S T O C H A S T I C C L A I M S R E S E R V I N G O F A S W E D I S H L I F E I N S U R A N C E P O R T F O L I O


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1 S T O C H A S T I C C L A I M S R E S E R V I N G O F A S W E D I S H L I F E I N S U R A N C E P O R T F O L I O DIPLOMA THESIS PRESENTED FOR THE. FOR SWEDISH ACTUARIAL SO CIETY PETER NIMAN 2007
2 ABSTRACT 2 ABSTRACT Traditionally, outstanding claims reserves were settled using deterministic methods which resulted in point estimates of the reserves. The main advantage of stochastic reserving models is that the point estimate of the reserve is completed with measures of precision of the estimate. The information retained is further enhanced if the full predictive distribution of the reserve is obtained. The aim of this thesis is to explore whether the methods and techniques used in P&C insurance for stochastic reserving would also be applicable for Life insurance given the differences between the two insurance lines. The conclusion drawn from this study based on the given portfolio is that the methods are indeed applicable even if some modifications may bee needed. Among the three different portfolios studied, a Life portfolio, an Accident portfolio and a Sickness portfolio, the Sickness risk is the most deviant from the others and even from P&C insurance mostly due to it s payout structure; The benefits are paid contingent on the policyholders working ability, i.e. as long as the claimant is sick or until he reaches the policy termination age. Due to that, the same methods and techniques which are used in P&C for stochastic reserving are not applicable. Apart from a different setup of the claims triangle, stochastic simulations of the recovery times need to be implemented which in its turn assumes knowledge about the distribution of the survival times. 2
3 3 STOCHASTIC CLAIMS RESERVING OF A SWEDISH LIFE INSURANCE PORTFOLIO TABLE OF CONTENTS Abstract... 2 Table of Contents Introduction... 5 Problem formulation... 5 Major differences between life and non life insurance 5 The Portfolio... 6 Life Insurance... 6 Personal Accident Insurance (Temporary Disability) 6 Sickness Insurance... 7 Data material... 7 Delimitations... 7 Disposition Prediction Error of the claims reserve... 9 The Smörgåsbord of Models... 9 Error types Chain Ladder Model Over Dispersed Poisson Model Estimation of the dispersion Parameter Parameter estimation Estimation of the prediction variance Practical implementation: Life insurance Predictive Distribution of the claims reserve Bootstrapping Practical implementation: Accident insurance Bayesian methods Stochastic reserving of sickness insurance Estimation of the Recovery Intensity Simulation of recovery times Inverse transform Acceptance rejection Validation of the simulated values Estimation of the reserve Results and Conclusions Index References Appendices
4 TABLE OF CONTENTS 4 Appendix 1. Prediction error of the accident portfolio Appendix 2. Predictive distribution of the Life portfolio Appendix 3. Sickness recovery estimation Figure 21 The covariance structure of the forecast observations Figure 22 Development triangle paid claims, Life portfolio Figure 41Split of the distribution function (illustration only) Figure 42 Acceptancerejection method Diagram 31Bootstrapped distribution of the reserves, Accident portfolio Diagram 41 Survival Distribution Function Diagram 42 Comparision of simulated recovery times vs. outcomes Diagram 43 Bootstrapped distribution of the reserves, Accident portfolio Diagram A201 Bootstrapped distribution of the reserves, Life portfolio Table 21Reserves, prediction error and coefficient of variation, Life portfolio 16 Table 31 Summary of results of the bootstrap method, Accident portfolio Table 32 Confidence intervals for different confidence limits, Accident portfolio Table 41 Summary of results, Sickness portfolio Table 42 Confidence intervals for different confidence limits, Sickness portfolio Table A201 Summary of results of the bootstrap method, Life portfolio Table A202 Confidence intervals for different confidence limits
