Construction Actuarial Model for Aggregate Loss under ... · PDF file8086 Osama Hanafy Mahmoud 1. Introduction Insurance companies need ... They include examples ... Construction actuarial

Applied Mathematical Sciences, Vol. 8, 2014, no. 162, 8085 - 8097

HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.410802

Construction Actuarial Model for Aggregate Loss

under Exponentiated Inverted Weibull

Distribution

Osama Hanafy Mahmoud

Department of Mathematics, Statistics and Insurance

Sadat Academy for Management Sciences, Egypt

Copyright © 2014 Osama Hanafy Mahmoud. This is an open access article distributed under

the Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

The problem of planning reinsurance policies in property and causality

insurance companies and how to estimate the loss reserve play an important role

for the results of the insurance company. Also, they have past or present data for

number of claims and its amounts which want to use in prediction future claim

frequency and claim severity. Many statistical distribution are fitting to find an

appropriate distribution to represent our data. In this paper, we introduce a

statistical distribution known as Exponentiated Inverted Weibull (EIW)

distribution to represent the claim amount and its characteristics for applying it in

actuarial studies. Second, we test the tail weight of the distribution and maximum

likelihood estimation for its parameters. we present our aggregate loss model

under collective risk theory when the claim frequency distribution is Poisson or

Negative Binomial distribution. Also, we present how to calculate the reinsurance

pure premium in case of stop loss reinsurance. Finally, The simulation numerical

example is given to represent our results.

Keywords: Exponentiated Inverted Weibull Distribution, Tail Weight of

Distribution, Maximum Likelihood Estimation, Aggregate Loss Model

8086 Osama Hanafy Mahmoud

1. Introduction

Insurance companies need to investigate claims experience and apply

mathematical techniques for many purposes such as ratemaking, reserving,

reinsurance arrangements and solvency.

Many papers have been presented to aggregate losses as: Heckman and

Meyers (1983) discussed aggregate loss distributions from the perspective of

collective risk theory for severity and count distributions. They include examples

for calculating the pure premium for a policy with an aggregate limit,

calculating the pure premium of an aggregate stop-loss policy for group life

insurance; and calculating the insurance charge for a multi-line retrospective

rating plan, including a line which is itself subject to an aggregate limit. Venter

(1983), Distribution functions are introduced based on power transformations of

beta and gamma distributions, and properties of these distributions are discussed.

The gamma, beta, F, Pareto, Burr, Weibull and loglogistic distributions are special

cases. The transformed gamma is used to model aggregate distributions by

matching moments. The transformed beta is used to account for parameter

uncertainty in this model.

Robertson (1992), Provided an application of the fast Fourier transform to

the computation of aggregate loss distributions from arbitrary frequency and

severity distributions. Papush el al (2001), addressed the question what type of

Normal , Lognormal and Gamma distributions is the most appropriate to use to

approximate aggregate loss distribution. Vilar et al. (2008), described a

nonparametric approach to make inference for aggregate loss models in the

insurance framework by assuming that an insurance company provides a historical

sample of claims given by claim occurrence times and claim sizes.

Bortoluzzo et al. (2009), aimed estimating claim size in the auto insurance by

using zero adjusted Inverse Gaussian distribution. Shevchenko (2010) reviewed

numerical algorithms that can be successfully used to calculate the aggregate loss

distributions. In particular Monte Carlo, Panjer recursion and Fourier

transformation methods are presented and compared. Also, several closed-form

approximations based on moment matching and asymptotic result for heavy-tailed

distributions are reviewed.

One of the most significant goals of any insurance risk activity is to achieve a

satisfactory model for the probability distribution of the total claim amount. In

this paper, we introduce a statistical distribution known as Exponentiated Inverted

Weibull (EIW) distribution to represent the claim amount and its characteristics

for applying it in actuarial studies.

