“Comparison of the claims reserves methods by analyzing the run-off error”

The variability of claim costs represents an important risk component, which should be taken into account while implementing the internal models for solvency evaluation of an insurance undertaking. This component can generate differences between future payments for claims and the provisions set aside for the same claims (run-off error). If the liability concerning the claims reserve is evaluated using synthetic methods, then the run-off error depends on the statistical method adopted; when it is not possible to study analytically the properties of the estimators, methods based on stochastic simulation are particularly effective. This work focuses on measuring the run-off error with reference to claims reserves evaluation methods applied to simulated run-off matrices for the claims settlement development. The results from the numerical implementations provide the authors with useful insights for a rational selection of the statistical-actuarial method for the claims reserve evaluation on an integrated risk management framework. The setting of the analysis is similar to that adopted in other studies (Stanard, 1986; Pentikainen and Rantala, 1992; Buhlmann et al., 1980), however, it differs for estimation and simulation methods considered and for the statistics elaborated in the comparison.


Introduction
The random claim settlement regarding the accident year i (i = 0, 1, …, t) is given by the sum of a random number of claims, each one subject to a single claim settlement, and it can be represented as follows: Since the settlement claimed for every accident usually requires two or more payments, which can take place during the accident year or the subsequent years, the aggregated claims cost for every accident year can be represented as follows: 0 01 t j= X i = X i, j , i = , , ..., t, (2) whereas j i X , ~ represents the amount paid for settlements regarding claims incurred during the accident year i and settled after j years; t represents the maximum number of deferment years considered for the total settlement of a single claim.
At the time of observation the recorded information from the company regards the amounts: 01 01 X i, j : i = , ,...,t; j = , ,...,t -i , (3) while a forecast of future amounts should be done: ,..., 1 ; ,..., The random amount required for future settlements regarding claims not yet settled or reported (and IBNR), for each accident year, is given by: 1 12 t j=t-i+ R i = X i , j , i = , , . . ., t . ( The aggregate amount required is then given by the sum:

The run-off errors
The statistical methods for the outstanding claims reserve evaluation consist in the formulation of a forecasted value of the necessary reserve, based on a projected analysis of the data obtained by the examination of relevant time series.In other words, an evaluation method provides an estimator 01 ˆ t R = f K ,K ,...,K of the expected value for the outstanding claims reserve 1 , which depends on the information at disposal 01 t K = f K ,K ,...,K for each accident year, and of which at the time t there 1 In general, for the distribution of the outstanding claims reserve, other than the expected value we can estimate moments of order higher than 1 or even particular quantiles.
The difference between future payments for claims settlements and the amount of the relative outstanding claims reserve, evaluated using a specific estimator, gives us the run-off error.The run-off error for each accident year can be represented as follows: 1 , 12 t j=t-i+ e i =R i R i = X i,j X i,j i = , , ..., t, ˆ3 (7) while the run-off error relative to the entire portfolio can be expressed by: The entity of the run-off error depends on the differences between the set of hypotheses on which the estimation model is based and the actual characteristics of the portfolio 4 ; such differences condition, evidently, the properties of the estimator of the claims reserve.
The formula measures the run-off error, at the observation period, for all the accident years taken into consideration, compensating the possible differences of opposite sign between the run-off errors of the various accident years.
The estimator R is called unbiased if the expected value of the estimator equals the expected value of the outstanding claims reserve for which the estimator is used:

