Pricing in the Multi-Line Insurer withDependent Gamma Distributed Risksallowing for Frictional Costs of Capital �

Zinoviy LandsmanDepartment of Statistics,Actuarial Research Centre,

University of HaifaHaifa 31905 Isreal

email: landsman@stat.haifa.ac.iland

Michael SherrisFaculty of Commerce and Economics,University of New South Wales,Sydney, NSW, Australia, 2052email: m.sherris@unsw.edu.au

September 27, 2005

Abstract

This paper considers the pricing of insurance contracts for a multi-line insurer in a single period model where insurance risks have depen-dent gamma distributions. The pricing takes into account the impactof default, as well as �nancial distress costs arising from the possibility

�This research supported by Australian Research Council Discovery Grant DP0345036,�nancial support from the UNSW Actuarial Foundation of The Institute of Actuaries ofAustralia and Caesarea Rothschild Institute and Zimmerman foundation at University ofHaifa. Corresponding author: Professor Michael Sherris.

1

of default by the insurer, and allocates these to line of business in aneconomically meaningful manner. The costs of capital include fric-tional costs arising from writing insurance business and are assumedto be proportional to insurer end of period capital. In practice thesemust be incorporated into by-line pricing by also allocating them toline of business. We develop closed form expressions for prices and theallocation of default and frictional costs of capital to lines of buisnessbased on assumptions used in the application of standard option pric-ing models to insurer balance sheets. These closed form expressionscan be used to implement the model in practice.JEL Classi�cation: G22, G13, G32IM Insurance Mathematics Subject Category: IM30, IM12Keywords: multi-line insurer pricing, frictional costs of capital,

dependent gamma distribution.

�

2

1 Introduction

Pricing in the multi-line insurer needs to allow for the distribution of theinsurance risks to be priced as well as the impact of insolvency of the insurerand the costs of capital. In order to price insurance risks it is necessary toallow for risk using a pricing principle or using �nancial economic theory.In a multi-line insurer, the capital of the insurer is available to support thepayment of claims for all lines of business. However, limited liability meansthat the capital of the insurer will not be able to guarantee payment of alloutstanding claims with certainty. Thus there will be a shortfall in the eventof insolvency and the cost of this shortfall has to be allocated to lines ofbusiness in order to derive fair prices since these costs, although contingent,will have to be met by the policyholders.The costs of capital include taxes, �nancial distress costs and agency

costs. In order for the insurer to attract capital to support the writingof insurance, these prices will have to cover the additional costs of capitalover and above those of the alternative marginal investment available to thiscapital. They will need to be allocated to line of business for fair pricing.Policyholder claims are collateralized by the premiums that they pay and thecapital subscribed by the shareholders. However policyholders pay for theguarantee in the premium rates they are charged in equilibrium since the fairpremium re�ects the payo¤s by line of business allowing for insolvency.We use the risk capital approach to an insurer balance sheet as in Merton

and Perold (1993) [8] and Myers and Read (2001) [10]. Our focus is onpricing in a multiline insurer rather than on allocation of costs of capitalto lines of business which is the main focus of Myers and Read (2001) [10].Pricing in the multi-line insurer is discussed in Phillips, Cummins and Allen(1998) [11] and Zanjani (2002) [15]. Sherris (2003) [12] discusses equilibriuminsurance pricing in a mean-variance model and Sherris (2004) [13] considersthe allocation of risk capital to line of business for the purposes of fair pricing.Closed form expressions for the allocation of the default option value of aninsurer for use in fair pricing in a multi-line insurer are derived in Sherrisand van der Hoek (2004) [14] on the assumption that claims for each lineof business are log-normal and that the ratio of assets to liabilities is alsolog-normal. Furman and Landsman (2004) [5] consider capital allocation fora model of insurer liabilities that have dependent gamma distributions. Seealso Hürlimann (2001) [6].In this paper we consider the pricing of insurance contracts in a risk

3

portfolio with dependent gamma distributions. This portfolio of risks can beregarded as consisting of di¤erent lines of business. Gamma distributions areused in practice to model insurance losses by line of business. The dependentgamma distribution model used has not been applied to model an insurerbalance sheet previously. We use a single period model and derive closed formexpressions for fair prices by line of business allowing for the insolvency risk ofthe insurer. We include assets as well as liabilities in our model formulation.We also show how costs of capital and �nancial distress costs can be allocatedin this model to lines of business. We use approximations where necessaryto derive the closed form expressions so that these pricing results can beimplemented in practice. The results for the gamma distribution used toderive the closed form expressions and the pricing model results are new.

