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In this paper, we have used an algorithm to fit the Burr XII distribution to a set of insurance data. As it is well known, the probability of ultimate ruin is obtained as a solution to an integro-differential equation and in case, the claim severity is distributed as Burr XII distribution, this equation has to be solved numerically to obtain an approximation to the probability of ultimate ruin. Two numerical algorithms, namely the stable recursive algorithm and the method of product integration have been used to obtain numerically an approximation to this probability of ultimate ruin. The use of these two numerical algorithms provides a scope for comparing the consistency in values obtained by them. The first two moments of the time to ruin in case of Burr XII distributed claim severity have also been computed using the probability of ultimate ruin obtained through the stable recursive algorithm as an input. All these computations have been done under the assumption of the classical risk model.

An actuarial risk model is concerned with the study of the mathematical aspects observed in the behavior of a collection of risks generated by an insurance portfolio. In general insurance risk modeling, the two quantities of paramount importance are the number of claims arriving in a fixed time period and size of each claim. Modeling of the former aspect is done in terms of a discrete distribution, more specifically a counting distribution whereas the claim severity is modeled through what is known as a loss distribution.

There are compelling reasons to use mathematical models to describe insurance loss amounts (claim severities). As specified, in [

A good introduction to the subject of fitting distribution to losses is given in [

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 [

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

2) To compute the probability of ultimate ruin when the claim severity is distributed as our fitted Burr XII distribution as well as a Burr XII distribution with a set of illustrative values for its parameters, using two numerical algorithms namely a stable recursive algorithm and the method of Product Integration.

3) To compute the first two moments of the time to ruin in case of Burr XII distributed claim severity.

The first part of the paper deals with the Watkins [

The illustrative Burr XII distribution that is being used is the one that is being fitted to the Property Claim Services (PCS) dataset covering losses resulting from natural catastrophic events in USA that occurred between 1990 and 1999 [

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

in which the positive parameters

The basic two parameter Burr XII distribution with shape parameters α and τ has the cumulative distribution function

An scale parameter

Letting

distribution for the Weibull distribution with shape parameter

If we consider a sample of “m” items

And the log likelihood function of the Burr XII distribution is given by

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

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

The argument in [

Let

where

distribution function F such that

Let

whereas

However it needs to be noted that the Classical Risk Model involves many simplication criterions which might make it deviate from real life situations. For example, the classical risk model assumes that the intensity parameter λ is independent of time, independence between claim severity distribution and claim number distribution, premium is received continuously in time, surplus earns no interest and is neither subjected to tax, no effect of inflation etc. Despite such assumptions, Classical Risk model still constitute the basis of many models in insurance mathematics.

Probability of Ruin is a very important component of the operational risk theory. It reflects the volatility in the business and can serve as a useful tool in long range planning for the use of insurer’s funds. Ruin, in some sense, corresponds to the insolvency of the insurance company although; solvency/insolvency of an insurance company involves many other complicated considerations.

Probability of ruin can be obtained as the solution of an integro differential equation [

According to [

with

Let

The usual approach is to apply a discretization technique to approximate the integral in (2.3.1) (see [

In Reference [

Reference [

1) First carry out the sub division of the interval

“n” the number of intervals is chosen to be sufficiently large.

2) For every

3) Then the upper bound and the lower bound to the probability of ruin is given by.

Upper bound is

and the lower bound is

with of course,

An upper bound for the error in the estimation of

The stability of this numerical procedure is justified from the fact that there is no cumulative effect of propa-

gation of error as it can be shown that if

For the derivation of the bounds and the full justification on the stability of this method, see [

We have from (2.3.2),

For the Burr XII distribution given in (2.2.1), we have

Therefore,

Now, it can be shown that,

Using this in Equation (2.3.5), we have,

And this can be computed using the pbeta function of R Software.

Also, we have,

Product integration which was traditionally used to numerically solve Volterra Integral equation of the second type (see [

As stated in Section (2.3), for calculating the infinite time ruin probability numerically, one has to solve the integral Equation (2.3.1) which can also be put in the form (of a Volterra integral equation of the second kind) as shown below

where

And

Since, the early 1980’s, the numerical methods for the evaluation of

The Volterra integral equation of the second kind is given by

where

We first factorize

where

The interval

A quadrature rule of the form

where

It is assumed that

Assuming,

And finally, the estimate of

with

where

For accelerating the convergence, as mentioned in [

We have used product integration to compute the Ultimate Probability of Ruin for Burr XII distributed claims taking an illustrative value of

We have considered the Burr distribution which has been fitted to our data as well as a Burr distribution with a set of illustrative values for its parameters.

Here

which gives

As derived in (2.3.7),

From (2.4.3), we have

As for this distribution, all the moments

Let

where

where

And

pbeta function of the R software.

