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1 Pricing and reserving in the general insurance industry Solutions developed in The SAS System John Hansen & Christian Larsen, Larsen & Partners Ltd 1. Introduction The two business solutions presented in this paper have been developed by Larsen&Partners, an independent Scandinavian actuarial consultancy based in the UK which provides SAS solutions for market leaders in the Insurance industry. Most insurance companies have realised that advanced analysis is of crucial importance in optimising profitability in a highly competitive market. This paper describes in further detail two business solutions, the Actuarial Claims Reserving System and The Rate Making System. These provide high quality and accurate estimates generated rapidly using the most up to date computer technology delivered by the SAS Institute. The applications are based on four SAS packages SAS/BASE, SAS/STAT, SAS/GRAPH and SAS/AF (FRAME). The basic issue that the Reserving system addresses is estimation of the future payments concerning claims incurred in a specific period in the past. The system deals with the following questions: How many claims have occurred that have not yet been reported to the company? What is the claim amount for these claims? What is the claim amount concerning claims reported but not settled yet? Of what size are the uncertainty/margins regarding these estimates? The basic issue that the Rate Making system addresses is optimising the premium rates concerning a specific portfolio based on the policy and claim file. The system deals with the following problems: How to create risk analysis and identification of important rating criteria/factors? How to identify profit and loss segments of statistical and financial significance? How to estimate the net present value (NPV) of a portfolio with a specific rating matrix? What is the relationship between price and lapses? How to create a Business Plan? 2. Actuarial Claims Reserving System This application projects and controls the company's claims reserves. It is a package that allows for a high degree of customisation to individual companies' requirements. The system can handle the original transaction data directly without any prior data manipulation such as summarising/triangulation. The advantage is improved cost efficiency, reduced dependency on resources from the IT department and increased flexibility. Data can be cleaned and claims capped, excluded or adjusted as required before summarisation/triangulation. The system includes the best current practice such as a Chain Ladder module as well as more advanced reserving methods. In addition to the forecasting of a point estimate for the outstanding claims reserve or IBNR-reserve the package includes calculation of security margins by using Bootstrap technique which is the latest Resampling method. Consequently margins related to each risk group can be estimated. The system also handles the calculation of security margins subject to Cross Funding assumptions. Some basic elements are outlined below: Data exploration. The system permits the user to explore data by chosen variables and to generate relevant triangles by, for example, transaction or claim code. In addition the system enables the user to view and subtract or exclude data, such as large claims or zero-claims. Triangulation. When the data has been selected, the accident period for which the reserves are to be estimated is selected. The data is then grouped automatically by accident and development period according to the selected

2 period unit i.e. month, quarter, half-year or year. The figures can also be calculated on an accumulated basis. Graphical descriptions and triangles can be printed. An example of a triangle is shown below: ACC. PERIOD Development Period Observed Payments The Chain Ladder/Link Ratio method. The Chain Ladder algorithm with the Original Weightings as described in The Institute of Actuaries Reserving Manual (UK) is the natural starting point. However, other weighting principles can be selected. The IBNR-Reserve and Outstanding Claims Reserve is immediately projected. The system calculates the Outstanding Reserve regarding Notified Claims and compares it to the company's current Case Reserve. The model assumption can be examined graphically and if necessary the assumption can be refined. Statistics regarding the stability (or lack of) in the speed of claims handling are calculated and displayed graphically. The Chain Ladder method can be based on the Number of Claims, The Average Cost per Claim, The Amount Paid or on the Incurred Claim Amount. The results of a Chain-Ladder analysis are outlined below, showing the outstanding claims reserves in the lower triangle: ACC. PERIOD Development Period Observed Payments Total Reserve Total Estimated Payment ALL Using graphical methods, it is easy to see whether this methodology is sufficient or more advanced statistical techniques should be used. Advanced Reserving Methods/Stochastic Modelling Methods. The Chain Ladder model involves a parameter for each accident and development period and the large number of parameters often leads to unstable projections, especially for the most recent accident periods. Decreasing the number of parameters by introducing families of run-off distributions such as Exponential Distributions improves the stability and quality of the fit. The Curve Fitting facilities in the system include Uniform, Beta, Exponential, Gamma/Hoerl, Craighead/Weibull, Extreme Value and Pareto curves and mixed curves. A further method of decreasing the number of parameters is to introduce assumptions concerning constant claim inflation and/or seasonal effects. The Bornhuetter-Ferguson model and more robust Loss-Ratio models are also included in the package. It possible to estimate the parameters in these advanced models using the NLIN procedure available in SAS/STAT. Uncertainty Estimation/Bootstrap. To a certain degree the observations are random. A specific claim is for example but it could also have been The Reserve is estimated on the basis of the observations and therefore the randomness in the observations is inherited by randomness in the estimated reserve. This type of uncertainty is known as the Estimation Uncertainty. The projected Outstanding Claims Reserve includes no margins for this randomness and consequently the estimate cannot be described as 'prudent' or 'secure'. However, using the Bootstrap technique/resampling, the variability (or distribution) of the reserve is calculated, including estimates for a 'prudent' and a 'secure' reserve/margin. Furthermore this distribution can be smoothed by curve fitting. An example is outlined below.