2 Model of the Multi-Line Insurer

We assume a single period model to derive our results. Since most non-life insurance contracts are annual contracts, this assumption, although nottotally realistic, will capture the main �nancial aspects we are interested inmodelling. The insurer writes a portfolio of n distinct insurable risks at thebeginning of the period. These can be regarded as lines of business or asindividual policies. We denote the end of period total claims for the ith riskby Li. Total claims at the end of the period will be L =

Pni=1 Li. The

insurer collects premiums at the start of the period for an amount Pi forline of business i. Pi allows for the expected losses, the risk loading, the riskof claims not being met due to the insolvency of the insurer and any costsof capital to be allocated to policyholders. Total premiums collected at thestart of the period are P =

Pni=1 Pi.

Since we wish to allow for the impact of limited liability and costs of cap-ital, we consider an insurer with shareholders who subscribe capital su¢ cientto ensure that the assets of the insurer will be a multiple of the value of thetotal policyholder liabilities ignoring the risk of insurer insolvency. Assets atthe start of the period will then equal V0 = (1 + s)L0 where L0 is the time 0actuarial value of the end of period claims, s > 0: The value of assets at theend of the period will be denoted by V .We will consider the costs of capital and their allocation to lines of busi-

ness later. To begin with we assume the costs of capital are zero. Our modelis arbitrage-free by assumption so that there exists a probability measure Q

4

such that the current values of the assets and liabilities are the discountedpresent value of end of period random payments using a risk free discountrate. We assume that there exists a risk free asset such that an investmentof 1 now will return er at the end of the period for certain. Denote by L0ithe time 0 price or fair value for line of business i given by

L0i = EQ�e�rLi

�for all i = 1; : : : ; n

and denote by V0 the time 0 price or fair value for the assets given by

V0 = EQ�e�rV

�where r is the risk-free continuous compounding rate of interest. We alsodenote the total value of the initial liabilities by L0 =

Pni=1 L0i. We assume

complete markets so that we can observe L0i for all i = 1; : : : ; n, and V0 froman asset market where these risks are traded. The model is based on marketdetermined values for the parameters of the distributions for these risks aswell as values for their prices of risk. This is a market consistent balancesheet approach.To determine the fair insurance premium we must determine the pay-

ments that the policyholder will receive from the insurer and allow for thee¤ect of limited liability in the event of insolvency. Payment of the totalliability for claims will only be made if the insurer is solvent at the end ofthe time period. From Sherris (2004) [13] and Sherris and van der Hoek(2004) [14], the payment of claims for line of business i at the end of theperiod, assuming equal priority for losses by line of business in the event ofinsolvency, will be

LiLV if L > V (or V

L� 1)

Li if L � V (or VL> 1)

or

Li

"1�

�1� V

L

�+#(1)

5

The premium for line of business i will be given by

Pi = EQ

"e�rLi

"1�

�1� V

L

�+##

= EQ�e�rLi

�� EQ

"e�rLi

�1� V

L

�+#

= L0i � e�rEQ

"Li

�1� V

L

�+#= L0i �D0i (2)

where we denote the line of business default value for line i by D0i.An insurance policy can then be viewed as a contingent claim since if

VL� 1 the shareholders can exchange the assets of the company for their

obligation to pay the outstanding claims. Thus policyholders have a contractfor payment of claims provided the assets of the insurer exceed the totaloutstanding claims. Otherwise they receive their share of the assets basedon the outstanding claims by line of business.Denote the insurer default option value by D0 so that

D0 = e�rnXi=1

EQ

"Li

�1� V

L

�+#=

nXi=1

D0i

Since

P =nXi=1

Pi = L0 �D0

The initial value of capital of the insurer will be equal to

C = V0 � P

= (1 + s)L0 � P

= sL0 +D0

The beginning of period assets are

V0 = (1 + s)L0

= sL0 +D0 + L0 �D0

= C + P

with C contributed by shareholders in capital and P from policyholders inpremiums.

