A Review of Berquist and Sherman Paper: Reserving in a Changing Environment

Transcription

1 A Review of Berquist and Sherman Paper: Reserving in a Changing Environment Abstract In the Property & Casualty development triangle are commonly used as tool in the reserving process. In the case of a change in the environment, common actuarial techniques should not be applied blindly. This paper intends to describe what kind of process can distort a loss triangle, explain how to detect shifts in the data and how to adjust the data to correct for those changes. In the process common actuarial method are described. The claim process is detailed. Berquist and Sherman two methods to adjust triangles for changes in case outstanding adequacy, or in payment pattern are presented. Keywords: Berquist and Sherman, Chain Ladder, loss reserving, Actual versus Expected, Mack, Venter, claim process 1

2 Acknowledgements I wish to express my sincere thank you to the people I had the chance to work with at PricewaterhouseCoopers. I would like to thank my coach Brittany Manseau. I am also grateful to Jeffrey Mayer for his guidance and suggestions. I would also like to thank Dan Thomas, Alistair Macpherson, Dorothy Woodrum, Gregory Moore. It was a pleasure working with them. 2

3 Table of contents Abstract... 1 Acknowledgements... 2 Introduction... 6 Chapter 1. The importance of reserving The necessity of reserving for insurance companies Main reasons In the United States Guidance and rules followed by reserving actuaries Actuarial Standard of Practice for reserving in the United States Tasks of an audit company Chapter 2. Common reserving methods Development triangles Remarks on insurance data A survey of claim reserving methods Deterministic versus stochastic methods Deterministic methods Stochastic methods Solvency The Development method Frequency / Severity techniques The Bornhuetter-Fergusson (BF) method Actuarial judgment Mack s assumptions The selection of factors

4 2.8.2 Three assumptions Testing for those assumptions Hypothesis 1: linear relationship Hypothesis 2: Independence of the columns Hypothesis 3: Key assumptions arising from data Chapter 3. Numerical example of unpaid losses ultimate selection The Data used Application of the different actuarial methods Selection of a method Introducing a change in the environment Chapter 4. Understanding the data The claim process Claim management Changes in case reserve adequacy Changes in paid settlement pattern The impact of external environment Tort reforms Other environmental changes Diagnostic tests The claim loss development triangles Changes in earned premium Average data Claim management evolution Chapter 5. Reserving in a changing environment

5 5.1 The Berquist and Sherman approach Adjustment to the reported loss triangle Berquist and Sherman method Sensitivity of the reserves to the selected trend Adjustment made to the paid triangle Berquist and Sherman method Sensitivity of the reserves to the selected parameters Influence of a change in development pattern on the case outstanding triangle Other adjustments Chapter 6. Testing the estimation Selection Criteria Pre-selection test Testing the forecasting abilities of the model Testing the reasonableness implied by the selected loss ultimate Post- selection test Actual versus Expected using the Chain Ladder method Actual versus Expected using the Bornhuetter- Fergusson technique Actual versus Expected using Berquist and Sherman method How to use the actual versus expected? Chapter 7. Conclusion References

6 Introduction More than thirty years ago, James R. Berquist and Richard E.Sherman published a paper entitled Loss Reserving Adequacy Testing: A Comprehensive, Systematic Approach (Berquist & Sherman, 1977). They thereby defined two methods to test and adjust estimate of unpaid losses to changes in case reserve adequacy or in settlement rates. After discussing with Jeffrey Mayer, a former Director in the Property & Casualty Actuarial Practice in PriceWaterhouseCoopers in New York City, reviewing Berquist and Sherman paper seemed a good subject for this thesis. This paper is a classic piece for a Property and Casualty (P&C) actuary in the U.S. Although it was written more than thirty years ago it is still part of the syllabus for Exam 6. Their paper focuses on Property and Casualty (P&C) reserving process in a changing environment. Actuaries can often encounter situation where changes have taken place or are said to have taken place and that might render the use of common projection methods inadequate to estimate the ultimate amount of loss. To reduce the distortions in historical patterns, the actuary can make some adjustments to the historical data and obtain a more adequate estimate of unpaid losses for some Property and Casualty claims. Berquist and Sherman really insisted on the importance of understanding the data, the underlying trends, and of knowing the processes in the underwriting and claim departments and be aware of any changes. They also designed a list of questions for different department executives in Appendix B of their paper. In July 2013, PricewaterhouseCoopers (PwC) conducted a survey (PricewaterhouseCoopers, 2013) of 61 companies in the United States. As of December 2012, they represented over a third of the P&C insurance market (measured using the amount of gross written premium). Regarding cross-function communication, almost 60% of the survey respondents acknowledged communicating at least monthly with the pricing and the claims department and 95% assessed contacting those functional areas at least quarterly. They would interact less often with the underwriting department. The quality of the information was not always deemed sufficient. 6

7 This paper intends to further explain and to examine the importance of cross-function communications in order to correctly assess the estimate of ultimate loss. 1. To do so the first chapter insists on the importance of reserving and on the regulatory context in the United States. 2. In a second chapter, different common actuarial techniques are developed and explained. Those present the traditional ways of using the historical data to come up with an actuarial central estimate. In case of a change in the environment of claims reporting and processing, those techniques are still needed and used, but applying them carelessly could be misleading. 3. The third chapter focuses on explaining what type of changes in the environment the actuary has to pay attention in order to properly detect changes in the environment. This chapter is also meant to encourage the actuary to talk to other department in the company so that he / she is well aware of what strategies are applied or what new process or new laws could affect the data that he / she is provided with. 4. Berquist and Sherman techniques are only described and explained in the fourth chapter. The examples are conducted on the data that they used in their own paper. Some formulas are given in more details and some criticism and improvements brought by other authors are stated and explained. 5. Chapter five is meant to give tools to conduct a direct analysis of the estimates produced and a method to evaluate the performance of the model with time as more data flow in. 7

8 Chapter 1. The importance of reserving This paper consists in detailing techniques to estimate the amount of unpaid losses used in changing environments. Before getting to the core of the subject, the first chapter is meant to explain what reserving is and what it is used for? What is the framework for reserving in the United States of America? 1.1 The necessity of reserving for insurance companies Main reasons Reserving is meant to enable an insurance company to pay its future obligations towards the insured. Two main parameters are unknown, what the amount of claims loss will be and when it will have to be paid. Unlike manufacturers, it may take several years for an insurer to know the true costs of the goods (insurance policies) sold. However the insurer needs to report its financial results regularly. Claims reserves amounts the insurer s estimate of unpaid claims that have already occurred. To whom is an appropriate estimation important? If the reserves are over-stated, the dividends received by the investors might be affected, if it is under-stated, investors might believe that the insurer is in a stronger state than it really is. Regulators also look at reserves to ensure a supervisory role, to protect the insured. An accurate estimation of unpaid claims is also essential for pricing purposes within the insurance company. A deficient estimate might lead the lowering of future rates and put the solvency of the insurer at risk. A redundant estimate could lead to an increase in the rates and loss of market share and revenue for the insurer. On top of influencing the pricing of future policies, it might also impact management decisions in underwriting strategies. A market can be exited on the wrong belief that it is underperforming. The reverse situation can also be envisioned, when an underperforming market is kept when it should not. 8

9 Reserving External Stakeholders Internal Stakeholders Shareholders Regulators Pricing Department Underwriting Department In the United States The role of the states In the United States, Congress adopted in 1945 the McCarran-Fergusson Act that declares that states regulate the business of insurance. This lead to the fact that the regulatory obligations can differ from on state to another as far as reserving is concerned. State laws require insurer to have a license to sell a specific type of policy. A company that is selling insurance products in the United States therefore need to have a license in all the states the products are sold. About half of the states require insurers to file rates and wait for them to be approved by the state before applying them. The regulator is trying to protect the public and ensure that the rates are affordable and thereby not too high, or on the other hand if they are low, the state needs to make sure that the insurance company will be able to pay for its future obligations toward the insured. The states role is to protect insured by making sure that the insurer is solvent and by making sure that the insured can find a way to protect itself. In the case of construction defects, construction companies are sometimes hindered by the fact that some jurisdictions interpret differently some liability policies. In some cases construction defect can be considered an occurrence and therefore be covered by liability insurance policies, in other cases it is considered a fault and thereby is not covered. Three outcome are possible in different courts: 9