3 Cross Funding. In practice the variability of the claims reserve for a specific group of business may be less important to management than the variability in the company's overall claims reserve and often an estimate for the total reserve including a prudent or sufficient margin is required by the management. The problem is that such margins are over estimated by using the simple sum of margins. However, under the 'Cross Funding Assumption' where the outcome of the future amount of payments on different groups of business are independent of each other, the estimates for the total portfolio can be calculated using a simulation method. Often the total Outstanding Claims Reserve can be reduced when taking the assumption of independence into consideration. Please find an example below. Group of Business Adequate Reserve Prudent Reserve 75% likelihood Prudent Margin Secure Reserve 95% likelihood Secure Margin HOUSEHOLD 2450K 2622K 172K 2889K 439K LIABILITY 3091K 3518K 427K 4198K 1107K Total, simple sum 5541K 6140K 599K 7087K 1546K Total, Cross Funding 5541K 6080K 539K 6623K 1082K Presentation of results. The system automatically generates Reserving reports and Cross Funding reports for the management as well as technical actuarial documentation reports. These reports can be modified and designed to the specific needs of the company. The figure below is a graphical example from a management report:

4 3. The Rate Making System The better performing companies in the years ahead will be those who have the best data bases and who combine traditional underwriting with the new advanced statistical analysis of the data. By omitting the analysis and the required premium adjustments, companies run the risk of losing profitable policies and attracting the unprofitable ones. The system (RMS) has been developed to support the Underwriter, Marketing Department and Actuary in determining pricing changes in Motor, Building and Contents and other business groups. An additional facility provides competitor analysis, and the financial consequences of adjusting premium rates can be estimated. The process of determining accurate premium rates can be a lengthy and cumbersome procedure involving IT departments, programmers, statisticians and underwriters. The use of RMS dramatically simplifies the process and the relevant risk analysis can be carried out in a matter of days. RMS uses the company's claims and policy data to identify profit and loss segments. The significance of the Criteria and interactions are analysed and the correlation between the Criteria and the risk is quantified. New potential criteria such as Lifestyle Rating Criteria can be analysed and the effects quantified. Taking the company's assumption of profit rate and costs into consideration model rates are generated from break even premiums. The financial consequence of introducing new rates is predicted incorporating price elasticity. The Business Planning module provides forecasts of Written Premium, Incurred Claims, Expenses, U/W Profit, Interest and Trading Profit. 3.1 Basic problems in premium calculation There are three essential problems in connection to premium calculation: 1. randomness in the data 2. the extent to which information concerning the past is applicable when making conclusions about the future 3. price elasticity The problem of randomness is a question of defining a suitable statistical model which describes the distribution of data. The system includes factor models with standard error distributions such as Poisson, Gamma etc. These models which are implemented using the GENMOD procedure available in SAS/STAT usually describe the insurance data adequately. However, the NLIN procedure is used when further limitations on the parameter space are needed, for example when the claim frequency or average claim is assumed to depend linearly on the policy holder's age. The second problem, that of representation, is dealt with by the introduction of the calendar period as a criteria. The third problem is to estimate the customer's reaction when premium changes are introduced. The system includes facilities to estimate the relationship between premium level and lapse ratio. This relationship can be taken into consideration when estimating the Net Present Value of a portfolio or a policy. 3.2 Basic features of the RMS system Data base handling easy access to data handles transaction data or summarised data viewing, exploration and management of data subtraction or exclusion of data, such as large claims capping claims