5 5 STOCHASTIC CLAIMS RESERVING OF A SWEDISH LIFE INSURANCE PORTFOLIO 1. INTRODUCTION PROBLEM FORMULATION Traditionally, outstanding claims reserves were settled using deterministic methods which resulted in point estimates of the reserves, i.e. the present values of the expected future costs of claims. Increasing demand for further insight about the variability of the reserves has lead to the development of stochastic models for the reserve calculations. These models have been developed for Property & Casualty (P&C) insurance. The aim of this thesis is to examine whether one of the methods frequently used in P&C insurance would be applicable for stochastic reserving purposes of a Swedish Life portfolio consisting of Life, Accident and Sickness risks. Stochastic reserving has long been ignored by many practitioners due to lack of easily implementable models; the ones available often required expensive specialist software. This situation is about to change due to the regulatory constraints in several countries where the regulators expect knowledge about not only the point estimate of the reserve but also about it s variability. Development of own models are encouraged and often incentivized by lower statutory capital requirement. The embedded value of having own models instead of using the so called standard model created according to the one size fits all principle is thus easily quantifiable. Furthermore, the theories have become more accessible thanks to some authors who apparently have laid a lot of emphasis on the pedagogical aspects when serving the often heavy theories in their papers which became thus fully digestible even for practitioners who have left school a long time ago. Also computers are significantly faster and even standard software are better qualified to cope with the complexity involved in the calculations. In fact, one of the constraints imposed on the stochastic model selection has been the possibility of performing the calculations without needing expensive software. MAJOR DIFFERENCES BETWEEN LIFE AND NON LIFE INSURANCE There are some major differences between non life and life insurance regarding the application of stochastic (and deterministic) reserving methods, specially for companies in which the business volumes (premiums and claims) have not yet reached the level where the variability in the reserves are to be considered low. The traditional triangulation methods (e.g. chain ladder technique) presume high volume business with relatively stable development between the accident years. This is obviously not the case for the smaller life offices who have to adopt their reserving methods accordingly which is in many cases done by adding a certain amount of subjectivity, often by using credibility models, e.g. Bornhuetter Ferguson. As the theory of stochastic reserving which incorporates expert opinion is still under development and is out of the scope for this thesis we confine ourselves with a a short discussion and with giving references for further studies. 5
6 INTRODUCTION 6 Another difference is that case reserves tend to be large in comparision with the paid claims. For accident insurance a typical paid claim amount is in the range of a few thousands of SEK while the case reserves have the magnitude of several tens of thousands. This increases the volatility in the data. In case of low claim payments and large reserve releases the incurred claims may be negative. Thus the chosen model has to be able to handle negative incurred claims. Last but not least, some life products pay an annuity (se the portfolio section) which de facto violates one of the key assumptions in the chain ladder technique, namely the independence of the claims between the development years. In order to tackle that a new way of setting up the data is needed. The reserves for this kind of annuities also depend on the claims recovery intensity which is not common in non life insurance where the payments are almost always done as a lump sum which are stochastic. Benefits in life insurance are often set out in the contract and are thus less stochastic. The literature on the subject of stochastic reserving is huge and the number of models and methods ever increasing. It is not the scope for the present paper to present all of them; as a matter of fast only one is examined closer, but hopefully the list of references will serve as a good source of inspiration for those interested to further explore this exciting field being in fashion. THE PORTFOLIO The very short description of the risks involved in the products consisting the studied portfolio is considered being necessary for a better understanding of the mechanism and the dynamics involved. Also the types of reserves used in the calculations are described. LIFE INSURANCE The insured event in the Life Insurance is the death of the insured and the benefit is the contractual insured amount which is paid as a lump sum to the beneficiaries. This product is a typical short tailed business which means that the period between the