10 - Construction defects is due to faulty workmanship it cannot constitute an occurrence / an accident and thereby cannot be covered; - Construction defect itself cannot be covered but the damage that results from the defect can since it was unplanned; - Construction defect is an occurrence and therefore is covered. In order to clarify the interpretation of the term occurrence in the case of construction defects some states (Colorado, Arkansas, South Carolina and Hawaï) have stepped up and passed laws. Every insurance company file annual reports with each state it does business in. About once every three years, the state in which the company is domiciled conducts an audit other states in which the insurer has clients may take part in the audit as well. The NAIC In order to coordinate the regulation of multistate insurers, the National Association of Insurance Commissioners (NAIC) was created in It promotes the development of uniform policies in the federation. They also provide standards and best practices. In 1990, most Property and Casualty insurers have been required to give a Statement of Actuarial Opinion signed by a qualified actuary. The actuaries are appointed by the Board of Directors of the insurance company and therefore bare the title of Appointed Actuary. This measure was implemented in some states in the 1980s already, following the insolvency of many property and casualty insurance companies in the 1970s. The NAIC s role is to assist state insurance regulators individually and collectively to ensure the protection of policyholder and maintain a stable and competitive insurance market. Protecting the insured population is granted by ensuring the solvency of insurers and making sure that insurance company can pay for reported claims. The NAIC Insurance Regulatory Information System (IRIS) monitors the solvency of insurers by calculating twelve different ratios based on the information communicated in the report that the company has to deliver every year. 10

11 Differences in US and European regulations The US regulatory environment is very different from the Solvency II system applied in Europe. M McCarty gave a few explanations supporting this aspect of insurance regulations, in a statement to the Senate (McCarty, 2015). According to him by comparing insurance companies to banks he claims that insurance companies better manage their asset and liability matching, ensuring that short term liabilities are covered by short term assets and long term liabilities by long term assets. Setting similar regulatory requirements to banks and insurance companies encourages new risk taking. New standards could increase costs and thereby harm US insurers as well as American insurance consumers. This seems to be a criticism of Solvency II which is very close to the Basel II regulations applied to banks in Europe. The GAAP accounting rules are different than the IFRS, they do not encourage market valuation. According to M McCarty, market valuation is focusing on the short term rather than on the long term. In addition to state-risk based capital standards, the Federal Reserve also has some rules applicable to insurance companies. The Insurance Capital Standards Clarification Act was meant to give the Federal Reserve some leeway to tailor specific rules to systematically important financial institutions. The international Association of Insurance Supervisors is developing capital proposal for insurance companies deemed too big to fail. An example of those developments is the basic group capital requirement. It is also developing Global insurance capital standards. 1.2 Guidance and rules followed by reserving actuaries Actuarial Standard of Practice for reserving in the United States Actuaries in the United States can follow the guidelines of the Actuarial Standards Board. They adopt Actuarial Standard of Practices (ASOP). In 2007, they defined ASOP 43, which addresses Property and Casualty Unpaid Claim Estimates. This text is meant to provide actuaries with guidance when estimating unpaid claims loss and loss adjustment expenses. 11

12 Since this paper focuses on how to deal with a changing environment, a quote from that ASOP is given. In this ASOP, section treats the problem of changing environment and conditions. The following is recommended: The actuary should consider whether there have been significant changes in conditions, particularly with regard to claims, losses, or exposures, that are likely to be insufficiently reflected in the experience data or in the assumptions used to estimate the unpaid claims. Examples include reinsurance program changes and changes in the practices used by the entity s claims personnel to the extent such changes are likely to have a material effect on the results of the actuary s unpaid claim estimate analysis. Changing conditions can arise from circumstances particular to the entity or from external factors affecting others within an industry. When determining whether there have been known, significant changes in conditions, the actuary should consider obtaining supporting information from the principal or the principal s duly authorized representative and may rely upon their representations unless, in the actuary s professional judgment, they appear to be unreasonable. Cross-function communication is recommended in the last sentence Tasks of an audit company Since 1974, the American Academy of Actuaries published some recommendations to guide the relationships between the actuaries of the insurance company and the auditors. Financial Reporting Recommendation 2 has been updated regularly and its title has been changed. Auditors are professional that examine a company financial statement in order to issue an opinion. In the case of an insurance company, auditors might want to check their reserves. In the case of a self-insured, the auditor will deliver an opinion on the reserves and on the discounting of those reserves. 12

13 The Actuarial Standard of practice number 21 addresses what role an actuary is expected to play when responding to or assisting auditors in their examination of their company s financial statement. In that ASOP, it is specified that one of task of the responding actuary is to communicate the changes in the operating environment. It means that if there is a change in the case outstanding adequacy or / and in the settlement rate, the responding actuary should share those aspects with the auditors. The actuary reviewing the process and numbers communicated by the responding actuary should provide the following documents: Proof that the reviewing actuary procedures have been planned and coordinated with the auditors; - All the data and documents that have been reviewed should be listed and described; - All the procedures used should be described; - The results should be communicated. 13

14 Chapter 2. Common reserving methods In the previous chapter, an explanation of the purpose of reserving has been given as well as a few guidelines. Chapter two presents some traditional methods used to project ultimate claims. Ultimate claims include cumulative paid claims, case outstanding and IBNR (incurred but not reported). Unpaid claim estimates are obtained by subtracting the cumulative paid claims loss at the time of the analysis from the ultimate claims loss. To give an example of the few techniques that can be used to estimate the ultimate amount of unpaid losses, one can imagine that the actuary wants to predict the ultimate number of baskets a team will score in a basketball game. After the break (the time remaining is supposed to represent 50% of the total time for the game) a few methods are available: - he / she can use past experience to determine the pattern that will be applied to the current score to obtain the ultimate score (development technique), - he /she can have an estimate in mind based on past games and not at all on what has just happened in the first part of the game (expected claims technique), - he /she could combine the first two techniques by taking into account the first part of the game and assuming that his / her prior prediction made using past data will be observable in the second part of the game. Therefore to the current score, he / she will add 50% of the initial estimation of the ultimate score (Bornhuetter-Ferguson method). Those three methods are used in actuarial science to estimate the ultimate amount of claims. The first and the last method are properly described below. 2.1 Development triangles In Property and Casualty liability insurance, when there is a sufficient loss history, the data is usually organized in development triangles. Those tables show changes in the value of a cohort over time. 14

15 The triangles can display cumulative or incremental data. It can be gross or net of reinsurance. And usually the loss data includes allocated loss adjustment expenses. Those constitute that is not directly paid to the insured but that is used to settle a specific claims. It can represent fees to pay an outside attorney or expert to investigate a specific claim. Different types of values can be analyzed using triangles: - The amount of paid losses - The amount of reported / Incurred losses - The number of open claims - The number of closed claims - The number of reported claims Each row corresponds to a cohort usually accident years, but they can also be reporting year, policy periods or in reinsurance underwriting years. Each column corresponds to a development period. In this paper triangles have development periods. The data is thereby displayed in the form of an upper-triangle. This way of displaying data is very convenient to determine an estimation of the ultimate payments for claims. The reserves are established by subtracting the amount of losses already paid to that of the estimate of ultimate payments 1. In the following paragraphs, two different methods are described to square the triangles. They are commonly used by actuaries to estimate the obligation of future payment resulting from claims due to past events. The figure below shows what a development triangle looks like. 1 This is a simplified explication of how the reserves are settled. Actually other elements are taken into account. To settle the reserves the amount of Unallocated Loss Adjustment Expense (ULAE) is taken into account. If the triangles were gross of reinsurance the amount of losses that will be paid by reinsurers need to be subtracted. Other adjustment can be made: in the case of a change in TPA (Third Party Administrator) for example. 15

16 i/j An Exemple of a development triangle m - 2 m - 1 m 1 C 1,1 C 1,2 C 1,3 C 1,m - 2 C 1,m - 1 C 1,m 2 C 2,1 C 2,2 C 2,3 C 2,m - 2 C 2,m C 3,1 C 3,2 C 3,3 C 3,m n - 2 C n - 2,1 C n - 2,2 C n - 2,3 n - 1 C n - 1,1 C n - 1,2 n C n,1 In this paper, the following notation is used: - refers to the accident year; ; - refers to the development period; ; - refers to the cumulative losses for accident year and development period. Those values are known for. Typically the values are available up to the current point in time. Squaring the triangle consists in finding an estimation for, for. Usually, it means that the number of columns is the same as the number of rows. This is explained by the fact that each additional calendar year adds a diagonal to the triangle. If the business is discontinued or in run-off, the triangle can include more columns compared to the number of rows. Indeed even if a policy is not sold anymore, past claims can still develop adding columns but no rows. 2.2 Remarks on insurance data A development triangle is mainly a tool used to analyze insurance data. The claim data is organized in homogeneous groups. The claims in each group are supposed to develop following the same pattern. One first sub-set of loss data is constituted by line of business. Since claims from one line of business do not develop in the same manner as some from another line of business. Indeed paid claims and case outstanding represent a high proportion of ultimate claims at early 16