Similarly, from (2.4.13), we have

The distribution of the time to ruin is an interesting quantity related to the probability of ruin and plays a vital role in warning the management for possible adverse situations. There is no closed form expression developed for the distribution of the time to ruin except for the Exponential distribution and the Erlang group of distributions [

Initial ideas on this aspect can be found in [

Reference [

where

Let L: the maximum of the aggregate loss process so that

In [

and

Formula (6.2.1) of [

Hence,

Similarly,

Data: Our data is a set of 160,000 claim amounts spread over a period of 6 months i.e. from April, 2013 to September, 2013 obtained from Bajaj Allianz General Insurance company, India 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. Since the inter arrival time of claim was difficult to track, the intensity parameter was estimated on the basis of the number of claims arriving per day during the period.

Summary statistics of the data as shown in

For finding the maximum likelihood estimators for the parameters of the Burr XII distribution, the use of the algorithm mentioned in [^{th} iteration thereby giving the estimated values of the parameters 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 | 10,583.00 | 19,374.25 | 188,209 | 3.576628 | 18.94972 |

Parameter | Estimates |
---|---|

1.670876e+05 | |

8.6572840e−01 | |

1.047651e+06 | |

Anderson Darling statistics | 5969.454 (0.002) |

Cramer Von statistics | 933.8827 (0.006) |

graphical displays.

Hence, we have little evidence to believe that the Burr XII distribution adequately describes the claim data. However, in the subsequent sections, we have used this fitted Burr distribution along with another Burr XII distribution with a set of illustrative values for its parameters mainly with the objective of depicting the computational methodologies associated with the Burr XII distribution in obtaining some of the important Actuarial Quantities.

In both the tables, it has been observed that the probability of ultimate ruin is decreasing with an increase in the initial capital which is as expected. In case of our fitted Burr XII distribution, the difference between the upper

Value of initial surplus u (Rs in Lakhs) | Lower bound to the probability of ruin | Upper bound to the probability of ruin | Probability of ultimate ruin | Upper bound to the error of estimation |
---|---|---|---|---|

10 20 30 40 50 60 70 80 90 100 200 500 1000 | 1.142307e−01 1.625645e−02 2.154203e−03 2.655454e−04 3.038352e−05 3.226737e−05 3.176367e−07 2.895590e−08 2.442391e−09 1.904733e−10 1.916410e−23 1.378756e−81 1.854485e−175 | 1.226913e−01 2.152154e−02 4.033019e−03 8.060850e−04 1.715703e−04 3.882505e−05 9.325516e−06 2.373527e−06 6.390523e−07 1.816993e−07 8.640573e−12 2.823172e−17 7.133608e−19 | 1.184610e−01 1.888890e−02 3.093611e−03 5.357697e−04 1.009769e−04 2.102589e−05 4.821576e−06 1.201242e−06 3.207473e−07 9.094491e−08 4.320286e−12 1.411586e−17 3.566804e−19 | 8.460656e−03 5.265093e−03 1.878815e−03 5.406304e−04 1.411867e−04 3.559831e−05 9.007879e−06 2.344572e−06 6.366099e−07 1.815089e−07 8.640573e−12 2.823172e−17 7.133608e−19 |

Value of initial surplus u (Rs in Lakhs) | Lower bound to the probability of ruin | Upper bound to the probability of ruin | Probability of ultimate ruin | Upper bound to the error of estimation |
---|---|---|---|---|

10 20 30 40 50 60 70 80 90 100 200 500 1000 | 7.692126e−01 7.691945e−01 7.691764e−01 7.691582e−01 7.691401e−01 7.691220e−01 7.691038e−01 7.690857e−01 7.690675e−01 7.690494e−01 7.688679e−01 7.683230e−01 7.674131e−01 | 7.692126e−01 7.691945e−01 7.691764e−01 7.691582e−01 7.691401e−01 7.691220e−01 7.691038e−01 7.690857e−01 7.690675e−01 7.690494e−01 7.688679e−01 7.683230e−01 7.674131e−01 | 7.692126e−01 7.691945e−01 7.691764e−01 7.691582e−01 7.691401e−01 7.691220e−01 7.691038e−01 7.690857e−01 7.690675e−01 7.690494e−01 7.688679e−01 7.683230e−01 7.674131e−01 | 8.904433e−12 3.561995e−11 8.015000e−11 1.424981e−10 2.226674e−10 3.206607e−10 4.364815e−10 5.701327e−10 7.216172e−10 8.909379e−10 3.565763e−09 2.231996e−08 8.947729e−08 |

and the lower bounds to the probability of ultimate ruin seems to be decreasing in the absolute sense which has lead to the decline in the upper bound to the error of estimation. In case of the Burr XII distribution with a set of illustrative values for its parameters (_{1}) more rapidly.