This paper is organized as follows: Section 2 we presentsThe Model claim

severity under Exponentiated Inverted Weibull Distribution and test the tail weight

Construction actuarial model for aggregate loss 8087

of a distribution. In Section 3 we discuss the problem of estimating the parameters

of distribution by using maximum likelihood method. Section 4 we present our

aggregate loss model under collective risk theory when the claim frequency

distribution is Poisson or Negative Binomial distribution. Section 5, how to

calculate the reinsurance pure premium in case of stop loss reinsurance. Finally,

The simulation numerical example is given to represent our results.

2. The Model under Exponentiated Inverted Weibull Distribution

Recently many studies in probability distributions and its applications presented

the Exponentiated Inverted Weibull distribution as:

Flaih et al (2012), Considered the standard exponentiated inverted weibull

distribution (EIW) that generalizes the standard inverted weibull distribution (IW),

the new distribution has two shape parameters. The moments, median, survival

function, hazard function, maximum likelihood estimators, least-squares

estimators, fisher information matrix and asymptotic confidence intervals have

been discussed. A real data set is analyzed and it is observed that the (EIW)

distribution can provide a better fitting than (IW) distribution. Aljouharah Aljuaid,

(2013), presented Bayes and classical estimators have been obtained for two

parameters exponentiated inverted Weibull distribution when sample is available

from complete and type II censoring scheme. Hassan (2013), dealt with the

optimal designing of failure step- stress partially accelerated life tests with two

stress levels under type-I censoring. The lifetime of the test items is assumed to

follow exponentiated inverted Weibull distribution. Hassan et al. (2014),

presented estimation of population parameters for the exponentiated inverted

Weibull distribution based on grouped data with equi and unequi-spaced grouping.

Several alternative estimation schemes, such as, the method of maximum

likelihood, least squares, minimum chi-square, and modified minimum chi-square

are considered.

If our claims amount of insurance portfolio nxxxx ,,,, 321 follow the

Exponentiated Inverted Weibull (EIW) distribution with parameters and in

the following form for the probability density function (pdf):

0,, )()( )1( xxexf x (1)

Therefore, its cumulative probability function (cpf) can be written in the form:

0,, )()( xexF x

(2)

The (right-) tail of a distribution is the portion of the distribution corresponding to


large values of the random variable. A distribution is said to be a heavy-tailed

distribution if it significantly puts more probability on larger values of the random

variable. We also say that the distribution has a larger tail weight. In contrast, a

distribution that puts less and less probability for larger values of the random

variable is said to be light-tailed distribution.

To test the tail weight of a distribution, we can use the Existence of Moments

method as follows:

A distribution )(xf is said to be light-tailed if 1)( rxE for all 0r

and the distribution )(xf is said to be heavy-tailed if either )( rxE does not

exist for all 0r or the moments exist only up to a certain value of a positive

integer r , Finan (2014).

The rth moments of the exponentiated inverted weibull distribution is given as

follows:

0,, )()(0

)1(

xdxxexxE xrr

This can be written as:

rx r

E(x

r

r

0,, )1() (3)

Proof: The pdf of the EIW distribution is:

0,, )()( )1( xxexf x

The rth moments function can be written in the form:

0

)()( dxxfxxE rr

dxxexxE xrr )()( )1(

0

By taking transformation xH

We can write the rth moments function as:

dHHH

eH

xE

Hr

r

1

11

)1(

0

)(

By simplification of the above equation ,we can get

dHeHxE H

rr

r

0

11

)(

This integral known as gamma function , therefore the rth moments function is:

r

xE

r

r 1)(


From the above equation, we can find to obtain the rth moment must the value of

greater than r to be exist.

Since the moments are not finite for all positive r; the exponentiated inverted

weibull distribution is heavy-tailed.

