RECURSIVE ESTIMATION OF THE CLAIM RATES AND SIZES IN AN INSURANCE MODEL

It is a common fact that for most classes of general insurance, many possible sources of heterogeneity of risk exist. Premium rates based on information from a heterogeneous portfolio might be quite inadequate. One way of reducing this danger is by grouping policies according to the different levels of the various risk factors involved. Using measure change techniques, we derive recursive filters and predictors for the claim rates and claim sizes for the different groups.


Introduction
All processes are defined on a measurable space (Ω,Ᏺ), with probability measure P. Consider a portfolio of L policyholders of, for instance, automobile insurance.Each policyholder belongs to one of a finite number G of risk level groups classified by age, sex, type of automobile owned, and so forth.
Under the two assumptions that the initial distribution of the rate of claims is Γ (α 0 ,β 0 ) and that the number of claims y and the number of policies N are Poisson random variables, it is easily seen that the posterior probability density of the rate of claims, given new data y, N, is Γ(α 0 + y,β 0 + N).
More precisely, we will be using the following notation and assumptions.
(i) Let N c n be the total number of new policies purchased by individuals classified in group c during the nth year and let y c n be the number of claims reported by the cth group during the same year.
(ii) The rate of claims reported by policyholders in the cth group during the nth year, δ c n , is a random variable with conditional Γ-distribution which is close to a normal distribution when α c n and β c n are large enough, where α c 0 , β c 0 are initial guesses and, for n ≥ 1, In this paper, we assume that = η n x c ,δ 1 ,...,δ G ,σ c n dx. (1.3) Here, d c i , c,i = 1,...,G, are real numbers expressing some dependence between claim sizes from the different groups and (1.4) (iii) The random variables y c n , N c n are Poisson random variables such that (1.5) We assume here that µ c n is either known or Ᏺ n -predictable.(iv) Let S c n be the mean claim size of group c by the end of year n.It is usually assumed that the lognormal distribution is suitable for claim sizes.(See, e.g., [4,5].)The central limit theorem suggests the following (conditional) normal distribution for S c n : Here, a c (X n ) = a c ,X n , where a c = (a c 1 ,...,a c K ) may represent the year index [5] which, for simplicity, belongs to the finite set of real numbers a c .The probability density function of S c n is modulated by an unobserved finite-state Markov chain X, that is, the mean number of policies purchased every year is changing from year to year due to many economical factors and the changes are modeled by a finite-state Markov chain X.Without loss of generality, let the state space of X be the standard basis {e 1 ,...,e K } of R K .
Write P = {p j,i }, i, j = 1,...,K, where K j=1 p j,i = 1 and (1.7) Then, we have the following dynamical representation [3]: where V n is a martingale increment with respect to the complete filtration generated by X.
Lakhdar Aggoun 247 (v) Credibility theory deals with adjusting insurance premiums as claim experience is obtained [4].The technique consists of using a credibility factor Z ∈ (0,1) to obtain a convex linear combination of some data obtained from past experience, which may not be very reliable, and data from recently reported claims.In this paper, we propose the following (conditional) normal distribution for S c n : (1.9) The parameter Z is reestimated in Section 6.
In Sections 3 and 4, recursive estimates for the rates of claims are derived under a suitable "reference" probability measure.
In Sections 4 and 5, recursive estimates of the claim sizes are derived under a different "reference" probability measure.The reason was to separate between the distributions of the claim rates and the claim sizes.Note that the changes in the economical environment, expressed by the jumps of the Markov chain X, link the claim sizes of the whole portfolio, therefore creating some dependence between the different risk groups.
In Section 6, the expectation maximization (EM) is used to update the parameters of the model.

Recursive estimation
In this section, we choose a probability measure P † , on the measurable space (Ω,Ᏺ), under which the processes y c , N c , c = 1,...,G, are sequences of stochastically independent and identically distributed (i.i.d.) random variables.The probability measure P is referred to as the "real world" measure, that is, under this measure, (1.5), (1.6), and (1.8) hold.

Predicting future claim rates
In this section, we wish to derive predictors for the rates of claims within the subgroups of policyholders.That is, we wish to compute the conditional probability of δ c n+1 given the history up to the nth year.Define the process Let f be a "test" function and write Proof.
assuming here that µ c n+1 is either known or predictable with respect to ᐅ n and using (3.2), this is The unnormalized density g n (u 1 ,...,u G ) is given recursively in (2.5).Since f is arbitrary, the result follows.

