tail conditional expectations for exponential dispersion …

TAIL CONDITIONAL EXPECTATIONSFOR EXPONENTIAL DISPERSION MODELS

BY

ZINOVIY LANDSMAN AND EMILIANO A. VALDEZ

ABSTRACT

There is a growing interest in the use of the tail conditional expectation as ameasure of risk. For an institution faced with a random loss, the tail condi-tional expectation represents the conditional average amount of loss that canbe incurred in a fixed period, given that the loss exceeds a specified value. Thisvalue is typically based on the quantile of the loss distribution, the so-calledvalue-at-risk. The tail conditional expectation can therefore provide a measureof the amount of capital needed due to exposure to loss. This paper examinesthis risk measure for “exponential dispersion models”, a wide and popularclass of distributions to actuaries which, on one hand, generalizes the Normaland shares some of its many important properties, but on the other hand, con-tains many distributions of nonnegative random variables like the Gammaand the Inverse Gaussian.

KEYWORDS

Tail value-at-risk, Tail conditional expectations, Exponential dispersion family.

1. INTRODUCTION

Insurance companies set aside amounts of capital from which it can draw fromin the event that premium revenues become insufficient to pay out claims.Determining these amounts needed is not an obvious exercise. First, it mustbe able to determine with accuracy the probability distribution of the lossesthat it is facing. Next, it has to decide on the optimal risk measure that can beused to determine the amount of loss to cover with a high degree of confidence.This risk measure which may be denoted for instance by ‡, is technically definedto be a mapping from the space of loss random variables L to the real line R.In effect, we have ‡ : L � X → ‡(X ) � R. Artzner, Delbean, Eber, and Heath(1999) developed the axiomatic treatment for a “coherent” measure of riskand in that seminal paper, the authors claimed that a risk measure must sharefour axioms: subadditivity, monotonicity, positive homogeneity, and translationinvariance. See their paper for definitions of these axioms. The tail conditionalexpectation of a continuous loss random variable X shares these axioms.

ASTIN BULLETIN, Vol. 35, No. 1, 2005, pp. 189-209

Assume for the moment that an insurance company faces the risk of los-ing an amount X for some fixed period of time. This generally refers to the totalclaims for the insurance company. We denote its distribution function by FX (x)= Prob(X ≤ x) and its tail function by F“

X (x) = Prob(X > x). Note that althoughour setting applies to insurance companies, it is equally applicable for any insti-tution confronted with any risky business. It may even refer to the loss facedby an investment portfolio. We define the tail conditional expectation of X as

TCEX (xq) = E(X |X > xq) (1)

and we can interpret this risk measure as the mean of worse possible losses.It gives an average amount of the tail of the distribution. This tail is usuallybased on the q-th quantile, xq, of the loss distribution, which is defined for 0 <q < 1 as

xq = inf(x |FX(x) ≥ q). (2)

For random variables with strictly monotonic distribution functions, it isuniquely defined as xq = FX

–1(q) and this happens, for example, for a randomvariable with continuous distribution function and nonnegative support. Theformula used to evaluate this tail conditional expectation is

XX ,TCE x

F xxdF x1

X qq xq

=3

#__

]ii

g (3)

provided that F“X (xq) > 0, where the integral is the Lebesgue-Stieltjes integral.

Tail conditional expectations for the univariate and multivariate Normal fam-ily have been well-developed in Panjer (2002). Landsman and Valdez (2003)extended these results for the class of elliptical distributions which contain thefamiliar Normal distributions. These results have been limited in scope becausethese authors considered only symmetric distributions. In this paper, we developformulas for tail conditional expectations of loss random variables belongingto the class of exponential dispersion models. This class of distributions has servedas “error distributions” for generalized linear models in the sense developed byNelder and Wedderburn (1972). This includes many well-known discrete dis-tributions like Poisson and Binomial as well as continuous distributions likeNormal, Gamma and Inverse Gaussian, which are, except for Normal, notsymmetric. Several of these distributions have nonnegative support and pro-vide excellent fit for modelling insurance claims or losses. It is not thereforesurprising to find that they are becoming popular to actuaries. For example,credibility formulas for the class of exponential dispersion models preserve theproperty of a predictive mean; see Kaas, Dannenburg and Goovaerts (1997),Nelder and Verrall (1997), Landsman and Makov (1998) and Landsman (2002).

The rest of this paper is organized as follows. Section 2 recalls definitionand main properties of the class of exponential dispersion models. We considerhere both the reproductive and the additive forms of exponential dispersionmodels. The distinction is necessary as there are exponential dispersion modeldistributions that can only be expressed in one of the two forms. In Section 3,

190 Z. LANDSMAN AND E.A. VALDEZ

we derive the main formula for computing tail conditional expectations forexponential dispersion models. In particular, we find that we can express the TCEin terms of a so-called generalized hazard form later defined in Section 3.Section 4 considers some familiar absolutely continuous distributions belong-ing to this class, such as the Normal, Gamma, and the Inverse Gaussian.In Section 5, we show that the “lack of memory” property of the Exponentialdistribution is equivalent to a characteristic property expressed in terms of thetail conditional expectation. The resulting expression provides for a uniquerepresentation of the TCE for exponential distributions. Section 6 providessome familiar discrete distributions belonging to the exponential dispersionfamily, and derive TCE formulas for these distributions. Section 7 brieflydescribes the computation formula for either the sum or the weighted sum ofrandom variables within the class. We conclude this paper in Section 8.

