MARKET PREMIUMS BY THE P-NORM PRINCIPLE Werner Hürlimann, Switzerland
Summary. Lp-space probabilistic techniques are applied to the pricing of risk-exchanges in Insurance and Finance, where the underlying claims have a finite mean but may have an infinite variance. A p-norm pricing principle is introduced and justified on the basis of four arguments. The prices of splitting components of a Pareto-optimal risk-exchange follow Lp-space CAPM relations of the type first proposed by Borch(1982) in the special case p=2. The present pricing model is derived using alternatively a first order Taylor approximation, the Lp-space Hölder inequality, a probabilistic induction procedure, and an utility theoretical approach. The p-norm pricing principle may be invoked to motivate a financial Lp-space CAPM, which should be useful in case return distributions have finite mean but infinite variance. We apply the p-norm pricing principle to experience rating contracts, which offer besides claim payments a bonus or dividend provided the financial gain is positive. A pricing methodology of the type considered first in Hürlimann(1994) is revisited, simplified and generalized to an environment of arbitrary p-integrable risks. Moreover, a Lp-space version of the equilibrium market relations (4.16), (4.17) in Hürlimann(1994) is obtained. It generalizes the previous result in two directions. It is valid for p-integrable random variables, and it can be applied to arbitrary Paretooptimal risk-exchanges, and not just to perfectly hedged experience rating contracts as in the previous work.
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1. Introduction. Most studies about insurance premiums and financial prices in a competitive equilibrium rely on Hilbert space theory, that is on linear spaces of random variables with finite first and second order moments. On the other side there exists a huge of financial literature, which support the use of stochastic models with infinite variance. At an early stage Mandelbrot(1963) and Fama(1963/65) have shown that stock market prices should follow Paretian stable distributions rather than normal distributions. Besides these empirical findings, there exist also rational arguments in favour of infinite variance risk models, e.g. Aebi et al.(1992) for non-life insurance and Hürlimann(1997) for finance. Despite strong evidence for such models, their practical use has not gained wide acceptance. However, the abundance of recent research in this area is likely to help change the attitude of practitioners. In the present paper, we apply Lp-space probabilistic techniques (a summary is given in Section 2) to the pricing of risk-exchanges in Insurance and Finance, where the underlying claims have a finite mean but may have an infinite variance. In Section 3, a p-norm pricing principle is introduced and justified on the basis of four different arguments. It acts on the splitting components of a Pareto-optimal risk-exchange according to Lp-space CAPM relations of the type first proposed by Borch(1982) in the special case p=2. The present pricing model is derived using alternatively a first order Taylor approximation (Section 3.1), the Lp-space Hölder inequality (Section 3.2), a probabilistic induction (Section 3.3), and an utility theoretical approach (Section 3.4). The p-norm pricing principle may be invoked to motivate a financial Lp-space CAPM, which should be useful in case return distributions have finite mean but infinite variance. Section 4 concerns the application of the p-norm pricing principle to experience rating contracts, which offer besides claim payments a bonus or dividend provided the financial gain is positive. A pricing methodology of the type considered first in Hürlimann(1994) is revisited, simplified and generalized to an environment of arbitrary p-integrable risks. In Section 5, a Lp-space version of the equilibrium market relations (4.16), (4.17) in Hürlimann(1994) is obtained. It generalizes the previous result in two directions. It is valid for p-integrable random variables, and it can be applied to arbitrary Pareto-optimal risk-exchanges, and not just to perfectly hedged experience rating contracts as in the previous work. Finally, Section 6 considers distribution-free diatomic conservative approximations to Lp-equilibrium market premiums. It is an attempt to generalize results obtained previously in Hürlimann(1994), Section 5.
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2. Lp-space probabilistic modelling. A statistical reason for the reluctance to apply infinite variance stochastic models is the difficulty to define adequate measures of dependence (in particular correlation) for distributions with infinite variance. The convenient mathematical structure of Hilbert L2-space is no more available. However, as a feasible substitute, it is possible to consider appropriate Banach Lp-spaces, 1≤p<2 (if p≥2 the Hilbert second order theory is applicable). Our setting has been partly inspired by the exposé of Gamrowski and Rachev(1994). In the following, let 1
p
< ∞ defined on some underlying
probability space ( Ω , A , P ) . The mean µ X = E X , X ∈ Lp , is assumed to be always λ
finite. For a positive real number λ let x λ = x ⋅ sgn ( x ) be the signed power function. If X , Y ∈ Lp one defines a p-product by (2.1)
X, Y
p
= E ( X − µ X ) ⋅ (Y − µ Y )
p −1
.
