The Stochastic Error Rate Estimation of Prediction Distributions

Hafedh Faires

Department of Mathematics and Statistics, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia.

DOI: 10.4236/jmf.2015.52016                

Abstract

The estimation of claims reserves is usually done by applying techniques called IBNR techniques within a stochastic framework. The main objective of this paper is to predict the partial reserve and to estimate the error rate of prediction distributions by using the stochastic model proposed in [1].

Keywords

Reserving Claims, EDS, Error Rate, Prediction Distributions

1. Introduction

The calculation of the provisions for disaster payments is intended to allow the integral payment of the commitments to the policy-holders and the recipients of the contract. The provisions measure the commitments that the insurer still has to honor. Nevertheless, this countable concept requires a subjacent probabilistic model since it allows one to define the ultimate claim, taking into account the disasters not yet declared, but which have occurred, the disasters not sufficiently funded. Reserves are given by evaluating the provisions for each contract, IBNR (sinister not yet declared) and IBNER (sinister not sufficiently funded). Traditional methods of provisioning (by triangulation) rest on the assumption that the data are homogeneous and in sufficient quantity to ensure a certain stability and a certain credibility. The purpose of this paper is to propose a stochastic extension of the Chain-Ladder model concerning the partial prediction reserve and to estimate the error rate of prediction distributions, which seems to be closer to reality for us than the existing methods of Schnieper [2] ; Mack [3] ; Liu and Verrall [4] ; Verrall and England [5] .

Models in which parameters move between a fixed number of regimes with switching controlled by an unobserved stochastic process, are very popular in a great variety of domains (Finance, Biology, Meteorology, Networks, etc.). This is notably due to the fact that this additional flexibility allows the model to account for random regime changes in the environment. In this paper we consider the prediction of partial reserve and consider the estimation of error rate of prediction distributions for a model described by a stochastic differential equation (SDE) with Markov regime-switching (MRS), i.e., with parameters controlled by a finite state continuous-time Markov chain (CTMC) [1] and [6] . Such a model was used, for example, in Deshpande and Ghosh (2008) [7] , to price options in a regime switching market. In such a setting, the parameter estimation problem posed a real challenge, mainly due to the fact that the paths of the CTMC were unobserved. A standard approach consists in using the celebrated EM algorithm (Dempster, Laird and Rubin, 1977) [8] as proposed, for example in Elliott, Malcolm and Tsoi (2003) [9] and Hamilton (1990) [10] , study this problem using a filtering approach.

The rest of the paper is structured as follows. In Section 2, we present the stochastic model for our problem. Section 3 is devoted to predicting the claims reserves variance. We conclude with a summary in the last section.

2. Hypotheses and Description of the Model

We suppose that the available data have a triangular form indexed by the year of accident, i, and the development time, t. Given a triangle, on T years, the goal is to consider models using a minimum of parameters, in order to envisage the best possible amounts of payments of future disasters. We note the evolution of the amounts of payments of the cumulated real disasters obtained by.

indicates the evolution of the cumulated real disasters indexed by the year of accident, i, and the time of development, t. We suppose that the increase in the disasters obtained is the sum of the disasters not sufficiently funded and of the disasters not declared yet claims. In the paper [1] the following relations between C, N and D is proposed.

(1)

We indicate for all the variables in the triangles D and N observed until the moment s.

To simulate the future claims, it is supposed that the not sufficiently funded claims, the stochastic differential equation of diffusion and the not yet incurred claims are governed by the stochastic dif-

ferential equation of Black and Scholes with jump. This assumption on the probability density function of ensures positivity and for ensures the membership, contrary to what is proposed by Liu and Verrall [4] .

Conditionally with, we simulate the distribution of as solution of the following stochastic differential equation:

(2)

where is a standard Brownian motion on; is the Poisson process of intensity; is a Markov process at continuous time;;; verifying

and is a positive constant.