6

3 The Distribution of Insurance and AssetRisks

An commonly used distribution assumption for insurance claims is the gammadistribution. We will develop a model of the underlying insurance and assetrisks of the insurer such that they have gamma marginal distributions andare dependent. The distributional assumptions we use are for the Q measureprobability distribution used for pricing. We follow the pricing approachdeveloped in Sherris and van der Hoek (2004) [14] and use the techniquesderived in Furman and Landsman (2004) [5] in order to determine fair pricesfor insurance premiums allowing for the default option value.The model for dependent gamma risks is based on results for the distrib-

ution for the sum of independent gamma random variables derived in Mathai(1982) [7]. The model applies an approach developed in Moschopoulos (1985)[9]. These are further developed for dependent gamma random variables inAlouini, Abdi, and Kaveh (2001) [2].Let (X0; ::::; Xn; Xn+1) be n+ 2 independent gamma distributed random

variables with shape parameters i and common rate parameter �, denotedby G( i; �); i = 0; :::; n+ 1: The probability density of the Xi is

fXi (xi) =� i

� ( i)e��Xix

i�1i ; xi > 0

Assume that the end of period claims for the n lines of business and the valueof the assets V are given by

Li =�0�X0 +Xi; i = 1; :::; n

V =�0�X0 +Xn+1 (3)

The intuition for this model is that there is a common factor X0 that impactsthe values of each line of business claims as well as the assets. This couldfor example capture the e¤ect of in�ation which has an impact on insuranceclaims and asset values. For each line of business and for the assets there is aseparate independent factor impacting the claims and asset payo¤s denotedby Xi, i = 1; :::; n for the line of business i; and Xn+1 for the assets.Under these assumptions the claims for each line of business L1; :::; Ln

are gamma distributed random variables, G(�i; �); i = 1; :::; n, where �i = 0 + i, and the assets V are gamma distributed as G( 0 + n+1; �); since

7

the sum of two independent gamma random variables with the same rateparameter is also gamma with shape parameter equal to the sum of theshape parameters.The total claims liability at the end of the period is

L =

nXi=1

Li =n�0�X0 +X�;

where

X� =

nXi=1

Xi

We have that X� is distributed G( �; �); with � =Pn

i=1 i: However, thetotal claims liability is a sum of 2 gamma random variables, n�0

�X0 and X�,

each with di¤erent rates and so the sum does not have a gamma distribution.However, using Furman and Landsman (2004), Proposition 2 [5], we can

represent L as a mixed gamma distribution with mixed shape parameter

L v G( 0 + � + �; �);

where � is a non negative integer random variable with probabilities

pk = C�k; k � 0; (4)

whereC =

1

n 0

�k = k�1 0

kXi=1

(n� 1n

)i�k�i; k > 0;

�0 = 1: (5)

We will use this model to develop approximations in the form of closedform expressions for the by-line price for lines of business in a multi-lineinsurer that can be readily implemented.A copula approach could also be used to model the dependence between

assets and lines of business. Such an approach requires numerical techniqueswhereas our aim is to develop a closed form approximation that can be im-plemented in practice.

8

4 The Q Measure

Under the Q measure the expected value of claims discounted at the risk freerate is equal to the price or fair value assuming claims are default free. Thevariance and covariances between lines of business and assets are estimatedfrom historical claims and asset price data.For the model of claims and assets that we have assumed, the Q measure

probability density of claims for line of business i is Gamma G(�i; �); i =1; :::; n so that

fLi (yi) =��i

� (�i)e��yiy�i�1i ; yi > 0;

with

EQ [Li] =�i�

and

V arQ [Li] =�i�2

In passing we also note that

Cov (Li; Lj) = 0�2

i 6= j

Cov (Li; V ) = 0�2; i = 1; ::; n

and� (Li; Lj) =

0p�i�j

; i; j = 1; :::; n

Now under Q we have the fair value of the liability for line of business igiven by