17 development periods for short-tail lines of insurance. Those lines can be for instance property insurance 2, commercial automobile liability 3, It takes a short amount of time for claims from those lines of business to be reported and settled. On the other hand, medical malpractice liability 4 or workers compensation liability 5 are long-tail lines of business. It takes many years for some of those claims to settle. The data can be analyzed by looking at specific actors of a certain line of business or specific geographic location. As it was explained in the previous chapter, states can have different types of regulations. For example, usually when analyzing workers compensation the actuary will not look at a countrywide triangle of data, some states will be analyzed separately (like New York or California). A study conducted biannually by the Department of Consumer and Business Services in Oregon compares the premium rates applied to workers compensation coverage by states. It was initiated in 1986, when Oregon s rates were among the highest in the country. In 2014, the study revealed that those rates were among the lowest. This change in rankings is primarily explained by state s reforms enacted in 1987, 1990 and 1995, and to the implementations of workers safety initiative. The fact that the rates decreased reflects the fact that aggregate losses decreased. It shows that states initiatives can influence the loss experience. Even within the states, claims can develop differently. For instance in the state of New York, court decisions in Brooklyn can be more plaintiff-friendly than decisions taken in courts located in Manhattan. The data can also be analyzed by claim size. Indeed small claims might develop differently than bigger claim. It might take longer for claim professionals to deal with larger claims than with smaller ones. A large claim implies a larger possible amount of loss, those types of claims are more probable to end up in court than smaller claims and thereby take longer to settle. 2 Property insurance provides protection against fire, theft or some weather damages like earthquake. There are different types of specialized property insurance. 3 Commercial automobile liability is a coverage that protects against the liability that can arise from business owned (hired or borrowed) vehicles. 4 Medical Malpractice coverage responds to the general liability exposure for individuals or organizations that offer medical care. 5 Workers compensation provides coverage for the employer against the illness or death of an employee as well as against employee that got injured on their workplace. It pays injured workers for lost wages and medical care for job related injuries. 17

18 2.3 A survey of claim reserving methods Deterministic versus stochastic methods Actuarial techniques can be classified in two large groups. Some methods are deterministic and others are stochastic. Whereas deterministic models only make assumptions about the expected value of future payments, stochastic models also model the variations of those payments. What may be the advantages of stochastic methods? One benefit is that the assumptions are more explicit than for deterministic models. For example when using a deterministic model it is difficult to know whether the difference between the actual payments and the expected payments is significant or not whereas with stochastic methods a range of possible outcomes can be defined. Stochastic reserving models enable to have a full predictive distribution of possible ultimate outcomes. In the last thirty years interests in stochastic reserving methods have been expressed especially as a tool in the estimation of the downside potential of claim reserves. By estimating the distribution of the reserves, the stochastic models provide the actuaries with more information than deterministic models. Deterministic models seem to still be preferred by practitioners. They indeed enable them to use judgment which is not always easy to incorporate in stochastic models. Deterministic models are also easier to perform since they involve less computing power. No random effects are added. Simple methods do not capture all the aspects of reality but it is easier to understand and perform them as well as to explain them to non-actuaries Deterministic methods In this paper a few methods are detailed but many techniques have been developed and used for decades. The most common deterministic methods are detailed in Estimating Unpaid Using Basic Techniques (Friedland, 2010). The oldest techniques described are development techniques. Using aggregate data grouped in the form of a triangle, future payments are obtained by estimating a factor for each development year. One of the main problems with the Chain- 18

19 Ladder is that the first two or three factors rely on little information and that the known cumulative claim amounts for the last calendar years might not be that reliable for the projection to the ultimate. In Measuring the Variability of Chain Ladder Reserve Estimates (Mack, 1994), Thomas Mack gave an easy example to illustrate the problem. If the cumulative claim amount for the most recent accident year happens to be zero but premium were collected for that year, the Chain Ladder method will not produce a relevant estimate of the ultimate payments for that year. Methods including expectation can be used to tackle this problem. In those cases it is considered that an A Priori can better estimate the ultimate unpaid claims than the information provided by the experience. To obtain an A Priori (an initial expectation of ultimate unpaid claims) the most commonly applied method is that of the claim ratio method. The selected claim ratio is multiplied by the actual earned premium to obtain the A Priori. The Bornhuetter Ferguson method is one of the most commonly used methods by practitioners. It appears as if it was first described in 1972 while the first mention of the Chain Ladder method (development technique) was made in 1934 as it was described by Tarbell. The Bornhuetter-Ferguson method estimates the outstanding losses by multiplying the A Priori by the percentage of outstanding loss with respect to the ultimate one. The Cape Cod method is a similar method but instead of using an A priori it estimates the loss ratio using the exposure and the experience. Other types of commonly used reserving methods are frequency-severity techniques. Those enable to see trends in the rate of claims emergence in the settlement of claims and in the average values of claims. Frequency refers to the number of claims per units of exposure. Severity refers to the average cost of claims. Development techniques can be applied to claim counts and average values in order to obtain the ultimate claim payments Stochastic methods Bayesian methods combine a priori information with the observations. An a priori distribution of a random quantity is used (it could be the distribution of the ultimate claim). The likelihood function is then used with the Bayes theorem to obtain the a posteriori distribution of the ultimate claim. A priori distributions such as the exponential dispersion family give analytic 19

20 expression of the a posteriori. If the predictive distribution is not of a known form it is possible to derive approximations by using Monte Carlo simulations. Although Bayesian methods are more commonly used in pricing, some authors have studied some models. Verrall (Verrall, 2004) used a Bayesian approach to prove the relationship between the Bornhuetter-Ferguson method and a Generalised Linear Model. Scollnik (Scollnik, 2001) described how to implement some Bayesian method using Markov Chain Monte Carlo technique. In cases where the Bayesian estimator cannot be derived, the class of possible estimators is restricted to the linear functions of the observations. The estimator minimizes the quadratic loss function among all the estimators that are linear combinations of the observations. This is known as the Bühlmann credibility theory set out in the late 1960s (Bühlmann, 1967). Kalman filters can be considered as an enhancement of credibility models since they add a dynamic aspect to the models, some of the parameters can evolve over time. Zehnwirth described some applications of Kalman filters to loss reserving (Zehnwirth, 1997). The Poisson model, the overdispersed Poisson model and the Gamma model belong to the exponential dispersion family and the parameters can be estimated with the Bayesian methods or with the maximum likelihood estimator method. They have a generalized linear model (GLM) basis. In reserving, Renshaw and Verrall (Renshaw & Verrall, 1998) were among the first to implement those techniques. Using an over-dispersed Poisson model, Renshaw and Verrall (Renshaw & Verrall, 1998) showed that a stochastic model could lead to the same estimates as the Chain Ladder technique. The bootstrap technique is also used as a stochastic method in loss reserving. Bootstrapping is a re-sampling procedure which uses observed data to create subsets. The samples are created randomly with replacement. The technique is easy to implement. Most of the time bootstrapping is applied to the residual of the chosen method since they are supposed to be independent and identically distributed (iid). With regression-type model, residuals are indeed supposed to be iid. For example when bootstrapping generalized linear models it is common to first define and fit the statistical model, then to find the residuals, resample them randomly with replacement. A pseudo triangle is thus obtained by inverting the definition of the residuals. This can be repeated many times and enables to obtain a distribution of parameter estimates. The aim 20

21 is to determine either the standard error of prediction using the Central Limit Theorem or to determine the prediction error (Pinheiro & al, 2003) Solvency The uncertainty measurement of the reserves called risk margin in the Solvency II framework does not have the same time horizon as the commonly calculated reserve risk. Reserve risk is the provision for outstanding claims; it goes to ultimate, whereas in the Solvency II framework the risk margin focuses on the next twelve months only. In the case of a Chain-Ladder estimation of the one year risk, a close formula was found (Wütrich, Merz, & Lysenko, 2007) similar to the model developed by Mack in This method considers the one year result as the difference between the estimations of the Ultimate at time I and at time I+1. Ohlsson and Lauzeningks described a simulation approach to the one year reserve risk (Ohlsson & Lauzeningks, 2009). Diers introduced the term re-reserving to name this simulation approach to the evaluation of the one-year reserve risk (Diers, 2009). With Solvency II, insurance companies are required to have a market consistent evaluation of their liabilities. Therefore part of the literature concentrates on the discounting of the reserves. Wüthrich and Bühlmann studied the volatility of discounted reserves taking into account the discounting risk (Wüthrich & Bühlmann, 2009). The next points will focus on some deterministic methods. 2.4 The Development method The development method assumes that ultimate claims for each accident year (rows of the triangles) can be obtained from past values, by determining future claims development from past claims development. In other words, claims reported to date are supposed to be developing in the same pattern as past claims. The claim processing is also considered to be consistent, the mix of claims as well and the environment policies and severity is also supposed to be constant. In Property and Casualty insurance, one of the main methods used to determine the reserves best estimate is the development method. It consists of a few steps: 21

22 1. Determining the age to age factors from a cumulative loss triangle (either paid amounts or reported amounts); 2. Computing averages: simple averages, weighted averages, averages excluding high and low factor. Those are either based on age to age factors or directly on the cumulative triangle; 3. Selecting loss development factors. The development factor for development age, is noted ; 4. Calculating cumulative loss development factors. The factor to ultimate for development age, is noted ; 5. Multiplying the latest diagonal by those cumulative factors to obtain the ultimate loss estimate. The ultimate loss estimate for accident year, is noted. The formula below indicates the value of. 2.5 Frequency / Severity techniques Those techniques are used when triangles of claim counts are available. The technique consists in projecting the ultimate claims by multiplying the estimate of the ultimate number of claims by the average amount of claim loss. This technique gives way to an analysis of trends and patterns in the rate of claims reporting and payments as well as in the severity of claims (average value of claims). The frequency is the number of claims per unit of exposure. The severity is the average cost per claim. As for loss development, claim count development for unmature accident years can be highly leveraged by development factors. That is why the Bornhuetter-Ferguson technique can be used. This method is described afterwards. 22