ER R= ER ER= ˆˆ(9)
For an unbiased estimator, the expected run-off error equals 0. You could say that an unbiased estimator provides estimates of the provision for claims that do not contain "loadings" (positive or negative, implicit or explicit).The amplitude of the distortion that characterizes the estimators is, however, only the first criterion of comparison.In fact, a method that can provide estimates with low 2 For the purpose of the evaluation reliability, the information on which the projected analysis is based should include sufficient, independent and homogeneous data. 3The formula measures the run-off error, at the observation period, for the entire accident year, compensating the possible differences of opposite signs during the forthcoming development years.Knowing the gap between expected and actual timing of settlements is of crucial importance for the reinsurance treaties that compensates the reserved claims of the cedent company. 4If the amounts of future settlements are discounted at the time of observation, to the error in the forecast of the cash flow of settlements you must add the error relative to the forecasted future rates of return.
distortion, but for which the individual forecasts differ considerably from the actual values, may not be an appropriate method for the estimation of reserves.It is useful then to consider other precise indicators such as the mean percentage error, ˆ R-R E R , and the mean square error, 2 ER -R ˆ 5 .Moreover, since it follows: a good estimation method must provide an estimator with high correlation with the reserve to estimate.

The claim reserve evaluation methods
Between the multiple procedures for the evaluation of outstanding claims reserve proposed in literature, four of them were chosen for this work, considering their widespread utilization in the professional environment: the Chain-Ladder method, the Separation Method (arithmetic and geometric), the Fisher-Lange method and the Bornhuetter-Ferguson method.We give below a concise representation of the content and how they were applied in the analysis.

The Chain-Ladder method.
The Chain-Ladder method considers the run-off triangle of cumulative payments of settlements: 01 01 C i, j : i = , ,...,t; j = , ,...,t -i , (11) The underlying hypothesis is that the distribution of the settlements is constant for each accident year, the development factors are estimated as: Then, assuming that development factors remain unaltered also for the future, the cumulative future payments are calculated: 5 Let's recall that, between two biased estimators, A R ˆ and B R ˆ, we will The difference between the ultimate cost and the cumulative cost until the year of observation will provide an estimate of the reserve for a single generation:

12
. ˆ R i = C i,t C i,t i , i = , , ..., t (14) The sum of these differences, for all generations, measures the estimated amount of total claims reserve.Among the variants of the method, the one based on the triangles of the relationship between the cumulative average costs theory was considered, which estimates development factors as weighted averages of the ratios observed with the weights obtained by calculating 2 1 i, j w=i + j + , that depends on the accident years of claims and the time span of settlement.

The Taylor separation method.
The method of separation (arithmetic) elaborates the triangle of the average costs of claims for the accident year, assuming that each of these costs, net of the random noise term, is, on average, expressed as a product of two factors: Factor r j , as a function of only years of development and varies between 0 and 1, is the way in which payments per claim are distributed in time, regardless of generation; while the second factor j 1 , that depends on both the year of development and the accident year, represents an index of exogeneity, with particular reference to the inflation, extrapolated by log-linear regression.The availability of an adequate information base allows the estimation of the factors ˆh r and ˆh (h = 0, 1, …, t), expressing then the "average cost per claim of generation", according to the product of the two factors mentioned above; while the factors ˆˆh (h = t +1, t + 2,…,2 t) are extrapolated from the factors ˆh by log-linear regression.It is estimated that in this way the future average costs per claim for each generation, multiplied by the corresponding number of claims, can predict the cumulative amounts of future claims and, subsequently, the total claims reserve.
Among the variants of the described method the socalled separation of geometric type was considered, with extrapolation of the index of inflation, using a log-linear regression.

The Fisher-Lange method.
The Fisher-Lange method is based on the average costs of claims paid in previous generations and their relative settlement speed.
The two key assumptions are: 1) the claim settlement speed is constant over time; 2) the average cost of claims paid is a function of the period between the accident date and the time of actual payment.Hence, starting from the triangle of run-off in the number of claims settled: 01 01 n i, j : i = , ,...,t; j = , ,...,t i .(16) The rate of settlement for development year is calculated as: with which the number of settled claims is estimated: 1 12 1 a i, j i, j j n = n v , j = , , ..., t; i = t j + , ..., t, ˆˆ (18) and the number of claims still outstanding: 12 1 aa i, j i, j-1 i, j n = n n , j = , , . . ., t ; i = t j + , . . ., t ˆˆˆ. (19)  Then, we consider the run-off triangle of the average cost paid 01 01 X i, j : i = , ,...,t; j = , ,...,t i that produces estimates of future average costs, 12 1 X i, j : i = , ,...,t; j = t i + ,...,t ˆ, by log-linear regression of the average costs for each development year.Next, multiplying the projected average costs corresponding to the number of claims that are expected to be settled, you get the estimates of the total amounts of claims still outstanding.The sum of all these future amounts represents an estimate of total claims reserve: .