L0i = EQ�e�rLi

�for all i = 1; : : : ; n

and similarly for the assets

V0 = EQ�e�rV

�So

erL0i = EQ [Li] =�i�

There are n+4 parameters that specify the Q measure, i for i = 0; : : : ; n+1;�0 and �. In practice these are estimated from the market price data and

9

historical data on losses. We have assumed that the � parameters are thesame for the gamma random variates used to construct our model of assetand liabilities. This means that the covariances are the same for all lines ofbusiness although the correlations di¤er by line. This assumption may beunrealistic in practice and it is possible to assume that the �0s are di¤erent.This will make the resulting pricing expressions more complex but adds littleto the �nal results.If we consider the continuous compounding discount rate for determining

the fair value of liabilities as the risk free rate plus a margin, r + li then

L0i = EQ�e�rLi

�for all i = 1; : : : ; n

= EP�e�(r+li)Li

�and therefore

li = ln

�EP [Li]

EQ [Li]

�= ln

��

�iEP [Li]

�This margin does not include any allowance for the (frictional) costs of cap-ital. The inclusion of these frictional costs is considered later in the paper.

5 Default Option Value By Line of Business

In order to price by line of business we need to determine the default optionvalue for each line. This is given by

D0i = e�rEQ

"Li

�1� V

L

�+#

= e�rEQ

"��0�X0 +Xi

��1�

�0�X0 +Xn+1

n�0�X0 +X�

�+#

= e�r�0�EQ

"X0

�1�

�0�X0 +Xn+1

n�0�X0 +X�

�+#

+e�rEQ

"Xi

�1�

�0�X0 +Xn+1

n�0�X0 +X�

�+#; i = 1; : : : ; n (6)

10

where Q �Yn+1

i=0G( i; �):

Consider the expectation in the �rst term in (6).

EQ

"X0

�1�

�0�X0 +Xn+1

n�0�X0 +X�

�+#

=

Z 1

0

: : :

Z 1

0

x0

�1�

�0�x0 + xn+1

n�0�x0 + x�

�+dQ

=� ( 0 + 1)

�� ( 0)

Z 1

0

: : :

Z 1

0

�1�

�0�x0 + xn+1

n�0�x0 + x�

�+dQ0

= 0�EQ0

"�1�

�0�X0 +Xn+1

n�0�X0 +X�

�+#(7)

where

� (z) =

Z 1

0

tz�1e�tdt

dQ = f (x0; ::::; xn+1) dx0 : : : dxn+1

=Yn+1

i=0

�� i

� ( i)e��xix i�1

�dx0 : : : dxn+1

and

dQ0 =

�� 0+1

� ( 0 + 1)e��x0x

00

� n+1Yi=1

�� i

� ( i)e��xix

i�1i � i

�dx0 : : : dxn+1

Similarly for the expectation in the second term in (6). We can write

EQ

"Xi

�1�

�0�X0 +Xn+1

n�0�X0 +X�

�+#

= i�EQi

"�1�

�0�X0 +Xn+1

n�0�X0 +X�

�+#(8)

where

dQi =

�� i+1

� ( i + 1)e��xix

ii �

i+1

� n+1Yj=0;j 6=i

� j

�� j�e��xjx j�1j

!dx0 : : : dxn+1

i = 1; :::; n:

11

Thus Qi � G( i + 1; �)Yn+1

j=0;;j 6=iG( i; �), i = 1; :::; n:

In order to evaluate the default option value we must evaluate the ex-pression

EQi

"�1�

�0�X0 +Xn+1

n�0�X0 +X�

�+#; i = 1; :::; n

Note that under Q0

V =�0�X0 +Xn+1 � G( 0 + 1 + n+1; �)

and under Qi, i = 1; :::; n

V =�0�X0 +Xn+1 � G( 0 + n+1; �)

We know that under Q0 and Qi; i = 1; :::; n; the distribution of n�0� X0 +X�will not be Gamma but its distribution can be represented as a mixture ofGamma random variables, so that again using Furman and Landsman (2004)Proposition 2, we have under Q0

L =n�0�X0 +X� � G( 0 + 1 + � + ~�; �)

where ~� is a non-negative integer random variable de�ned in

~pk = ~C~�k; k � 0;

~C =1

n 0+1

~�k = k�1( 0 + 1)

kXi=1

(n� 1n

)i~�k�i; k > 0;

