^{1}

^{1}

^{*}

In this paper, we have fitted two heavy tailed distributions viz the Weibull distribution and the Burr XII distribution to a set of Motor insurance claim data. As it is known, the probability of ruin is obtained as a solution to an integro differential equation, general solution of which leads to what is known as the Pollaczek-Khinchin Formula for the probability of ultimate ruin. In case, the claim severity is distributed as the above two mentioned distributions, and Pollaczek-Khinchin formula cannot be used to evaluate the probability of ruin through inversion of their Laplace transform since the Laplace Transforms themselves don’t have closed form expression. However, an approximation to the probability of ultimate ruin in such cases can be obtained by the Pollaczek-Khinchin formula through simulation and one crucial step in this simulation is to simulate from the corresponding Equilibrium distribution of the claim severity distribution. The paper lays down methodologies to simulate from the Equilibrium distribution of Burr XII distribution and Weibull distribution and has used them to obtain an approximation to the probability of ultimate ruin through Pollaczek-Khinchin formula by Monte Carlo simulation. An attempt has also been made to obtain numerical values to the probability function for the number of claims until ruin in case of zero initial surplus under these claim severity distributions and this in turn necessitates the computation of the convolutions of these distributions. The paper makes a preliminary effort to address this issue. All the computations are done under the assumption of the Classical Risk Model.

Modeling of the uncertainty prevalent in the domain of insurance in terms of the number of claims arriving in a particular period and the size of the claim severity has been a major research goal in Actuarial science since decades. In this paper, we are concerned with the modeling of the claim severity by the use of two heavy tailed distributions and subsequently for these distributions, we have evaluated the probability of ultimate ruin through Monte Carlo simulation by the use of the Pollachez-Khinchin formula.

In a general insurance portfolio, two quantities of interest characterizing the uncertainty involved in the underlying Risk scenario are modeled in terms of random variables, specifically, a counting distribution is used to model the claim arrival pattern whereas a continuous distribution is used to model the claim severity. This is the basic essence of loss modeling in the domain of general insurance, which constitutes an important ingredient of Risk modeling in this aspect. The proper modeling of these two components determine the base for the computation of the some of the other related actuarial quantities of interest like the probability of ultimate ruin, pure premiums, reserves to be maintained etc.

The distribution fitting which is synonymous to loss modeling in insurance implies choosing an appropriate model to describe the claim arrival pattern and the claim severity. Indispensable to the theme of distribution fitting is the task of estimating the unknown parameters involved in the model and then testing the goodness of fit of the fitted model in describing the observed trend. Some good references for the subject of fitting distributions to losses are [

The data used in this paper is extracted from a motor insurance portfolio where existence of a high positive skewness is a typical characteristic [

The three-parameter Burr XII distribution was originally used in the analysis of lifetime data and is becoming increasingly useful in the context of actuarial science [

In applying the Pollazek-Khinchin formula for the computation of the probability of ultimate ruin, when the claim severity is distributed as the Burr XII or Weibull, we need to generate random observations from their equilibrium distributions and hence, we have derived methodologies to simulate observations from the equilibrium distributions of the Weibull and Burr XII. We have also tried assessing the consistency of these simulation schemes by comparison of the results generated by them with some standard results. The importance of the probability function of the number of claims until ruin is justified from the fact that it renders some insight into the potentiality of a claim to cause ruin. The computation of this function necessitates the convolution of these distributions. Hence, we have found some of the lower order convolution of the Burr XII and the Weibull distributions and have used them as input in computing the probability function of the number of claims until ruin for these distributions.

Much of the literature on ruin theory is concentrated on the Classical Risk Theory. Classical Risk model is one of the models to study the evolution of the Surplus process of an insurance company continuously over time. Classical Risk model provides the basic frame work in which the probability of ultimate ruin is defined and so also, constitutes the assumption under which, an expression for the probability function for the number of claims until ruin is derived.

Briefly our objectives for this paper are:

1) To fit the Burr XII distribution to a set of insurance data through an algorithm mentioned in [

2) To simulate from the Equilibrium distributions of the Burr XII and Weibull distributions and to use them in obtaining an approximation to the probability of ultimate ruin from the Pollaczek Khinchin formula through simulation.