2. DEFINITION AND PROPERTIES OF EXPONENTIAL DISPERSION MODELS

The early development of exponential dispersion models is often attributed toTweedie (1947) although a more thorough and systematic investigation of itsstatistical properties was done by Jorgensen (1986, 1987). This class of modelstogether with their properties is extensively discussed in Jorgensen (1997).For the properties discussed below and for an in-depth investigation of theseproperties, we ask the reader to consult this excellent source of reference ondispersion models.

Let Q(x) be some s-finite measure on the real line R. Define the log Laplacetransformation of this measure by

,log expk x dQ xq qR

= #] ] ]g g g

and define the parameter set

Q = {q ∈R | k(q) < ∞}. (4)

The family of probability measures

dPq = eqx – k(q)dQ(x), q ∈Q

defines the Natural Exponential Family (NEF), where k(q) is called the cumu-lant. NEF, being a one-parameter model, generates two-parameter models inthe following manner. Let L be a subset of R+ = (0,∞) such that

> log expk x dQ xl l q qL 0R

l= = *#] ] ]g g g& 0 (5)

for some measure Ql*. It is clear that L is not empty, because 1 ∈L (when l = 1,

Ql* = Q ). From (5) we get the family of probability measures

TAIL CONDITIONAL EXPECTATIONS FOR EXPONENTIAL DISPERSION MODELS 191

dP *q,l = e[qx–lk(q)]dQl

*(x), q ⊂ Q, l ∈L, (6)

which clearly contains two parameters: the parameter q which is named thecanonical parameter, and the parameter l which is called the index parame-ter. This model is called the additive Exponential Dispersion family (EDF).The random variable Y is said to belong to the additive Exponential Disper-sion family (EDF) if its probability measure can be represented in the form (6)and we write Y + ED*(q,l).

The transformation X = Y/l reduces to the so-called reproductive EDF whichhas the form

dPq,l = e l [qx – k(q)]dQl(x), (7)

and we shall denote by X + ED (q,l). For further details, please see Jorgensen(1997).

If the measure Ql in (7) is absolutely continuous with respect to a Lebesguemeasure, then the density of X has the form

fX (x) = el[qx – k(q)]ql(x). (8)

The same can be said about the additive form of EDF, in which case, Y hasdensity of the form

fY (y) = e[qy – lk(q)]q*l(y). (9)

We now briefly examine some basic and important properties of the class ofexponential dispersion models.

Consider the reproductive form of EDF. We note that its cumulant gener-ating function can be derived as follows:

X

/ .

log log

log

log

K t E e e e dQ x

e dQ x

e e dQ x

tl k q l k q

/

/ / /

Xt xt x

R

t x

R

t t x t

R

l q k ql

l q l k ql

l k q l k q l q l k q ll

= =

=

=

= + -

-

+ -

+ - + - +

#

#

#

] `]

]

] ]]

] ] ] ]]

] ]

g jg

g

g gg

g g g gg

g g

5

5

5 5

6

?

?

? ?

@

&

!&

! !&

0

+ 0

+ + 0

It follows that its moment generating function can be written as

MX(t) = exp{l [k(q + t /l) – k(q)]}. (10)

From these generating functions, it becomes straightforward to derive the meanand the variance.

By letting the index parameter l = 1, we get a one-parameter family whichreduces to the NEF, as already clearly described above. As pointed out, NEF


generates EDF. An important subclass of NEF is called regular if the para-meter set Q, defined in (4), is an open set [see Jorgensen (1997), Section 3.1].The regular EDF has several interesting properties. In particular, the cumulant,k(q), is a differentiable function of q, and one can write that the mean is

XE X tK t

m k q�t 0

22

= = ==

]]

]gg

g (11)

and the variance is

X .Var Xt

K tl k q�

t

2

2

0

1

2

2= =

=

-]

]]g

gg (12)

Moreover, the function t(q) = k�(q) is a one-to-one and increasing mappingfrom Q → W, where W ⊂ R is the set containing all possible values of m, so thatthe function

q = t –1(m), m ∈ W

is clearly well-defined. Notice that the above-mentioned interesting properties forthe mean and the variance hold even for a weaker condition on NEF which is

,x dP ,S

q l 3=# for q ∈ Q \ int(Q),

where S is the support of the distribution and int(Q) is the set of all internalpoints of Q. Such NEF satisfying this condition is said to be steep [see detailsin Brown (1986), Chapter 3].

By defining the unit variance function

V(m) = k�(q) = k�(t–1(m)),

the variance in (12) can be expressed as

Var (X ) = s2V(m) (13)

where s2 = l–1 is called the dispersion parameter. This reparameterization leadsus to write X + ED(m,s2) in terms of the mean and dispersion parameters.