From the above equation, we can find the mean and the variance of EIW

distribution as follows:

By putting r=1

rx E(x

0,, )1

1()

1

(4)

And the second moment by putting r =2 in the form:

rx E(x

0,, )2

1()

2

2 (5)

Thus the variance is:

22

22

)1

1()2

1(

))(()()

xExEV(x

3. Maximum Likelihood Estimation (MLE) for parameters

Suppose that we have postulated a probability model, such as the Exponented

Inverted Weibull distribution, to describe a given loss amount distribution. The

next step in our procedure should be to estimate values for the parameters of the

model.

We use the maximum likelihood method (MLE) for estimating the

unknown parameters and of Exponented Inverted Weibull distribution, as

follows:-

N

1i

)( ) , ( ixfL

Then the likelihood function is as follows,

N

i

i

xNN

N

i

i

x

xe

xeL

N

ii

1

)1()(

1

)1(

)( )(

)(),(

1

(6)

by taking the natural logarithm for the likelihood function , we get


N

i

i

N

i

i xxNNL11

ln)1(lnln),(ln (7)

So, we need to estimate the two parameters and . The first derivatives

for the natural logarithm of the likelihood function with respect to and , are

given by

N

i

ixNL

1

),(ln

(8)

N

i

i

N

i

ii xxxNL

11

ln)ln(),(ln

(9)

The maximum likelihood estimators of and could be obtained by

equating the equations (8) and (9) by zero, and solving them simultaneously using

an iterative technique. We obtain the approximate variance covariance matrix by

replacing expected values by their maximum likelihood estimators and inverting

the Fisher – information matrix, defined by:

2

22

2

2

2

lnln

lnln

LL

LL

I

Where, the second derivatives of the natural logarithm of likelihood function

defined in equation (6) are given as follows:

22

2 ),(ln

NL

(10)

N

i

ii xxL

1

2

)ln(),(ln

(11)

N

i

ii xxNL

1

2

22

2

))(ln(),(ln

(12)

The MLE ˆ and ˆ have an asymptotic variance covariance matrix

obtained by inverting the Fisher – information matrix.

4. Aggregate loss model under collective Risk Theory

Suppose that portfolio has N claims in the past period of time in our experience

and each unit has ix is the claim size which is independent identical distributed


exponented inverted Weibull with parameters and its p.d.f in the equation

(1) and cumulative probability function in the equation (2) .

Then the aggregate losses is S where the sum of claim amounts as:

NxxxxS 321

Suppose also that the individual loss amounts ix are independent on the annual

loss frequency N.

Then it follows that:

The probability density function (pdf) of aggregate losses is

0

* )()()(k

k

xs sfkNprSf (13)

Where )(* sf k

x is called the k th fold convolution of Nxxxx 321

the k th fold convolutions are often extremely difficult to compute in practice and

therefore one encounters difficulties dealing with the probability distribution of S:

An alternative approach is to use various approximation techniques. We consider

a technique known as the Panjer recursive formula.

The mean and variance of aggregate loss distribution can get as:

)()()()()(

)()()(

2NVxExVNEsV

xENEsE

(14)

The pricing problem usually reduces to finding moment of S . A common pricing

formula is )( )( svksEprice

Where the price is the expected payout plus a risk loading which k times the

variance of the payout for some k.

The expected payout )(sE is also known as the pure premium and it can be

shown to be )()( xENE .

Estimation the Mean and the Variance of Aggregate losses Distribution:

We will consider Poisson and Negative Binomial distributions for the frequency

distribution of losses as follows:

I. Poisson Distribution

Suppose that the annual frequency of losses from a portfolio follows a Poisson

distribution with parameter .

In this case the mean of the loss distribution is:


sE )1

1()(

1

(15)

sV )2

1()(

2

(16)

II. Negative Binomial Distribution

Suppose that the annual frequency of losses from a portfolio follows a Negative

Binomial distribution with parameters pr and .