23 In order to use this method, the process used to put together the triangle of claim counts need to be consistent over time. Indeed, trivial claims can be included or not, same with claim without or with payments,... If those types of claims used not to be included but then are because of a change in process, it can be misleading. A shift in the relative presence of those can also be misleading. 2.6 The Bornhuetter-Fergusson (BF) method That method is used as often as the development method. It adjusts the development approach by taking into account an expectation in the ultimate loss. This method is particularly relevant for early development periods since it offers more stability compared to the development method when the data is sparse. The Chain Ladder method can be affected by the presence of claims of unusual size. Development factors to ultimate for early development periods are highly leveraged. When is the Bornhuetter-Fergusson method used? The Bornhuetter-Fergusson method gives more weight to the experience for more developed accident years and more weight to the expectation for early development ages. The expectation of ultimate claims loss can be obtained from pricing data, underwriting data or by assuming a loss ratio applied to the earned premiums. This claim ratio method consists in coming up with an A Priori Ultimate Claims for an experience period. This estimation can be set equal to a selected expected claim ratio multiplied by an exposure. The later is not always the earned premiums. Self-insured organizations do not collect premiums; they therefore need another exposure to run the Bornhuetter-Ferguson method. Usually, for Workers Compensations, selfinsured companies in the United States use payroll. For Automobile Liability, the number of vehicles can be used. Description of the method The Bornhuetter-Ferguson method can be expressed as follow: 23

24 - is the accident year. To determine the expected future losses, the following formula is often used: - is the accident year, - is the cumulative factors used in the development method to obtain the ultimate loss. Since the and are often obtained from the paid loss triangle and the reported loss triangle. The Bornhuetter-Ferguson method can also be applied to those two different set of data. Using this method makes the estimation of unpaid loss more independent from case reserve philosophy. Sometimes it can be like playing case adjuster without beneficiating from the detailed information gathered from actually handling claims. That is also why this method is mostly used for most recent accident years. The Chain Ladder and the Bornhuetter-Ferguson methods are the two most used methods by actuaries. Usually, actuaries do not use one or the other, they combine different methods. 2.7 Actuarial judgment In the selection of the parameters Actuarial judgment can be exercised with all the previous methods. For the Chain Ladder method actuarial judgment is used to select development factors. Different averages based on historical data from the triangle can be used to justify or help the actuary make a selection. The actuary can chose to smoothen the pattern indicated by the history. To help with his selection, the actuary can use industry benchmark. SNL financial is a good tool to create those benchmarks. For long-tailed lines of business it is also important to select a tail factor. In those cases it is assumed that the development of claims will not stop after n development periods (n being the number of columns and rows available in the triangles). 24

25 For the Bornhuetter-Ferguson method, the actuary used his judgment to come up with an expected ultimate amount for an accident year. In that case, industry pattern can also be used to obtain a benchmark of loss ratios. Other methods can help calculate implied loss ratios for older accident years. By assuming that past accident year loss ratios are a good representation of more recent accident years loss ratios, those can help selecting an A Priori ultimate loss for unmature accident years. In the selection of the methods For each accident year the actuary has to determine the total ultimate loss amount. To do so different methods can be used: - For each accident year, the paid or the incurred loss development method can be selected, - So can the paid or the incurred Bornhuetter-Ferguson method, - In some cases (mostly for unmature accident years), the actuary can select the expected ultimate loss, - An average of all or some of those methods can be selected. 2.8 Mack s assumptions Thomas Mack introduced a distribution free method to measure the variability of the Chain Ladder 6 reserve estimates (Mack, 1994). By doing so, he also defined conditions in which it is appropriate to apply the Chain Ladder method. Hence, this point in the development of this paper describes mathematical methods that can be used to select one actuarial method from another The selection of factors In 1994, Thomas Mack defined the Chain Ladder selected factors as shown in the equation below. is defined for. 6 The Chain Ladder method is another term for the Loss Development Method. 25

26 This formula corresponds to the volume weighted average of all the cumulative values shown in the triangle. It can be applied to a triangle of loss payments, of reported losses or of claim counts. The same weight is given to all the accident years. The following formulas enable to estimate the unknown amount for and that of the ultimate loss for accident year number, with Three assumptions Thomas Mack presented three assumptions needed to achieve optimality when using the Chain Ladder method in the way specified above (Mack, 1994). Those three assumptions are given and explained below: 1., with and. The expected value of the cumulative loss of the following development period is proportional to the cumulative loss already emerged. Since the development factor for each development period does not depend on (the accident year), it implies that the equation can be identified to a liner relationship for each development periods. There is no constant in this method. 2. are independent for. This assumption assumes that there is no diagonal that would be stronger, or weaker than the others. 3., with and. The variance of the cumulative value for the next development period is proportional to the 7 and take values from 1 to n. They thereby designate row number 1, 2, n and columns number 1, 2, n. If, then the value of the amount is not known yet. 26

27 previous cumulative amount to date and a constant defined for each development period. Under those three assumptions, Thomas Mack proved that with the Chain Ladder method a minimum variance unbiased linar estimator of future emergence can be obtained. The formulas below prove that under the first hypothesis stated above, the estimator of the factor of development is unbiased. The formula below shows the estimator of the variance developed by Thomas Mack: 2.9 Testing for those assumptions In Testing the Assumptions of Age-To-Age factors, Gary G. Venter (Venter & Chu, 1998) analyzed different method to test the above assumptions. Depending on the results to certain tests, he also proposed alternative actuarial method that would be deemed more suitable to produce an optimal reserve estimates Hypothesis 1: linear relationship For the first assumption, Gary Ventor suggested to test for different models and to choose the one that best fits the data. It might not necessary lead to the use of the Chain Ladder method. 27

28 As explained by Gary Ventor, hypothesis 1 cannot directly be checked, but its implications can. Therefore other alternatives can be tested and if one of them is deemed more suitable, hypothesis 1 will not hold, and a method different from the Chain Ladder might be chosen. Hypothesis 1 can be tested by doing a linear regression and estimating factors. The goodness of fit 8 of the regression can be used to compare that model to other alternatives: i. if the constant is not null, then the Chain Ladder model is not the most adequate model. In that case another method would be preferred; ii., being a parameter that differs from an accident year to another; iii., would represent calendar year effects. If equation i. with no constant is deemed to be the best at modeling the relationship between two consecutive columns of a triangle, then the Chain-Ladder should be further examined as a good estimation model. The model that best fit the data could be accepted as a good alternative. One thing to keep in mind when comparing two models is that more parameters can give an advantage in fitting but not in prediction. In other word, the regression analysis does not necessary test for the predictive abilities of a model. If the constant in the regression model is statistically significant, then the Chain-Ladder model cannot be applied as such, a constant is necessary. The Chain-Ladder has one parameter for each development period, which is less than the two other methods listed above. Another aspect that should be examined is the relevance of the regressions. The estimate of the parameters of the equation obtained with the regression should be statistically significant. It means that the development factor should be significantly different from zero if the regression is obtained from the incremental triangle of losses and different from one if it the cumulative 8 Since the number of parameters estimated in the different regressions is not the same, other statistics can be looked at to motivate a choice. 28

29 triangle is used. The Chain-Ladder is not optimal for development period for which this is not verified. If the regression model indicates that a factor times the losses emerged so far is a better model, then a Bornhuetter-Ferguson method could be more adequate than a Chain-Ladder model for the accident year to which the development factor obtained with the development method would have been applied. Usually this is true for the more recent accident years. If a linear regression seems to fit, the actuary should also check that the residual plot shows a random pattern. The residuals are obtained by subtracting the predicted value from the observed value. Both the sum and the mean of the residuals should be close to zero Hypothesis 2: Independence of the columns Hypothesis 2 states that the columns of incremental losses within the triangle should be independent. Hypothesis 2 can be tested by looking at the diagonal of the triangle. If one looks abnormally stronger or weaker, it implies that the triangle might need some adjustments. It might mean that there was a change in case reserves adequacy or settlement rates. Even after adjusting the triangles for changes in settlement rates and / or case reserves adequacy, a correlation test for adjacent columns can be run. Mack developed a correlation test for consecutive columns. He also introduced a high-low diagonal test. Those two tests enable to test for Hypothesis 2. The correlation test The correlation test that he developed is detailed in Appendix G of Measuring the Variability of Chain Ladder Reserve Estimates (Mack, 1994). The following steps describe how to get to the correlation test: a) Calculate the triangle of age to age factors. b) For each column rank the factors from smallest to highest. c) Do the same but this time by dropping the factor for the last available accident year. d) For each development period, each element of the column obtained after step b) (but the last one) is subtracted with the element from the column of the same length in step 29

