desirable properties
of a premium principle. We consider a premium principle for risks of
any sign, and prove a representation of premium principle without some
property which involves the distribution of a risk. Later we introduce
this property as a corollary.
1. Introduction
Insurance contract can be seen as a risk-exchange between two parties, the
insurer and the policyholder. The insurer promises to pay for the financial
consequences of the claims produced by the insured risk. In return for
this coverage, the policyholder pays a fixed amount, called the premium.
Observe that the payments made by the insurer are random, while the
payments made by the policyholder are non-random.
The pure premium of the insured risk is defined as the expected value of
the claim amounts to be paid by the insurer. In practice the insurer will
add a risk loading to this pure premium. The sum of the pure premium
and the risk loading is called the net premium. Adding acquisition and
administration costs to this premium, one gets the gross premium.
In this paper we will investigate the problem of determining the net pre-
mium. We will assume that the insurer adopts some economic decision rule
to determine the amount of the net premium for each risk in his portfolio.
Such a principle is called a premium principle, see e.g. Kaas et al. (2001).
Several premium principles have been presented in the actuarial literature.
Wang (1996) introduced a new class of premium principles which, roughly
speaking, compute the net premium as the expectation of the risk under an
adjusted measure.
In Goovaerts & Dhaene (1997) we can see some desirable properties for
a premium principle. They assume that the net premium is not lower than
1Invited lecture.
2000 Mathematics Subject Classification. Primary 62P05.
Key words and phrases. Premium principle, risk, premium, comonotonicity, stochastic
dominance.
26
ON THE CHARACTERIZATION OF PREMIUM PRINCIPLE 27
the pure premium. We consider premium principle without this property,
and than introduce premium principle for risks of any sign. Also we show
that Greco’s Representation Theorem, see Dennenberg (1996), do not follow
from our statements, and vice versa.
2. Basic definitions for non-negative risks
We fix a probability space (Ω, �, P ) . A risk is a non-negative real-valued
random variable.
Definition 1. Two risks X and Y are 1-comonotonic if the set
AXY := {(X (ω) , Y (ω)) : ω ∈ Ω}
is comonotonic in R2.
Let Γ be a set of risks with such properties:
a) X ∈ Γ, d ≥ 0 ⇒ min (X, d) ∈ Γ, (X + d) ∈ Γ, and dX ∈ Γ;
b) X ∈ Γ, X (Ω) ⊂ [0, b] , b > 0 ⇒
for Xn :=
{
i
2n b, if i
2n b < X ≤ i+1
2n b, i = 0, 2n − 1
0, if X = 0
,n = 0, 1, 2, ...,
holds: Xn ∈ Γ.
c) A ∈ � ⇒ IA ∈ Γ.
Hereafter
IA = IA (ω) :=
{
1, if ω ∈ A
0, if ω ∈ Ω\A,
and
X (Ω) := {X (ω) : ω ∈ Ω} .
Definition 2. A premium principle is a functional H : Γ → [0,∞] . For
X ∈ Γ, H (X) is called the premium.
Remark 1. For risks with the same distribution, the premiums can be
different.
3. Properties of a premium principle
Property 1 X, Y ∈ Γ, X ≤ Y ⇒ H (X) ≤ H (Y ) .
Property 2 If X, Y ∈ Γ, X + Y ∈ Γ, and X, Y are 1-comonotonic, then
H (X + Y ) = H (X) + H (Y ) .
Property 3 H (1) = 1.
Property 4 X ∈ Γ ⇒ lim
d→+∞
H [min (X, d)] = H (X) .
Remark 2. Properties 1-3 imply that H (aX + b) = aH (X) + b, for all
a, b ≥ 0.
4. Layers and distortions
Definition 3. Let 0 ≤ a < b. A layer at (a, b) of a risk X is defined by
L(a,b) =
⎧⎨
⎩
0, if 0 ≤ X ≤ a
X − a, if a < X < b
b − a, if X ≥ b.
28 J. DHAENE, A. KUKUSH, AND M. PUPASHENKO
Definition 4. A function g : � → [0, 1] is called a distortion function if :
a) g (∅) = 0, g (Ω) = 1,
b) A, B ∈ �, A ⊂ B ⇒ g (A) ≤ g (B) .
Lemma 1. Let H be a premium principle satisfying Property 1-3, and
g (A) := H (IA) , A ∈ �.
Then g is a distortion function.
Proof.
g (∅) = H (I∅) = H (0) = 0,
see Property 2.
g (Ω) = H (IΩ) = H (1) = 1,
see Property 3.
A, B ∈ �, A ⊂ B ⇒ IA ≤ IB ⇒ g(A) ≤ g(B), see Property 1.