Suppose X has an additive form, i.e. dP *q,l = e [qx–lk(q)]dQl

*(x) as given in (6).Then

KX(t) = l [k(q + t) – k(q)]

and

MX(t) = exp{l [k(q + t) – k(q)]}. (14)

It immediately follows that for the additive version of the EDF, we have the mean

m = E (X) = lk�(q) (15)


and the variance

Var (X ) = lk�(q) = V(m) /s2 (16)

where the unit variance function is given by

V (m) = k�(t –1(m))

with again

t(q) = k�(q).

We will similarly write X+ED*(m,s2) whenever X is represented in the additiveEDF form. Provided no confusion arises, we will sometimes write ED (m,l) orED (q,l) or ED (q,s2) for the family of reproductive exponential dispersionmodels. Similar notations can be used for the additive form, except that asuperscript * is used to emphasize the form.

The EDF has been established as a rich model with wide potential applica-tions. It extends the natural exponential family (NEF) and includes many standardcontinuous and discrete distribution models such as Normal, Gamma, InverseGaussian, Poisson, Binomial and the Negative Binomial. We consider these mem-bers of the EDF in a later section.

3. TCE FORMULA FOR THE EXPONENTIAL DISPERSION FAMILY

Consider the loss random variable X belonging to the family of exponentialdispersion models in either the reproductive or the additive form. Let q besuch that 0 < q < 1 and let xq denote the q-th quantile of the distribution of X.To keep the notation simple, we denote the tail probability function as F“ (· |q,s2) emphasizing the parameters q and s2.

Theorem 1. Suppose that the NEF which generates the EDF is regular, or at leaststeep. Then we have the following results:

• For X + ED (m,l), the reproductive form of EDF, the tail conditional expecta-tion of X is

TCEX (xq) = m + s2h, (17)

where s2 = 1/l and

h q22

= logF“ (xq | q,l) (18)

• For X + ED*(m,l), the additive form of EDF, the tail conditional expectationof X is

TCEX (xq) = m + h. (19)


Proof. To prove (17), first note that because EDF is regular or steep, the cumu-lant function k(q) is a differentiable function and one can therefore differentiatein q under the integral sign the tail function

, .F x e dQ xq lqx

x

l q k ql=

3-

q#_

]]i

gg5 ?

We can then write

) )

,,

,

,,

( ( / ,

logF xF x

e dQ x

F xx e dQ x

F xxdP F x

TCE x TCE x

q q lq l q

q ll k q

q ll k q q l

l k q m s

1

1 �

�

�

,

qq

x

x

qx

x

qq

x

X q X q

l q k ql

l q k ql

q l

2

22

22

=

= -

= -

= - = -

3

3

3

-

-

q

q

q

#

#

#

__

]]

_]

]]

_] _

]

ii

gg

ig

gg

ig i

g

5

6 5

<

7 7

?

@ ?

F

A A

# -

after noting that the mean is k�(q). After a re-arrangement of the terms, theresult should immediately follow. Suppose now X +ED*(m,s2), the additivecase. It can similarly be proven that

) )

,,

,

,,

( ( ,

logF xF x

e dQ x

F xx e dQ x

F xxdP F x

TCE x TCE x

q q lq l q

q llk q

q ll lk q q l

lk q m

1

1 �

�

�

,

qq

x

x

qx

x

qq

x

X q X q

q lk ql

l q k ql

q l

22

22

=

= -

= -

= - = -

3

3

3

-

-

*

*

*

q

q

q

#

#

#

__

]]

_]

]]

_] _

]

ii

gg

ig

gg

ig i

g

5

6 5

<

?

@ ?

F

# -

with m as defined in (15). A re-arrangement will lead us to the desired result. ¬

The only difference with (19) and (17) is the additional factor of s2 or equiva-lently, 1/l.

Let us notice that the function

h x q22

=] g logF“ (x | q,l)

can be considered as a generalization of the hazard rate. In fact, if q is a loca-tion parameter, i.e.


F“ (x | q,l) = F“ xs

q-b l

for some distribution function F having the density f (x) (for example, Normaldistribution), then

,h xF x

f x

sq

s sq1

=-

-

]

b

b

g

l

l

is exactly hazard rate. Clearly, the constant h in (18) is the generalized hazardrate evaluated at xq, i.e. h = h(xq).

4. EXAMPLES – ABSOLUTELY CONTINUOUS

We now consider some examples of distributions belonging to the class of expo-nential dispersion models. In particular, we give as examples the familiar Normal(an example of a symmetric distribution), Gamma and Inverse Gaussian (exam-ples of nonsymmetric distributions with nonnegative support), for the absolutelycontinuous case, and Poisson, Binomial, and Negative Binomial for the dis-crete case. For other examples, we suggest the reader to consult Jorgensen (1997).