~�0 = 1:

Under Qi; i = 1; :::; n;

L =n�0�X0 +X� � G( 0 + 1 + � + �; �);

where � is a non-negative integer random variable de�ned by (4) and (5).In order to determine a closed form expression to approximate the default

option value allocated to line of business i; we will determine expressions forthe moments of ln � = lnV � lnL:

12

Lemma 1 IfX � Gamma ( ; �) then E [lnX] = ( )�ln� and V ar [lnX] =�00( )�( )

� ( )2 where ( ) = �0( )�( )

= dd ln � ( ) is the Psi (Digamma) func-

tion, �0 ( ) = dd � ( ), and 0 ( ) = d

d ( ) is the Trigamma function (refer

Abramowitz and Stegun, 1972 [1]).

Proof. We have

E [lnX] =�

� ( )

Z 1

0

lnxe��xx �1dx (9)

=

Z 1

0

ln� y�

� 1

� ( )e�yy �1dy

=1

� ( )

Z 1

0

ln (y) e�yy �1dy � ln�

=�0 ( )

� ( )� ln�

= ( )� ln� (10)

where the second line follows from the change of variable y = �x and thelast line by noting that

�0 ( ) =d

d

Z 1

0

e�yy �1dy

=

Z 1

0

ln (y) e�yy �1dy

For the variance we have

V ar [lnX] = E�(lnX)2

�� E [lnX]2

and

E�(lnX)2

�=

Z 1

0

(lnx)21

� ( )e��xx �1� dx

=1

� ( )

�Z 1

0

ln (y)2 e�yy �1dy �Z 1

0

2 ln (y) ln�e�yy �1dy

�+ ln�2

=�00 ( )

� ( )� 2 ( ) ln�+ (ln�)2 (11)

13

Hence

V ar [lnX] = E�(lnX)2

�� E [lnX]2

=�00 ( )

� ( )� 2 ( ) ln�+ (ln�)2 � [ ( )� ln�]2

=�00 ( )

� ( )� 2 ( ) ln�+ (ln�)2 �

� ( )2 � 2 ( ) ln�+ (ln�)2

�=

�00 ( )

� ( )� ( )2 (12)

De�ne the function

( ; a) =1

�( )

Z 1

0

ln(a+ x)x �1 exp(�x)dx:

where ( ; 0) is simply the digamma function ( ):

Lemma 2 IfX � Gamma ( X ; �) ; Y � Gamma ( Y ; �) ; Z � Gamma ( Z ; �)are three independent gamma variables, then the covariance of the dependentrandom variables U = ln(�X+Y ); � > 0 and W = ln(X+Z) takes the form

cov(U;W ) = E[ ( Y ;��X) ( Z ;�X)]� E ( Y ;��X)E ( Z ;�X); (13)

and is approximated by

cov(U;W ) =� X

( Y � 1)( Z � 1)+o(

( Z)

Y � 1)+o(

( Y )

Z � 1); Y ; Z !1: (14)

Proof. Notice that conditional on X = x the random variables U and Ware independent. From (9) we have

E(U �W jX = x) = E(U jX = x)E(W jX = x)

=� Y

� ( Y )

Z 1

0

ln(�x+ y)e��yy Y �1dy

� � z

� ( Z)

Z 1

0

ln(x+ z)e��zz z�1dz

= ( Y ;��x) ( Z ;�x);

14

and applying the expression for covariance gives (13). Now, using a Taylorexpansion for the function ( ; a) with respect to a we get

( Z ;�x) = ( Z) +1

( Z � 1)�x+

1

2�2x2RZ ; (15)

where

jRZ j =1

� ( Z)

Z 1

0

1

(� + z)2e�zz z�1dz � 1

� ( Z)

Z 1

0

e�zz z�3dz

=1

( Z � 1)( Z � 2); Z > 2; (16)

and � 2 [0; �x] and �x > 0: Similarly

( Y ;��x) = ( Y ) +1

( Y � 1)��x+

1

2�2�2x2RY ; (17)

wherejRY j �

1

( Y � 1)( Y � 2); Y > 2: (18)

From (15) and (17) it follows that

E ( Y ;��X) = ( Y ) +� X

( Y � 1)+O(

1

2Y); Y !1;