5 Modelling the analysis assesses and allows for time trends risk analysis by riskcode finds best model using financial or statistical significance conditional fit the system quantifies the relationship between lapse ratio and price it compares premium rates with those of competitors Presentation of results formatted technical reports and summaries generated automatically graphical examination of model assumption graphical presentation of results 3.3 Areas of focus The system focuses on the following areas: Identification of risk criteria The statistical or, more importantly the financial significance of the examined Criteria and Levels can be analysed simultaneously and randomness eliminated. The statistical models used are the Generalised Linear Models which are implemented using the GENMOD procedure in SAS/STAT. An example is Vehicle classification and No Claim Discount in Motor Insurance, Accident History and Lifestyle Rating Factors. The system is not confined to personal line premium rating. The correlation between the Rating Criteria and Claim Frequency, the Average Claim and the Loss Rates are estimated. The significance of a specific Criteria or the interaction between two Criteria is calculated automatically by the system. Identification of profit and loss segments of financial or statistical significance. The system provides management information such as loss ratio and profit concerning subsegments defined by the relevant Criteria/Levels. In addition estimated loss ratio and profit can be calculated on subsegments. The problems connected with summaries of large data sets, e.g. the time involved and space used to produce them, have been reduced dramatically. Estimating the consequences of a new rating matrix. Taking the company's assumption of costs and profit allocation into consideration, model rates are generated from the break even premiums. The financial consequence of introducing the model rates or any other pricing strategy is predicted. Model rates can be estimated under assumptions of changing the insurance cover or the value of excess. Also the financial implication of 'redlining' a specific sub portfolio is estimated. Competitor analysis and identification of profitable niche strategies. Comparing premium rates with competitors' and highlighting segments where profit margins are inexpedient. How to create a Business Plan. Based on analyses of the claim frequency, claim costs, lapses, incepts, price elasticity and cash flow, the Business Planning module provides forecasts of Written Premium, Incurred Claims, Expenses, U/W Profit, Interest and Trading Profit. The number of Lapses, Inceptions and Renewals can also be estimated. The projections for a specific period selected by the user can be made on a Daily, Monthly, Quarterly or Yearly basis. 3.4 The application CRITERIA - Manipulating class variables and formats The criteria/class variables for the analysis are selected by clicking on the variables. The dimensions, i.e. the number of different levels for the variables are shown and also the current format related to the variable. A format can now be associated/generated in four different ways:

6 1. Generation of a new format by joining levels together. 2. A format can also be selected from the list of already existing formats. By clicking on 'FORMAT' a list of available formats will be shown. 3. Generation of a new format for a numeric variable can also be made by grouping the data into intervals of approximately the same size (measured by exposure or number of policies) by increasing value of the variable. 4. A format can be defined externally. Example of creating a format for joining levels together: All levels for criteria MAXAGE below 30 are gathered in one group : 30-, and all levels above 55 are grouped in : 55+. This is done by clicking on the levels in an extended table. RATE MAKING - The analysis The model for the analysis (FREQUENCY AND AVERAGE CLAIM or LOSS RATE) has to be selected. If the model FREQUENCY AND AVERAGE CLAIM is chosen, then the claim frequency and the average claim are estimated separately. The loss rates are estimated as the product of the estimates regarding Claim Frequency and Average Claim: 1. Claim Frequency = Number of Claims/Exposure 2. Average Claim = Claim Amount / Number of Claims 3. Loss Rate = Claim Frequency * Average Claim The screen controls the flow through the chosen method giving access to tables and graphics when appropriate. The colours of the buttons are used as a guide for the next selection. FACTOR MODELS - Generalised Linear Models Select the criteria which may have an impact on the claim frequency (or average claim) as main effects and the potential interactions and reestimate the factors and the statistical level of significance. The model used to estimate the INTERCEPT and the FACTORS is, that the claim frequency (or average claim) for a specific combination of the levels is equal to the product of an intercept and factors regarding the levels. If a level has the factor for example, it means that the claim frequency (or average claim) for that level is 12.3% higher than for the level with a factor equal to 1. Criteria Level Frequency Factor Average Claim Factor Loss Factor INTERCEPT HBAND A B C MAXAGE PLAN GD SR The nature of the random fluctuation is initialised to be POISSON for frequency models which is the most frequently used distribution to describe the variation in the number of claims (and GAMMA for other models, but other error distributions can be selected). When the model/analysis has been run the levels of the statistical significance and financial impact indicate whether or not the model should be corrected and rerun.