It induces the notion of p-norm (2.2)
X
p p
= X,X
p
[
= E X − µX
p
].
A p-scalar product is obtained by setting (2.3)
( X , X ) p = µ X ⋅ µY +
X,X
p
.
The p-product, often called p-covariation, is a simple generalization of the notion of covariance obtained in case p=2. It is especially useful in the study of symmetric stable processes (e.g. Kanter(1972), Cambanis and Miller(1981), Samorodnitsky and Taqqu(1991/94)). The p-product, p>1, satisfies the following properties, which will be freely used throughout : (P1)
(P2)
1 ∂ λX + µY ⋅ λ = 0 , µ =1 p ∂λ (directional derivative representation) X, Y
p
=
λX + µY , Z
p
= λ X, Z
p
(linearity in first argument)
+ µ Y, Z
p
4
X , λY
(P3)
= λ p −1 ⋅ X , Y
p
p
(p-homogeneity in second argument)
X, Y
(P4)
p
≤ X p⋅ Y
p −1 p
(inequality of Hölder) (P5)
X, Y
p
=0 ⇔
λX + Y
p
≥ Y
p
for all λ ∈ R
(James orthogonality, e.g. Singer(1970), Theorem 1.11) X ≠ Y,
(P6) (P7)
X − Y, Y
p
=0 ⇒
X
p
> Y
p
Any finite dimensional space of p-integrable random variables of dimension n is has a basis X1, ..., Xn for which the matrix of p-products X i , X j p
invertible triangular. (P8)
If X1, ..., Xn is a family of linearly independent p-integrable random variables, then the real-valued function f ( λ 1 ,..., λ n ) =
(P9)
∑ i =1 λ i X i n
p
is strictly convex. p
Given a linear space of p-integrable random variables and ϕ(⋅) a continuous (positive) linear functional on it, there exists a random variable W0 ∈ Lp such that ϕ(⋅) = ⋅, W0 p . Alternatively there exists W ∈ Lp such that ϕ (⋅) = (⋅, W ) p . (Representation Theorem of Riesz(1909) : see for example Cambanis and Miller(1981), Prop. 2.1. In the classical case p=2 : see Feller(1971), chap. IV.5, p.120).
3. The p-norm principle. Random variables are assumed to be p-integrable, p>1. Consider an economy in which n claims X1, ..., Xn have to be covered by n agents. Suppose the agents want to improve their level of security through a risk exchange treaty represented by Y1=Y1(X1,...,Xn), ..., Yn=Yn(X1,...,Xn), such that n
(3.1)
n
∑Y = ∑X i =1
i
i =1
i
= X (closed condition)
Assume that the risk exchange is Pareto-optimal and, for mathematical convenience, that the Yi's are differentiable functions. By the Theorem of Borch(1962) (see also Lemaire(1991), Theorem 3), the Pareto-optimal treaty depends on the Xi's only through their sum X. In particular, the total claim X may be thought of as split into n
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transformed components Yi=fi(X), i=1,...,n, such that Y1 + ... + Yn = X. The pricing principle is denoted by H ⋅ . To avoid arbitrage opportunities, this must be a positive linear functional on Lp. From Riesz Representation Theorem, that is property (P9) of Section 2, there exists W ∈ Lp (depending on X) such that the following relations hold :
(3.2)
H X = E X ⋅ E W + X, W p , H Yi = E Yi ⋅ E W + Yi , W p , i = 1,..., n .
What are now appropriate choices for W ? In the following Subsections, four different arguments are presented to justify the use of the p-norm pricing principle defined by (3.3)
H X = α⋅E X + γ
p −1
⋅ X p , α, γ ∈R , p
where the splitting components Yi=fi(X) are priced according to the Lp-space CAPM relations of the type first proposed by Borch(1982) in the special case p=2 : (3.4)
H [Yi ] = α ⋅ E [Yi ] +
Yi , X X
p p
⋅ (H [ X ] − α ⋅ E [X ]), i = 1,..., n .
p
Remarks 3.1. (i) The case p=2, n=2, has been also considered in Hürlimann(1994), where probabilistic induction as in Subsection 3.3 is applied. (ii) If one normalizes the choice of W such that E W = 1, then α=1 and the pnorm pricing principle depends only on one free parameter γ. Can normalization always be assumed? This is not the case in general, but can be assumed if competitive equilibrium prices are required (e.g. Aase(1993) in the special case p=2).