E ( Z ;�X) = ( Z) + X

( Z � 1)+O(

1

2Z); Z !1;

and therefore

E ( Y ;��X)E ( Z ;�X) = ( Y ) ( Z) + X ( Y )

( Z � 1)

+� X ( Z)

( Y � 1)+

� 2X( Y � 1)( Z � 1)

+o( ( Z)

Y � 1) + o(

( Y )

Z � 1): (19)

On the other hand

15

( Y ;��x) ( Z ;�x) = ( Y ) ( Z) +��x ( Z)

( Y � 1)+�x ( Y )

( Z � 1)

+��2x2

( Y � 1)( Z � 1)+R; (20)

where

R =1

2 ( Z)�

2�2x2RY +1

2

1

( Z � 1)�2�3x3RY

+1

2 ( Y )�

2x2RZ +1

2

1

( Y � 1)��3x3RZ +

1

4�2�4x4RYRZ (21)

Then from (20) and (21) we have, taking into account (16) and (18),

E [ ( Y ;��X) ( Z ;�X)] = ( Y ) ( Z) +� X ( Z)

( Y � 1)+ X ( Y )

( Z � 1)

+� X( X + 1)

( Y � 1)( Z � 1)+ o(

( Z)

Y � 1) + o(

( Y )

Z � 1):

(22)

Substituting (19) and (22) into (13) we get (14).From Lemma 2 it follows that we can use the following approximate

formula for the covariance between U and W for large Y ; Z

cov(U;W ) t� X

( Y � 1)( Z � 1): (23)

Using Lemma 1 we have that under Q0

�0� = EQ0 [ln �]

= EQ0 [lnV ]� EQ0 [lnL] (24)

= � 0 + n+1 + 1

�� 1Xk=0

( 0 + 1 + � + k) ~pk

!and under Qi; i = 1; :::; n

�i� = EQi [ln �]

= EQi [lnV ]� EQi [lnL] (25)

= � 0 + n+1

�� 1Xk=0

( 0 + 1 + � + k) pk

!

16

Also under Q0

�2�0 = V arQ0 [ln �]

=�00� 0 + n+1 + 1

��� 0 + n+1 + 1

� � � 0 + n+1 + 1

�2+

1Xk=0

��00 ( 0 + 1 + � + k)

� ( 0 + 1 + � + k)� ( 0 + 1 + � + k)2

�~pk

�2CovQ0 [lnV; lnL] (26)

From Lemma 2, putting � = n; X = �0�X0 v G( 0+1; �); Y = X� v G( �; �);

Z = Xn+1 v G( n+1; �); gives

CovQ0 [lnV; lnL] = E[ ( �;n�0X0) ( n+1;�0X0)]�E ( �;n�0X0)E ( n+1;�0X0)

and from the approximation formula (23) we have for large � and n+1

CovQ0 [lnV; lnL] � n( 0 + 1)

( � � 1)( n+1 � 1):

Under Qi i = 1; :::; n; we have

�2�i = V arQi [ln �]

=�00� 0 + n+1

��� 0 + n+1

� � � 0 + n+1

�2+

1Xk=0

��00 ( 0 + 1 + � + k)

� ( 0 + 1 + � + k)� ( 0 + 1 + � + k)2

�pk

�2CovQi [lnV; lnL] ; i = 1; :::; n: (27)

With X = �0�X0 v G( 0; �); Y = X� v G( � + 1; �) we have from Lemma 2

that

CovQi [lnV; lnL] = E[ ( �+1;n�0X0) ( n+1;�0X0)]�E ( �+1;n�0X0)E ( n+1;�0X0)

or using the approximation gives

CovQi [lnV; lnL] � n 0 �( n+1 � 1)

; i = 1; :::; n:

17

Having determined the mean and variance of the log ratio based on theunderlying Gamma distributions, we follow Sherris and van der Hoek (2004)[14], and develop an approximation by assuming that V

Lis lognormal with the

mean and variance derived for the log ratio. We will then be able to determinean approximate closed form expression for the by line default option value.The accuracy of this approximation is based on our assumption that thelognormal distribution for the ratio of the assets to the liabilities is a goodpractical approximation for our Gamma distribution assumptions for lines ofbusiness and asset values. In practice the log-normal assumption made hereis an often used assumption and is the basis for the application of standardoption pricing models to insurer balance sheets.We assume that � = V

Lis lognormal with

ln � = lnV � lnL � Normal���; �

2�

�(28)

where�� = E [ln �] = E [lnV ]� E [lnL]

and�2� = V ar [ln �] = V ar [lnV ] + V ar [lnL]� 2Cov [lnV; lnL]

under the relevant distributions for Q0 and Qi were obtained before.