7 The level of statistical significance expresses the probability that the criteria (or the level) has no impact on the claim frequency (or average claim). If the statistical level of significance is 1% for example, then there is a 99% probability that the criteria has an impact. The financial impact of a criteria describes the impact of removing the criteria from the set of criteria. If the financial impact of a specific criteria is 10% for example, it means that 10% of the premium would be reallocated on the portfolio if the specific criteria was neglected. This key figure often contains more relevant information than the statistical significance. ADVANCED MODELS - Non-Linear Regression If the factors for a specific numeric criteria (such as AGE or SUM INSURED) show a trend then this can be estimated and the factors smoothed accordingly by a specific curve. The factors regarding the chosen numeric criteria will then be reestimated using a non linear regression analysis. This facility is not available in PROC GENMOD and PROC NLIN (Non-Linear Regression) is used to identify the relationship. The curve has been smoothed by an exponential curve, reducing the number of parameters and producing a curve that in this example is constantly increasing (avoiding the decrease in the parameters for AGE around 40). QUALITY CONTROL Frequency / Average Claim or Loss Rate (Tables) The exposure, the estimated and observed Loss Rate and the estimated and observed number of claims are tabulated by criteria level or by combinations of levels. Example of table on Criteria AREA (using PROC TABULATE and reading the output to an extended table in FRAME): Criteria AREA Exposure in years Number of Claims Claim Amount Observed Loss Rate MODEL Loss Rate A B C D E ALL

8 Model Control The Model Control can disclose if the model, for example, under-estimates the loss rate for the low risk segments and/or over-estimates the loss rate for the high risk segments. For each cell, i.e. for each combination of criteria levels, the observed loss rate is plotted against the estimated loss rate. A good model will give plots on or around the line where the observed and estimated frequency is equal. A regression line with confidence intervals as well as the line y=x is shown. TREND ASSUMPTIONS It is possible to estimate the time trend and to calculate an estimate of the loss rates for a specific period in the future. The exposure is indicated on the top of the graph to indicate the level of weight on the error and the confidence intervals on the observed values are calculated and shown on the graph (PROC GCHART). Furthermore a regression (using PROC REG) on the estimated figures is carried out and the result implemented in the graph (using the ANNOTATE facilities) :

9 NEW PREMIUMS When the analysis is finished and the profits and costs are decided then the new premium matrix can be generated as shown in the following screen. It is possible to manipulate the order of the variables in the table and to have 1-4 variables shown at the same time. Premiums Number of Bedrooms Type of House Rate Area 1 A B C D E A B C D E When a criteria is not shown in the table a specific value must be chosen. This is done by putting all the different levels into a list and using the SCL-function POPLIST when clicking on one of the criteria not shown in the tables. IDENTIFY PROFIT This screen keeps information about the profitability of the business in the period under consideration. It is used to identify profit and loss segments of financial significance. For each combination of the levels of the criteria key figures are created and sorted by the sort-variable if this has been selected: PROFITIDENTIFICATION Claim Amount Number of Claims Actual Premium PREMIUM New Premium NEWPRM Profit Loss Rati o Profit per Insurance Year Est. Claim Amount Exposure in years Type of House Rate Area 1 A B C D E A B C D E ALL WHAT IF This is used to identify the consequences of redlining/eliminating a specific (unprofitable) sub-segment by clicking on the segments to redline. By manipulating the table (selecting different criteria) it is possible to see if eliminating a sub segment from the portfolio would affect other segments. For each combination of the levels of the criteria the following key figures are created : 1. Actual Variable (Premium, Loss Ratio or Profit) 2. New Variable 3. Adjustment, i.e. the difference New Variable minus Actual Variable 4. Relative Adjustment, i.e. the relative difference between New Variable and Actual Variable 5. Exposure in years 6. Claim Amount 7. Profit per Insurance year 8. Estimated Loss Ratio i.e. Estimated Claim Amount in percentage of Reference Premium 9. Profit i.e. Reference Premium minus Claim Amount (Contribution 1) 10. Estimated Profit i.e. Reference Premium minus Estimated Claim Amount (Contribution 1) COMPARE PREMIUM

10 This is used to answer questions such as: 'What is the proportion of the portfolio that will have a premium increase if the Reference Premium is changed to The Office Premium?' 'What are the consequences of using a new rating matrix?' Comparison of premiums is necessary to avoid some segments in the portfolio being given too high an increase or decrease when using the new rating matrix and is also used to discover if the company is competitive. This is done by grouping the business by the size of the difference or relative difference (using a FORMAT) between the rating matrices as shown in the following example: Group defined by (NEWPREM-EARNED)/EARNED. Exposure Ref. Comp. Ref. Comp. Premium Premium Loss Loss Claim % EARNED NEWPREM Diff. Diff.% Ratio Ratio Amount Group <= <= <= <= <= <= <= <= <= <= <= > ALL For further information please contact Larsen & Partners Ltd 20 Prior Street Greenwich London SE10 8SF UK Tel&Fax: +44 (0) Trademarks : SAS is registered trademark of SAS Institute Inc., Cary, NC.

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