3.1. First order Taylor approximation. Suppose W can be developed in a Taylor series around the mean total claims deviation ( X − E [ X ]) such that (3.5)
∞
γk
k =2
k!
W = α + γ ⋅ ( X − E[ X ]) + ∑
⋅ ( X − E[ X ]) . k
Neglecting terms of higher order, a first order approximation implies a "linear world" for which
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W = α + γ ⋅ ( X − E [ X ]) .
(3.6)
Since E W = α one has W − E[W ] = γ ⋅ ( X − E[ X ]) . Then (3.3) follows from (3.2) using the property (P3) of the p-product. Similarly one has p −1 Yi , W p = γ ⋅ Yi , X p , i=1,...,n, and thus H Yi = α ⋅ E Yi + γ
(3.7) Eliminating γ
p−1
p −1
⋅ Yi , X p , i = 1,..., n .
using (3.3) one gets the desired formulas (3.4).
3.2. Conservative property. From Hölder's inequality (property (P4) of Section 2) and (3.2), one gets (3.8)
H X = E X ⋅ E W + X, W
p
≤ E X ⋅E W + X p ⋅ W
p −1 p
,
where equality holds if and only if W is linear in X, that is W = α + γ ⋅ ( X − E [ X ]) . Therefore, the linear form (3.6) is also a most conservative choice of W in (3.2).
3.3. Probabilistic induction. Suppose a pricing principle of the form (3.2) is already characterized by the values it takes on the set of all diatomic claims X with fixed mean µ X = E X . Then the p-norm principle (3.3), (3.4) necessarily holds. The basic step needed to show this result is the following simple observation.
Lemma 3.1. Let X be a diatomic claim with mean µ X , and let Z=g(X) be a diatomic transform of X. Then one has the p-product formula : (3.9)
X,Z
p
x −x p = 2 1 ⋅ Z p . z2 − z1
Proof. By assumption X has support {x1 , x2 } and probabilities (3.10)
p1 =
{p1, p2 }
such that
x2 − µ X µ − x1 , p2 = X . x 2 − x1 x 2 − x1
Similarly Z has support {z1 = g ( x1 ), z2 = g ( x2 )} with the same probabilities. By definition of the p-product one has
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(3.11)
X,Z
p
= p1 ( x1 − µ X )( z1 − µ Z )
X,Z
p
+ p2 (x2 − µ X )( z2 − µ Z )
x1 − µ X = p2 ( x1 − x2 ) ,
But µ X = p 1 x 1 + p 2 x 2 , hence (3.11) can be rewritten as (3.12)
p −1
{
= p1 p2 (x2 − x1 ) (z2 − µ Z )
p −1
p −1
.
x2 − µ X = p1 ( x2 − x1 ) . Thus
− ( z1 − µ Z )
p −1
}.
On the other side Z is diatomic, hence similarly to (3.10) one has
p1 =
(3.13)
z2 − µ Z µ − z1 , p2 = Z . z 2 − z1 z 2 − z1
Inserting in (3.12) one gets easily (3.9). ◊ Let us apply Lemma 3.1 to (3.2). Let W=g(X) for some function g(x) and let X be an arbitrary diatomic claim with mean µ X and support {x1 , x2 }. Denote the support of W by {w1 , w2 } and that of Yi=fi(X) by {yi ,1 , yi , 2 }, i=1,...,n. From (3.9) one has
X ,W
(3.14)
p
x −x = 2 1 ⋅ W w2 − w1
p
.
p
, one sees that the pricing principle Setting α = µ W and γ p is of the form (3.2). Apply (3.9) two more times to get the further relations p −1
(3.15)
Yi ,W =γ
p −1
p
x −x W = 2 1 ⋅ w2 − w1 X
p
y − yi ,1 ⋅ W = i , 2 w2 − w1 ⋅ Yi , X
p
p p
p
x − x W = 2 1 ⋅ w2 − w1 X
p p
p
yi , 2 − yi ,1 ⋅ X ⋅ x2 − x1
p p
, i = 1,..., n.
Inserting into (3.2) one sees that (3.7) holds. Elimination of γ yields (3.4) as desired.