3) To obtain the convolution of Burr XII distribution and the Weibull distribution up to the fourth order and to use them in obtaining the probability function for the number of claims until ruin.

The first part of the paper deals with the Watkins (1999) algorithm for obtaining the MLE for the parameters of the Burr XII distribution, which also leads to the estimation of the parameters of the Weibull distribution. This is followed by testing the goodness of fit of these distributions through some statistics based on the empirical distribution function (EDF). The second part deals with the simulation from the equilibrium distributions of Burr XII and Weibull distribution and with the evaluation of the probability of ultimate ruin through Pollaczek-Khinchin formula. This is being followed by the section on the convolution of the Burr XII distribution and Weibull distribution and their application in the evaluation of the probability function for the number of claims until ruin. The concluding section deals with results and discussions.

However, it needs to be mentioned that in case of the Burr XII distribution, in computing the interested actuarial quantities like the probability of ultimate ruin and the probability function for the number of claims until ruin, instead of the fitted Burr XII distribution, use has been made of an illustrative Burr XII distribution since the fitted Burr XII distribution led to some complexities in determining these quantities. The illustrative Burr XII distribution that is being used is the one which was fitted to the Property Claim Services (PCS) dataset covering losses resulting from natural catastrophic events in USA that occurred between 1990 and 1999 [

The cumulative distribution function for the two parameter Weibull distribution is given by

in which the positive parameters

And the corresponding probability density function is given by

If we consider a sample of “m” items

where

We have used the Multi Parameter Newton Raphson Method for estimating the parameters of the Weibull distribution. In the appendix: (1) of [

The pdf of the three parameter Burr XII distribution is given

by

The algorithm for finding the maximum likelihood estimators (MLE) for the parameters of the Burr XII distribution is taken from [

The basic two parameter Burr XII distribution with shape parameters

An scale parameter

Letting

with shape parameter

For a sample of “m” items

where

The main steps of the algorithm are:

Step 1: First, we find the maximum likelihood of the parameters

Step 2 Then, we rescale the original data by

the MLEs of the parameters of the Burr XII distribution, the utilized values are the rescaled values

The argument in [

In the appendix: (1) of [

Let

where

claim process and we have

Let

Classical Risk model, despite the fact that it is considered to be the basis of many models in insurance mathematics involves many simplication criterions which make it deviate from the real life situations. For example, the assumptions like the independence between the claim severity and the claim number distributions, the intensity parameter

As given in [

If L is the maximal aggregate loss random variable, then it can be shown that

where

And the number of drops K is geometrically distributed with the parameter

[

It is evident that ruin never occurs or that the company survives if starting with an initial surplus of u, the maximal aggregate loss random variable L never exceeds u i.e. the probability of ultimate survival is:

Let

of the K-fold convolution of the distribution of Y with itself.

Then the general solution to Equation (2.4.1) is given by (see [

Equation (2.4.4) is known as the Pollaczek-Khinchin formula for the Probability of ultimate ruin.

An explicit expression for the probability of ultimate ruin can be derived through the use of the Pollaczek-Khinchin formula for those claim amount distributions whose Laplace transforms have closed form expressions [

In the subsequent section, we give a brief description to the computation of the approximation to the probability of ultimate ruin from the Pollaczek-Khinchin formula through Monte Carlo simulation. However, one of the main objectives of this paper is to lay down a methodology to simulate observations from the Equilibrium distributions of Burr XII and Weibull and to use these observations to obtain an approximation to the Probability of Ultimate Ruin from the Pollaczek-Khinchin formula. In fact, the simulation from the Equilibrium distribution in case the claim severity is heavy tailed, constitutes one of the main challenges in the application of the above simulation procedure to obtain an approximation to the Probability of ultimate ruin.

In [

Step 1: Generate

Step 2: Generate

Step 3: If

Step 4: Repeat steps 1 to 3n (the number of times the simulations is to be carried out) times.

