Academic journal article International Advances in Economic Research

A Stochastic Minimax Model to Calculate Outstanding Claims

Academic journal article International Advances in Economic Research

A Stochastic Minimax Model to Calculate Outstanding Claims

Article excerpt


Correct estimation of the Outstanding Claims Reserve, an item that includes Incurred But Not Reported Claims (IBNR) as well as Incurred But Not Enough Reserved Claims (IBNER), is one of the most important issues currently facing actuarial science, due to its effect on the technical and financial stability of insurance companies. The purpose of this paper is to calculate the reserve in a decision-making environment, so that estimates can be made according to accurately defined and previously established rational criteria. Specifically, the estimating process enables a company's particular situation to be taken into account, by incorporating its approach to the consequences arising from estimation errors into the model. The proposed calculation method gives rise to optimum link ratio estimators that can also be interpreted from a Bayesian perspective, with the advantages associated to such methodology. (JEL G22)


Determining of the outstanding claims amount and appropriate calculating of reserves allocated to such is an essential part of general insurance company activities. In particular, among insurance companies, link ratio based methods stand out substantially.

The first attempt to stochastically generalize this type of method was with the de Vylder [1978] least squared model, the starting point for the work done by Mack [1994] in which the estimation of link ratios is carried out by using a linear regression model, the original stochastic model later being updated and modified. This is the case of the work by Halliwell [1996] that performs estimates of stochastic model parameters with the generalized least squares technique, and that of the Australians, Barnett and Zehnwirth [1998] with their "extended link ratio family." A different line of thought was that of Kremer [1982] who provided the basis for the Verrall [1994] two-way factor variance analysis model, which was also later subject to certain improvements using Bayesian methods, state-space models, and generalised linear or additive models, by England and Verrall [1999].

Essentially, these stochastic generalisations are linear models with different variables that are dealt with differently. In this context, it is possible to approach the problem from a new perspective by calculating an Outstanding Claims Reserve using the above-mentioned link ratios. The decision framework from which the issue is approached enables decision makers to intervene in such structure, mainly materialised in the minimax hypothesis, which reflects a sensible attitude on their part, moderated by the restrictions imposed on the range of claims accumulated over development periods. The consequences of choices made are expressed by a loss function construed as the arithmetic sum of squared estimation errors in each origin and development year. This means granting priority to the behaviour of claims in each of these periods in detriment to overall or joint behaviour.

General Stochastic Model

The majority of methods used to calculate Outstanding Claims are based on the so-called claims triangle or run-off triangle (Table 1), in which claim amounts are represented by matrices, with two variables: year of origin (i) and development year (j). (1)

In what follows, it is assumed that the [X.sub.i,j] variable, which contains the previous matrix, represents the accumulated amount until the development year j of claims originating in year i, in other words, those recorded before the end of year i + j - 1.

[X.sub.i,j] with j = n - i + 1 represent the last available data, being the existing information in the current year, n, on claims originating in year i. The right side area of the matrix of the diagonal contains data that are still unknown.

In addition, the accumulated amount of claims tends to stabilise over time. In practise, this is equivalent to considering that for a large enough development year, that is, j = n, we get [X. …

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