30 c) The square of the difference is calculated and all those are added to calculated Spearman s rank correlation coefficient. The Spearman s ratio is defined by Under the null hypothesis (no correlation among the columns): e) A test statistics is defined to test for the correlation. It is put together by calculating the weighted average of the, with the following weights: which correspond to the number of elements that were added to obtain. Under the null hypothesis, the are uncorrelated: f) A significance test is conducted the statistic T can be assumed to follow a Normal distribution. Thomas Mack (Mack, 1994) recommended to run the test with a 50% confidence interval. If T is within the interval, the null hypothesis of having uncorrelated development factors is not rejected. 30

31 The high and low diagonal test This test was designed to test for calendar year effects. Mack justified it by the fact that a calendar year effect will affect one of the diagonal and thereby the corresponding development factors. Those factors can therefore be larger or smaller compared to those obtained at the same development period If the elements on the diagonal are larger than the other diagonal, then the corresponding factors will be larger, if the diagonal is smaller, the factors will be smaller as well. Mack s technique was described in Appendix H of his paper (Mack, 1994). It consists in looking at all the development factors and to classify them into two categories: low factors and high factors : a) As in the previous technique, the factors of each column are ranked. b) The factors higher than the median are the high factors, those that are lower the median are the low factors. For each column, if there are an odd number of factors, the factor equal to the median is neither classified in the high or low factors. c) For each diagonal, the number of low and high factors should be the same if there are no changes in calendar year period. If the number of low factors is higher than the number of low factors, then a change in the specific calendar year can be suspected. A formal test is described below. Intuitively, if there is no calendar year effect, each factor should have a 50% chance of belonging to the low factors category and 50% of belonging to the high factors category. The number of low factors (or high factors) per diagonal should follow a Binomial distribution, is the number of factors on the diagonal minus the number of factors that were neither classified in the low or high factors classes, is the probability of observing a low factor (or a high factor) for a given factor of the diagonal. 31

32 is the number of low factors on the diagonal. is the number of high factors on diagonal. and should follow a Binomial distribution with parameter.. The expected value of and is defined by: The variance should be: Thomas Mack proposed to test for the Binomial distribution by looking at the minimum. The following characteristics are given: Demonstration of the expected value designates the floor of, it is the largest integer equal or less than. If is an even number. If, it means that its value cannot be higher than. If n is an even integer:. Since in that case. If 32

33 and since then. This equality makes it easier to calculate. Since then: The following equalities enable to continue the demonstration: Since then: Demonstration of the variance 33

34 Since and The statistical test The null-hypothesis of no calendar effect is tested by looking at the statistic M. Since the are supposed to be independent under the null- hypothesis, the variance of M is easy to calculate. M is supposed to have a Normal distribution. Thomas Mack suggested to test for the nullhypothesis of un-correlation between calendar years with a probability error of 5%. The nullhypothesis is rejected if: 34

35 2.9.3 Hypothesis 3: To test for the third hypothesis, a plot of the residuals can be looked at. The plot should indicate a random pattern Key assumptions arising from data To correctly apply the different development techniques, the triangle of loss data should have been constituted in a stable environment. The list below corresponds to some of the assumptions that constitute a stable environment. - Claims settlement patterns are assumed to be unchanged: it might be problematic if there was a speed-up or a slow-down in payments pattern; - Case reserving practices are also assumed to be constant in times: the introduction of new reserving practices, or the implementation of a new way of reserving might change the case reserves adequacy; - The claim processing is assumed to be unchanged; - So are the policy limits - Loss costs trend are assumed to be constant: no variation in inflation, no increase in litigation or new court rulings; - The mix of business is supposed to be constant; - No underwriting cycles 35

36 Chapter 3. Numerical example of unpaid losses ultimate selection This chapter intends to use the methods developed in the previous chapter. Both the Chain Ladder method and the Bornhuetter-Ferguson method are presented. 3.1 The Data used This example uses Schedule P information pulled from SNL financial. Some data was collected from a company that sales Medical Malpractice liability coverage. In 2014 that company net written premiums was constituted by 95% by Claims-made coverage and by 5% by Occurrence coverage. Most of the direct premiums in 2014 were written in California. The Medical Malpractice claims made liability segment was considered. Two types of coverage are usually available to medical professionals. They can either get a Claims-made or an Occurrence coverage. The Claims-made one is the most commonly offered coverage. There are two main differences between those two coverages: - The timing of claim filling is different. In the case of Occurrence coverage, the insured is covered from any accident that occurred during the policy period regardless of whether the claim was reported during the policy period or not. In the case of the Claims-made coverage, the claim has to be reported during the policy period. The policy period in that case comes to an end once the insured stops paying for the premiums. - The limits are also different under the two different coverages. In the case of the Occurrence coverage, the limit is restored every year while in the case of the Claimsmade coverage, the limit is kept as such as long as the policy is in force. Schedule P is a large section of the Annual Statement. The document can provide loss triangles for different line of business and include 10 accident years. The following data was used: 36

37 - Cumulative Net of Reinsurance Paid Loss Triangle for accident years 2005 to 2014, from scheduled P Part 3. - Cumulative Net of Reinsurance Incurred Loss Triangle for accident years 2005 to 2014, calculated from Schedule P Part 2 minus Schedule P Part 4. - Earned Premiums on a net basis for accident years 2005 to 2014, from Part 1 of Schedule P. A benchmark was put together using the data from six comparable companies offering Medical Malpractice Claims-made coverage in the United States. 3.2 Application of the different actuarial methods The methods developed in the previous chapter, are applied to the data. In Exhibit 1, Sheet 1, the first triangle shown is the triangle of cumulative paid losses. The second is the triangle with the age to age factors. The averages below those two tables are intended to help the actuary select the age to age development factors shown on line Selection. The line Cumulative indicates the factors to ultimate associated to the selection. The cumulative factor for development k to ultimate is obtained by multiplying age-to-age factors k to n-1 and the tail factor. The last line indicates the estimation of the ultimate for each accident year using the development method. The factor for age 108 to ultimate was selected so that the estimation of the Ultimate for accident year 2005 matches the one obtained with the incurred loss development method. On the paid analysis, the benchmark seems to fit, correspond to the age-to-age factors and the averages showed. This seem to justify the usage of that benchmark to select the development factors 84-96, and the tail factor (108 to ultimate). The incurred loss development method is presented on the next exhibit. The same template was used to present the results of that development method. In that triangle, downward development is noticeable after a few development periods. 37

38 The next exhibit shows the estimation of unpaid losses using the Bornhuetter-Ferguson methods applied to the cumulative paid and the cumulative incurred losses. The exposure used is the earned premiums shown in column (A). The first columns enable to select an A Priori that will be used in both the paid and the incurred Bornhuetter-Ferguson methods. Columns (B) and (C) show the loss ratio implied by the paid and the incurred development methods. In Column (D), the loss ratios obtained from with the Benchmark. Column (E) corresponds to the selected loss ratio, for accident year 2005 to 2011 the selection is set equal to the average of the loss ratios shown in columns (B) and (C). For the latest accident years, the loss ratios were set equal to the average of column (B), (C), (D). The A Priori (the expected ultimate loss) is shown in column (F). The next step consists into deciding what percentage of the ultimate loss has already been paid column (H) and has been reported, column (K). Those columns are set equal to, with the cumulative factor associated to the ith accident year. Eventually, for each accident year, the ultimate losses are calculated using the Bornhuetter Ferguson method. Column (I) indicates the ultimate obtained from the paid data. Column (G) corresponds to the cumulative paid loss obtained from the last diagonal of the paid triangle. In a similar fashion, column (J) corresponds to the cumulative incurred loss obtained from the last diagonal of the incurred triangle. 3.3 Selection of a method By running the Chain Ladder method and the Bornhuetter-Ferguson method on both the incurred and the paid development triangle, four different estimates of ultimate losses are obtain for each accident year. The chart below represents the value of the estimates given by each method. For accident year 2005 to 2011 (except for 2010) the estimates are relatively close to one another. The following ratio is used to compare the differences in estimate of ultimate between accident years: 38

39 The notations designate different actuarial methods for accident year i. is for the paid loss development method. designates the incurred loss development method. is used for the paid Bornhuetter-Ferguson method and for the incurred Bornhuetter- Ferguson method. Most of the ratios obtained are below 7% for accident years 2005 to 2011, except for accident year 2010 which is equal to 11%. For latter accident years, the ratios are above 12%. To determine what method to rely on, Mack s assumptions as described in the previous chapter are first tested Paid Development Method Paid BF Method Incurred Development Method Incurred BF Method Test of assumption 1 The expected value of the cumulative loss of the following development period should be proportional to the cumulative loss already emerged. Since the development factor for each development period should not depend on a specific accident year the accident year, it implies 39