5. Characterization of premium principle
Let Q (ω) be a certain property of an elementary event ω ∈ Ω, which
can be satisfied or not. For a distortion function g we write for brevity
g {Q} = g {ω ∈ Ω : Q (ω)} holds. E.g., for a risk X and x > 0 we write
g {X > x} = g ({ω ∈ Ω : X (ω) > x}) .
Lemma 2. Let H be a premium principle with Properties 1-3. Then there
exists a unique distortion function g, such that for all discrete risky X ∈ Γ
with only finitely many mass points, we have that
(1) H (X) =
∫ ∞
0
g {X > x} dx.
Proof. Let X be discrete risk with finitely many mass points. Then for
certain n ≥ 0,
X =
n∑
0
xiIAi
,
where 0 = x0 < x1 < ... < xn, Ai ∈ �, {Ai} form a partition of Ω. Then (we
consider the case n ≥ 1 only)
L(x0,x1) = (x1 − x0) IA0
,
L(x1,x2) = (x2 − x1) IA0∪A1
,
. . .
L(xn−1,xn) = (xn − xn−1) IA0∪...∪An−1
.
We have
X =
n−1∑
i=0
L(xi,xi+1),
ON THE CHARACTERIZATION OF PREMIUM PRINCIPLE 29
and the layers L(xi,xi+1), i = 0, n − 1, are pairwise mutually 1-comonotonic
risks. Then by Property 2
H (X) =
n−1∑
i=0
H
(
L(xi,xi+1)
)
.
But
L(xi,xi+1) = (xi+1 − xi) I{X>xi}.
Then by Remark 2,
H
(
L(xi,xi+1)
)
= (xi+1 − xi) g {X > xi} ,
where
g (A) := H (IA) , A ∈ �.
Then
H (X) =
n−1∑
i=0
(xi+1 − xi) g {X > xi} =
n−1∑
i=0
∫ xi+1
xi
g {X > x} dx
=
∫ ∞
0
g {X > x} dx,
since
g (∅) = H (0) = 0
and
{X > x} = ∅
for x ≥ xn. By Lemma 1 g is a distortion function.
In representation (1) the function g : � → [0, 1] is unique, since for A ∈ �
and any g satisfying (1),
H (IA) =
∫ ∞
0
g (A) I[0,1) (x) dx = g (A) .
Theorem 1. Assume that the premium principle H satisfies Properties
1-4. Then there exists a unique function g : � → [0, 1] , such that for all
risks X ∈ Γ we have that
(1a) H (X) =
∫ ∞
0
g {X > x} dx.
Proof.
1◦ Let X ∈ Γ, X (Ω) ⊂ [0, b] , b > 0. For n ≥ 0, we use Xn defined in
Section 1. We have by Lemma 2
(2) H (Xn) =
∫ b
0
g {Xn > x} dx,
where g is a uniquely defined distortion. Now, Xn ≤ Xn+1 ≤ X,
and ∀ω ∈ Ω : Xn (ω) → X (ω) , as n → ∞. Therefore by Property 1,
H (Xn) ≤ H (Xn+1) ≤ H (X) .
30 J. DHAENE, A. KUKUSH, AND M. PUPASHENKO
Moreover
X ≤ Xn +
b
2n
⇒ H (X) ≤ H (Xn) +
b
2n
,
we used here Property 1 and Remark 2. Thus
(3) lim
n→∞
H (Xn) = H (X) .
Next,
{Xn > x} ↑ {X > x} ,
then
(4) g {Xn > x} ↑ g∗ (x) , g∗ (x) ≤ g {X > x} .
We used here part b) of definition 4. Next, the function
g∗ (x) := lim
n→∞
g {Xn > x} , x ≥ 0,
is non-increasing, therefore it has at most countable number of dis-
continuous points.
Let x be a point of continuity of the function g∗ (x) , x > 0. Then we will
show that g∗ (x) = g {X > x} .
Indeed, for 0 < ε < x, we have
{Xn > x} ⊂ {X > x} ⊂
{
Xn > x − b
2n
}
⊂ {Xn > x − ε} ,
for n ≥ nε. Therefore
g∗ (x) ≤ g {X > x} ≤ g∗ (x − ε) .
But due to our assumption g∗ (x − ε) → g∗ (x) , as ε → 0, therefore g∗ (x) =
g {X > x} , and we proved this equality.
Now, return to (4). We have
lim
n→∞
g {Xn > x} = g {X > x}
for all x ≥ 0 a.e. with respect to Lebesgue measure.
From this fact, (2) and (3) we obtain finally by the dominated convergence
theorem:
H (X) =
∫ b
0
g {X > x} dx.
2◦ Now let X be an arbitrary risk from Γ. For any d > 0 we have by
part 1 of the proof that
H [min (X, d)] =
∫ d
0
g {X > x} dx.
The desired result follows now from Property 4.