In this case the mean of the loss distribution is:

p

prsE )

11(

1)(

1

(17)

p

p

p

prsV

22

)1

1(1

)2

1(1

)(

(18)

5. Stop Loss Reinsurance

Gauger and Hosking (2008) and Finan (2014) presented the stop loss reinsurance

as:

When a deductible D is applied to the aggregate loss S over a definite period, then

the insurance payment will be

DSDS

DSDSSDSMaxDS

,

,0^0,

the reinsurer will pay the insurer an amount equal to DS

. The insurer's

retained loss is thus DS^ .

The main problem is how to calculate the reinsurance pure premium?.

the reinsurance pure premium is payment as the stop-loss re insurance. Its

expected cost is called the net stop-loss premium and can be computed as:

dxxfdxdxxFDSE S

DD

S )()(1

(19)

6. Numerical Results In this section, we will present a numerical investigation of the maximum

likelihood estimation for the parameters of and .

We need to estimate the two parameters and by using the maximum

likelihood method. So, we will need to solve the three non-linear equations of


logarithm likelihood function (8) and (9) simultaneously using Newton-Raphson

method .The iterative technique, can be applied as follows:

mmmm ACxx 1

1

where

and

2

22

2

2

2

lnln

lnln

mm

mmmm

LL

LL

C

Assuming initial values for each of and , the Newton-Raphson iterative

procedure is continued until either the number of iterations will be ( 200 ) or

when |Xm – Xm+1 | < 5 10-5 .

In the following table, the estimates of unknown parameters, the relative bias

which is the absolute difference between the estimated parameter and its true

value divided by its true value.

-ˆ

Bais Re0

0

ltive

And the mean square error (MSE) which is the mean square of the difference

between the estimated parameter are presented for all the estimated parameters

considering different initial points of the parameters.

ˆM

2

0

NSE

where N is the number of experiments carry out .

m

mm

m

mm

m

mm L

L

Axx

ln

ln

,ˆ

ˆ,

ˆ

ˆ

1

11


Table (1)

Estimators for parameters of EIW distribution, Relative Bias and MSE

MSE Relative Bias Estimator Parameters 0

0

0.02393 0.003084 50.1547 5 50

0.007265 0.01676 5.085238

0.140869 0.006217 60.37536 5 60

0.00727 0.016772 5.085291

0.140825 0.006215 60.37527 6 60

0.010474 0.016771 6.102346

0.3848482 0.008784 70.62036 6 70

0.0104615 0.0167612 6.102282

0.8045189 0.011087 80.89695 7 80

0.0142606 0.01677358 7.119418

Table (1), shows The Estimators of the parameters and of the model,

Relative Bias, and MSE. We can notice that the absolute value of the difference

between the true value of the parameter and its estimator is small value converges

to zero, so these estimators are said to be consistent estimators.

Estimate the mean and the variance of EIW distribution

By substituting the estimated values of the parameters ˆ and ˆ in equations (4)

and (5), we can get the mean and the variance of EIW distribution as shown in

Table (2):

Table (2)

The estimated mean and variance of EIW distribution

)( xE )( 2xE )( xV

50.1547 5.085238 2.26264 5.60568 0.486147

60.37536 5.085291 2.34661 6.029741 0.5229219

60.37527 6.102346 2.013053 4.382492 0.3301104

70.62036 6.102282 2.065445 4.613583 0.3475178

80.89695 7.119418 1.881189 3.798702 0.2598307


From Table (2), we can notice there is direct relationship between the value of and the value of the mean and the variance of distribution. Also, there is inverse

relationship between the value of and the value of the mean and the variance

of distribution.

Estimation the mean and the variance of Aggregate losses distribution:

When the annual frequency of losses from a portfolio follows a Poisson

distribution with parameter 8 by substituting in equations (15) and (16), or

the Negative Binomial distribution with parameters 6.0 and 10 pr by

substituting in equations (17) and (18) in table (3) as follows:

Table (3)

Estimation the mean and the variance of Aggregate losses distribution

Concluding remarks:

In this study, we address the exponentaited inverted weubil distribution issue and

its empirical application of aggregate losses. By testing the tail of the EIW

distribution, we find it has heavy tail. The maximum likelihood method was

applied for estimating the parameters of distribution. Under collective risk model,

we estimate the mean and the variance of the aggregate losses distribution where

the frequency distribution for claim counts is Poisson or Negative Binomial. If we

identify the aggregate losses distribution, we can depend on it to ratemaking,

arrangement for stop of loss reinsurance and estimate the needed loss reserve.