Example 4.1. Normal. Let X+N(m,s2) be Normal with mean m and variance s2.Then we can express its density as

.

exp

exp

exp exp

f xx

x x

x x

p s sm

p s sm m

p s s sm m

21

21

21

21 2

21

21 1

21

2

2

2 2

22

22

= --

= -- +

= - -

] d

e

b b

g n

o

l l

>

>

;

H

H

E

(20)

Thus, we see that it belongs to the reproductive ED family by choosing q = m,l = 1/s2, k(q) = 2

1 q2 and ql(x) = ( p s2 )–1 exp(–x2/s2). Hence, the unit variancefunction for Normal distribution is

V(m) = k�(m) = 1. (21)

Now denoting by f(z) = ( p2 )–1exp (– 21 z2) and F(z) =

3-fz

# (x)dx, respectively,the density and distribution functions of a standard Normal, we see that

,

.

log logF x x

x

xq q l q l q

l q

l f l q

F

F

1

1

q q

q

q

22

22

= - -

=- -

-

_ __

__

__

i ii

ii

ii

8 B


Reparameterizing back to m and s2, we find from (17) that the TCE for the Nor-mal distribution gives

)(

/

/ /.

TCE xx

x

x

x

m l l q

l f l q

mm s

s f m ss

F

F

1

1

1

1

X qq

q

q

q 2

= +- -

-

= +- -

-

__

__

_

] _

ii

ii

i

g i

8

8

B

B

(22)

This gives the same formula as that derived in Panjer (2002) and Landsman andValdez (2003).

Example 4.2. Gamma. Let X+Ga (�,b) be Gamma distributed with parame-ters � and b. We express its density as follows:

,�

f x x eb

G

�� xb1= - -

]]

gg

for x > 0. (23)

With re-arrangement of the terms we get

.exp log�

�f x x xb bG

� 1

= - +-

]]

^gg

h

Thus, we see that it belongs to the additive ED family by choosing q = –b,l = �, k(q) = – log(–q) and ql(x) = x�–1(G(�))–1. Because t(q) = k�(q) = –1/q,the unit variance function for the Gamma distribution is

V(m) = k�(t–1(m)) = m2. (24)

Note that both the Exponential and the Chi-Square distributions are specialcases of the Gamma distribution. By choosing � = 1 in (23), we have the Expo-nential and by choosing � = n /2 and b = 1/2, we end up with the Chi-Squaredistribution with n degrees of freedom. From the parameterization

� = l and b = – q,

the Gamma density in (23) becomes

, ( ) ,exp logf x x xq l l q l qG

l 1

= + --

] ]g g6 @

here emphasizing the parameters q and l. Therefore, we have

, ( ) exp logF x x x dxq q l q l q l qGqx

l 1

q22

22

= + -3 -

#_ ]i g6 @( 2


, /

( ) , / ,

, ,

f x x dx

f x dx F x

F x F x

q l l q

q ll

q l l q q l

ql q l q l

GG1 1

1

1

x

xq

q q

q

q

= +

= -+

+ +

= - + -

3

3

#

#

] ]

]] ] _

_ _

g g

gg g i

i i8 B

and

,,

,.logF x

F x

F xq q l q

lq l

q l 11q

q

q

22

= -+

-__

_i

i

iR

T

SSS

V

X

WWW

Since m = – l /q, we have the TCE formula for a Gamma distribution:

)(,

,

,

,��

�

�

�TCE x

F x

F x

F x

F xb b

bm

b

b1 1X q

q

q

q

q=

+=

+

_

_

_

_

i

i

i

i(25)

after reparameterizing back to the original parameters. In the special case where� = 1, we have the Exponential distribution and

)(,

,

/ ,

TCE xF x

F x

ex e e

x x

b b

bb

b

b m

1

1

2 1

1

X qq

qx

qx x

q q

b

b b

q

q q

= =+

= + = +

-

- -

_

_

^

i

i

h

which interestingly represents the lack of memory, a property well-knownabout the Exponential distribution.

Example 4.3. Inverse Gaussian. Let X+ IG (l, m) be an Inverse Gaussian withparameters l and m. The density for an Inverse Gaussian is normally written as

2

,expf xx x

xpl

ml m

2 23 2=- -

]^

gh

R

T

SSS

V

X

WWW for x > 0. (26)

Clearly, we can write this density as

.expf xx

x xpl l

m m2 21

21

3 2

1

= - - +-

] dg n= G

See Jorgensen (1997) for this form of the density. Thus, we see that it belongs tothe reproductive ED family by choosing q = –1/(2m2), k(q) = –1/m = – (–2q)1/2


and ql(x) =xp

l2 3 exp (– x

l2 ). Now observing that the m parameter can be expressed

as

m = (–2q) –1/2,

then

ddqm

= (–2q) –3/2 = m3,

and so the variance function is

V(m) = m3. (27)

It can be shown for the Inverse Gaussian that its distribution function can beexpressed as

, / / ,F x x x e x xm l m l l m l lF F1 1/l m2= - + - -^ b bh l l

where F (·) denotes the cdf of a standard Normal. See, for example, Jorgensen(1997), p. 137 and Klugman, et al. (1998). Thus, we have

,

/ /

/ /

/ .