3.4. The utility theoretical approach. From the point of view of an economic theory under uncertainty, one argues that W=u'(X), where u'(x) represents the marginal utility of the market as a whole, or u'(x) is the marginal utility of some representative insurer (e.g. Aase(1993) in the
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classical case p=2). In a first approach, it is useful to disregard from exact technical assumptions needed for the existence of a competitive equilibrium in Lp-space implying that W=u'(X), and just look at meaningful consequences of making this assumption. Similarly to the classical case p=2, the linear form (3.6) follows from the assumption of a quadratic utility function with all its shortcomings. Combined with the diatomic characterization of Section 3.3, it is sometimes possible to determine the unknown parameters α, γ of the p-norm principle. Indeed, if X has support {x1 , x2 } with probabilities {p1, p2 }, and W=u'(X), then one must have according to Section 3.3 : α = E u ' ( X ) = p 1 u ' ( x 1 ) + p 2 u '( x 2 )
(3.16)
γ (3.17)
p −1
=
X , u' ( X ) X
p p
p
u ' ( X ) x2 − x1 ⋅ = u ' ( x2 ) − u ' ( x1 ) X p
p
p
p1 (u ' ( x1 ) − α )p + p2 (u ' ( x2 ) − α ) p x2 − x1 ⋅ = p ( x − µ )p + p ( x − µ ) p u ' ( x ) − u ' ( x ) 2 1 X X 1 1 2 2
Example 3.1 : Lp-space version of the Capital Asset Pricing Model Let us apply the p-norm principle to the class of symmetric α-stable distributions, where 1 < α < 2 is a fixed characteristic exponent. Combining results of Kanter(1972), Cambanis and Miller(1981), and Cambanis et al.(1988) (see Rutkowski(1995), p.177), one obtains the relationship Y, X
(3.18) X
p p
p
=
Y, X X, X
α
for all p such that 1 ≤ p < α ,
α
where the square bracket ⋅, ⋅ α is the α-covariation measure of dependence. In particular, (3.18) is the "beta" factor of the financial stable CAPM of Gamrowski and Rachev(1993). Generalizing the observation made by Borch in the classical case p=2, there should be an analogy between the Lp-space CAPM (3.4) and a financial Lpspace CAPM, expressed by the market conditions (3.19)
E [Ri ] = rf + β i p ⋅ (E [RM ] − rf ),
β ip =
Ri , RM RM
p p
p
, i = 1,..., n,
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where rf is the risk-free rate of interest, RM is the random market return, and Ri is the random return of the i-th financial asset. The interested reader should try to give a rigorous derivation of this useful model, which is applicable to any return distribution with finite mean but infinite variance.
4. Experience rating contracts with minimum p-norm financial loss. An insurance contract is a pair {X , P} consisting of an insurance risk X and a risk premium P = H X , where H ⋅ is some premium calculation principle. An experience rated contract is a triple {X , P, D} consisting of an insurance contract {X , P}, which besides claims payment X offers a bonus or dividend D = D X ≥ 0, paid out in case the financial gain P−X is positive.
Definition 4.1. The experience rating premium P of an experience rated contract {X , P, D} is H-compatible if P = H X . The liability of an experience rated contract is X+D. Suppose its financial valuation, also called risk premium, equals P = P X + D , and is thus a functional depending on X and D through their sum. Compared to an ordinary insurance contract (without bonus or dividend payment) with financial loss X−P, the financial loss X+D−P of an experience rated contract may be much more important. To improve his level of security, suppose the insurer concludes a risk-exchange treaty, denoted REX, with some agent on the (re)insurance market. The REX consists of a pair (Y,Z) such that Y+Z=X, where Y = Y X is the retained amount of the insurer and Z = Z X is the random payment from the agent to the insurer. Suppose the insurer chooses a REX from a feasible set S(X) of possible REX's. Feasible sets often considered are S(X)=POREX(X), the set of feasible Pareto-optimal REX's, or S(X)=Com(X) the set of feasible reinsurance contracts described by the class of comonotonic random variables Com(X)={ (Y=f(X),Z=g(X)) : f(x), g(x) are nondecreasing functions such that f(x), g(x) ≤ x and f(x)+g(x)=x }. In the latter situation, a REX is restricted to those compensation functions for which neither the cedant nor the reinsurer will benefit in case the claim amount increases. Three main questions are: (Q1) How does the insurer choose (Y,Z) ∈ S(X) and D ? (Q2) What is an adequate price P for an experience rated contract ? (Q3) Which properties must the premium principle H ⋅ satisfy ?
