Loss Reinsurance Research Articles

Sur la determination d'un contrat optimal de reassurance

SummaryOhlin [7] showed optimality properties of Stop Loss reinsurance when the ceding insurer uses a continuous loss function to evaluate his risks. We generalize this property to the range of the distribution; we then show that Stop Loss reinsurance is no longer the best form when the company uses a percentile parameter. Finally we prove an optimality theorem concerning chance games which allows us to determine the retention as a function of the size of the portfolio.

Excess of Loss Reinsurance: Review of standard clauses

Excess of Loss Reinsurance: Review of standard clauses

Correlations between excess of loss reinsurance covers and reinsurance of the n largest claims baruch berliner

E. Franckx [1] has established the distribution function of the largest individual claim of a portfolio. By assuming the number of claims to be Poisson distributed, H. Ammeter was able to develop the distribution function of the total loss excluding the largest individual claim [2] as well as the distribution function of the nth largest claim [3].Of course, the nth largest claim is dependent on the largest claim, second largest claim and so on, down to the (nth — 1) largest claim. If we assume the number of claims to be Poisson distributed and the amount of the individual claim to be Pareto distributed, the correlation between the mth largest and the nth largest claim can be expressed by an analytical formula which is susceptible to numerical computation.With this knowledge we shall be able to compute the variance of the sum of the n largest claims and moreover the correlation between the sum of the n largest claims and the total loss amount. Although an excess of loss reinsurance treaty and a treaty reinsuring the n largest claims are very different in their construction, this paper will show that from a practical point of view there exists a similarity between the two treaties. The correlation coefficient between the sum of the n largest claims and the sum of all claims exceeding a certain limit enables us to assess the degree of similarity.The correlation coefficient and thus the degree of similarity will prove to be high even in case of the reinsurance of only a small number of largest claims.Finally, the knowledge of the two first moments of the sum of the n largest claims allows us to compute the premium and the security or variance loading for the reinsurance of the n largest claims.

An approach to the analysis of claims experience in motor liability excess of loss reinsurance

In this paper motor liability insurance is considered from the viewpoint of an excess of loss reinsurer.The reinsurer pays portions of claim amounts in excess of a contractually agreed limit, further referred to as excess point.Aspects of practical interest are the rating of new contracts and cost-projections on claims occurred in past years which have not yet been reported.What causes a major difficulty to a rating approach is the timelag between the occurrence of a claim and its settlement.During this timelag, which may extend to ten years or even longer, the size of a claim will usually increase considerably. This is due to such factors as inflation and more victim oriented legal procedures.As a result of this phenomenon, a claim, initially not involving the reinsurer, may confront him many years later, when its size overtakes the excess point. Hence, each risk year produces a generation of claims stretching far into the future.The problem facing the reinsurer trying to compute premiums is that he must look, say ten years into the future, on the basis of incomplete data.The table below is a schematic presentation of the time history of a contract that started k years ago.The experience is assumed to be expressed in numbers of claims exceeding a constant excess point.Successive rows show the generations of excess claims down to the last completed risk year k.In the pages that follow we will attempt to project expected numbers of excess claims into the future, using information as schematized in this table.

Stop Loss Reinsurance. A New Look

This paper analyzes the prevailing theoretical excess of loss ratio reinsurance rating model, noting the limitations of this basic model. Improvements in stop loss reinsurance rating are proposed to reflect the dynamic changes now taking place in underlying insurance premium structures. The authors propose that a theoretical claim distribution curve, composed of: potential changes in the class of risk assumed, increases or decreases in loss payments, managerial attitudes towards claims settlements, and testing the validity of the reported data, be superimposed upon the traditional (ceteris paribus) loss history model. An actuarial model for estimating the expected value of reinsurance is developed. The authors attempt to show in their model that traditional static loss ratio theory is ineffective in forecasting future trends. Reinsurance, as both a practical and an academic subject, is considerably more involved than ordinary forms of insurance. This may result from the fact that there is no set formula in matters of reinsurance practice and, as such, it frequently enters into the arena of theoretical and empirical