F x

x x e x x dd

x x x e x x

e x x x

q m q l

m m l l m l l qm

m l f m l l m l m l l

m l f m l l

F F

F

1 1

12

1

1

/

/

/

q

q q q q

q q q q q

q q q

l m

l m

l m

2

2

2

$

$

$

22

22

= - - + - -

= - + - -

- - -

]_

b b

b b

b

g i

l l

l l

l

; E

Denoting by z*q = /x xl lq qm

1 - so that – z*q – 2 / /x x xl l lq q qm

1= - - , wehave

q

q

q

,,

/

/.logF x

F xx z e

z x

x z xq m lm l

ml f

l l

l f l

F2 2

2

/q

qq

q

q q

l m2

22

= +- -

- - -*

*

*_

_`

a

ai

ij

k

k

R

T

SSS

V

X

WWW

* 4

Using the reproductive version of Theorem 1, we have

) q

q

q

(,

/ /

/.TCE x

F xx z e

z x

x z xm

m lm l

l fl l

l f l

F2 2

2

/X q

qq

q

q q

l m2= + +- -

- - -*

*

*_`

a

aij

k

k

R

T

SSS

V

X

WWW

* 4


FIGURE 1: Tail Conditional Expectations of Normal, Gamma, and Inverse Gaussian.

Let us notice that from (21), (24) and (27), the Normal, Gamma, and theInverse Gaussian distributions have unit variance function of the form

V(m) = m p, (28)

for p = 0,2,3, respectively. Members of the EDF with variance function of theform in (28), where p ∈R, are sometimes called Tweedie models. In Figure 1,we compare the resulting tail conditional expectations of these three membersof the Tweedie family. The parameters in each distribution have been selectedso that they all have mean E(X) = 10 and variance Var (X) = 100. We can seefrom the graph that starting from some level q, larger p in the Tweedie modelleads to larger TCE.

5. A CHARACTERIZATION PROPERTY OF THE EXPONENTIAL DISTRIBUTION

In the previous section, we have shown that for the exponential distribution,the tail conditional expectation can be expressed as

TCEX (xq) = xq + m, (29)


and we associated this with the “lack of memory” property of the exponen-tial distribution. In this section, we show that (29) is indeed equivalent to this“lack of memory” property because (29) holds only for exponential distribution.See Kagan, Linnik, and Rao (1973) for the importance of characterization prop-erties in Mathematical Statistics. To evaluate this characterization problem, weneed the following lemma which gives a convenient representation of the TCE.

Lemma 1. The existence of the expectation of X guarantees the following repre-sentation of the TCE:

))X

X

((

,TCE x xF x

F x dx

X q qq

xq= +

3

# ] g

(30)

provided that F“X (xq) > 0.

Proof. Because the expectation of X exists and assuming F“X (xq) > 0, we have

))

)

)

X

X

X X

X

X

((

(

(.

TCE xF x

xdF x

F x

xF x F x dx

xF x

F x dx

X qq

Xx

q

x x

qq

x

q

q q

=

=- +

= +

3

3 3

3

q

#

#

#

]

] ]

]

g

g g

g

Thus, the result follows. ¬

Theorem 2. Suppose FX (x) is a continuous distribution function in the internalpoints of its support (a,b) with either finite or infinite endpoints. Then the repre-sentation

TCEX (xq) = xq + �, for any q ∈(0,1), (31)

where � ! 0 is some constant not depending on xq, holds if and only if X has ashifted exponential distribution and in which case, we have

� = E (X) = m (32)

Proof. First notice that condition (31) automatically requires that b, the rightend of support of distribution FX(x), should equal b = ∞. In fact, if b < ∞, bythe definition of quantile in (2), there exists 0 < q0 ≤ 1 such that

xq = b, for q ≥ q0 .


Then, strictly speaking, TCEX(xq) is undefined for such q. Of course, one canalways naturally extend the definition for q ≥ q0 as

X

X

X

X X

,lim limF x

xdF x

F x

xdF x x F b F bx b

0 0

p q p

x

p q p

q Xx

b

q0 0

p p

0 0

0

0=

- + - -

= =" "

3

- -

# #

_

]

_

] ] ]^

i

g

i

g g gh

if q0 < 1. In that case we would have � = 0. Thus, b = ∞. Now define U (z) =

XFz

3# (x)dx. As F“

X(x) is continuous, U(z) is differentiable. We can then write,using Lemma 1, that for z ∈ (a,∞), we have

TCEX (z) = z – U zU z

�]]

g

g .

Now using the representation in (31), we then have

U zU z

�]]

g

g = – � for any z ∈ (a,∞) ,

or equivalently,

U zU z�]

]

g

g = – �1 = – b

for any z ∈ (a,∞). This simple differential equation leads to the only solutionof the form

U (z) = ke–bz, z ∈ (a,∞).

for some constant k ! 0. Since

– U �(z) = kbe–bz = F“X (z), z ∈ (a,∞), (33)

it immediately follows that b > 0. Based on the restriction on the left end of thesupport, FX (a – 0) = 0, and (33) follows that a > –∞, that is, a is finite and sowe have

kb = eba.

This gives us

F“X (x) = e – b (x – a) for all x ≥ a,

which implies X has a shifted exponential distribution. Equation (32) auto-matically follows.