occurrence of the insured event and the claim settlement is short, usually less then 6 month. There are no case reserves for this product, the reserve is equal to the IBNR (Incurred But Not Reported). PERSONAL ACCIDENT INSURANCE (TEMPORARY DISABILITY) An accidental injury is a bodily injury caused by a sudden, external and involuntary event, occurring at an identifiable time and place. The insurance contract covers loss of income, medical costs, dental injury costs, travel costs, costs for psychologist services, rehabilitation and medical aids costs etc.. This product is a long tailed business since the period between the occurrence of the insured event and the claim settlement may be very long, in some cases several years. Reserves for the claims being assessed by the claims handling personnel as large claims are called case reserves. The incurred claims for Accident Insurance are the sum of the paid claims, the case reserves and the IBNR. 6
7 7 STOCHASTIC CLAIMS RESERVING OF A SWEDISH LIFE INSURANCE PORTFOLIO SICKNESS INSURANCE The benefit from the Sickness Insurance is an annuity based on the insured amount and paid after a certain waiting period when the insured loses it s working ability due to sickness. Even if claims are usually reported shortly after occurrence, the lengths of the annuity payments may be very long, in the worst case until the insured reaches the age of 65 which is the ordinary pension age in Sweden. The incurred claims in this case is the sum of claims paid, the RBNS (Reported But Not Settled) reserve and the IBNR. Sickness Insurance is a typical long tailed product. DATA MATERIAL All data used in this study is real yearly data disposed in the traditional way according to accident year and development period or notification period respectively from a small size insurance carrier with medium long history in the Swedish market. For the sake of data integrity, the data has been distorted by multiplying all the values with the same constant. The values of the recovery times used in the Kaplan Meier estimation, the Survival Distribution Function and when presenting the results of the simulated recovery times, the time values on the x axis of the chart are deleted. DELIMITATIONS The following delimitations apply to the data and the models chosen: The reserves are undiscounted (in other words the discount rate is nil) and unadjusted for inflation. No other financial aspects, e.g. different economic scenarios have been taken into consideration as in e.g. (Haberman, Booth, Chadburn, Cooper, & James, 1999) where different stochastic economic scenarios are generated in order to study the volatility of the prediction errors. The reserves are gross of reinsurance; net and ceded data should be analyzed separately as the claims distributions may have other characteristics (specially in the case of excess of loss treaties). Sickness inceptions with a shorter duration than the waiting period are not taken into consideration. The term reserve is used as a synonym for the sum of the RBNS and the IBNR, i.e. no UPR (Unearned Premium Reserve) or Unexpired Risks are taken into account. The notions of risk and portfolio may be interchanged throughout this paper as their meaning is the same: Essentially, each of the portfolios contains only one risk, i.e. there are no products within a portfolio which are the combination of several risks. No emphasis is laid on explanations of the deterministic method, i.e. the reader is presupposed to be familiar with the chain ladder technique. There are no tails in the studied triangles, i.e. the oldest accident years are being considered fully developed with no need for reserve. Even if not stated or commented in the paper, the quality of the fitting to the data were controlled by residual analysis. 7
8 INTRODUCTION 8 DISPOSITION Throughout in this paper the presentation of the theoretical background is followed by the practical implementation on the available data. Even though all the calculations are carried out for all risks the result for only one risk is presented and commented in detail in the disposition part of the paper whereas the results for the other risks can be found in the Appendix. The remainder of this paper is structured as follows; In the first of the three main chapters the methods that lead to the second moment of the reserve are discussed; a short exposé about the most commonly used methods is given narrowing down the range to the so called chain ladder models. The brief introduction to what the prediction error (variance) is and how it is calculated is