Step 5: Estimate E(z) by

An approximation to

We have used the rejection method for generating random observations from the equilibrium distributions of Burr XII and Weibull distributions. As outlined in [

The algorithm for generating observations from

Step 1 Generate Y having density g.

Step 2: Generate a random number U.

Step 3: If

where c is a constant such that

Here X can be considered to be a random observation generated from the density

If

(

The density of the Equilibrium distribution of Burr XII is given by

where

The inverse transform algorithm (see [

Therefore,

Our goal is to generate observations from

Let U be a random number lying between 0 and 1 and the cumulative distribution function of the Burr XII is given by

Hence the transform equation is

Solving for y gives

Now, we need to find c such that

Let

To maximize,

Therefore,

Þ Either

Now

Hence,

which gives

It can be shown that for this value of y,

Therefore,

Therefore,

Hence the algorithm for generating random observations from the Equilibrium distribution of Burr XII is

Step 1: Generate a random number

Step 2: Generate a random number

Step 3: If

Here, x can be considered to be a random observation generated from the Equilibrium distribution of Burr XII.

Repeat the above steps as many times as the number of random observations required from the Equilibrium distribution of the Burr XII distribution.

The density of the Equilibrium distribution of Weibull is given by

Here

Here unlike the situation in Burr XII distribution,

bution because in that case

tribution given in Equation (2.1.2), we choose the shape parameter

nential distribution with parameter

If U is a random number, using inverse transform algorithm it can be shown that an observation generated

from the exponential distribution with parameter

Next, we need to find c such that

i.e.

let

Therefore,

It can be shown that for this value of y,

Therefore,

and

And

Hence the algorithm for generating random observations from the Equilibrium distribution of Weibull is:

Step 1: Generate a random number

Step 2: Generate a random number

Here X can be considered as a random observation generated from the equilibrium distribution of Weibull and the process is repeated as many times as the number of random observations required to be generated from the equilibrium distribution.

In assessing the efficiency of our simulation scheme, we have adopted the following procedure.

As given is Equation (2.4.2),

If L is the maximal aggregate loss random variable, then it can be shown that

where

And the number of drops K is geometrically distributed with the parameter

Also as shown in [

Using our simulation schemes, we have simulated 20 values of L for each of the cases, when the claim severity is Burr XII and when the claim severity is Weibull. The mean of L obtained through simulation is compared with

In this section, we have attempted to carry out the convolution of the Burr XII distribution with itself and the convolution of the Weibull distribution with itself and illustrated their applications in computing the Probability function of the number of claims until ruin in case of zero initial surpluses. The convolution of the Burr XII distribution and the Weibull distribution can be carried out only numerically. It can be noted that evaluation of the

Here we have obtained the convolution of the Burr XII distribution and Weibull up to the fourth order in the form of integrals which were evaluated numerically using R program.

First convolution of the Burr distribution is its pdf itself.

1) Second Convolution of the Burr Distribution

The second convolution of the Burr XII distribution is the distribution of

The pdf of Z is given by

It is not possible to find an explicit expression for it and it has to be computed only numerically.

2) Third Convolution of the Burr distribution

The third convolution of the Burr XII distribution is the pdf of

The pdf of

Similarly, the n^{th} convolution of the Burr XII distribution is given by

It is to be noted that in determining the

First convolution of the Weibull is its pdf itself.

1) Second Convolution of Weibull

The second convolution of Weibull is the distribution of

The pdf of Z is given by

As in the case of Burr XII, this also can only be evaluated numerically.

Likewise, the third and the higher order convolutions of Weibull can be defined and as in the case of Burr XII, they can be evaluated only numerically and as stated earlier, the

The distribution of the number of claims until ruin has been studied by a number of authors over the year. Reference [

In this paper, we have used the results derived in [

We state the results from [

Probability function for the number of claims until ruin for zero initial surplus is given by

for

where

Here, we note that in obtaining

Inserting

we have

and

Data: Our data is a set of 160,000 claim amounts spread over a period of 6 months i.e. April, 2013 to September, 2013 from a General Insurance company from its motor insurance portfolio covering all its branches in India. No adjustment was made for inflation for the time horizon is narrow. It needs to be mentioned that the data is utilized more for the illustration of the various methodologies rather than for the extraction of any concrete meaningful conclusion.