Now to complete the proof, consider the case where q0 = 1 and b < ∞. Thenone defines

XU z F x dxz

b= #] ]g g


and F“X(b) = 0. This then contradicts with (33), which holds for z ∈(a,b]. ¬

Of course among nonnegative random variables with support (0,∞), the onlydistribution applicable to the result in the theorem above is the canonical expo-nential distribution with

F“X (x) = e – bx, x ≥ 0.

6. EXAMPLES – DISCRETE

In this section, we consider some examples of discrete distributions belongingto the class of exponential dispersion models.

Example 6.1. Poisson. Let X + Poisson(m) be Poisson distributed with meanparameter m. We express its probability function as follows:

! ,p x xe mxm

=-

] g for x = 0,1, ...

A re-arrangement of the terms gives us

!p x x1

=] g exp(x log m – m).

Thus, we see that it belongs to the additive ED family by choosing q = log m,l = 1, k(q) = eq and ql(x) = 1/x!. Therefore, we have

X

, !

!

!

! ,

, ,

,

exp

exp

exp

exp

F x x x e

x x e

x x e x e

e y y e F x

e F x F x

e p x

q q q q

q q

q

q q

q q

q

11

1

1

11

1 1 1

1

qx x

x x

x x

qy x

q q

q

q

q

q q

q q

q

q

1

1

1

q

q

q

q

$

22

22

22

= -

= -

= - -

= - -

= - -

=

3

3

3

3

= +

= +

= +

=

!

!

!

!

_ _

_

_ _

_ _

_ _

_

i i

i

i i

i i

i i

i

R

T

SSS

R

T

SSS

:

:

8

V

X

WWW

V

X

WWW

D

D

B

and

,,

, .logF xF x

e p xq qq

q11

11q

qq

q

22

=__

_ii

i


Since m = eq, we have the TCE formula for a Poisson distribution:

)))

(((

.TCE xF xp x

m 1X qq

q= += G (34)

where F“ (xq) = F“ (xq | q,1). Notice also that for the Poisson distribution, m =Var (X), so we can also write

)))

(((

.TCE xF xp x

Var XmX qq

q#= + ] g

In that sense, the Poisson distribution is analogue of Normal for a discrete case.

Example 6.2. Binomial. Let X + binomial (p,n) be binomial with parametersp and n. Then we can express its probability function as

n x n- .p x P X xnx p p

nx p

pp1

11x

x

= = = - =-

-] ] c ^ c d ^g g m h m n h

By defining q = log[ p / (1 – p)], we can write this as

n-

,exp log

p xnx e e

nx x n eq

1

1

xq q

q

= +

= - +

] c _

c _

g m i

m i8 B

(35)

for x = 0,1, ...,n. Thus, we see that it belongs to the additive ED family with l = n,k (q) = log (1 + eq) and ql (x) = n

xc m. Notice that we can actually write

ql(x) = !nx n x x

nG

G1

1=

+ -

+c

]

]m

g

g . We therefore see that for xq < n,

, !

!

!

! ,

! ,

, , .

exp log

exp log

F x p n n x xn

x n e

n x xn

x ne

e x n e

n x xn

x np p p

n x xn

p p npF x p n

npn y y

np p npF x p n

npF x p n npF x p n

q q q q

q

G

G

G

G

G

G

G

G

G

G

11

1

11

11

11

1

1 11

1

1

1 1

qx x

n

x x

n

x x

nx n x

x x

nx n x

q

y x

ny n y

q

q q

q

q

qq

1

1

1

1

11

q

q

q

q

q

22

22

=+ -

+- +

=+ -

+-

+- +

=+ -

+- -

=+ - -

+- -

=-

- -

= - - -

= +

= +

= +

-

= +

-

=

-- -

!

!

!

!

!

]_]

]_

]

]d _

]

]^ ^

] ]

]^ _

^

]^ _

_ _

g ig

gi

g

gn i

g

gh h

g g

gh i

h

gh i

i i

8

8

B

B

$ .


Noting that m = np for the binomial and reparameterizing back to m and s2,we find from (19) that the TCE for the binomial distribution gives

)(,

,

,

,.

TCE xF x p n

F x p n

F x p n

F x p n

m q

m

1

1 1

X qq

q

q

q

22

= +

=- -

__

_

_

ii

i

i(36)

Example 6.3. Negative Binomial. Let X + NB(p,�) belong to the NegativeBinomial family with parameters p and �. Its probability function has the form

x

! ,�

�p x x

xp pG

G1�=

+-]

]

]^g

g

gh for x = 0,1, ...

A re-arrangement of the terms leads us to

! .exp log log�

��p x x

xx p pG

G1=

+- +]

]

]^g

g

gh6 @

By choosing q = log(1 – p), l = �,k(q) = – log(1 – eq) and ql(x) = !��

xx

GG +

]

]

g

g, we seethat it belongs to the additive ED family. We therefore see that

, !

!

!

! ,

! ,

, , .

exp log

exp log

��

�

��

��

�

��

� �

��

��

� � �

F x p xx

x e

xx

xe

e x e

xx

x pp

p p

xx

xp p pp

F x p

pp

yy

p p pp

F x p

pp

F x p F x p

q q q q l

l q l

G

G

G

G

G

G

G

G

GG

1

11

11

11

1

11

11

1

11 1

�

�

�

qx x

x x

x x

x

x x

xq

y x

yq

q q

q

q

qq

1

1

1

1

1

q

q

q

q

q

22

22

=+

+ -

=+

--

+ -

=+

--

-

=-

+- -

-

=-

+

+ +- -

-

=-

- + -

3

3

3

3

3

= +

= +

= +

= +

=

+

!