ON THE CHARACTERIZATION OF PREMIUM PRINCIPLE 31
6. Connection with Wang’s class
Introduce stronger property than Property 1.
Property 1’ X, Y ∈ Γ, X ≤st Y ⇒ H (X) ≤ H (Y ) .
Hereafter X ≤st Y means that X is stochastically dominated by Y.
Denote
P (�) = {P (A) : A ∈ �} .
Corollary 1. Assume that the premium principle H satisfies Properties 1’
and 2 to 4. Then the function g : � → [0, 1] in Theorem 1 has representation
(5) g (A) = g0 (P (A)) , A ∈ �,
where g0 : P (�) → [0, 1] is non-decreasing, with g0 (0) = 0, g0 (1) = 1.
Proof. Property 1’ implies Property 1, therefore the statement of Theorem
1 holds true for the premium principle H , and the function g : � → [0, 1]
in (1a) exists and unique.
Now let P (A) = P (B) . Then IA ≤st IB and IB ≤st IA. By Property 1’
we have H (IA) = H (IB) ,therefore g (A) = g (B) . Thus representation (5)
holds, and the properties of g0 follow from distortion properties of g.
Consider stronger property than Property 2.
Property 2’ If X, Y ∈ Γ, X + Y ∈ Γ, and X, Y are comonotonic (i.e.
∃Ω0, P (Ω0) = 1 : A0
XY := {(X (ω) , Y (ω)) : ω ∈ Ω0} is comonotonic in R2),
then H (X + Y ) = H (X) + H (Y ) .
Corollary 2. Assume that the premium principle H satisfies Property 1’
and 2 to 4. Then it satisfies Property 2’ as well, and for all risks X ∈ Γ we
have
(6) H (X) =
∫ ∞
0
g [SX (x)] dx,
where SX (x) := P {X > x} , g : P (�) → [0, 1] is non-decreasing, g (0) = 0,
g (1) = 1. Moreover the function g in representation (6) is unique.
Representation (6) and uniqueness of g follow from Corollary 2. Property
2’ then follows from (6), see Wang (1996).
7. Inverse statement
The inverse conclusion of Theorem 1 holds in the next theorem:
Theorem 2. The premium principle H : Γ → [0,∞] fulfills the Properties
1-4 if, and only if, there exists a distortion function g : � → [0, 1] , for
which:
(7) H (X) =
∫ ∞
0
g {X > x} dx, for any risk X ∈ Γ.
Moreover g (A) = H (IA) , A ∈ �.
Proof. If H fulfill Properties 1-4 then there exists a distortion function
g : � → [0, 1] , for which representation (7) is true, by Theorem 1.
32 J. DHAENE, A. KUKUSH, AND M. PUPASHENKO
Now let premium principle H : Γ → [0,∞] is defined by
H (X) =
∫ ∞
0
g {X > x} dx
for some distortion function g : � → [0, 1] . Then H fulfills the Properties
1-4.
Indeed, the proofs of Properties 1,3 and 4 are straightforward. Now, we
prove the key Property 2.
Let X, Y be 1-comonotonic risks from Γ. Then these exists a non-negative
r.v. Z and two non-decreasing functions f, h : [0,∞) → [0,∞), such that
X = f (Z) , Y = h (Z) . We have to prove that
(8)
∞∫
0
g {(f + h) (Z) > x} dx =
∞∫
0
g {f (Z) > x} dx+
∞∫
0
g {h (Z) > x} dx.
We do it in several steps.
1◦ It is enough to show (8) for bounded r.v. Z only. Indeed, for d > 0
define
Zd = min (d, Z) .
Then f (Zd) = min (f (d) , f (Z)) ∈ Γ, and h (Zd) ∈ Γ in a similar way.
Suppose that (8) holds for each bounded risk. Then
∞∫
0
g {(f + h) (Zd) > x} dx =
∞∫
0
g {f (Zd) > x} dx +
∞∫
0
g {h (Zd) > x} dx,
or
(f+h)(d)∫
0
g {(f + h) (Z) > x} dx =
f(d)∫
0
g {f (Z) > x} dx+
h(d)∫
0
g {h (Z) > x} dx.
Tending d → ∞, we immediately obtain (8).
2◦ Thus we suppose that Z if a bounded non-negative r.v. Assume that f
and h are strictly increasing. Then f +h is also strictly increasing. Consider
(9) I (f) :=
∫ ∞
0
g {f (Z) > x} dx.
Let a = f (0) , b = f (+∞) . Then
(10) I (f) = f (0) +
∫ b
a
g {f (Z) > x} dx.
Let 0 ≤ t1 < t2 < . . . be the set of all points of discontinuity of f ( it is
a finite or countable set).
Denote x−
i = f (ti−) , x+
i = f (ti+) , xi = f (ti) .