Poisson distribution Negative Binomial

distribution

)( sE )( sV )( sE )( sV

50.1547 5.085238 18.1011 44.84544 33.93958 199.2747

60.37536 5.085291 18.77309 48.23793 35.19992 214.3496

60.37527 6.102346 16.10442 35.05993 30.19579 156.916

70.62036 6.102282 16.52356 36.90866 30.98168 165.1902

80.89695 7.119418 15.04951 30.38961 28.21783 136.6051


References

[1] A. Aljuaid, Estimating the Parameters of an Exponentiated Inverted Weibull

Distribution under Type-II Censoring, Applied Mathematical Sciences, 7 (2013),

35, 1721 – 1736.

[2] A. B. Bortoluzzo, D. P. Claro, M. A. Caetano and R. Artes, Estimating Claim

Size and Probability in the Auto-insurance Industry: the Zero adjusted Inverse

Gaussian (ZAIG) Distribution, Insper Working Paper WPE: 175 (2009).

[3] M. B. Finan M. B., An Introductory Guide in the Construction of Actuarial

Models, Arkansas Tech University, 2014.

[4] A. Flaih, H. Elsalloukh, E. Mendi and M. Milanova. The Exponentiated

Inverted Weibull Distribution, Applied Mathematics & Information Sciences, 6

(2012), 2 , 167 - 171.

[5] M. Gauger and M. Hosking, Construction of Actuarial Models, Bpp

Professional Education, Inc, (2008).

[6] A. S. Hassan, On the Optimal Design of Failure Step-Stress Partially

Accelerated Life Tests for Exponentiated Inverted Weibull with Censoring,

Australian Journal of Basic and Applied Sciences, 7 (2013), 1, 97 - 104.

[7] A. Hassan, A. Marwa ,H. Zaher and E. Elsherpiny, Comparison of

Estimators for Exponentiated Inverted Weibull Distribution Based on Grouped

Data, Int. Journal of Engineering Research and Applications, 4 (2014), 4, 77 - 90.

[8] Heckman, Philip E. Meyers, Glenn G. (1983), "The Calculation of Aggregate

Loss Distributions from Claim Severity and Claim Count Distributions",

Proceedings of the Casualty Actuarial Society Casualty Actuarial Society -

Arlington, Virginia 1983: LXX 22 - 61.

[9] Papush D. E., Pateik, G. S. and Podgaits F., (2001), "Approximations of the

aggregate distributions", CAS Forum, pp. 175 - 186.

[10] Robertson John, (1992), "The Computation of Aggregation loss

distributions", PCAS, pp. 57 - 133.

[11] Shevchenko P. V., (2010), "Calculation of aggregate loss distributions", The

Journal of Operational Risk 5 (2), pp. 3 - 40.


[12] Venter G. (1983), "Transformed Beta and Gamma distributions and the

aggregate losses" PCAS, LXX, pp. 156 – 193.

[13] Vilar J., Cao R., Ausín M. C. and C. González-Fragueiro C., (2008),

"Nonparametric analysis of aggregate loss models", 2nd International Workshop

on Computational and Financial Econometrics (CFE'08), Neuchatel (Suiza), 19 a

21 de junio de 2008.

Received: October 17, 2014; Published: November 18, 2014

Documents

Construction Actuarial Model for Aggregate Loss under ... · PDF file8086 Osama Hanafy Mahmoud 1. Introduction Insurance companies need ... They include examples ... Construction actuarial