Lemma 3 If � is lognormal with

ln � = lnV � lnL � Normal���; �

2�

�then

E�(1� �)+

�= E

�(1� �) I(1��>0)

�= �(0)� e��+

12�2��

�� (�� + �2�)

��

�=

1

2� e��+

12�2��

�� (�� + �2�)

��

�where I(1��>0) = 1 if 1� � > 0 and 0 otherwise.

Proof. Follows from standard properties of the normal distribution.

18

The value of the default option value for line of business i 2 f1; :::; ng is

e�r�0 0�2

EQ0

"�1� V

L

�+#

+e�r i�EQi

�1� V

L

�+;

and under our log normal approximating assumption it is given by

D0i = e�r�0 0�2

"1

2� e��0+

12�2�0�

����0 + �2�0

���0

!#

+e�r i�

"1

2� e��i+

12�2�i�

����i + �2�i

���i

!#; i = 1; :::; n: (29)

These expressions provide closed form expressions for computation of by-line prices based on our dependent gamma distribution assumptions andlog-normal approximations that are commonly used in the application ofstandard option pricing models to insurer balance sheets.

6 Frictional Costs of Capital and Allocationto Lines of Business

There are various costs of capital that by-line pricing should include arisingfrom the frictional costs associated with capital. These include the trans-actions costs of raising the capital, any additional insurer tax costs, agencycosts as well as �nancial distress costs. We adopt the assumptions used inEstrella (2004) [4] for costs of capital and show how these can be allocatedto line of business. We assume that total costs are determined based on theinsurer balance sheet and we then allocate these to lines of business based onthe allocation of the default option cost. This approach takes into accountthe total balance sheet frictional costs rather than the marginal approach ofMyers and Read (2001) [10]. Since these costs are assumed to be contingenton the total balance sheet, this approach is considered realistic for practicalapplication of the model.Frictional costs such as those arising from transactions costs and other

costs such as additional taxation and agency costs are assumed to occur as

19

a percentage of the end of period surplus and are only incurred provided theinsurer is solvent. If we denote the value of these as Cc then

Cc = EQ�cce

�r (V � L)+�

where cc is the costs of capital as a percentage of the surplus, provided it ispositive. We then have

Cc = EQ

"cce

�rL

�V

L� 1�+#

=nXi=1

EQ�cce

�rLi (�� 1)+�

If we denote the costs of capital allocated to line of business i 2 f1; :::; ng byCic we then have

Cic = cce�rEQ

�Li (�� 1)+

�= cce

�rEQ�Li�(1� �)+ � (1� �)

��= ccD0i � cce

�rEQ [Li (1� �)]= ccD0i � ccL�i (30)

Consider the last term in this expression. Changing measure in the man-ner of (7) and (8) we get

L�i = e�rEQ [Li (1� �)]= e�r

h 0�EQ0 (1� �) + i

�EQi (1� �)

i= e�r

h 0�(1� EQ0�) +

i�(1� EQi�)

i; i = 1; :::; n:

Under assumption (28) we have

EQi [�] = exp(�i� +1

2�2�i); i = 0; 1; :::; n;

where �i�; �2�i ; i = 0; 1; :::; n; were derived in formulas (24), (25), (26), and

(27).We derive an approximation for the expectation of � so that the assump-

tion (28) of log normality of � is not required De�ne the function

�( ; a) =1

�( )

Z 1

0

1

a+ xx �1 exp(�x)dx; > 1

so that �( ; 0) = 1 �1 :

20

Lemma 4 IfX � Gamma ( X ; �) ; Y � Gamma ( Y ; �) ; Z � Gamma ( Z ; �)are three independent gamma variables, then for � > 0;