followed by the presentation of the model which plays the starring role in this thesis: the Overdispersed Poisson Distribution (ODP). The data of the Life portfolio is used to give the illustration of how the model works. In the following chapter the level of complexity is raised further with the aim to derive the full predictive distribution of the reserves. A short description of the method of bootstrapping is given before used in the practical implementation on the Accident data. Without going into details, advice and references are given about how a credibility model could be used for stochastic reserving purposes. The final chapter is devoted to the risk which is the most complex of the three studied ones, the Sickness insurance. This part uses almost all the prerequisites which were presented in the previous chapters. Even though it is out of the scope of this thesis, a short presentation of the method used for the estimation of the recovery intensity is given which is believed to be useful to understand how the survival times are generated with Monte Carlo methods. The thesis is wind up with a section dedicated to conclusions and discussions. 8
9 9 STOCHASTIC CLAIMS RESERVING OF A SWEDISH LIFE INSURANCE PORTFOLIO 2. PREDICTION ERROR OF THE CLAIMS RESERVE THE SMÖRGÅSBORD OF MODELS The work which probably had the most influence at the early stage of development of stochastic reserving is the paper of Mack, (Mack, 1993), which presents one of the earliest attempts at formalizing a stochastic model for claims reserving. The proposed nonparametric distributionfree model that reproduces the chain ladder values and gives the standard deviation of the reserve estimates is now called the Mack model. According to Taylor et. al. (Taylor, McGuire, & Greenfield, 2003) the number of publications about reserving published during the last two decades is higher than the total volume published prior to that. A significant part of the published works deal with stochastic reserving. One of the most cited and definitely the one which gave the most inspiration to the present thesis is the paper of Engalnd and Verrall, (England & Verrall, 2002) which can also be used as a source of reference over the most important models used in stochastic reserving. According to the authors, the models used in stochastic reserving can be divided into the following groups: Chain Ladder models which replicate the estimates of the chain ladder technique. This can be achieved in two ways; by specifying distributions for the data or specifying only the first two moments of the distribution. Models belonging to the first category are the OverDispersed Poisson Model (ODP), the Negative Binomial Model and the Normal Approximating to the Negative Binomial Model. It has been shown that these models produce essentially the same results. Models in the second category are the Log Normal Model and the Gamma Model amongst others. Other Parametric Models have arisen to tackle the problem of the chain ladder models being overparameterized (one parameter is used for each development period and accident year). One model belonging to this category is the Hoerl Curve which fits a parametric curve to the run off pattern of the triangle. The disadvantage with the Hoerl Curve is that it seldom fits well to the entire range of development times. Wrights Model is another model in this category which gained a lot of attention. The model specification is close to that of the well known one in which the number of claims are Poisson distributed and the claim size is modeled as a random variable with Gamma distribution. The last category is the NonParametric Smoothing Models. If the (flexible) chain ladder model is considered at one end and of the extreme and the (rigid) Hoerl curve at the other than there can be created a number of (NonParametric) Smoothing Models which move seamlessly between these two extremes. These models are implemented by using generalized additive models, abbreviated by GAM and often require the use of specialized software. In the following analysis we will exclusively use the chain ladder model approximated by the Over Dispersed Poissson distribution. 9