ON THE CHARACTERIZATION OF PREMIUM PRINCIPLE 33
If t1 = 0 then x−
1 := x1 = f (0) . We have from (10):
(11)
I (f) = f (0)
+
∑
i≥1
[(
xi − x−
i
)
g {f (Z) ≥ xi} +
(
x+
i − xi
)
g {f (Z) > xi}
]
+
∫
Af
g {f (Z) > x} dx,
where
Af := (a, b) � ∪
i≥1
[
x−
i , x+
i
]
.
Now we define a right-continuous modification of f ,
frc (t) =
{
f(t), t ≥ 0, t �= ti, i ≥ 1
f(t+), t = ti,i ≥ 1.
Denote
Ic(f) =
∫
Af
g {f (Z) > x} dx.
Then
(12) Ic(f) =
∫
Af
g {frc (Z) > x} dx =
∫
Af
g
{
Z > f−1
rc (x)
}
dx.
Here f−1
rc is inverse mapping for strictly increasing function frc. Change
of variables in Lebesgue integral (12) , t = f−1
rc (x) , leads to the following
representation
(13) Ic (f) =
∫
[0,∞) {ti,i≥1}
g {Z > t} dλfrc (t) ,
where λfrc is Lebesgue-Stiltjes measure on Borel σ-field ß([0,∞)) ,generated
by the function frc. Then
I(f) = f(0) +
∑
i≥1
[(
xi − x−
i
)
g {Z ≥ ti} +
(
x+
i − xi
)
g {Z > ti}
]
+
∫
[0,∞) {ti,i≥1}
g {Z > t} dλfrc (t) .(14)
But
Ic (f) =
∫
[0,∞)
g {Z > t} dλfrc (t) −
∑
i≥1
g {Z > ti}
(
x+
i − x−
i
)
.
We have(
xi − x−
i
)
g {Z ≥ ti} +
(
x+
i − xi
)
g {Z > ti} −
(
x+
i − x−
i
)
g {Z > ti}
=
(
xi − x−
i
)
(g {Z ≥ ti} − g {Z > ti}) .
34 J. DHAENE, A. KUKUSH, AND M. PUPASHENKO
Finally
I (f) = f (0) +
∑
i≥1
(
xi − x−
i
)
(g {Z ≥ ti} − g {Z > ti})(15)
+
∫
[0,∞)
g {Z > t} dλfrc (t) ,
xi = f (ti) , x−
i = f
(
t−i
)
.
Now we are able to show (8) for strictly increasing f and h. Let
0 ≤ u1 < u2 < . . . be the points of discontinuity of f + h. Then by (15)
I (f + h) = (f + h) (0) +
∑
i≥1
[
(f + h) (ui) − (f + h)
(
u−
i
)] ×(16)
(g {Z ≥ ui} − g {Z > ui}) +
∫
[0,∞)
g {Z > t} dλfrc+hrc (t) .
But λfrc+hrc = λfrc + λhrc,therefore (16) immediately implies
I (f + h) = I (f) + I (h) ,
since the set of discontinuity of f + h includes both sets of discontinuity of
f and h.
3◦ Now, Z is a bounded non-negative r.v., and f, ḣ are non-decreasing.
Let
fn (t) = f (t) +
t
n
, hn (t) = h (t) +
t
n
, t ∈ [0,∞) , n ≥ 1.
Then fn, hn are strictly increasing, and by part 2◦,
I (fn + hn) = I (fn) + I (hn) .
We show first that
lim
n→∞
I (fn) = I (f) .
Since Z is bounded, all these integrals equal the corresponding intervals on
[0, b] , with large enough b. Thus we have to show that
(17) lim
n→∞
∫ b
0
g
{
f (Z) +
Z
n
> x
}
dx =
∫ b
0
g {f (Z) > x} dx.
We have {
f (Z) +
Z
n
> x
}
↑ {f (Z) > x} .
Let
g∗ (x) = lim
n→∞
g {fn (Z) > x} .
The function g∗ is non-increasing. And similarly to part 1◦ of Theorem 1
we have the following: if g∗ is continuous at point x0 then
g∗ (x0) = g {f (Z) > x0} .
ON THE CHARACTERIZATION OF PREMIUM PRINCIPLE 35
But g∗ is continuous a.e. with respect to Lebesgue measure, since it is
monotone. Then
lim
n→∞
g {fn (Z) > x} = g {f (Z) > x} ,
x ∈ [0.b] , a.e. with respect to Lebesgue measure. Thus (17) holds from the
dominated convergence theorem. By part 1◦ relation (8) follows now for
any non-negative r.v. Z.
Example 1. Let {Pα, α ∈ I} be a family of probability measures on (Ω, �).