!

!

!

!

]_]

]_

]

]d _

]

]d ^

] ]

]^ _

]

^^ _

_ _

g ig

gi

g

gn i

g

gn h

g g

gh i

g

hh i

i i

8

8

8

B

B

B

$ .

Note that for a NB (p,�) random variable, its mean is m = �(1 – p) /p so thatreparameterizing back to m and s2, we find from (19)

)(,

,.

�

�TCE x

F x p

F x pm

1 1X q

q

q=

- +

_

_

i

i(37)


Comparing the TCEX(xq) for Negative Binomial with Gamma as shown in(25), we may conclude that the Negative Binomial is the discrete analogue ofGamma.

7. TAIL CONDITIONAL EXPECTATION FOR SUMS

Consider the case where we have n independent random variables X1, X2, ..., Xncoming from the same EDF family having a common parameter q but differ-ent l’s. Consider first the additive case, that is, Xk +ED*(mk, lk) for k = 1,2,...,n.The density is thus

pk(x | q, lk) = exp[qx – lk k(q)] qlk(x), for k = 1, 2, ..., n. (38)

Denote the sum by

S = X1 + X2 + ··· + Xn

and denote by TCEX(xq | m,l) the tail conditional expectation of X belongingto the EDF family either in the reproductive or additive form with mean para-meter m and index parameter l.

Theorem 3. Suppose X1, X2, ..., Xn are n independent random variables from theadditive family (38). Then the tail conditional expectation of the sum is

TCES(sq) = TCEX1(sq | mS,lS)

where mS = k 1=mk

n! , lS = k 1=lk

n! , and sq is the q-th quantile of the distributionof S.

Proof. By independence and using (14), the moment generating function of thesum can be expressed as

MS (t) = exp{lS [k(q + t) – k(q)]}

where we have expressed lS = k 1=lk

n! . Thus, we see that the sum also belongsto the additive Exponential Dispersion family with

S + ED*(mS,lS).

(This is similarly mentioned in Jorgensen (1997), Section 3.2, as a convolutionformula.) It becomes straightforward from Theorem 1 to prove that the tail con-ditional expectation for the sum can be expressed as

TCES (sq) = mS + h (39)

where the generalized hazard function ,logh F sq q lq S22

= _ i and sq is the q-thquantile of the distribution of S. ¬


To illustrate, consider the Gamma distribution case in Example (4.2). Let X1,...,Xn be n independent random variables such that

Xk + Ga(�k, b) for k = 1,2, ...,n.

From (25) and Theorem 3, taking into account lk = �k, we see that the TCE forthe sum is of the form

)(,

,,

�

�TCE s

F s

F sm

b

b1S q S

q S

q S=

+

_

_

i

i

where �S = k 1=�k

n! .Next, we consider the reproductive form of the EDF family, that is, Xk +

ED(m,lk) for k = 1,2, ...,n. The density is thus

pk(x |q,lk) = exp[lk(qx – k(q))]qlk(x), for k = 1,2, ...,n.

Unfortunately, we can only derive explicit form of TCE for a weighted sum ofXi’s. As a matter of fact, we have the following result.

Theorem 4. Suppose lS = k 1=lk

n! and Yk = wkXk where Xk +ED( m,lk) and wk =lk /lS for k = 1,2, ...,n. Define the weighted sum

kk .Y w XSk

n

kk

n

1 1

= == =

! !

Then its tail conditional expectation is given by

TCES (sq) = TCEX1(sq | m,lS)

where sq is the q-th quantile of the distribution of S.

Proof. From (10), we have the moment generating function of Yk

MYk(t) = MXk

(wkt) = exp{lk [k(q + t /lS) – k(q)]}

so that the moment generating function of S is given by

MS(t) = exp{lS [k(q + t /lS) – k(q)]}.

Thus, by taking into account (10), we see S + ED(m,lS). (See also Jorgensen(1997), Section 3.2, the reproductive form of the convolution formula.) Thus,the tail conditional expectation of S immediately follows from Theorem 1. ¬

Notice that the usefulness of Theorems 3 and 4 immediately comes from thefact that the TCE of the sum S (in the additive form) or the weighted sum S(in the reproductive form) can be evaluated using the TCE of only one of the X ’s.


To illustrate Theorem 4, consider the Normal distribution case in Exam-ple (4.1). Let X1, ...,Xn be n independent random variables such that

Xk + N(m, s2k ) for k = 1,2, ...,n.