Summary statistics of the data as shown in

Sample size | Mean | Standard deviation | Min | 25% Quantile | Median | 75% Quantile | Max | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|---|

160,000 | 1.78834e+04 | 22,805.81 | 523 | 6043.00 | 10,583.0 | 19,374.2 | 188,209 | 3.576628 | 18.94972 |

the right, which in a way, justifies the use of these heavy tailed distributions for modeling our data.

For finding the maximum likelihood estimators for the parameters of the Weibull distribution, the use has been made of the Multi parameter Newton Raphson method. ^{th} iteration thereby giving the estimated values of the parameters as shown in

In making a comparative assessment of the fit as to judge which of the two distributions is providing a better fit to the data, values of the log-likelihood indicate that compared to Weibull, Burr XII is modeling the data in a better way since the log-likelihood for the sample in case of Burr XII is more than that for Weibull (Log-like- lihood for the sample under the fitted Weibull was found to be −1726599 and that for the fitted Burr XII was found to be −162475.8).

Hence, we have little evidence to believe that either the Burr XII distribution or Weibull distribution adequately describes the claim data. In the subsequent sections, we have used this fitted Weibull distribution mainly with the objective of depicting the computational methodologies associated with the Weibull distribution in obtaining some of the important Actuarial Quantities viz the Probability of ultimate ruin and probability function for the number of claims until ruin. However, the fitted Burr XII distribution was excluded from being used in the subsequent computational methodologies for the current limitations encountered in our computing schemes, with the occurrence of numerical error being indispensable, it leads to some absurd results which were difficult to interprete in the normal framework. Instead, an illustrative Burr XII distribution was used for displaying the complexity associated in determining these actuarial quantities in case of Burr XII claim severity distribution.

One of the main objectives of this paper was to lay down methodologies to simulate from the equilibrium distributions of Weibull and Burr XII distributions and as indicated earlier, these simulations constitute a vital component for the evaluation of the probability of ultimate ruin through the Pollaczek-Khinchin formula.

Parameter | Estimate |
---|---|

18,058.838357 | |

1.0196673 | |

Anderson Darling statistics | 4123.742 (0.04) |

Cramer Von statistics | 655.1592 (0.07) |

Parameter | Estimate |
---|---|

1.670876e+05 | |

8.6572840e−01 | |

1.047651e+06 | |

Anderson Darling statistics | 5969.454 (0.002) |

Cramer Von statistics | 933.8827 (0.006) |

Sample size | Random observations generated | ||||
---|---|---|---|---|---|

31,678.10 | 43,124.48 | 120,051.29 | 224,368.08 | 41,248.71 | |

192,876.91 78,806.61 | 128,269.12 112,627.39 | 148,702.50 128,295.76 | 133,815.45 37,733.32 | 384.87 61,205.60 |

of Weibull whereas

Sample size | Random observations generated | ||||
---|---|---|---|---|---|

12,482.550 | 3049.276 | 31,127.129 | 23,829.245 | 9455.99 | |

1491.491 20,713.703 | 3352.752 24,037.673 | 14,719.378 5606.029 | 14,056.884 21,762.111 | 4901.109 6159.868 |

to appraise the efficiencies of the simulation schemes, the direct expressions for the mean and variance (theoretical) of these equilibrium distributions were not available and therefore, we have used an indirect way to assess the efficiencies of these schemes.