Let gα : [0, 1] → [0, 1] be non-decreasing function, for which gα (0) = 0,
and gα (1) = 1. Introduce two premium principles
H1 (X) :=
∫ ∞
0
sup
α∈ I
gα (Pα {X > x}) dx,
H2 (X) :=
∫ ∞
0
inf
α∈ I
gα (Pα {X > x}) dx; x ∈ Γ.
Then due to Theorem 2, both principles fulfill the Properties 1-4. We
mention that in general these principles do not have the form (6), i.e. they
depend not only on the distribution of X under the basic probability measure
P , but on the events {ω : X (ω) > x} , x > 0, as well.
Example 2. Another example of this kind could be
H3 (X) :=
∫
I
[∫ ∞
0
gα (Pα {X > x}) dx
]
dμ (α) ,
where μ is a probability measure on (I, �I) , where �I is a σ-field on I.
In this case we have to demand that for any X ∈ Γ the function
h (x, α) := gα (Pα {X > x}) , α ∈ I, x > 0, is measurable with respect to
the σ-field σ (S × �I), where S is Lebesgue σ-field on (0, +∞) .
For both examples, Property 2’ holds as well, if all the probabilities Pα
are absolutely continuous w.r.t. P . But Property 1’ need not hold for the
examples.
8. Greco’s Representation Theorem
Given a family Γ′ of functions X : Ω → R (here R = R ∪ {−∞, +∞})
and a functional H ′ : Γ′ → R, we list the properties of Γ′ and H ′ which play
a role in Greco’s Representation Theorem (GRT), stated in Dennenberg
(1996).
For Γ′ those are
a’) X ≥ 0 for all X ∈ Γ′;
b’) X ∈ Γ′, d ≥ 0 ⇒ min(X, d) ∈ Γ′, X − min(X, d) ∈ Γ′, and dX ∈ Γ′.
For H ′ the following conditions are relevant:
(i) X, Y ∈ Γ′, X ≤ Y ⇒ H ′(X) ≤ H ′(Y );
(ii) If X, Y ∈ Γ′, X + Y ∈ Γ′ and X, Y are 1-comonotonic, then
H ′ (X + Y ) = H ′ (X) + H ′ (Y ) ;
(iii) X ∈ Γ′, X ≥ 0 ⇒ lim
d↘0
H ′[X − min(X, d)] = H ′(X)
36 J. DHAENE, A. KUKUSH, AND M. PUPASHENKO
(iv) X ∈ Γ′ ⇒ lim
b→+∞
H ′[min(X, b)] = H ′(X)
Theorem 3(GRT). Given a family Γ′ of functions on Ω with properties
a’) and b’) and given a functional H ′ : Γ′ → R with properties (i)-(iv), then
there exists a monotone set function γ : 2Ω → R, such that for all X ∈ Γ′
we have that
(18) H ′(X) =
∫ ∞
0
γ{X > x}dx.
Remark 3. In GRT H ′(X) ≥ 0, for all risks X ∈ Γ′. This follows from
properties (i) and (ii).
Now we demonstrate with two examples, that properties of Γ in Definition
1 do not follow from properties of Γ′, listed in GRT, and vice versa.
Example 3. Let Ω = {ω}, � = 2Ω and Γ′ consists of one risk X = X(ω) =
0 ∈ Γ′, then Γ′ satisfies properties a’),b’) of GRT. But I{ω} = IΩ = 1 /∈ Γ′,
therefore Γ′ does not satisfy properties a)-c), listed in Definition 1.
Example 4. Let Ω = {ω1, ω2, ω3}, � = 2Ω. Introduce designation for risk
X = (X(ω1), X(ω2), X(ω3)) – well-ordered vector of values of risk X in
points ω1, ω2, ω3. First, ∀A ∈ �, IA ∈ Γ therefore (0, 0, 0), (1, 0, 0), (0, 1, 0),
(0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1) ∈ Γ. Then we get all risks X ∈ Γ,
so that Γ fulfills properties a) and b), listed in Definition 1. In that way
we get risks with one or two different values, because ∀d ≥ 0 all risks:
min(X, d), (X + d), dX, Xn (from Definition 1) have less or equal different
values then risk X. Later we add to Γ risk X0 with three values, for example
X0 = (x1, x2, x3), where 0 < x1 < x2 < x3 < ∞ and those risks, which a
necessary by properties a) and b) in Definition 1. It is easy to prove, that
for some d : x1 < d < x2, risk X1 = (0, x2 − d, x3 − d) does not belong to
Γ(use that we can get risk X1 only from risks with three values, and that for
getting X(ω1) = 0 we can use only min(X, 0) or 0 ·X, which do not lead us
to risk X1). But by property b’) from GRT we should have, that X1 ∈ Γ′,
because X1 = X0 − min(X0, d). Thus Γ fulfills properties from Definition 1
and does not fulfill properties from GRT.