Bornhuetter-Ferguson method.
As part of Bornhuetter-Ferguson Method, the run-off triangle of accumulated payments 01 01 C i, j : i = , ,...,t; j = , ,...,t i is considered, from which we estimate development factors: Benchmark values are determined according to the cost of generation by multiplying the premiums of each generation for a suitable loss-ratio (Bornhuetter and Ferguson, 1972).In this case, the benchmark value was calculated using the following formula6 : The estimates of the ultimate cost for each generation are obtained by applying the factors of Bornhuetter-Ferguson to the benchmark values: Then, the estimate of the reserve for each generation:

The simulation methods of the run-off matrix
In practice, the run-off error can be measured only after the completion of the claims settlement process.In this work, we will quantify the run-off error, simulating the claims settlement process until we obtain all the members of the run-off error formula For this purpose, we represent the random settlements, in each cell of the run-off matrix, with the following (collective) model: whereas Ni , j represents the total number of claims for the accident year i, settled during the development year j; k Yi , j represents the random settlement for the claim k incurred during the accident year i and settled after j years.For simplicity we will exclude the possibility of settlement in installments over several years of development.
For the simulation of the amounts Xi , j we have considered four methods, which are distinguished for the development rule concerning the claims settlement inside the run-off triangle and are based on probabilistic assumptions regarding both the distribution of the number of claims j Ni= Ni , j and the distribution of the random settlement of each claim k Yi , j .

Method of random development factors.
The temporal distribution of the settlements inside the run-off matrix is governed by the development factors, as described in the Chain-Ladder method framework with the exception that the main hypotheses on which the Chain-Ladder method is based upon are not respected in this case 7 .This method simulates the run-off matrix through the following steps (Narayan and Warthen, 1997): R i = C i,t C i,t i .
(30) 7 The assumptions implicit in the Chain-Ladder model are: 1) the development of settlement is made according to unknown development 8. For each accident year steps 1 to 7 are repeated, multiplying the cost of settlements for each accident year by the factor: whereas i infl is an annual inflation rate.

Method of single settlements.
This simulative method is derived from the estimation models proposed by Stanard (1986) and by Buhlmann, Schnierper and Straub (1980), which consider the settlement of a single claim as a stochastic process depending on three parameters: the incurring year, the reporting year and the settlement year.
In this framework, the simulating model assumes an exponential distribution for the deferment periods regarding the reporting year and the settlement year (McCleanahan, 1975;Weissner, 1978).
Furthermore, the settlement amount varies with the variation of the deferment period between the settlement year and the incurring year.
The method simulates the run-off matrices through the following steps:  So, for each of the n (i) claims an amount of settlement is associated with whereas t is the smallest integer greater than or equal to the sum t 1 + t 2 + t 3 , while p is generated by a random variable with an uniform distribution on (0, 1). 4. Cumulating the settlements observed in each cell, we obtain the aggregate amount X (i, j) while the cumulative payment for the generation i, until the development year j, is given by C (i, j). 5. Cumulating until the last year of development the amount of the final cost for the generation considered C (i, t) is obtained; while the loss reserve is R (i) = C (i, t) -C (i, ti).6.For each accident year steps 1 to 5 are repeated, inflating the cost of settlements for each accident year at the annual inflation rate i infl .

3.4.
Pentikainen-Rantala method.This method simulates the development of the aggregated settlements for claims incurred during a given year, assuming that the structure function and the inflation rate follow an autoregressive process.This method operates through the following steps: 1.For accident claims during the generation of the most remote (base year, i = 0), we choose arbitrarily the average number claims, n, and the first three moments from the origin of the single cost distribution, respectively, m = a 1 , a 2 and a 3 .