The maximal aggregate loss random variable is related to the simulated observations by Equation (2.4.2) and a direct expression for the mean of L is given by Equation (2.4.18). Hence a comparison of the mean of L obtained through our simulation schemes with that of

Serial No. | Value of K simulated from the geometric distribution with r = 0.7692308 | K number of random observations generated from the equilibrium distribution | Value of L |
---|---|---|---|

1 | 0 | ---- | 0 |

2 | 2 | 9498.68 91,875.943 | 101,374.6 |

3 | 0 | ---- | 0 |

4 | 1 | 139,696.3 | 139,696.3 |

5 | 0 | ---- | 0 |

6 | 0 | ---- | 0 |

7 | 0 | ---- | 0 |

8 | 1 | 144,745.2 | 144,745.2 |

9 | 1 | 49,135.06 | 49,135.06 |

10 | 0 | --- | 0 |

11 | 0 | --- | 0 |

12 | 0 | --- | 0 |

13 | 0 | --- | 0 |

14 | 2 | 40,129.82 104,107.86 | 144,237.7 |

15 | 0 | --- | 0 |

16 | 0 | --- | 0 |

17 | 1 | 31,469.31 | 31,469.31 |

18 | 0 | --- | 0 |

19 | 4 | 77,518.48 48,798.00 208,826.83 77,272.90 | 412,416.2 |

20 | 0 | --- | 0 |

Therefore, the simulated mean of L based on 20 simulations is (101,374.6 + 139,696.3 + ∙∙∙ +412,416.2)/20 = 51,153.72.

Serial No. | Value of K simulated from the geometric distribution with r = 0.7692308 | K number of random observations generated from the equilibrium distribution | Value of L |
---|---|---|---|

1 | 1 | 54,042.1 | 54,042.1 |

2 | 1 | 17,896.88 | 17,896.88 |

3 | 1 | 13,860.43 | 13,860.43 |

4 | 1 | 31,367.25 | 31,367.25 |

5 | 0 | --- | 0 |

6 | 0 | --- | 0 |

7 | 0 | --- | 0 |

8 | 1 | 15,205.51 | 15,205.51 |

9 | 1 | 10,586.62 | 10,586.62 |

10 | 1 | 4092.258 | 4092.258 |

11 | 0 | --- | 0 |

12 | 2 | 4334.18 28,415.49 | 32,749.67 |

13 | 0 | --- | 0 |

14 | 0 | --- | 0 |

15 | 0 | --- | 0 |

16 | 0 | --- | 0 |

17 | 2 | 4068.676 20,641.919 | 24,710.59 |

18 | 0 | --- | 0 |

19 | 0 | --- | 0 |

20 | 0 | --- | 0 |

Therefore, the simulated mean of L based on 20 simulations is (54,042.1 + 17,896.88+ ∙∙∙ +24,710.59)/20 = 10,225.57.

The algorithm for the evaluation of the probability of ultimate ruin through the Pollaczek Khinchin formula as described in section (2.4.1) shows how the simulated observations are to be used in evaluating the Probability of ultimate ruin in case of Weibull and Burr XII distributed claim severity.

Computation of the convolutions of a distribution with itself is very challenging, specially, when the distribution does not have a closed form expression for its Laplace transform (Moment generating function) and since neither Weibull nor Burr XII has closed form Laplace transform, their convolutions can be determined only

Value of the initial surplus u (in Rs) | |
---|---|

10 100 1000 2000 10,000 20,000 30,000 50,000 100,000 200,000 | 0.2136 0.2136 0.2134 0.2130 0.2080 0.1961 0.1837 0.1557 0.1023 0.0381 |

Value of the initial surplus u (in Rs) | |
---|---|

10 100 1000 2000 10,000 20,000 30,000 50,000 100,000 200,000 | 0.2321 0.2307 0.2218 0.2130 0.1510 0.0969 0.0628 0.0257 0.0031 0.0001 |

numerically.

Z (point at which the convolution is determined) | 2^{nd} Convolution | 3^{rd} Convolution | 4^{th} Convolution |
---|---|---|---|

10 | 0.001825971 | 0.005829559 | 0.01329008 |

100 | 2.411275e−24 | 8.967633e−24 | 3.153996e−23 |

200 | 3.38285e−44 | 1.025027e−43 | 2.929347e−43 |

1000 | 1.100867e−176 | 2.649168e−176 | 6.010733e−176 |

10,000 | 0 | 0 | …0… |

Z (point at which the convolution is determined) | 2^{nd} Convolution | 3^{rd} Convolution | 4^{th} Convolution |
---|---|---|---|

10 | 4.200736e−09 | 4.123718e−13 | 2.691496e−17 |

100 | 4.252141e−08 | 4.351768e−11 | 2.965936e−14 |

200 | 8.529473e−08 | 1.752377e−10 | 2.397906e−13 |

1000 | 4.288571e−07 | 4.428114e−09 | 2.872162e−11 |

10,000 | 4.308908e−06 | 4.466750e−07 | 2.726931e−08 |

cost of reducing some accuracy in the values obtained through these numerical integrations.