A Combination of Surplus and Excess Reinsurance of a Fire Portfolio

Reinsurance forms can roughly be classified into proportional and non-proportional. The authors of this paper had planned to investigate the “efficiency” of two different reinsurance forms, one from each of these categories. Efficiency is here understood as reduction in the variance of the annual results of the risk business achieved per unit of ceded reinsurance risk premium. This investigation may be carried out in full later.This note will only deal with the interplay between surplus and excess of loss reinsurance; more specifically the effect of changes in the volume of surplus cessions on the excess of loss risk premium.The study came out of a practical Fire Reinsurance rating problem and will be carried through under very simplified assumptions. Thus we will ignore the conflagration hazard and the possibility of a wrongly taxed PML. This means that if amounts above a PML of M are ceded on a surplus basis the highest loss per event will be M, and an excess cover above a priority m will never pay more than M-m per event.The following notations will be usedR(M) ceded risk premium volume on surplus basis, the PML retention being M.πM(m) excess of loss risk premium if priority is m and surplus cessions are made above a PML of M.

On Optimal Properties of the Stop Loss Reinsurance

Let F(x) be the distribution function of the total amount ξ of claims of an insurance company. It is assumed that this company reduces its liability by means of one or several reinsurance arrangements. Let the remaining part of claims amount be equal to η, which is another random variable. Reinsurance arrangements are supposed to fulfil the following consistency condition: if ξ = x, then almost certainly o ≤ η ≤ x. Presumably all reinsurance arrangements occuring in practice can be supposed to fulfil this requirement.The problem is to find an optimal reinsurance arrangement, i.e. an optimal random variable η, in the sense that, the net reinsurance premium being given, the variance of η reaches its minimum. In other words, a variable η is looked for, which givesE{η} = P = const.; V{η} = minimumIn the sequel we use conditional expectations in the sense defined by DOOB ([I]).Let η be an arbitrary random variable satisfying the consistency condition. IfR(x) = E{η|ξ =x},then evidently ≤ R(x) ≤ x. Further we have For the variance the inequality holds true, since Clearly the arrangement E {η|ξ} gives also to the reinsurer a smaller variance than the original arrangement.

On the Use of Extreme Values to Estimate the Premium for an Excess of Loss Reinsurance

I. Suppose that the claims experienced by a portfolio could be represented as independent random variables with a distribution function F(x). The net premium per claim for an excess loss cover above an amount of L is thenIf we have no information about F(x) except a number M of independent claims, we might compute the observed “staircase” distribution function SM (x) which is for every x an unbiased estimate of F(x), and could thus compute an unbiased estimate for P(L) with the variance [7])2. In real life we have some qualitative knowledge of F(x) and very limited information about the claims. In his introduction to this subject Beard treats the case where the only information about F(x) consists of the largest claim xi and the number of claims ni (i = 1, 2, …, N) observed during N periods (Reference No 2). It is known from the theory of extreme values [3] that for large ni the distribution of xi depends mainly on the parameters uni and αni( defined byBeard further assumes that F(x) belongs to what is called by Gumbel “the exponential type” of distribution functions, which have an unlimited tail and finite moments. This class is strictly defined by Gnedenko's necessary and sufficient condition [4, p. 68]:

Some Notes on the Statistical Theory of Extreme Values

In preparing the notes on the subjects for discussion at the 4th ASTIN Colloquium at Trieste [1] I used some of the material which formed the basis of a talk given to the Scandinavian Actuarial Societies in September 1962. The papers presented in Trieste have established on a firm mathematical footing the formula for the excess loss premium derived in my note but the discussions also showed that some of the other 1962 material would be of interest.It will be appreciated that the question originally proposed, i.e. to calculate an excess of loss reinsurance premium when the only information available is the largest claim experienced in each of a succession of periods, was deliberately phrased in this form as being the most troublesome case likely to arise in practice. Clearly if other information is available it would not be rejected in arriving at a premium, but this immediately extends the problem to one of finding the best methods of combining information of different kinds. For example if, say, the largest 5 claims in each period are known, what technique will make the maximum use of the data? Such extensions of the problem are not discussed in this note. Furthermore, I am not unmindful of the valuable comment by Jung, “that there is a natural law which states that you can never get more out of a mincing machine than what you have put into it” [2].