40 that there is a linear relationship between the cumulative loss of a certain development age and the following development age. This assumption can be tested for all the columns of the triangle. The estimate of ultimate losses for accident year i is obtained by multiplying the current cumulative loss amount for that accident year by the factor to ultimate associated to that accident year. The latter is obtained by multiplying all the age to age factors selected for all the following development factors (including the tail factor). Therefore to check the validity of the Chain Ladder method for accident year 2014, the linear regression assumptions has to be tested for all the columns. For accident year 2013 only seven regressions need to be tested. For accident year 2007 only one regression needs to be checked. The first assumption cannot be tested for accident years 2006 and The nine possible regressions for both the paid and the incurred triangle are displayed in Exhibit 1, Sheets 5 & 7. As suggested in Venter s paper (Venter & Chu, 1998) the regression is done for. is the incremental loss for i,j. In all the cases, and for both the incurred and paid losses, the goodness of fit does not seem to indicate that the regressions are a good way of estimating the incremental losses using the previous development period cumulative losses. Test of assumption 2 Accident years should be independent. To test for that the correlation test and the calendar year effect test. Both of those tests were described in the previous chapter. The results of those tests are presented in the Appendix. The correlation test using the Spearman s ratio is conducted in Exhibit 1, Sheets 9 & 10. As suggested in the paper written by Thomas Mack (Mack, 1994), the null hypothesis of having uncorrelated development factors is tested by checking whether the statistic T is inside a 50% interval. The test is run using the cumulative paid triangle and the cumulative incurred triangle. In the case of the paid triangle, the test validates the null hypothesis. In the case of the incurred loss, the hypothesis of having uncorrelated development factors is rejected. 40

41 The high and low test is used to check for calendar year effects. The null-hypothesis of un-correlation between calendar years is not rejected at a 5% level for both the paid and the incurred triangle. Those tests are displayed in Exhibit 1, Sheets 11 & 12. Test of assumption 3 The variance of the cumulative value for the next development period should be proportional to the previous cumulative amount to date and a constant defined for each development period. One way to test for the third hypothesis is to look at a plot of the residuals and see whether they look random. In the case of both the paid and the incurred triangles, the residuals seem to show a random pattern. Testing for the Bornhuetter-Ferguson As suggested by Gary Venter (Venter & Chu, 1998), to test for the Bornhuetter-Ferguson method, an iterative technique can be used to estimate which is the percentage of loss that are expected to be added to the triangle on an incremental basis and which is the estimated ultimate losses.. is the incremental losses for accident year and development period. To determine and a few iterations were necessary for the value to converge. The table below shows the values obtained by iteration from the paid triangle. h(i) i= First iteration Last iteration f(j) j= First iteration 0, , , , , , , , , ,00220 Last iteration 0, , , , , , , , , ,00219 To obtain the following formula was used: 41

42 Once all the are calculated, is recomputed: This is done until the values converge. In total the sum of square error is lower for the Bornhuetter-Ferguson method than for the Chain-Ladder method. The problem with Venter s technique is that the comparison is only done on a global basis not on an accident year basis. In practice, those tests are rarely done. Usually the Bornhuetter-Ferguson is adopted for the latest accident years when the factors selected for the Chain-Ladder are above a certain threshold. For some actuaries, the threshold is 2 (for others it can be 1.3). For development periods with factors superior to that number, the Bornhuetter-Ferguson is preferred. For the other accident years, the Chain-Ladder method is preferred. 3.4 Introducing a change in the environment The triangle used to obtain the previous analysis is now modified to reflect a possible change in the environment. It is assumed that the claim process has changed between accident year 2013 and 2014, the last diagonal of the triangles is thereby modified. The same methods has described above are still used. In theory the change in the process should not affect the ultimate loss. However the estimates obtained are very different than those expected as described below. The table below shows the estimates of different ultimate for all the accident years. The estimates are different for accident years 2011 to

43 Incurred Incurred Incurred Dev elopment Incurred BF Accident Dev elopment Method BF Method Y ear Method Modified Method Modified Total

44 Chapter 4. Understanding the data In previous chapters, a few actuarial techniques have been defined and a numerical example as been given. This chapter will explain what impact an unstable environment may have on development methods and on the estimation of ultimate loss. A few questions can be address regarding the impact of an unstable environment: how can the claim process change and why? What may impact the data from outside the insurance company? A few diagnostics are then defined to test for changes in the environment. 4.1 The claim process One of the key assumptions of a stable environment is linked to the stability of the claim process. Two main changes in that process can impact loss triangle. One if the case reserve adequacy, the other one is the settlement rate. This first part explains what the claim process is, how it can be changed and why. One part of the unpaid claim estimate is constituted by the case outstanding. This is an estimate provided by the claim department to the reserving professionals of an insurance company. Actuaries rely on the case outstanding as estimated by the claims professional to determine the reserves. Therefore it is important to understand what the case outstanding is and how it affects the unpaid claims estimates. Claims examiners can be part of an internal department especially in large insurance companies, or they can work for a third party claims administrators (TPAs), as it is often the case for self-insured. Sedgwick Claims Management Services Inc. is an example of TPA, it is the U.S. largest workers compensation third party administrator. Its clients are employers like AT&T, General Electric, Delta Airlines, Those companies offer workers compensation to their employees but they let Sedgwick handle the claim management process. 44

45 The life of a claim starts as soon as an insured reports a claim. First the claim adjuster will determine whether the claim is covered by the policy. If he / she recognizes that there is a relevant liability (the incident occurred within the effective dates of the policy; met the reporting requirements and respects the terms and conditions of the policy), then the case outstanding estimate is to be determined. To do so the claim adjuster has to consider the extent of the damages, the policy deductible, and the policy limit. The estimated value of the claim changes as more information is gathered. Case outstanding estimations can include estimations for: - the case outstanding for claims loss only; - the case outstanding for claims loss and claim allocated adjustment expenses; - the case outstanding for claims loss and adjustment expenses including unallocated adjustment expenses. Insurance companies can have different techniques to evaluate the case outstanding of a claim. One might set the initial case outstanding amount equal to the policy limit minus the deductible. That would equal the maximum losses the insurance company could pay for that claim (not including loss adjustment expenses). Another technique could be to get the counsel of a specialist and let him estimate the claim loss amount. The case outstanding value could also be estimated using tables made using the historic of claims of the company for claims with specific characteristics. 4.2 Claim management A claim can take years to settle, and many payments can be made for the same claim at different time. The estimated value of a claim can change from one period to another. The ultimate value of the claim is definitive once the claim closes (if it is not reopened). The development techniques can both be used on the paid claim loss triangle and the reported claim loss triangle. Those two types of triangles can be affected by changes in claim management techniques. 45

46 4.2.1 Changes in case reserve adequacy The reported loss triangle is the sum of the paid loss and the case outstanding loss estimates triangles. Therefore it can be affected by changes in case outstanding adequacy. If the claim department decides to implement a new system to have a better assessment of the case outstanding, this change will only affect the last diagonal of the triangle and not the previous ones. However, the development technique has described in part 2 of this paper rely on past experience to determine what future patterns will be. Therefore applying actuarial development technique to a reported claim loss triangle with changes in case reserves adequacy could lead to an inaccurate assessment of the reserves estimates. For example, an insurance company used to set up the initial estimation of case outstanding for all medical malpractice claims to the maximum loss possible (it could be the policy limit minus the deductible). In order to have a more accurate case outstanding estimate 46

47 management could decide to implement a new system using its claims history to settle a table with an estimate on a medical malpractice claim. It could be estimated by putting together averages of claim loss and adjustment expenses for some specific category of claim. Those claims could be defined by parameters such as the service that is concerned (neurosurgery, gynecology, cardiology, ), the type of claim (improper treatment, prescription errors, failure to monitor, ), the district of the hospital. This might reduce the amount of case outstanding showed on the last diagonal of the medical malpractice liability triangle, especially for early development periods. If the actuary was to apply an historic loss development factor to the new diagonal, he / she might underestimate the reserves for that segment, especially for more recent accident years, since it is expected that the case outstanding estimates for mature accident years is more accurate than that of un-mature accident years. The diagram above gives an example of the lifecycle of a claim. This might fit the lifecycle of a claim for a short-tailed business line like a personal automobile liability Changes in paid settlement pattern The paid loss triangle can also be affected by changes in the claims management process. Indeed a speed up or a slowdown in the settlement of claims can affect the payment pattern and therefore the paid triangle. Claim management can assess that settling claims faster reduces the amount of allocated and unallocated loss expenses more than it increases the claim loss payments. Therefore it can prefer to settle cases at a higher price rather than go to court and on average save money. These changes are reflected on the paid loss triangles. On the latest diagonal of the triangle, the early development period paid losses might be relatively higher than historical ones. Selecting a loss development factor based on historical values and applying it to that latest diagonal might lead to an overstatement of the reserves for that specific segment. On the other hand if the payments were to slow down that technique might lead to an underestimation of the reserves for that segment. 47