10 PREDICTION ERROR OF THE CLAIMS RESERVE 10 ERROR TYPES Before we are ready to take care of the chain ladder model approximation by the ODP, we need to introduce the different error types which play an important role in our analysis. Since mathematical models are only an idealizations of the real world, these are associated with uncertainties or errors. According to (Daykin, Pentikäinen, & Pesosnen, 1994), this errors can be divided into the following three categories: Model errors arise due to the fact that models are not known with certainty and are only approximations to the real world phenomena which they intend to model. Parameter errors are due to that the observations are limited in quantity so parameters are not known with certainty, and finally Process error (stochastic error) which arises due to the random fluctuations of the target quantities even in a ideal situation where the model and the parameters are correct. In spite of it s importance, model error is omitted in this paper and focus is laid entirely on the two latter errors or variances; the parameter error, even called estimation variance and the process errors. The sum of the process variance and the estimation variance is called prediction variance and is a measure of the variability of the prediction calculated as the root mean squared error of the prediction (RMSEP). Let y be a random variable with the expected value y. The mean squared error of prediction (MSEP) is: E y y 2 = E y E(y) (y E(y) 2. (1) Expanding this expression and assuming inserting y instead of y we get: E y y 2 E y E(y) 2 2E (y E(y))(y E(y) + E y E(y) 2. (2) As future observations are assumed to be independent of past observations (2) gives: i.e. E y y 2 E y E(y) 2 + E y E(y) 2. (3) Prediction variance = process variance + estimation variance 10
11 11 STOCHASTIC CLAIMS RESERVING OF A SWEDISH LIFE INSURANCE PORTFOLIO CHAIN LADDER MODEL The heuristic and deterministic  according to Taylor s notation 1  chain ladder model is probably the most used technique for claims reserving and is based on setting up a so called claims triangle with the data to be analyzed (paid claims, incurred claims, number of claims or whatever the target of interest may be) according to a convenient setup. In most of the cases this setup has the accident years in the rows and the development years in the columns. Using the standard notations, the elements of this triangle can be described as {C ij j = 1,, n 1 + 1; i = 1, n} (4) Where C ij denotes the incremental claims in a triangle with the row indicator i, i.e. the accident year i and the column index j i.e. the delay period  measured in years throughout the whole paper. Analysis of more frequent data (or less frequent for that matter) is a straightforward extension of the present analysis. Consider further the cumulative claims denoted by: D ij = j C ik k=1 (5) The fundamental idea of the chain ladder method is to estimate the development factors {λ j j = 2,, n} by taking the weighted average of the individual development factors f ij as follows: (6) λ j = n j +1 i=1 D ij n j +1 i=1 D i,j 1 = n j +1 D ij i=1 D ij 1 D i,j 1 n j +1 i=1 D i,j 1 = n j +1 D i,j 1 f ij i=1 n j +1 i=1 D i,j 1 (7) These development factors are then applied to the latest cumulative claims in each row which give the forecast of future values of cumulative claims: D i,ni +2 = λ n j +2 D i,ni +2 (8) D i,j = λ j D i,j 1 (9) 1 (Taylor, McGuire, & Greenfield, 2003) 11
12 PREDICTION ERROR OF THE CLAIMS RESERVE 12 This is the formal description of the chain ladder model. As we wish to estimate the prediction error of the reserve estimate given by the chain ladder model, we have to find a suitable model which approximates the data in the triangle  both past and future data and hopefully result in the same or very similar estimate as that given by the chain ladder model. This is where the ODP model enters the scene. OVER DISPERSED POISSON MODEL There are a number of different models that can be used to reproduce the results of the Chain Ladder technique. One particular well known one is the Over Dispersed Poisson (ODP) and which assumes that the incremental claims have an over dispersed Poisson distribution. An ODP looks like a Poisson distribution but it s variance is not equal to the mean but proportional to it where the proportionality factor is called the over dispersion parameter. We have C ij ~ IID ODPo μ ij (10) where IID denotes independent, identically distributed with E C ij = μ ij and Var C ij = ΦE C ij. Let logμ ij = η ij = μ + α i + β j (11) which is recognized as a generalized linear model in which the responses C ij are modelled with a logarithmic link function and linear predictor, η ij. This model is linear in the parameters and is thus suitable for fitting the chain ladder model since we have one parameter for each row i and each column j. Due to this overparametrisation of the model, we have as many parameters as values to fit, we apply the corner