Conditionally with, we suppose that the evolution of is governed by the following diffusion:

(3)

where is a standard Brownian motion on; is a Markov process at continuous time; and such that and. The process is assumed to be a continuous time Markov process taking values in the set The transition probabilities of this chain are denoted by and the transition rate matrix is with

For any state consider priors and defined as follows

(4)

(5)

(6)

where denotes a Gamma distribution with shape parameter and scale parameter.

Let us define the log-returns for the Equation (2) and for the second Equation (3),

Given a path Let be the time spent by the path X in state j in the time interval. Define

(7)

(8)

Then, conditional on the path X, the solutions of the Equations (2) and (3) are respectively given by :

・ the solution of the first Equation (2) is

(9)

and,

・ are i.i.d., for the second Equation (3).

3. Predicted Claims Reserves Variance

The main reason for using stochastic models is to estimate the error rate of prediction distributions. It is useful for solvency, capital modelling and measurement of risk. We begin by proving how the predicted error rate can be calculated. The predicted error rate is obtained from the predicted variance of the partial reserve of loss. We

remember that the variables in triangles D and N observed until a time s. Let. The future claims IBNR and IBNER estimators are given respectively by, and. Our objective is to predict the partial reserve. We denote

the partial reserve by:

(10)

is its estimator.

The mean square error prediction of can be written as follows

Theorem 3.1 Let and, respectively the future claims IBNR and IBNER of the i^th contract, verifying Equations (2) and (3), then the mean square error prediction of partial reserve is

(11)

Proof. The mean square error prediction of the partial reserve can be written as follows

(12)

Using the fact that

and

As a first step, we calculate

We calculate the second term (12), taking into account

and

Then

4. Conclusion

We have studied the Bayesian approach for the regime switching geometric Brownian motion proposed by [1] . It has been observed empirically that sinisters fluctuate among periods of high, moderate and low volatilities, so in this paper the estimation of the error rate of prediction distributions is proposed. For the future research, our developments raise an interesting axe when the Markov chain is hidden.

Acknowledgements

We would like to thank the reviews for the comments.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Faires, H. (2014) Non-Life Claims Reserves Using Dirichlet Random Environment. International Journal of Engineering Research and Applications (IJERA), 4, 63-77.
[2]	Schnieper, R. (1991) Separating True IBNR and IBNER Claims. ASTIN Bulletin, 21, 111-127.
[3]	Mack, T. (2000) Distribution-Free Calculation of the Standard Error of Chain Ladder Reserve Estimates. ASTIN Bulletin, 23, 214-225.
[4]	Liu, H.J. and Verrall, R. (2002) Predictive Distributions for Reserves which Separate True IBNR and IBNER Claims. http://www.actuaries.org/ASTIN/Colloquia/Orlando/Papers/Liu.pdf
[5]	Verrall, R.J. and England, P.D. (2000) Comments on: “A Comparison of Stochastic Models That Reproduce Chain Ladder Reserve Estimates”, by Mack and Venter. Insurance: Mathematics and Economics, 26, 109-111. http://ideas.repec.org/a/eee/insuma/v26y2000i1p109-111.html
[6]	Faires, H. (2012) SDEs in Dirichlet Random Environment. International Journal of Statistics and Systems, 1, 55-66.
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[8]	Dempster, A.P., Laird, R.M. and Rubin, D.B. (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39, 1-38.
[9]	Elliott, R.J., Malcolm, W.P. and Tsoi, A.H. (2003) Robust Parameter Estimation for Asset Price Models with Markov Modulated Volatilities. Journal of Economics Dynamics and Control, 27, 1391-1409. http://dx.doi.org/10.1016/S0165-1889(02)00064-7
[10]	Hamilton, J. A. (1990) Analysis of Time Series Subject to Changes in Regime. Journal of Econometrics, 45, 39-70. http://dx.doi.org/10.1016/0304-4076(90)90093-9