48 Note that a change in payment patterns can also impact case outstanding estimates and therefore case outstanding adequacy. An example might help to understand that case. If management of an insurance company decided that for a specific line of business, claims had to be settled faster. Under the previous management process a specific claim would have been settled three years after the occurrence of the incident, and the case outstanding for this case would have been evaluated a couple of times. A speed up in payments under the new management process might encourage the settlement of the claim one year after the occurrence of the incident. This example sheds some light on the fact that both the reported and the paid loss triangles might be impacted by a change in settlement pattern. The reverse does not seem true. A change in the evaluation of case outstanding does not impact the payments of the claim. 4.3 The impact of external environment As explained in the two previous points, changes in the claim management process can distort reserves projections. Indeed, the case reserves adequacy and / or the payment settlement patterns are not consistent for the entire experience shown in a loss development triangle. Management decisions are not the only factors that can cause a change in case outstanding adequacy or in payment patterns. The external environment can also affect it Tort reforms New regulations can change the evaluation of case outstanding or the payments of claims. Tort reforms in some states can, for instance, cap the total claim damages, or change the statute of limitations, reducing the severity of some claims or limiting the number of years given to the plaintiff to sue a person or a company after the accident occurred. For example, the Medical Injury Compensation Refom Act (MICRA) of 1975, introduced a damage cap in the state of California. Non-economic damages are now limited to $250,000. This means that a plaintiff cannot recover more than $ if he / she goes to trial after a medical malpractice case that caused him / her pain, or made him / her suffer from emotional distress. This reform was meant to reduce the severity and the frequency of claims directed towards health care providers. Medical Malpractice insurance coverage was becoming 48

49 unaffordable. A RAND research determined that 30 percent of the defendants liabilities were reduced by that tort reform. This surely impacted loss development triangles for medical malpractice liability in California. It also implies that if a state was to implement a similar tort reform tomorrow and the losses of that state used to be analyzed amid losses coming from different states with laws in favor of the plaintiffs, then it might be necessary to analyze the losses of that state separately to come up with a better ultimate loss estimation Other environmental changes Changes in the results of an industry can be function of environmental factors. For example, the Private Passenger Auto Liability witnessed great improvement in the 90s. This can partially be explained by the increase of safety items in cars such as air bags, seat belts. In addition to safer cars, communication campaigns were made to encourage drivers not to drink before driving (or at least to limit the use of alcohol before driving). 4.4 Diagnostic tests One of the prerequisite to estimating ultimate losses consists into understanding the data provided and running diagnostics to test for the consistency in the case reserve level adequacy and in settlement rates. As explained in the previous point changes in the insurer s operations and the external environment have an influence on claim data. The question is then how to detect those changes and measure their impacts to properly estimate the unpaid claims amount. One of the first thing to do when estimating the ultimate loss is to talk to underwriters, claim professionals and pricing teams to ask if there has been any change in the way claims are handled, or changes in the book of business. James Berquist and Richard Sherman (Berquist & Sherman, 1977) really insisted on the need for cross-function communications, they dedicated one Appendix to a list of potential questions to ask to different contacts in different departments. To claims executives, the reserving actuary can ask how the case reserves are set: is it set equal to the most likely settlement amount or to the minimum possible amount or any other standard. For 49

50 the underwriters an example of questions could regard the mix of business, what percentage of a certain coverage is sold in each state, how has the mix changed in time. An example of question that could be asked to the pricing team regards the evolution of the pricing, how can a change in earned premiums be explained. Development triangles can be used as a tool to check management assertions and select a proper reserving method. Claim professionals can report changes in claims settlement or / and in the strength of its case outstanding. Underwriters can assert that policy holders are now of better quality and that the severity and frequency of claims should drop which could motivate a decrease in pricing. Those changes do not necessarily have the impact that those professionals were expecting. The effects of the new system implemented by claim management can take longer than expected and its impact may not be as important as they might have assessed. The underwriters can have overestimated the potential of a new market, and led the pricing team to reduce the price of a product. To test for those, the actuary needs to run some diagnostics before squaring the triangles The claim loss development triangles A diagnostic could consist into looking at the paid and the reported claim loss triangles. By looking down each column of those triangles, the actuary is comparing the experience of different accident years at the same age of development. In a stable environment the number shown should look similar. Development triangles for reported claims over on level premium and paid claims over on level premium can also be used to check for the stability of the claim data environment. To examine the consistency of paid claims to reported claims, the triangle of paid to reported losses can be computed. It enables to test for changes in settlement pattern or / and in case outstanding adequacy. However if the ratios look stable down each column, it does not necessarily mean that there has been no change: case outstanding strengthening can be offset by a speed up in settlement rate. 50

51 4.4.2 Changes in earned premium One of the first data that can be looked at is the earned premium history. What were the year to year changes? This analysis may indicate if there has been a significant increase / drop in business, if the evolution is consistent year after year. This might be an indication of the consistency in the mix of business. Questions might raise such has how is the frequency / severity of claims impacted? Is there an increase or a decrease in the loss ratio for the more recent accident years? Or what trends were used by the pricing team to set up an increase or a decrease in the policy rates? Average data To conduct further investigation, the actuary needs to talk to the claim department and learn more about the management of claims. Triangles for paid, reported and closed claim count can also be requested and analyzed. It is important to understand the type of data included in those triangles. Are claims with no payments included in the count? Are re-opened claim considered as a new claim? What if the claim has no payment to the beneficiary but expenses payments for the insurance company? The ratio of closed to reported claim counts can be used to check for a change in settlement rate. If there is a change in the settlement of claims within the company does it impact all the accident years, or only the last few? Is a change in the computer system affecting the reported claim count only or also the closed claim count? A change in handling procedures could result in a speed up or a slow down in settlement patterns. The new system could be much more efficient right away or sometimes it may take time for it to be fully operational. The average case outstanding triangle can be used to test for changes in case outstanding adequacy. If the environment is stable, the average case outstanding is increasing down each column following the annual trend rate 9. A decreasing pattern down each column could indicate a weakening in case outstanding, an increase could indicate a strengthening. Before jumping to 9 This rate reflects change in the severity of claims not necessarily in inflation. This rate is different from one line of business to another. For example, the Insurance Research Council found that the average claim payment per insured home across the U.S. rose from $229 in 1997 to $625 in 2011 and fell to $442 in 2013 (Research, 2015). This corresponds to an increase of an average annualized rate of 5% compared to an average inflation rate of 2.4%. 51

52 conclusions, the actuary needs to check if this is really due to a change in the case outstanding practices within the insurance company or if this is due to changes in the portfolio of claims (new mix of business), or any other reasons that could explain a shift in the average case outstanding. Before drawing conclusions, the actuary should look at more than one diagnostic tools and communicate with other departments to check if his observations are confirmed. In practice, a survey conducted by PwC (PricewaterhouseCoopers, 2013), in the United States, revealed that almost 60% of the companies surveyed interact with the pricing and the claims department monthly. 95% responded that they reach out to those departments at least quarterly. The quality of those communications could be improved. In order to properly understand the changes in internal processes, the external environment and trends in the data, the reserving actuaries need to be provided with more information. 4.5 Claim management evolution The claim process his regularly improved and changed. Some of the factors that might explain it are the advances in technology that enabled carrier to recognize and adjust for claims more quickly, the willingness to cut unnecessary costs led insurers to try to settle claims at their right value more quickly. Another aspect of the claim system improvement can also be by changing the way insurer deal with fraudulent claims and how those are detected. Changing the claim process for one of those reasons can distort the historical actuarial data. Therefore the application of one of the method listed in the previous chapter would not lead to an adequate estimation of the unpaid claims amount. Internal changes in the claim environment usually tend to distinguish between different types of claims that will be handled differently. Besides considering the different line of business such as personal automobile liability or commercial automobile liability, claims are also separated into different sets. For example, claims with suits versus non-suit or fraudulent claims versus non-fraudulent claims could be dealt with differently by different teams of employees who are trained to handle those specific claims. 52

53 Such a segmentation is also better for management control. It enables to set evaluation metrics per type of claim handling and evaluate the performance of one type of claim handling. Indeed, if one performance criterion is the rapidity at which a claim adjuster settle a claim, claim handler dealing with claims belonging to a long-tailed line of business should not have the same objective as those dealing with short-tailed line of business. New systems are supposed to offer more confidence and consistency in the determination of the claim liability and settlement of claims should be fastened. New technology can also be expected to help settle claim faster. United Services Automobile Association (USAA) has for example sought permission from the U.S. Federal Aviation Administration to use drones to process insurance claims faster and more safely in areas hit by catastrophes. 53