Minimum Variance Reinsurance

In a paper entitled “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance” (XVIth Int. Congr. Act. Bruxelles 1960) Karl Borch has shown that, if the reinsurance premium is given, the smallest variance of the cedent's payments is obtained by a stop-loss reinsurance contract. Paul Markham Kahn, in “Some Remarks on a Recent Paper by Borch”, a paper read to the 1961 Astin Colloquium, has given an elegant proof of this theorem which appears to apply also to cases not considered by Borch. In this paper we study the problem from the reinsurer's point of view and it will be seen that, under natural conditions which are also used in the proof of the Borch-Kahn theorem, the minimum variance of the reinsurer's payments is obtained by a quota contract. This focusses attention on a peculiar opposition of interests of the two partners of a reinsurance contract. However, we do not enter any further into the investigation of a possible resolution of this conflict.We study a problem concerning the division of risk between a cedent and his reinsurer. The risk may refer to a whole portfolio (in which case one might consider a Stop-Loss contract), or to a single contract (when an Excess-Loss contract is a possibility). We shall here use the nomenclature of a portfolio reinsurance.Let it be assumed that a function F(x) is known which gives the probability of a total claim not exceeding x. We have then in Stieltjes integral notationThe two partners to a reinsurance arrangement agree that the reinsurer reimburses m(x).x out of a claim of x, where m(x) is a continuous and differentiate function of x and o ≤ m(x) ≤ 1.

On the correlation in results from different layers in excess of loss reinsurance

Abstract It becomes more and more usual to arrange a desired excess of loss cover in two, three or more layers (intervals), which are often offered on partly separate markets and through different channels. From the point of view of the ceding company and the broker this is rational, since certain reinsurers seem to have a rather marked preference for the low layers, the so called working covers, whereas other prefer to stake on the high layers, the so called catastrophe covers. These are—wrongly— onsidered as less susceptible to inflation than the lower layers. The reinsurer bases his decision of which one of a number of layers with given conditions is interesting on a comparison between the said conditions. This, however, presupposes a certain knowledge of the distribution function of the claims [1, 2].

Some Remarks on a Recent Paper by Borch

In his recent paper, “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, presented to the XVIth International Congress of Actuaries, Dr. Karl Borch considers the problem of minimizing the variance of the total claims borne by the ceding insurer. Adopting this variance as a measure of risk, he considers as the most efficient reinsurance scheme that one which serves to minimize this variance. If x represents the amount of total claims with distribution function F (x), he considers a reinsurance scheme as a transformation of F (x). Attacking his problem from a different point of view, we restate and prove it for a set of transformations apparently wider than that which he allows.The process of reinsurance substitutes for the amount of total claims x a transformed value Tx as the liability of the ceding insurer, and hence a reinsurance scheme may be described by the associated transformation T of the random variable x representing the amount of total claims, rather than by a transformation of its distribution as discussed by Borch. Let us define an admissible transformation as a Lebesgue-measurable transformation T such thatwhere c is a fixed number between o and m = E (x). Condition (a) implies that the insurer will never bear an amount greater than the actual total claims. In condition (b), c represents the reinsurance premium, assumed fixed, and is equal to the expected value of the difference between the total amount of claims x and the total retained amount of claims Tx borne by the insurer.

Excess of Loss and Stop Loss Reinsurance

Excess of Loss and Stop Loss Reinsurance

A Method for the Estimation of the Risk Premiums in Stop Loss Reinsurance

The reinsurance to be treated in this note shall cover the excess over a certain limit Q of the total amount of claims for each accounting period paid by the ceding company. Regardless of the rule for the determination of the limit Q such a reinsurance shall in this context be called stop loss reinsurance.Generally, such a reinsurance is called either stop loss or excess of loss reinsurance depending on the rule for the determination of Q. The use of the term stop loss regardless of this rule is preferred here in order to avoid confusion with an excess of loss reinsurance which refers to the excess of the amount of one or more claims caused by an individual event.The total amount of claims paid by the ceding company c during the period t shall be denoted cXt. This random function constitutes for a fixed value of c a stochastic process with the discontinuous parameter t > o.(It may be remarked that for a fixed pair of values of c and tcXt can be regarded as a particular value of a sample function pertaining to the process of the continuous paramater τ generally considered in the collective theory of risk to which the total amount of claims up to time τ is attached, in this case o < τ ≤ 1. cXt can also be considered as the increments during the time interval t of the total amount of the claims up to τ.)