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

10 20 30 40 50 60 70 80 90 100 200 500 1000 | 1.184138e−01 1.873948e−02 2.966592e−03 4.693954e−04 7.427929e−05 1.175717e−05 1.867170e−06 3.020832e−07 5.284120e−08 1.176783e−08 4.626994e−11 1.456966e−14 2.089981e−17 |

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

10 20 30 40 50 60 70 80 90 100 200 500 1000 | 0.7692126 0.7691945 0. 7691764 0.7691582 0.7691401 0.7691220 0.7691038 0.7690857 0.7690675 0.7690494 0.7688679 0.7683230 0.7674130 |

of ultimate ruin through both the algorithms is that in case of our fitted Burr XII distribution with an increase in the value of the initial capital, the values for the probability of ultimate ruin are decreasing at a significantly high rate whereas this declined at a moderate rate in case of the illustrative Burr XII distribution.

In computing the moments of the time to ruin, in contrast to [

the first moment as an input. For numerical integration, we have used Simpson’s

tegration. All of the computations have been done using the R software [

From

1) Mean (in years) of the time to ruin for the illustrative Burr XII i.e. Burr XII with

2) Mean (in years) of the time to ruin for the fitted Burr XII distribution i.e. Burr XII with

In obtaining the mean of the time to ruin, an illustrative value of λ has been taken as λ = 32.78 and the same value is retained in obtaining the second moment of the time to ruin.

Considering the fact that heavy tailed right skewed distribution like Burr XII arises frequently in case of risk modeling in general insurance, our work may be useful for insurance practitioners and experts from the financial industry. Our main objective was to compute the probability of ultimate ruin in case the claim severity is distributed as Burr XII distribution and this was implemented through the application of two numerical algorithms.

Value of the initial surplus u (in unit of Rs 1 lakh) (1) | Mean (in years) of the time to ruin for Burr XII with | Mean (in years) of the time to ruin for Burr XII with |
---|---|---|

10 | 16.066847 | 0.09771947 |

20 | 10.207575 | 0.09772715 |

30 | 8.3243480 | 0.09773484 |

40 | 7.8522660 | 0.09774253 |

50 | 7.7381620 | 0.09775021 |

60 | 7.7089970 | 0.09775790 |

70 | 7.7007810 | 0.09776559 |

80 | 7.6982140 | 0.09777328 |

90 | 7.6973290 | 0.09778098 |

100 | 7.6969920 | 0.09778866 |

Value of the initial surplus u (in unit of Rs 1 lakh) | Second order moment of the time to ruin for Burr XII with |
---|---|

10 | 0.09799326 |

20 | 0.09800281 |

30 | 0.09801237 |

However, no effort has been made to judge which method gives the better estimate to the probability of ultimate ruin as there is no exact expression for it in case of Burr XII claim severity distribution. Although, there are instances in literature [

Also, in obtaining the moments of the time to ruin, it was found that in case of the illustrative Burr, the first moment (mean) of the time to ruin is increasing with an increase in the value of the initial surplus. This is to be expected in practice, because the induction of larger surpluses tends to prolong the time to ruin (if it ever happens). However in case of our fitted Burr XII distribution, there was deviation from this intuitive logic, implying that the mean of the time to ruin was found to be decreasing with an increase in initial surplus. The numerical error accumulated via the two numerical algorithms namely the numerical computation of the value of ultimate ruin probability through the stable recursive algorithm and then inserting it as an input into another numerical algorithm to compute the mean of the time to ruin might be the cause of this deviation. The executing time for computing the second moment was too high thereby limiting us just to the computation of this moment for a very few values of the initial surplus.

Extension of this work can be directed towards the computation of other actuarial quantities like aggregate claim models, number of claims until ruin etc in case of Burr XII claim severity. Further analysis is required to give more explicit error bounds to the solutions generated via the two numerical algorithms and the control of error in the numerical computation of the moments of the time to ruin.

The authors would like to thank the anonymous referees of the Journal for imparting valuable suggestions that helped us to improve the paper.

JagritiDas,Dilip C.Nath, (2016) Burr Distribution as an Actuarial Risk Model and the Computation of Some of Its Actuarial Quantities Related to the Probability of Ruin. Journal of Mathematical Finance,06,213-231. doi: 10.4236/jmf.2016.61019

One of the most used methods for optimization in the Multi Parameter situation in Statistics is the Newton- Raphson method which is described briefly as given below:

Assume

bution involving

Let us now define what is known as the gradient matrix and the Hessian matrix given by.

The gradient matrix is given by

And the Hessian matrix is given by

where

Then the iterative relationship for the multi parameter Newton Raphson method is given by

where

The log likelihood of the Weibull distribution is given by (2.1.5).

The Gradient matrix for Weibull is given by

where

and the Hessian matrix is given by

where

and

The Log likelihood of Burr XII distribution is given by (2.1.6).

Its Gradient matrix is given by

where

The hessian matrix is given by

where

A statistics measuring the difference between the Empirical

A class of measures of discrepancy given by the Cramer-Von Mises Family is

where

When

size [