9. Premium of risk of any sign
We fix a probability space (Ω, �, P ) , and in this section we call risk to
be any real-valued r.v. with finite mean. Thus negative risks are allowed.
Definition 1 is still valid.
Let Γ be a set of risks with such properties:
a1) X ∈ Γ, d ≥ 0 ⇒ (X + d) ∈ Γ, dX ∈ Γ
a2) Let d > 0. If X ∈ Γ, X ≥ 0 then min (X, d) ∈ Γ; if X ∈ Γ, X ≤ 0
then max (X,−d) ∈ Γ.
b1) Coincides with b) from Section 2.
b2) X ∈ Γ, Γ (Ω) ⊂ [−b, 0] , b > 0 ⇒ − (−X)n ∈ Γ,where (−X)n is
defined for non-negative r.v. (−X) as in property b) of Γ, see Section
2, where X is replaced to (−X) .
ON THE CHARACTERIZATION OF PREMIUM PRINCIPLE 37
c1) A ∈ � ⇒ IA, (−IA) ∈ Γ.
d1) X ∈ Γ ⇒ X+ ∈ Γ, (−X−) ∈ Γ.
Hereafter we use common notations X+ = max (X, 0), X− = −min (X, 0) .
Definition 5. A premium principle is a functional H : Γ → (−∞, +∞] .
We have to change Property 4.
Property 4’. If X ∈ Γ, X ≥ 0 then
lim
d→+∞
H [min (X, d)] = H (X) ,
and if X ∈ Γ, X ≤ 0 then
lim
d→+∞
H [max (X,−d)] = H (X) .
Introduce condition on a distortion function:
Condition 1. ∀X ∈ Γ :
∫ 0
−∞
(1 − g{X > x})dx < ∞.
Theorem 4. The premium principle H : Γ → (−∞, +∞] fulfills the Prop-
erties 1 to 3, and 4’ if, and only if, there exists a distortion functions
g : � → [0, 1] which satisfies Condition 1 and for all risks X ∈ Γ,
(19) H (X) = −
∫ 0
−∞
(1 − g {X > x}) dx +
∫ ∞
0
g {X > x} dx.
Moreover g (A) = H (IA) , A ∈ �.
Proof.
1◦ Let H fulfills the Properties 1 to 3, and 4’. We prove the represen-
tation (19).
Let X ∈ Γ. Then X = X+ + (−X−), and both X+ and (−X−)
belong to Γ. Moreover
X+ = f1 (X) ,−X− = f2 (X) ,
with non-decreasing functions f1 and f2. Therefore X+, (−X−) are
1-comonotonic. Then by Property 2
H (X) = H (X+) + H(−X−).
Now, X+ ∈ Γ+ := {Y+ : Y ∈ Γ} . The set of risks Γ+ satisfies prop-
erties a) and c) from Section 2, due to the properties of Γ listed
in this section. And H restricted to Γ+ satisfies Properties 1 to 4.
Therefore by Theorem 2 there exists a distortion g : � → [0, 1] ,
such that
H (X+) =
∫ ∞
0
g {X+ > x} dx,
But ∀x > 0 : {X+ > x} = {X > x} , therefore
H (X+) =
∫ ∞
0
g {X > x} dx.
Moreover g (A) = H (IA) , A ∈ �.
38 J. DHAENE, A. KUKUSH, AND M. PUPASHENKO
2◦ Next, consider the class of risks
Γ− := {Y− : Y ∈ Γ} .
Define
H1 (Y−) := −H (−Y−) , Y− ∈ Γ−.
The class Γ− satisfies properties a) to c) from Section 2. A premium
principle H1 : Γ− → [0, +∞) satisfies the following properties.
-Property 1 follows from Property 1 of H. Indeed, X− ≤ Y− ⇒
H (−X−) ≥ H (−Y−) ⇒ H1 (X−) ≤ H1 (Y−) .
-Properties 2 and 3 follow from Properties 2 and 3 of H.
-Property 4 follows from the second part of Property 4’ for H.
Then by the statement similar to Theorem 2 there exists a distor-
tion h, such that
H1 (U) =
∫ ∞
0
h {U > x} dx =
∫ ∞
0
h {U ≥ x} dx,
since h {U ≥ x} = h {U > x} a.e. with respect to Lebesgue measure,
see the proof of Theorem 1. Moreover h (A) = H1 (IA) , A ∈ �.
Then define g1 (A) = 1 − h
(
A
)
, A ∈ �. It is a distortion as well,
and
g1 (A) = 1 − H1 (IA) = 1 + H (−IA) = H (1 − IA)
= H (IA) = g (A) .