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Having concluded a REX, the required premium P = P X + D of an N N experience rated contract is the sum of the net retained premium P = P Y + D of the insurer plus the price H Z paid to the agent for the REX treaty, hence P = P N + H Z . To answer question (Q1), suppose the insurer applies a minimum pnorm financial loss principle as decision criterion. Thus, one has to minimize the pnorm of the difference between assets and liabilities of the insurer, that is one considers the optimization problem (4.1)
R ( p) = P N − Y − D
p p
= E PN − Y − D
p
= min.
over all (Y,Z) ∈ S(X), all D from some set of dividend formulas. Questions (Q2) and (Q3) are related. Clearly the premium functional P = P Y , Z , D = P N Y + D + H Z and H ⋅ must satisfy some desirable properties. In the following, only three plausible properties will be relevant : (P1)
The no unjustified loading property : If X=c is a constant risk, then H c = c .
(P2)
The S(X)-additive property : If (Y,Z) ∈ S(X) then H Y + H Z = H X .
(P3)
The fair property for the retained business : The net retained premium of the insurer is a fair premium in the sense that P N = E Y + D .
The property (P2) says that no arbitrage profit can be made from concluding a REX treaty. It is satisfied by the p-norm principle (3.3), (3.4). In the important special case S(X)=Com(X), (P2) is the comonotonic additive property. It is in particular satisfied by the class of quantile premium principles, which include the absolute deviation principle and the Gini principle (e.g. Denneberg(1985/90), and Wang(1996)). Note that the comonotonic property is also essential in Chateauneuf et al.(1996). Another premium principle satisfying (P2) is the distribution-free principle derived in Hürlimann(1994), Theorem 5.1. The fair property (P3) is satisfied by the "fair premiums" required to cover the perfectly hedged experience rated contracts considered in Hürlimann(1994), Section 4, which correspond to the solutions of the minimization problem R(p=2)min=0 in (4.1) with S(X)=Com(X). This property implies that the insurer does not take any risk in this situation. In general, under (P3), the premium of an experience rated contract {X , P, D} equals P = E Y + D + H Z and may be decomposed in three components as follows : (4.2)
P
= E[X]
+ E[D]
+
expected expected claims bonus
(H[Z] - E[Z]). loading in risk-exchange price
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If one requires additionally characterization holds.
H-compatible premiums, then the following general
Theorem 4.1. Let {X , P, D} be an experience rated contract, which offers a claims dependent bonus under the help of a REX treaty (Y,Z) ∈ S(X), with premium P = E Y + D + H Z (fair property (P3)). Let H ⋅ be a premium principle, which satisfies the properties (P1) and (P2). Then the experience rated premium is Hcompatible, that is P = H X = H Y + H Z , for all (Y,Z) ∈ S(X),
(4.3)
if and only if H ⋅ acts as follows on the retained business : H Y = E Y+D .
(4.4)
Proof. This is an immediate consequence of the formulas (4.2) and (4.3). ◊ Definition 4.2. An experience rated contract with H-compatible premium as in Theorem 4.1 will be called a H-fair experience rated contract. To justify the validity of the pricing methodology proposed in Theorem 4.1 for general p-integrable insurance risks, e.g. Pareto claims with infinite variance (index 1
S d = (Y = f ( X ), Z = g ( X )) : (Y , Z ) ∈ Com( X ) and d = sup{ f ( x)} < ∞ x∈R
such that D=d−Y is the bonus or dividend and d+Z=X+D with probability one. Indeed, the premium of such a contract is given by P= P X+D = P d+Z =d+H Z (under the assumption of the very plausible translation-invariant property). It follows that P N = P − H Z = d = Y + D = E Y + D , which shows the required property (P3). The class Sd contains a lot of interesting reinsurance structures, of which (iv) of the examples below has not yet been considered so far.
Examples 4.1. (i)
A stop-loss contract Z=(X-d)+ has (maximum) deductible d.
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(ii)
A linear combination of proportional and stop-loss reinsurance Z=(1-r)X+r(X-T)+ has a maximum deductible d=rT.
(iii)
A linear combination of stop-loss contracts in layers Z=r(X-L)++(1-r)(X-M)+, M>L, has a maximum deductible d=rL+(1-r)M.