From Theorem 4 and the first item of Theorem 1, we see that the TCE for theweighted sum S has the form

TCES (sq) = m +/

/ /,

m s

s f m ss

F s

s

1

1

q

q

S

S S

S2

- -

-

_

^ _

i

h i

8

8

B

B

where s2S = 1/lS = 1/ k 1=

/ s1 kn 2! _ i, the harmonic mean of the individual vari-

ances. Note that for Normal distribution, we can also derive result for thesum S since the Normal distribution N(mk, s2

k ) can also be considered as amember of ED*(mk, lk) where lk = s2

k . The well-known TCE formula for sum S(see for example, Panjer, 2002) given by

TCES (sq) = m +/

/ /

s

s

m s

s f m ss

F1

1

q S

S q S

S2

- -

-

_

^ _

i

h i

8

8

B

B

immediately follows from Theorem 3.

8. CONCLUDING REMARKS

This paper examines tail conditional expectations for loss random variablesthat belong to the class of exponential dispersion models. This class of distri-butions has served as “error distributions” for generalized linear models in thesense developed by Nelder and Wedderburn (1972). This class extends manyof the properties and ideas developed for natural exponential families. It alsoincludes several standard and well-known discrete distributions like Poisson andBinomial as well as continuous distributions like Normal and Gamma, and thispaper develops tail conditional expectations for these members. We find anappealing way to express the tail conditional expectation for the class of Expo-nential Dispersion models; this TCE is equal to the expectation plus an addi-tional term which is the partial derivative of the logarithm of the tail of thedistribution with respect to the canonical parameter q. We observe that this par-tial derivative is a generalization of the hazard rate function. The results arefurther extended for sums or weighted sums of random variables belonging tothe exponential dispersion family.

ACKNOWLEDGEMENT

The first author wishes to acknowledge financial support provided by the Schoolof Actuarial Studies, University of New South Wales, during his visit at theuniversity. The second author wishes to acknowledge financial support provided


by the Australian Research Council through the Discovery Grant DP0345036 andthe UNSW Actuarial Foundations of the Institute of Actuaries of Australia.

REFERENCES

ARTZNER, P., DELBAEN, F., EBER, J.M., and HEATH, D. (1999) Coherent Measures of Risk. Mathe-matical Finance 9, 203-228.

BARNDOFF-NIELSEN, O.E. (1978) Hyperbolic Distributions and Distributions on Hyperbolae.Scandinavian Journal of Statistics 5, 151-157.

BOWERS, N.L., GERBER, H.U., HICKMAN, J.C., JONES, D.A., and NESBITT, C.J. (1997) ActuarialMathematics 2nd edition. Society of Actuaries, Schaumburg, Illinois.

BROWN, L.D. (1987) Fundamentals of Statistical Exponential Families. IMS Lecture Notes –Monograph Series: Vol. 9, Hayward, California.

Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. John Wiley,New York.

JOE, H. (1997) Multivariate Models and Dependence Concepts. Chapman & Hall, London.JORGENSEN, B. (1986) Some Properties of Exponential Dispersion Models. Scandinavian Journal

of Statistics 13, 187-198.JORGENSEN, B. (1987) Exponential Dispersion Models (with discussion). Journal of the Royal

Statistical Society, Series B, 49, 127-162.JORGENSEN, B. (1997) The Theory of Dispersion Models. Chapman & Hall, London.KAAS, R., DANNENBURG, D., GOOVAERTS, M. (1997) Exact Credibility for Weighted Observations.

Astin Bulletin, 27, 2, 287-295.KAGAN, A.M., LINNIK, Y.V. and RAO, C.R. (1973) Characterization Problems in Mathematical

Statistics. John Wiley, New York.KLUGMAN, S., PANJER, H. and WILLMOT, G. (1998) Loss Models: From Data to Decisions. John

Wiley, New York.LANDSMAN, Z. (2002) Credibility Theory: A New View from the Theory of Second Order Opti-

mal Statistics. Insurance: Mathematics & Economics, 30, 351-362.LANDSMAN, Z. and MAKOV, U. (1998). Exponential Dispersion Models and Credibility. Scandi-

navian Actuarial Journal, 1, 89-96LANDSMAN, Z. and VALDEZ, E.A. (2003) Tail Conditional Expectations for Elliptical Distribu-

tions. North American Actuarial Journal, 7(4), 55-71.NELDER, J.A. and VERRALL, R.J. (1997) Credibility Theory and Generalized Linear Models.

Astin Bulletin, 27, 71-82.NELDER, J.A. and WEDDERBURN, R.W.M. (1972) Generalized Linear Models. Journal of the

Royal Statistical Society, Series A, 135, 370-384.PANJER, H.H. (2002) Measurement of Risk, Solvency Requirements, and Allocation of Capital

within Financial Conglomerates. Institute of Insurance and Pension Research, University ofWaterloo Research Report 01-15.

TWEEDIE, M.C.K. (1947) Functions of a Statistical Variate with Given Means, with Special Refer-ence to Laplacian Distributions. Proceedings of the Cambridge Philosophical Society, 49, 41-49.

ZINOVIY LANDSMAN

Actuarial Research CenterDepartment of Statistics – University of HaifaMount Carmel, Haifa 31905, Israel.E-mail: [email protected]

EMILIANO A. VALDEZ

School of Actuarial StudiesFaculty of Commerce & Economics – University of New South WalesSydney, Australia 2052E-mail: [email protected]


tail conditional expectations for exponential dispersion …

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