constraints as follows: η ij = μ + α i + β j α 1 = 0 β 1 = 0 logμ ij = η ij (12) i.e. the first two parameters are zeroised. β j, the column parameter determines the run off structure of the data. Since we have one parameter for each column, we assume that there is no particular shape of the run off pattern, which is in line with the general assumptions imposed on the traditional chain ladder model. The ODP model is robust for a small number of negative incremental claims. Indeed, this is an important feature specially for the product lines where case reserves are set by the claims handlers who quite often tend to be over conservative in their judgments. Adjusting for this overestimation at a later stage may lead to negative incremental values. 12
13 13 STOCHASTIC CLAIMS RESERVING OF A SWEDISH LIFE INSURANCE PORTFOLIO ESTIMATION OF THE DISPERSION PARAMETER The next step is to estimate the parameters of the ODP model. We begin with the estimation of the dispersion parameter Φ. Given that 2 C ij φ μ ij φ 2 χ n p (13) μ ij φ which is a result of that standard residuals, i.e. the quotient which is squared in the expression above is N(0,1) distributed and the sum of squared N(0,1) variables are χ 2 distributed with the degree of freedom equal to the number of observations less the number of parameters. The method of moments gives: C ij μ ij φ μ ij 2 = n p (14) since Φ is constant it is readily available from (14): φ = 1 n p C ij μ ij μ ij 2 (15) One may interpret Φ as the average claim size while C ij is the number of claims. PARAMETER ESTIMATION The estimation of the other parameters of the model is done by calculating the loglikelihood function, l, and maximizing it: l = n i=1 n i+1 1 C ij logμ ij μ ij + = j =1 (16) = 1 n n i+1 i=1 j =1 C ij (μ + α i + β j e μ+α i+β j + (17) 13
14 PREDICTION ERROR OF THE CLAIMS RESERVE 14 ESTIMATION OF THE PREDICTION VARIANCE The estimation variance for each estimated value is: Var(μ ij ) = Var e η ij = eη ij η ij 2 Var(η ij ) (18) Taylor estimation of (18) gives: Var(μ ij ) = e η ij 2 Var(η ij ) = μ ij 2 Var(η ij ) (19) According to (3) the prediction variance for each value in a cell is: PE = φμ ij + μ ij 2 Var(η ij ) (20) In the overall prediction variance even the covariances should be taken into account: PE = φμ ij + μ ij 2 Var(η ij ) + 2 Cov( η ij, η ik ) μ ij μ ik (21) The indexes under the summations above have been omitted. It is not hard to realize, that calculation of the prediction error is quite cumbersome. An alternative which is relatively easy to implement in a spreadsheet was presented by professor R.J.Verrall at a seminar held in Stockholm in September The method is best understood when setting up the model in matrix form. The design matrix is denoted by X, the parameter vector by β and the fitted values by μ : the covariance matrix of β is μ = e (Xβ ) (22) where Σ = φ X T W X 1 (23) W = diag 1 V(μ ij ) μ ij η ij 2 (24) 14
15 15 STOCHASTIC CLAIMS RESERVING OF A SWEDISH LIFE INSURANCE PORTFOLIO If the design matrix of future values is denoted with F then the covariance matrix of the linear predictors can be written in analogy with 23 as F Σ F T and the covariance matrix of fitted values as diag(μ ) F Σ F T diag(μ ). The data for the forcast vector is being placed in the same column (only one column). The covariance matrix of the forecast observations will then have the form presented in figure 21 below. The data consist of 7 accident and development years which are marked with light grey in the figure. Figure 21 The covariance structure of the forecast observations In this section the method of fitting the chain ladder triangle with an overdispersed Poisson model was presented. The goal has been to calculate the prediction error of the reserve. The procedure can be summarized by the following scheme: 1. Set up the chain ladder triangle of cumulative data 2. Calculate the linear predictors 3. Calculate the mean for each cell 4. Calculate the loglikelihood for each cell 5. Sum up the loglikelihood 6. Estimate the parameters by minimizing the loglikelihood above 7. Calculate the Pearson residuals 8. Calculate the overdisperion parameter 9. Calculate the prediction error using the matrix set up. The parameter estimation in step 6 is done with Solver in Excel by giving the constraints as in (12). PRACTICAL IMPLEMENTATION: LIFE INSURANCE In this section the theory presented above is implemented on the Life data described in the portfolio section on page 6. Development year Acc. Year D1 D2 D3 D4 D5 A A A A A Link ratios 1,832 1,008 1,013 1,007 Cumulative link ratios 1,883 1,028 1,020 1,007 15