E(X + Z

�X + Y) = ( X + Z)E�( Y ; ��X) (31)

= ( X + Z)(1

Y � 1� � X( Y � 1)( Y � 2)

+ o(� X

( Y � 1)( Y � 2)));

Y ! 1:

Proof. Conditional on X = x, the nominator and denominator in (31) areindependent so we can write

E(X + Z

�X + YjX = x) = E(X + ZjX = x)E(

1

�X + YjX = x)

= �(x+ Z�)�( ;��x):

Then the full expectation has the form

E(X + Y

�X + Z) = ( X + Z)E�( Y ; ��X):

Using a Taylor expansion for the function �( ; a) with respect to a we get

�( Y ; �ax) =1

Y � 1� �ax

1

( Y � 1)( Y � 2)+1

2�2a2x2RY ;

where

0 < RY =2

� ( Y )

Z 1

0

1

(� + y)3e�yy Y �1dy � 2

� ( Z)

Z 1

0

e�yz Y �4dz

=2

( Y � 1)( Y � 2)( Y � 3); Y > 3; (32)

where � 2 [0; ��x] and ��x > 0: Then

E(X + Z

�X + Y) = ( X + Z)(

1

Y � 1� �a

( Y � 1)( Y � 2)EX +

1

2�2a2E(X2RY ))

= ( X + Z)(1

Y � 1� � X( Y � 1)( Y � 2)

+ o(� X

( Y � 1)( Y � 2)));

Y ! 1;

21

because from (32) one obtains

1

2�2a2E(X2RY ) �

�2 X( X + 1)

( Y � 1)( Y � 2)( Y � 3)

Using Lemma 4 and the appropriate expressions for X; Y and Z givenafter Lemma 2 we immediately obtain

EQ0 [�] t ( 0 + 1 + n+1)(1

� � 1� n( 0 + 1)

( � � 1)( � � 2));

and

EQi [�] t ( 0 + n+1)(1

�� n( 0 + 1)

�( � � 1)); i = 1; :::; n

In practice for an insurer balance sheet of reasonable size � =Pn

i=1 i wouldbe expected to be large so that the approximations will be reasonable forpractical applications.Following Estrella (2004) [4], we assume that �nancial distress costs are

a percentage of the absolute value of the shortfall of assets over liabilities inthe event of insolvency. The value of these costs will then be related to thedefault option value by line of business. Denoting the value of these �nancialdistress costs by Cf we have

Cf = EQ�cfe

�r (L� V )+�

= EQ�cfe

�rL (1� �)+�

where cf is the �nancial distress costs as a percentage of the asset shortfallin the event of insolvency.We can determine the allocation of these to line of business i as

Cif = cfe�rEQ

�Li (1� �)+

�= cfD0i (33)

where Cif is the allocation of the �nancial distress costs to line of business i.An expression for the price by line of business, including the frictional

costs of capital, for our multi-line insurer is given by

Pi + Cic + Cif = e�r�i�� ccL�i � (1� cf � cc)D0i

= e�r�i�� cc (L�i �D0i)� (1� cf )D0i (34)

22

Closed form expressions for L�i and D0i are given in this paper under adependent gamma distribution model.

7 Conclusion

We have developed a pricing model for an insurer with multiple lines ofbusiness where we assume that lines of business and assets have dependentgamma distributions and where we allow for the default option value andits allocation to line of business. We include the �nancial distress costs ofcapital in the pricing by an allocation to line of business using the defaultoption value. In order to do this we consider frictional costs of capital andhow they relate to the variables in the pricing. On the assumption that�nancial distress costs are a percentage of the asset shortfall in the event ofinsolvency these can be uniquely allocated to line of business. Frictional coststhat arise as a percentage of the end of period surplus, assumed to includetaxation, moral hazard and adverse selection costs, agency costs (managerialperquisites) and transactions costs, can also be allocated to line of businessin the same manner. We derive closed form expressions that are used todevelop by-line prices allowing for dependent gamma distributed risks andan assumption that is the basis of the application of standard option pricingmodels to an insurer balance sheet. Closed form expressions are derived forthe allocation of frictional costs to lines of business in the model.

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