It simulates the number of claims incurred in
the base year, n (0), (using the inverse of the Anscombe tranformation) and its aggregate cost of claims X (0, 0) (using the formula of Wilson-Hilferty, applicable to a compound Poisson random variable).3. The number of claims for subsequent generations is calculated using the following: ), ( ) 0 ( ) while i n measures the annual rate of growth of the portfolio.4. Represented, then, the structure function (function that modifies the average frequency of claims paid annually) with the autoregressive process of the first order: 1 qq q q i,j =a +b q i,j + . (44) It simulates q q N ; 0 ~ for each cell (i, j) and then calculates: 12 1 q i, j : i = , ,...,t; j = t i + ,...,t , (45) assuming 01 qi , = , .
5. The temporal distribution of the number of claims for each accident year is determined by the following: whereas g n (j) is the function of the temporal distribution of the number of claims (measures the probability an incurred claim in the year i is liquidated after j years).The values of the probability g n (j) (j = 0, 1, .., t ) have been hypothesized independent to the generation year and assumed equal to the components of the vector:

Numerical application
A comparative analysis was set up for the examination of the run-off error amplitude regarding each estimating method, considering different sets of parameters, which were recursively modified predicting: a different level of inflation, a higher volatility of the settlement amount, a higher volatility of the disturbing factors characterizing the settlement process, various temporal profiles for the claims development.
For each set of parameters 4.100 settlement matrices were generated with each one of the described simulation techniques.The inferior triangle of the future settlements was obtained from the superior triangle of every simulated matrix.Therefore, gap indicators between estimated reserves and effective (simulated) reserves were calculated.
Table 1 shows the statistics elaborated to analyze the method.These statistics allow to know the sign of the error, and then the tendency of the evaluation methods to overestimate or underestimate the value of the reserve.Considering all the accident years, all the methods for outstanding claims reserve prediction provide more or less biased estimators, while showing a restrained mean percentage error (with the exception of the Bornhuetter-Ferguson method).According to the mean square error criterion, the Fisher-Lange method, whose estimator is characterized by an adequate correlation level with the estimated reserves, presents a higher level of preferability.
Analyzing the single accident years, we deduce that the Chain-Ladder method provides a less biased estimator with a lower mean square error, with the relevant exception of the last accident year, which compromises, more than for any other method, the overall efficiency of the estimator.
The following table shows the analysis of the statistics calculated for each generation:   Table 5 shows the statistics elaborated for the second method.The Fisher-Lange method estimator shows the lower mean square error for both the single accident year estimation and the whole portfolio estimation.The Chain-Ladder estimator results to be the less biased estimator and shows the lower mean percentage error.
The separation methods are characterized by a systematic underestimation of the reserve, which results to be a discriminating characteristic for a method utilized in controlling the reserves set aside by an insurance company.The Bornhuetter-Ferguson method, whose outstanding claims reserve prediction is based upon a benchmark value, which depends on the ultimate cost for each accident year, presents a systematic overestimation (underestimation) of the reserve concerning the first (last 3) accident years, as it is evident from the following analysis carried out for single generation:   inflation rate: i infl = 4%; deferment: t2 =2; t3 =2.
Table 9 shows the statistics elaborated for the third method.The simulating technique appears to be rather coherent in structure, with the claims development model upon which the Fisher-Lange method is based, thus, resulting in an estimator with the lowest estimation gap for both the single accident year es-timation and the whole portfolio estimation.All methods provide estimators with high levels of correlation with the estimated reserve.
In addition for more information, the following are the analysis for single generation:

Pentikainen-Rantala method.
The numerical values attributed to the parameters were the following: number of claims: = 1000; a q = 0.4; b q = 0.6; q = 0.05; i a =1%; settlement: 1 = 0.006; 2 = 0.001; 3 = 0.0001; inflation rate: i infl =4%; i min =2%; b infl =0.7; infl = 0.015.Table 13 shows the statistics elaborated for the last method.In this case, according to both the mean percentage error criterion and the dispersion criterion, the geometric separation method presents the higher level of preferability.Moreover, the analysis of individual generations still shows the characteristic underestimation of methods based on the separation.
The following show the analysis for single generation: The method proposed by Pentikainen and Rantala has been used to test the sensitivity of the estimators when we change the temporal distribution of the settlements.Figure 2 and Figure 3 show that, under the scenarios described in Figure 1, the precision obviously is higher when the settlements are concentrated in the early development years.The precision becomes very low in the methods when the settlements occur during longer time spans.
The level of precision of the estimators based on geometric separation is superior in all the scenarios outlined.The mean percentage error and the mean square error of the estimators provided by the methods Fisher-Lange and method Bornhuetter-Ferguson achieve elevated values when the temporal distribution of settlements does not assume the canonical forms (scenarios A2 and B1).

Conclusion
The numerical implementation results point out the following: the estimating methods produce a lower run-off error if applied to a development matrix which, despite not respecting some of the probabilistic hypothesis of the method, provide a settlement distribution according to the mechanism considered by the estimating model; the sign of the run-off error may differ from generation to generation and, as a result of compensation, between the individual generations and the entire portfolio; some of the estimating methods, despite showing a minor distortion of the reserve estimation for the entire portfolio, result imprecise in the prediction of the run-off for single accident years; the preferability of the estimating methods did not show particular sensibility to the choice of numerical values attributed to the parameters.
The analysis has suggested that there isn't a better applicable method for each data set and for each line of business.Therefore, before selecting the more coherent method, it is necessary to examine the dataset, the run-off triangle, the underlying dynamics of the data and the different evolution of the settlement mechanism of different lines of business.
settlement for the claim k incurred during the accident year i; t symbolizes both the time of observation of the portfolio and the total number of generations still open.
whereas: a (j) = a a -b a j is the shape parameter and 1 annual inflation rate i infl .The model used to represent the dynamics of the parameters of the Pareto distribution generates values with increasing settlement parallel to deferment period, this ensures that the cumulative amounts of settlement of a generation, along the rows of the matrix of development, have a positive trend.In practice, the assessment of the amount of a claim can in time have either an increase or a decrease, resulting in a non-monotonic cumulative settlement.

Fig. 1 .Fig. 2 .Fig. 3 .
Fig. 1.Scenarios of the temporal distribution of settlements the random variable Ni, number of claims, is generated from a Poisson distribution with a preset parameter .2. n (i) values, k yi , of the random variable Yi U = T +T +...+T .
Buhlmann et al. (1980)butions associated with the random quantities are chosen in an arbitrary manner, but they are consistent with the actuarial literature.For the hypothesis of claims frequency with the Poisson distribution, seeBuhlmann et al. (1980); for the assumption of lognormal distribution of the amount of settlements -Hewitt and Lefkowitz (1979), Hewitt (1970). k.8 1.A value n (i) of the random variable Ni, number of claims, is generated from a Poisson distribution with a preset parameter .2. For each claim n (i), it is necessary to simulate: 2.1.the deferment of the time of accident, t 1 , respect to the beginning of the year of generation; with t 1 an uniform random variable (0, 1); 2.2. the amplitude of the deferral period from the time reported, t 2 , measured from the time

Table 1 .
Method of random development factors Total claims reserve: mean = 16.037.707;standard deviation = 669.881.

Table 2 .
Bias method of random development factors

Table 3 .
Mean square error method of random development factors

Table 3 (
cont.).Mean square error method of random development factors

Table 4 .
Mean percentage error method of random development factors

Table 5 .
Method of backward calculated random development factors

Table 6 .
Bias method of backward calculated random development factors

Table 6 (
cont.).Bias method of backward calculated random development factors

Table 7 .
Mean square error method of backward calculated random development factors

Table 8 .
Mean percentage error method of backward calculated random development factors

Table 9 .
Method of single settlements

Table 10 .
Bias method of single settlements

Table 11 .
Mean square error method of single settlements

Table 12 .
Mean percentage error method of single settlements