16 PREDICTION ERROR OF THE CLAIMS RESERVE 16 Figure 22 Development triangle paid claims, Life portfolio The figure above presents the development of the paid claims for each accident year. There are no case reserves for this portfolio why the incurred claims equal the paid claims. There is a strong evidence for short delay between occurrence and payout which is characteristic for Term Life Insurance. Basic Chain Ladder Overdispersed Poisson Latest Ultimate Prediction Acc. Year Period Forecast Reserve Reserve Error C.V A A % A % A % A % Total % Table 21Reserves, prediction error and coefficient of variation, Life portfolio The prediction error for the ultimate reserve for the Life portfolio is 50% which is a relatively high amount and is a consequence of the volatility in the portfolio. Even if it cannot be read from the data presented here, the size of the portfolio is to small in comparision with the paid claims. This becomes evident when comparing the results for the Life portfolio with that of the Accident portfolio presented in Appendix 1. The Accident portfolio is a high volume business with a relatively long history and is thus more suited for the chain ladder technique. However, even for this portfolio the prediction errors for the early development years are relatively high whereas the outstanding reserves are small. The most recent year s higher prediction error is explained by the high number of parameters to be estimated. We can notice that the reserve estimates are very similar to that of the chain ladder result for both portfolios. The difference is due to round up differences in the calculation. 16
17 17 STOCHASTIC CLAIMS RESERVING OF A SWEDISH LIFE INSURANCE PORTFOLIO 3. PREDICTIVE DISTRIBUTION OF THE CLAIMS RESERVE In this section we seek to further increase our insight about the uncertainties associated with the reserves. Instead of only obtaining the second moment for the estimate we strive for finding the whole predictive distribution of the reserve. If the underlying distribution of the data is not given than it is not possible to obtain the predictive distribution without further assumptions. Here comes simulation in, a method which becomes more and more used and accepted. There are two main methods which are currently used for this purposes: bootstrapping and Bayesian methods using MCPC (Marcov Chain Monte Carlo) techniques. In this paper only the former method is used even if the later is also discussed briefly in the discussion section at the end of this chapter with references for further studies. BOOTSTRAPPING Bootstrapping is a technique of obtaining information from a sample of data without any requirement on knowledge about the analytical features of the underlying data. The methodology is based on sampling with replacement from the observed data to create a new data set called pseudo data which are consistent with the underlying distribution. Iterating this enough times a large set of pseudo data is obtained which can than be investigated for further insight. The target of interest is often the mean and standard deviation of the data along with the shape of the distribution. This is even applicable in our case. In the next section the bootstrapping technique is described. The resampling of independent and identically distributed data is done from the data itself, i.e. the new data is a subset drawn with replacement from the original data set. This is, however not suitable for our problem which is of regression type. The standard procedure for regression type problems is to bootstrap the residuals since these are approximately independent and identically distributed or if not, they can be made so. For generalized linear models the so called Pearson residuals are used for bootstrapping. Dropping the indexes the Pearson residuals r P are defined as: C μ r P = μ (25) where μ are the fitted incremental claims given in the equation (11). The pseudo data, C created in this way is simply obtained by C = r P μ + μ (26) Resampling with replacement gives a new triangle of past claims payments which is then refitted and forecast incremental claim payments can be calculated in the same manner as presented in the previous chapter. 17