Therefore g1 = g, and
H1 (U) =
∫ ∞
0
(1 − g {U < x}) dx
=
∫ 0
−∞
(1 − g {−U > x}) dx.
Finally, for X ∈ Γ,
H (X) = H (−X−) + H (X+) = −H1 (X−) + H (X+)
= −
∫ 0
−∞
(1 − g {−X− > x}) dx + H (X+) ,
H (X) = −
∫ 0
−∞
(1 − g {X > x}) dx +
∫ ∞
0
g {X > x} dx,
since for x < 0, {X > x} = {−X− > x} .
3◦ Now, let H : Γ → (−∞, +∞] has representation (19), with a distor-
tion function g : � → [0, 1], satisfying Condition 1. Then equality
H (IA) = g (A) , A ∈ �, follows immediately, and the properties 1, 3
and 4’ are easily verified.
ON THE CHARACTERIZATION OF PREMIUM PRINCIPLE 39
Prove the Property 2 for H. The right hand side of (19) is defined
for all risks, there fore we can assume now that H is defined by (19)
for all risks, not only for the risks for Γ.
Let X, Y be two 1-comonotonic risks. First consider the case
X ≥ −d, Y ≥ −d, with fixed positive d. Then since (19) implies
H (Y + c) = H (Y ) + c, c ∈ R, for any risk Y , we have
H (X + Y ) = H ((X + d) + (Y + d)) − 2d.
But X + d and Y + d are non-negative 1-comonotomic risks. For
any non-negative risk Y,
H (Y ) =
∫ ∞
0
g {Y > x} dx =: G (Y ) .
But the functional G fulfills Property 2 for non-negative risks, due
to Theorem 2. Then
H (X + Y ) = H (X + d) + H (Y + d) − 2d
= H (X) + d + H (Y ) + d − 2d
= H (X) + H (Y ) ,
and we showed Property 2 for bounded from below risks.
Let now X, Y be arbitrary 1-comonotonic risks. Then X = f (Z),
Y = h (Z) , where Z is a r.v., and f, h are non-decreasing. We fix
c ∈ R and for any r.v.U define Uc = max (U, c) . Now, f (Zc) and
h (Zc) are bounded from below 1-comonotonic risks, then
H (f (Zc) + h (Zc)) = H (f (Zc)) + H (h (Zc)) ,
(20) H ((f + h) (Zc)) = H (f (Zc)) + H (h (Zc)) .
Let α (Z) be a risk, with non-decreasing function α : R → R. Then
α (Zα) = (α (Z))b , with b := α (c) . And due to the integral repre-
sentation (19),
lim
c→−∞
H (α (Zc)) = H (α (Z)) .
Now, in (20) we tend c → −∞ and obtain
H ((f + h) (Z)) = H (f (Z)) + H (h (Z)) .
Now, instead of Property 1 we consider stronger Property 1’. Introduce
condition on function g0 : P (�) → [0, 1] much as Condition 1 on distortion
function.
Condition 2. ∀X ∈ Γ :
∫ 0
−∞
(1 − g0(SX(x)))dx < ∞.
Corollary 3. The premium principle H : Γ → (−∞, +∞] fulfills the
Properties 1’, 2, 3 and 4’ if, and only if, these exists a function
g0 : P (�) → [0, 1] ,
40 J. DHAENE, A. KUKUSH, AND M. PUPASHENKO
which satisfies Condition 2, g0 (0) = 0, g0 (1) = 1, g0 is non-decreasing, and
such that for any risk X ∈ Γ,
(21) H (X) = −
∫ 0
−∞
(1 − g0 (SX (x))) dx +
∫ ∞
0
g0 (SX (x)) dx,
where SX (x) := P {X > x} , x ∈ R. Moreover g0 (q) = H (Bq) , q ∈ P (�) ,
where Bq is Bernoulli r.v. with parameter q.
Proof. Let H satisfies the properties listed above. Then weaker Property 1
holds as well, and representation (19) holds, with
g (A) = H (IA) .
But IA
d
= Bq, with q := P (A) . Thus g (A) depends only on q = P (A) ,
g (A) = g0 (P (A)) , and (21) follows. The desired properties of g0 follow
from corresponding properties of g.
Next, let representation (21) holds for the principle H . Then (19) holds,
with g (A) := g0 (P (A)) , A ∈ �. The distortion properties of g follow from
the properties of g0. Then by Theorem 4, H fulfills Properties 1 to 3, and
4’. The Property 1’ follows now from Property 1, since H (X) in (21) is
determined by the distribution of X.