A second change of measure
In this section, we choose a probability measure P, on the measurable space (Ω,Ᏺ), under which the processes S 1 ,...,S G are sequences of stochastically i.i.d.random variables with the standard normal distribution.Define where γ 0 = 1 and Here, ψ c is the density function of the standard normal distribution and Define the measure-valued process Proof.In view of (4.1), (4.2), and (1.8), we have

Predicting future claim sizes
In this section, we wish to derive one-year-ahead predictors for the claim size.That is, we wish to compute the joint conditional probability of S 1 n+1 ,...,S G n+1 given the history up to the nth year.Define the process (5.1) Let f be a "test" function, S n+1 = (S 1 n+1 ,...,S G n+1 ), and write (5.2) Lemma 5.1.The one-step (unnormalized) predictor for the claim sizes is given by the measure (5. 3) The unnormalized density ζ n (i) is given recursively in Theorem 4.1. Proof. ( This finishes the proof.

The EM algorithm
The EM algorithm (see [1,2]) is a widely used iterative numerical method for computing maximum likelihood parameter estimates (MLEs) of partially observed models such as linear Gaussian state-space models.For such models, direct computation of the MLE is difficult.The EM algorithm has the appealing property that successive iterations yield parameter estimates with nondecreasing values of the likelihood function.Suppose that we have observations y 1 ,..., y K available, where K is a fixed positive integer.Let {P θ ,θ ∈ Θ} be a family of probability measures on (Ω,Ᏺ), all absolutely continuous with respect to a fixed probability measure P 0 .The log-likelihood function for computing an estimate of the parameter θ based on the information available in and the MLE is defined by Let θ0 be the initial parameter estimate.The EM algorithm generates a sequence of parameter estimates { θj }, j ≥ 1, as follows.
Each iteration of the algorithm consists of two steps.
Sufficient conditions for convergence of the EM algorithm are given in [6].We briefly summarize them here: assume that (i) the parameter space Θ is a subset of some finite-dimensional Euclidean space R r ; (ii (iii) ᏸ K is continuous in Θ and differentiable in the interior of Θ (as a consequence of (i), (ii), and (iii), clearly ᏸ K ( θj ) is bounded from above); (iv) the function ᏽ(θ, θj ) is continuous in both θ and θj .
Then, by [6, Theorem 2], the limit of the sequence of EM estimates { θj } has a stationary point θ of ᏸ K .Also, {ᏸ K ( θj )} converges monotonically to ᏸt = ᏸ t ( θ) for some stationary point θ.To make sure that ᏸt is a maximum value of the likelihood, it is necessary to try different initial values θ0 .
Here, we wish to update the parameters from ..,G, be a G × G nonsingular matrix and δ n = (δ 1 n ,...,δ G n ).Definition 6.1.Given two (column) vectors X and Y , the tensor or Kronecker product X ⊗ Y is the (column) vector obtained by stacking the rows of the matrix XY , where is the transpose, with entries obtained by multiplying the ith entry of X by the jth entry of Y .
Now the expression for ᏽ(θ, θ) is derived.
To implement the M-step, set the derivatives ∂ᏽ/∂θ = 0.This yields Define the measure-valued processes (6.9) Then, for any "test" function g : R G → R, write n (x)g(x)dx.
To update the parameters in (1.6), let A(X n ) be a G × G diagonal matrix with diagonal entries a 1 (X n ),...,a G (X n ), that is, on the event [X n = e j ], A(e j ) = diag(a 1 j ,...,a G j ).We assume that a c j = 0 for all c and all j.
To implement the M-step, set the derivatives ∂ᏽ/∂θ = 0.This yields p t e ε j n−1 ,e t + e j ζ n ( j).
To implement the M-step, set the derivatives ∂ᏽ/∂θ = 0.This yields  Proof.The proof is similar to that of Theorem 4.1.

Conclusion
In this paper, using hidden Markov model techniques, recursive filters for various quantities of interest related to an insurance model were derived.Formulae to predict future claims were established.The EM algorithm was used to update the parameters of the discussed model.

Remark 6 . 3 .Theorem 6 . 4 .
,e j and define the processΥ j n = E[Λ n T j n | n ].The following recursive filters are derived under P which is defined in Section 4. A closed-form finite-dimensional recursion is only possible for the conditional joint distributions of T j n and X n .That is, we will consider recursive filters forE[Γ n T j nXn | n ] = ε i n .However, Υ Let ε j 0 be the initial joint density function of T j 0 , X 0 and, for n ≥ 1, .20) 258 Recursive estimation in an insurance model Remark 6.5.Since Z ∈ (0,1), it is clear that S same sign.So we may assume that they are both positive.Hence we can use the Cauchy-Schwartz inequality to see that0 < Z c (n) ) 2 .To replace the parameters p ji by pji (n) in the Markov chain X, we define dP θ /dP θ | Ᏻn = ᏸ n .Then one can show[3] that the new estimates of the parameter pji (n), given the observations up to time n, are given by pji (n) = −1 ,e i X m ,e j .Remark 6.6.The following recursive filters are derived under P which is defined in Section 4. A closed-form finite-dimensional recursion is only possible for the conditional joint distributions of i j n and X n .That is, we will consider recursive filters for E[Γ n i j n X n | n ] = ρ i j (n).However, γ n ( ,e t + p ji e j ζ n−1 (i) .