18 PREDICTIVE DISTRIBUTION OF THE CLAIMS RESERVE 18 In order to capture the process variance the claim payments for each future cell in the run off triangle is simulated with the bootstrapped value as the mean value and using the process distribution assumed in the underlying model, i.e. the ODP. The whole procedure is repeated a large number of times and the relevant statistics are pertained as these will build the predictive distribution of the reserve. A summary stepbystep description of the bootstrap procedure similar to that given for the prediction error in the previous chapter is following the two important formulas incorporated in the description: φ = P r 2 i,jn i+1 ij (27) 1 2 n n + 1 2n + 1 where Φ is the Pearson scale parameter, i.e. the sum of the Pearson residuals as in (25) divided with the number of freedom which is the number of observations minus the number of parameters estimated. A bias adjustment of the Pearson residuals is done by: r ij adj = n P r 1 ij (28) 2 n n + 1 2n Set up the chain ladder triangle of cumulative data 2. Obtain the development factors from the cumulative data 3. Calculate fitted values for the past triangle by backwards recursion 4. Calculate incremental fitted values, μ, for the past values by differencing 5. Calculate the Pearson residuals as in (25) 6. Calculate the Pearson scale parameter as in (27) 7. Bias adjust the Pearson residuals by using (28) 8. Resample the adjusted residuals with replacement and create a new past triangle and apply (26), i.e. create new pseudo incremental data in the past triangle 9. Fit the chain ladder model to the pseudo cumulative data 10. Project the future cumulative payments 11. Calculate the incremental payments by differencing 12. For each cell in the future triangle simulate a payment from the process distribution (ODP) with the mean obtained in the step Store the results for future analysis 14. Loop from step 1 a number of times. Finally, the stored data is analyzed to get the desired statistics of the predictive distribution. The estimate of the predictive error is the standard deviation of the stored results. Confidence intervals can be constructed, e.g. with the confidence level of 95%, form a sample of 1000 by taking the 25 th and the 975 th largest value from the sample. 18
19 19 STOCHASTIC CLAIMS RESERVING OF A SWEDISH LIFE INSURANCE PORTFOLIO In this section the method of deriving the predictive distribution of the ultimate claim reserve by applying the bootstrap method and simulation was presented. The goal with this method has been to enhance the information obtained in the traditional deterministic way. PRACTICAL IMPLEMENTATION: ACCIDENT INSURANCE In this section the theory presented above is implemented on the Accident data described in the portfolio section on page 6. As mentioned previously, there are no tails in the triangle, i.e. the oldest accident year is considered fully developed. The development triangle is presented in figure A1 in the Appendix 1. Acc Year Chain Ladder Bootstrap P.E. C.V. A % A % A % A % A % A % Total % Table 31 Summary of results of the bootstrap method, Accident portfolio The table 31 above presents the results from iterations of the bootstrap method for the Accident portfolio. It is noteworthy that the results (point estimates) of the two methods are so different; the relative difference is 4% while theoretically we should have arrived to the same results. However, in exchange we have now the full predictive distribution for the ultimate outstanding reserve which a lot more valuable than only the point estimate. As a matter of fact, the point estimate has by now lost it s importance in the sense that the communication henceforth will only be in terms of confidence intervals. The diagram 31 below displays the distribution of the reserves resulting from the bootstrap simulation. As expected, the distribution is skewed with most of its mass on the left hand side of the distribution. Comparing the prediction error (P.E.) with those of the table A1 we can note a quite substantial reduction when the bootstrap method is applied. 19
20 PREDICTIVE DISTRIBUTION OF THE CLAIMS RESERVE 20 Diagram 31Bootstrapped distribution of the reserves, Accident portfolio In order to get a grasp of the width of the confidence interval given some certain confidence levels (symmetrical in this case but these can easily be changed in the calculation) the table 32 was created and presented below. L.C.L (%) Lower Limit U.P.L (%) Upper Limit Interval Lenght 2,5% ,5% ,0% ,0% ,5% ,5% ,0% ,0% ,0% ,0% Table 32 Confidence intervals for different confidence limits 2, Accident portfolio Bootstrapping is easier than the calculations in the previous chapter since we skip the cumbersome matrix operations in order to get the measures of variability. 2 L.C.L: Lower Confidence Limit; U.C.L.: Upper Confidence Limit. 20
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