Remark 4. For the premium principle (19), defined for all risks X, we
show the property:
(22) H (X − d) = H (X) − d, d > 0,
which was used in part 3◦ of the proof of Theorem 4. We have
H (X − d) = −
∫ 0
−∞
(1 − g {X > x + d}) dx +
∫ ∞
0
g {X > x + d} dx
= −
∫ d
−∞
(1 − g {X > t}) dt +
∫ ∞
d
g {X > t} dt
= −
∫ 0
−∞
(1 − g {X > t}) dt +
∫ ∞
0
g {X > t} dt
−
∫ d
0
(1 − g {X > t}) −
∫ d
0
g {X > t} dt
= H (X) − d.
Thus (22) is proven.
10. Corollary
Introduce an important property of a premium principle:
Property 5 X ∈ Γ ⇒ H(X) ≥ EX.
This property means that the net premium is not lower than the pure pre-
mium.
ON THE CHARACTERIZATION OF PREMIUM PRINCIPLE 41
Lemma 3. Let H be a premium principle satisfying Property 1-3,5, and
g(A) := H(IA), A ∈ �.
Then g is a distortion function, moreover g(A) ≥ P (A), for all A ∈ �.
Proof. Distortion properties of g follow from Lemma 1.
Let A ∈ �, then
g(A) = H(IA) ≥ EIA = P (A).
Now we have the next corollary of Theorem 2.
Theorem 5. The premium principle H : Γ → [0,∞] fulfills the Properties
1-5 if, and only if, there exists a distortion function g : � → [0, 1] , which
satisfies:
(1) H (X) =
∫ ∞
0
g {X > x} dx, for any risk X ∈ Γ,
(2) g (A) ≥ P (A) , for any A ∈ �.
Moreover g (A) = H (IA) , A ∈ �.
Now return to Examples 1,2. Introduce property of {Pα, α ∈ I} :
Pα (A) ≥ CαP (A) ,
for all α ∈ Γ, A ∈ �.
Here 0 < Cα ≤ 1. We mention that if Cα0 = 1 then Pα0 = P.
Let gα : [0, 1] → [0, 1] be non-decreasing function, for which
gα (0) = 0, gα (1) = 1,
and gα (q) ≥ min (1, q/Cα) for all q ∈ [0, 1] . Then due to Theorem 5, prin-
ciples H1, H2, H3 fulfill the Properties 1-5.
Now we have the following corollary of Theorem 4.
Theorem 6. The premium principle H : Γ → (−∞, +∞] fulfills the Prop-
erties 1, 2, 3’, 4’, and 5 if, and only if, there exists a distortion functions
g : � → [0, 1] which satisfies:
(23) a) H (X) = −
∫ 0
−∞
(1 − g {X > x}) dx +
∫ ∞
0
g {X > x} dx,
for any risk X ∈ Γ,
(24) b) g (A) ≥ P (A) ,
for any A ∈ �. Moreover g (A) = H (IA) , A ∈ �.
Finally we modify Corollary 3.
Corollary 4. The premium principle H : Γ → (−∞, +∞] fulfills the
Properties 1’, 2, 3’, 4’, and 5 if, and only if, these exists a function
g0 : P (�) → [0, 1] ,
g0 (0) = 0, g0 (1) = 1, g0 is non-decreasing, g0 (q) ≥ q, q ∈ P (�) , such that
for any risk X ∈ Γ,
(25) H (X) = −
∫ 0
−∞
(1 − g0 (SX (x))) dx +
∫ ∞
0
g0 (SX (x)) dx,
42 J. DHAENE, A. KUKUSH, AND M. PUPASHENKO
where SX (x) := P {X > x} , x ∈ R. Moreover g0 (q) = H (Bq) , q ∈ P (�) ,
where Bq is Bernoulli r.v. with parameter q.
References
1. Wang, S.S., Premium calculations by transforming the layer premium density.
ASTIN Bulletin, 26, (1996), 71–92.
2. Goovaerts, M.J. and Dhaene, J., On the characterization of Wang’s class of
premium principle. DTEW Research Report 9740, K.U.Leuven, (1997).
3. Denneberg, D., Non-Additive Measure and Integral. Kluwer Academic Publish-
ers, Boston, (1996).
4. Kaas, R., Goovaerts, M.J., Dhaene, J. and Denuit, M., Modern Acturial Risk
Theory. Kluwer Academic Publishers, Boston, (2001).
Department of Applied Economics, K.U.Leuven, Naamsestraat 69, B-3000
Leuven, Belgium
E-mail address: jan.dhaene@econ.kuleuven.be
Department of Probability Theory and Mathematical Statistics, Kyiv
National Taras Shevchenko University, Vladimirskaya st.64, 01033 Kyiv,
Ukraine
E-mail address: alexander kukush@univ.kiev.ua
Department of Mechanics and Mathematics, Kyiv National Taras Shev-
chenko University, Vladimirskaya st.64, 01033 Kyiv, Ukraine
E-mail address: myhailo.pupashenko@gmail.com
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