The computation of the probability function of the number of claims until ruin requires the computation of the convolution of the underlying claim severity distribution, for example, the computation of the probability function for 3 number of claims until ruin would require second convolution as an input, 4 number of claims until ruin would require third convolution and so on.

We give some insight into the numerical integration underlying the computation of

From Equation (2.5.9), we have

1) We have first evaluated

Here we have used the function for 3^{rd} convolution of Burr XII as required in

2) Now consider the integrand

respect to t in the interval

0.1029602 | 0.05060141 | 0.003102334 |

0.4559831 | 0.4044081 | 0.7318781 |

number of intervals of values for t and was found to have significant value only in the interval [1e−06, 22], its value being zero beyond it. The final value was obtained by numerical integration of ^{rd} rule in the interval [1e−06, 22].

Interestingly, it may be noted from Equation (2.5.6), that to find the probability function of “m” number of claims until ruin (in case of zero initial surplus), it is required to use the

This paper has chance to provide themes for further exploration if means are devised to eliminate the following limitations.

1) Neither of the distribution was found to qualify the goodness of fit tests, as judged, from the EDF statistics, though we proceeded with the use of the estimated values of the parameters of the fitted Weibull distribution as input for the computational methodologies, targeted at the evaluation of the actuarial quantities under consideration.

2) The fitted Burr XII distribution had to be excluded from being used as an input for the computational methodologies for it led to some inconsistent results. It needs further scrutiny to identify the cause for these inconsistencies.

3) The simulation schemes need to be further improved for the values of the probability of ultimate ruin for the Burr XII distribution, it yielded are inconsistent to the values computed earlier [

4) To avoid the complexity of having to evaluate a number of nested integrals numerically, we had to remain content with the evaluation of just the lower order convolution of these distributions, which, in turn, enabled us to compute the probability function for the number of claims until ruin, for a lower ranking whose significance to reality is not as important as the function, computed for a higher ranking of the order at which the claims arrive.

Modeling of the insurance data through these two heavy tailed distributions is quite a challenge in the realm of statistical computational theory and considering the fact, that highly skewed data which can be adequately modeled only through heavy tailed distributions, occurs frequently in the domain of General insurance, our work might be useful for insurance practitioners concerned with statistical modeling in Risk analysis.

Methodologies suggested for the simulation from the Equilibrium distributions of Burr XII and Weibull were reasonably efficient and when applied to the algorithms for the evaluation of the probability of ultimate ruin through simulation using the Pollaczek-Khinchin formula, they led to fairly good approximations to the probability of ultimate ruin and these approximations were also consistent with practical rationalism. However, further exploration is needed to improve these simulation schemes, for example, by implementing conditional Monte Carlo algorithms and by using some properties of the family of the sub-exponential distributions (of which Weibull and Burr XII are members) to improve the algorithms.

The paper has made an attempt to address the complex issue of evaluating the convolution of the Weibull and the Burr XII distributions. Further investigation is needed to identify if any method other than numerical integration exists for the evaluation of the convolution of these heavy tailed distributions. Control of error in evaluating these numerical integrals, reduction in the execution time etc. can be the themes for further probe. Further extension of this work can also be directed towards the evaluation of the convolutions of higher order for they owe much relevance to the assessment of the chances of ruin (insolvency) for the insurance company.

Dilip C. Nath,Jagriti Das, (2016) Modeling of Insurance Data through Two Heavy Tailed Distributions: Computations of Some of Their Actuarial Quantities through Simulation from Their Equilibrium Distributions and the Use of Their Convolutions. Journal of Mathematical Finance,06,378-400. doi: 10.4236/jmf.2016.63031