(iv)
Let be given a compound Poisson risk N
(4.6)
X = ∑ Ui , i =1
where N is a Poisson random variable, and the Ui's are independent and identically distributed random variables, which are independent from N. Then the reinsurance payment N
(4.7)
Z = b ( N − c) + + ∑ ( U i − b ) + i =1
defines a feasible reinsurance contract with maximum deductible d=bc, which might be attractive in the framework of the classical collective model of risk theory.
5. Lp equilibrium market premiums for experience rated contracts. If the insurance market is in equilibrium, in the sense that it offers no arbitrage opportunities, the p-norm principle can be used to model market premiums. Consider an experience rated contract {X , P, D} , which satisfies the assumptions of Theorem 4.1, and suppose the experience rated premium is H-compatible, where H ⋅ is the pnorm principle. As observed in the Remarks 3.1, it can be assumed for a competitive equilibrium that the following relations hold : H X =E X +γ (5.1)
H [Y ] = E [Y ] +
H [Z ] = E[Z ] +
p −1
Y, X X
p
p p
⋅ (H [ X ] − E [X ]),
p
Z, X X
⋅ X p , γ ∈ R,
p p p
⋅ (H [X ] − E [X ]),
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where say (Y,Z) ∈ S(X)=POREX(X). The H-compatible property is satisfied if and only if H Y = E Y + D by Theorem 4.1. Using (5.1), the unknown consant γ can be eliminated, and one gets the required Lp-market premium
(5.2)
P=H X =E X +
X
p
⋅E D .
p
Y, X
p
The corresponding Lp-reinsurance premium is (5.3)
H Z =E Z +
Z, X
p
Y, X
p
⋅E D .
These formulas extend the relations (4.16), (4.17) of Hürlimann(1994) in two directions. First of all, they are valid for arbitrary p-integrable random variables, allowing models with infinite variance. Second, they are valid for arbitrary Paretooptimal risk-exchanges, and not just for perfectly hedged experience rated contracts as in our first study. In particular, even in the classical case p=2, a useful generalization has been obtained. In the next Section, an attempt to extend beyond p=2 the distribution-free results of Hürlimann(1994), Section 5, is made.
6. Diatomic conservative approximations to Lp equilibrium market premiums. In the Euclidean case p=2, it is well-known that a diatomic risk with given mean µ, variance σ2, and range [a,b], is uniquely characterized by its support {x1, x2 }, a ≤ x1 < x2 ≤ b , and probabilities {p1, p2 } such that the following properties hold (e.g. Kaas et al.(1994), Theorem X.2.1) :
(6.1)
p1 =
x2 − µ µ − x1 , p1 = , x 2 − x1 x 2 − x1
a ≤ µ ≤ b , σ 2 = ( x 2 − µ )( µ − x 1 ). The class of all such diatomic risks is denoted shortly by D 2 ( a , b ): = D 2 ( a , b , µ , σ 2 ) . More generally, if 1≤ p < ∞ one considers the class of diatomic risks D 2p ( a , b ): = D 2p ( a , b , µ , σ p ) with given mean µ, p-norm σp, and range [a,b]. In particular, one has D 22 ( a , b ) = D 2 ( a , b ) . Generalizing the special case p=2, a diatomic risk from D 2p ( a , b ) is uniquely characterized by its support {x1, x2 }, a ≤ x1 < x2 ≤ b , probabilities {p1, p2 } and the following properties (derivation is left to the reader) :
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(6.2)
p1 =
x2 − µ , x2 − x1
σp =
( x2 − µ )( µ − x1 ) ⋅ ( µ − x1 ) p −1 + ( x2 − µ ) p −1 ( y − x)
p1 =
µ − x1 x2 − x1
, a ≤ µ ≤ b,
{
}
= p1 ( µ − x1 ) p + p2 ( x2 − µ ) p . Consider now an experience rated contract {X , P, D} priced according to the Lp-equilibrium market premium functional (5.2), (5.3). Suppose that X=Y+Z with Y=u(Z), Z=v(Z). Assume only incomplete information about X is available, for example X belongs to some class of models with infinite variance. In this situation, the results of Hürlimann(1994), Section 5, cannot be applied. In practice, it is often desirable to have a conservative estimate of the expected reinsurance payment E[Z], which lies possibly on the safe side. A popular and practical method consists to take as conservative approximation the diatomic risk Z*=v(X*) with maximum expected value over all diatomic risks such that E Z * = max E v( X) . p
(6.3)
X ∈D 2 ( a , b )
Let {x1 = x, x2 = y} be the support of the maximizing diatomic distribution. To calculate (6.3), consider the Lagrange function L ( x , y , λ ) = p 1 v ( x ) + p 2 v ( y ) + λ ⋅ ( σ p − p 1 ( µ − x ) p + p 2 ( y − µ ) p ). Its partial derivatives are
y− x Lx = v( x) − v( y ) + ( y − x)v ' ( x ) + λ ⋅ ( y − µ ) p − ( µ − x) p + p ( y − x)( µ − x) p −1 p 1
{
}
y− x Ly = v ( x) − v( y ) + ( y − x)v' ( y ) + λ ⋅ ( y − µ ) p − ( µ − x) p − p ( y − x)( y − µ ) p −1 p 2
{
}
A calculation shows that L x = L y = L λ = 0 is satisfied provided x, y solves the system of equations
p ( µ − x) p −1 ⋅ {v( y ) − v( x) − ( y − x)v' ( y )}
(6.4)
+ p ( y − µ ) p −1 ⋅ {v( y ) − v( x) − ( y − x)v' ( x)}
{
= {v' ( y ) − v' ( x)}⋅ ( y − µ ) p − ( µ − x) p
(6.5)
}
( y − x)σ p = ( y − µ )( µ − x) ⋅ {( µ − x) p −1 + ( y − µ ) p −1}.
15
To guarantee a maximum, the Hessian of L must be negative semidefinite and x, y ∈ a , b . If x≤a or x≥b, or the Hessian of L is not negative semidefinite, the maximum is attained on the boundary with x=a or x=b. However, even the simplest stop-loss case cannot be solved in closed form as shown in case 2 of the Example 6.1 below. Though a two-dimensional Newton iteration procedure can be applied to find numerical solutions, it appears instructive to get analytical approximate solutions through linearization of the p-norm function in (6.2). For this, approximate the fractional power functions by second order binomial expansions as follows :
(6.6)
p( p − 1) 2 (y − µ)p ≈ y − pµ y + µ 2 ⋅ µ p − 2 2 p( p − 1) 2 ( µ − x) p ≈ x − pµx + µ 2 ⋅ µ p − 2 2
A tedious but straightforward calculation implies the following approximate linear pnorm equation (6.7)
σ p ≈ 21 p ( p − 1) µ p − 2 ( µ − x )( y − µ ) + 21 ( p − 1)( p − 2 ) µ p ,
which can be rewritten as (6.8)
σ 2 ( p , µ ): =
σ p − 21 ( p − 1)( p − 2 ) µ p 1 2
p ( p − 1) µ p − 2
≈ ( µ − x )( y − µ ) ,
and which one can interpret as a p-adjusted variance parameter. In this situation, the results obtained previously in Hürlimann(1994), Section 5, apply directly. The obvious advantage is applicability for classes of stochastic models with infinite variance, for which the p-adjusted variance can be calculated.
Example 6.1 : stop-loss reinsurance If Z = ( X − d ) + one can assume x ≤ d ≤ y and set v(x)=0, v'(x)=0, v(y)=y-d, v'(y)=1. Then the system of equations (6.4), (6.5) can be put in the form
{
}
{
(6.9)
( p − 1) ⋅ ( y − µ ) p − ( µ − x) p = p (d − µ ) ⋅ ( µ − x) p −1 + ( y − µ ) p −1
(6.10)
( y − µ ) ⋅ {( µ − x) p − σ p }= ( µ − x) ⋅ {σ p − ( y − µ ) p }
To simplify, use the standardized values
}
16
(6.11)
X=
µ−x y−µ d−µ , Y = 1/ p , D = 1/ p , X , Y > 0 , d ∈ a , b . 1/ p σp σp σp
Then the system (6.9), (6.10) is equivalent to
{
}
{
}
(6.12)
( p − 1) ⋅ Y p − X p = pD ⋅ X p −1 + Y p −1 ,
(6.13)
( X p − 1) ⋅ Y = (1 − Y p ) ⋅ X .
We distinguish between two cases: Case 1 : D=0 One has necessarily Y=X=1, hence x = µ − σ 1p/ p , y = µ + σ 1p/ p , p 1 = p 2 = 21 . The exact solution to the optimization problem (6.3) equals max E ( X − µ ) + =
(6.14)
X∈D 2p ( a , b )
1 1/ p σp . 2
The approximate solution obtained from (6.8) has the value 21 σ ( p , µ ) . In particular the p-adjusted variance is a good approximation of the p-norm provided σ 2 ( p , µ ) ≈ σ 2p/ p . Case 2 : D≠0 From (6.12) one gets immediately
Y X
(6.15)
p −1
=
p X +Z D ≠ 0 . , Z := Y −Z p − 1
Y p −1 Rewrite (6.13) as X Y ⋅ 1 + = X + Y and insert (6.15). One obtains X p
(6.16)
Y=
Z . 1− Xp
Divide now (6.12) by Yp and insert (6.16) to get the fixed-point equation p
(6.17)
1− X p , X = ( X + Z ) ⋅ Z
which can be solved numerically through application of Newton's iteration algorithm.
17
References. Aase, K.K. (1993). Equilibrium in a reinsurance syndicate; existence, uniqueness and characterization. ASTIN Bulletin 23, 185-211. Aebi, M., Embrechts, P., Mikosch, T. (1992). A large claims index. Bulletin of the Swiss Association of Actuaries, 143-56. Borch, K. (1962). Equilibrium in a reinsurance market. Econometrica 30, 424-44. Borch, K. (1982). Additive insurance premiums : a note. The Journal of Finance, 129598. Reprinted in Borch(1990), 192-97. Borch, K. (1990). Economics of Insurance. Advanced Textbooks in Economics 29. Elsevier Science Publishers, Amsterdam. Cambanis, S., Miller, G. (1981). Linear problems in p-th order and stable processes. SIAM Journal of Applied Mathematics 41, 43-69. Cambanis, S., Hardin, C.D. Jr., Weron, A. (1988). Innovations and Wold decompositions of stable sequences. Probability Theory and Related Fields 79, 1-27. Chateauneuf, A., Kast, R. and A. Lapied (1996). Choquet pricing for financial markets with frictions. Mathematical Finance 6(3), 323-30. Denneberg, D. (1985). Valuation of first moment risk for decision purposes in Finance and Insurance. In : Göppel, H., Henn, R. (ed.) 3. Tagung Geld, Banken, und Versicherungen. Verlag Versicherungswirtschaft, Karlsruhe, 855-69. Denneberg, D. (1990). Premium calculation : why standard deviation should be replaced by absolute deviation. ASTIN Bulletin 20, 181-90. Fama, E.F. (1963). Mandelbrot and the Stable Paretian Hypothesis. Journal of Business 36, 420-29. Fama, E. (1965). The behaviour of stock market prices. Journal of Business 38, 34105. Feller, W. (1971). An Introduction to Probability Theory and its Applications, vol. 2. John Wiley, New York. Gamrowski, B., Rachev, S.T. (1993). Stable laws in testable asset pricing. Technical Report 257, University of California at Santa Barbara. Gamrowski, B., Rachev, S.T. (1994). The implementation of stable laws in financial models : a practical approach. Technical Report 268, University of California at Santa Barbara. Hürlimann, W. (1994). Splitting risk and premium calculation. Bulletin of the Swiss Association of Actuaries, 167-197. Hürlimann, W. (1997). Is there a rational evidence for an infinite variance asset pricing model? Proceedings of the 7-th International AFIR Colloquium, Cairns. Kaas, R., Heerwaarden, A.E. Van, Goovaerts, M.J. (1994). Ordering of Actuarial Risks. CAIRE Education Series 1, Brussels. Kanter, M. (1972). Linear sample spaces and stable processes. Journal of Functional Analysis 9, 441-59.
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Lemaire, J. (1991). Risk, Information and Insurance. In : Loubergé, H. (editor). Essays in the Memory of Karl H. Borch. Kluwer Adademic Publishers. Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business 36, 394-419. Riesz, F. (1909). Sur les opérations fonctionelles linéaires. Comptes Rendus de l'Académie des Sciences de Paris, 974-76. Rutkowski, M. (1995). Left and right linear innovations for a multivariate SαS random variable. Statistics and Probability Letters 22, 175-84. Samorodnitsky, G., Taqqu, M.S. (1991). Conditional moments and linear regression for stable random variables. Stochastic Processes and their Applications 39,183-99. Samorodnitsky, G., Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman and Hall. Singer, I. (1970). Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer, New York. Wang, S. (1996). Premium calculation by transforming the layer premium density. ASTIN Bulletin 26(1), 71-92.