54 Chapter 5. Reserving in a changing environment The actuarial methods previously presented assume consistency in the claim process. Historical data is supposed to be representative of the conditions that will drive future development. As introduced in the previous chapter, changes in case outstanding adequacy and / or in settlement rate can impact the development triangles and are not taken into account by traditional development techniques. A few diagnostics have been described so that such changes can be detected. The question is now how to come up with an unpaid claim loss estimate once a change in the environment has been detected and confirmed by another department within the company? The first part of this chapter details Berquist and Sherman method for reserving. The different methods and assumptions are tested for variability. The second part discusses some concerns or improvement that can be made to that technique so that it is not applied blindly. 5.1 The Berquist and Sherman approach In 1977, Berquist and Sherman published a pivotal paper entitled Loss Reserve Adequacy Testing: A Comprehensive, Systematic Approach (Berquist & Sherman, 1977). There were two main motivations for treating the subject of reserving in a changing environment. 1. In a discussion (Schwartz, 2002), James Berquist explained that at the time of the publication of the paper, reserving actuaries were threatened by accountants who could have become the new reserving professionals. James Berquist and Richard Sherman therefore intended to prove that estimating the ultimate unpaid losses was not a mechanical task, actuarial judgment was needed. 2. Richard Sherman gave another motivation for their paper in a presentation (Sherman, 2010). In early 1976, Geico (Government Employees Insurance Company), one of the U.S. s major writers of automobile insurance was close to collapsing. The financial difficulties arose because of the inadequate recognition 54

55 of loss costs through poor reserving techniques. It all started in 1971 when the head of claims became president of Geico. He decided to change the case reserving standards by selecting the smallest loss amount a claim could settle for instead of the most likely one. Therefore the adequacy of case outstanding dropped. At the time no adjustment was made to the loss development triangles to reflect those changes. The incurred development technique gave very favorable results. It seemed as if the loss results of the company were remarkable and it impacted the pricing of the products. For six years there were no rate increases when the severity trend was high. The low rates encouraged the coming of new customers. Twenty eight auto insurers took part in the effort to rescue Geico. After five years, the financial position of the company was decent again. This case study encourages taking into account changes in case reserve adequacy and more generally any changes in claims handling. Berquist and Sherman therefore developed two main methods to adjust for changes in case reserve adequacy and in settlement rate. They also insist on the need of keeping an open dialogue between the different actuarial departments within an insurance company. Indeed as revealed by the case above the decisions made in the claims department, have an impact on the reserving practices; the pricing department relies on the loss results of the company to adapt its pricing and the underwriters benefit from good rates to develop their customers list. In the first part of the paper the two authors insist on the type of questions a reserving actuary should ask the claim department and the pricing department in the company. By encouraging cross-function communication, the aim is to detect if the company has undergone changes in operations and procedures. Two alternatives are suggested to address those changes: one implies data selection and rearrangement and the other focuses on data adjustment. The first option consists in substituting the data affected by the changes to some that was not. It could also be done by dividing data into smaller more homogeneous groups. A last suggestion was to analyze separately claims of larger size and smaller size. The main point of that paper was to introduce two techniques to adjust data. The first one consists into adjusting the case outstanding triangle in case of a change in the adequacy level of 55

56 case reserve. The second is a method to adjust the paid triangle in case of a change in claims rate settlement. In this chapter, the data and the example are the same as the one used in the 1977 paper by Berquist and Sherman. They studied two cases the first one is a case of medical malpractice claims using data between 1969 and The second used automobile bodily injury claims for the same experience period. The data is obviously outdated but the approach remain today. 5.2 Adjustment to the reported loss triangle Berquist and Sherman method In the first example the development of reported claims and paid claims lead to two very different estimations of what the reserves should be. There is a 20% difference between the two estimates as shown in Table 1 below. The estimate in column (B) was obtained by taking the simple average of the age to age factors from the cumulative reported loss triangle. Those were used to project ultimate losses. To get to the reserves estimates the cumulative paid by accident years are to be subtracted from that projections. Column (C) was obtained using the same pattern applied to the cumulative paid triangle. There are significant differences between those two estimates by accident years and in total as shown in columns (D) and (E). In the Appendix Exhibit 2, Sheets 1 & 2 the reported and the paid triangles are shown, so are the selected factors to ultimate and the projected ultimate. 56

57 Table 1 Medical Malpractice Summary Of Reserv es Estimates Amounts In ($000) (A) (B) (C) (D) (E) Reported Paid Difference Accident Cumulativ e Reserv es Reserv es in % Y ear Paid Projection Projection Reserv es Difference ,0% ,8% ,3% ,4% ,8% ,5% ,4% ,7 % Total ,4% One of the underlying assumptions of the reported loss development method is that reported losses will continue to develop in the same pattern as in the past. Thus, the adequacy of case reserve is assumed not to change over time except from inflation 10. The claim count reporting pattern is assumed to be consistent from year to year. To verify those assumptions different diagnostics can be run as described in part 3 of this paper. In Berquist and Sherman paper, the two authors looked at the Case Reserve per Open Claims and Paid Losses per Closed Claims. Thereby, they compared the average case outstanding to the average paid claims to check for changes in the case outstanding adequacy. Those diagnostics are shown in the Appendix in Exhibit 2, Sheet 3. To adjust the reported loss triangle to reduce the effect of changes in the case reserves adequacy, Berquist and Sherman recommend a few steps: - Identify a mathematical curve which approximates the severity trend. - Select a severity trend based on judgment. - Detrend the average case outstanding up each column with that selected rate and calculate the adjusted case outstanding triangle by using the open claim count triangle. 10 As explain earlier, the term inflation in that context is linked to the specific severity trend observed in a certain line of business. 57

58 - Adjust the reported loss triangle by adding the adjusted case outstanding triangle to the paid loss triangle. Step 1: fitting of a mathematical curve The fitting of an exponential curve was used as suggested in Berquist and Sherman paper. Other curves might fit better for other sets of data. The only way to know is to try different hypothesis and select the one that seems to give the best fitting. To find the trend associated to an exponential curve, the following equation has to be solved for each column:, with j the jth column. Then the empirical trend is obtained from the following formula:. The last two diagonals on the first diagnostic (case outstanding per open claims) are higher than the other diagonals. The calculated trend ranges between 27% and 34%, except for the first twelve months of development. The goodness of fit for those estimations is above 80%. The second diagnostic shows the Paid losses per closed claims. The trend ranges from 6.7% to 14.3%. The equations with a trend higher than 14% have a goodness of fit higher than 80%, the other estimates have a goodness of fit lower than 40%. Step 2: selection of a trend The next step consists into adjusting case outstanding per open claims. Two judgment calls are to be made. What trend to select? What diagonals to modify? Berquist and Sherman selected a 15% trend, which was the industry benchmark at the time according to their paper. They also decided to keep the last diagonal unchanged and to adjust the others. Exhibit 2, Sheet 3 shows how to adjust the case outstanding triangle. This supposes that the trend is the same at each evaluation period. This trend is supposed to reflect the inflation or deflation in claim size. If the triangles are net of reinsurance, it can also be affected by a change in retention levels. 58

59 By adjusting all the diagonals using the latest one, it is assumed that the reserves level reflected by the latest calendar year data are correct. This means that using those would lead to a good estimation of the unpaid losses. Step 3: adjusting the case outstanding triangle Exhibit 2, Sheet 4 shows three triangles: the adjusted case outstanding per open claims triangle, the open claims count triangle, and the adjusted case outstanding triangle. The first of those three triangles was obtained by de-trending the last diagonal with the selected 15% rate. The equation below shows the relationship between the averages of two consecutive accident years for the same development period. represents the case outstanding per open claims for accident year number and development period number. and and. represents the selected rate in that case it is 15%. The adjusted case outstanding triangle is then obtained by multiplying the number of open claim for a specific row and a specific column of the open claim count triangle, by the adjusted case outstanding per open claims for that same row and column. Step 4: adjusting the reported loss triangle The reported loss triangle was obtained from the previously adjusted case outstanding triangle added to the paid losses triangle. This enables to get a second projected ultimate. Table 2 below compares the two estimates obtained without adjustment in column (A) and with adjustment in column (B). By adjusting the triangle with a 15% trend instead of a 30% the total estimate of ultimate is decreased by more than a third. 59

60 Table 2 Medical Malpractice Ultimates Using the Reported Losses Triangle Amounts In ($000) (A) (B) (C) (D) Initial Adjusted Accident Ultimate Ultimate % Y ear Projection Projection Difference Difference ,0% ,7 % ,7 % ,8% ,0% ,6% ,6% ,0% Total ,8% Sensitivity of the reserves to the selected trend Joseph Thorne, one year after the publication of Berquist and Sherman paper produced a chart in his discussion paper (Thorne, 1978). Chart 1 below is similar to that produced by Joseph Thorne. It intends to show the sensitivity of the reserves estimates to the selected severity trend. Chart 1 does not only test the sensitivity of the selected trend it also tests the sensitivity linked to the selected adequacy level. The blue line with squares on the chart was obtained by adjusting the case outstanding triangle to the adequacy level of the latest diagonal of the triangle. The red line with circles was obtained by adjusting the triangle to the level of adequacy of the second diagonal (the last diagonal was left unchanged in that case). Those two methods give similar estimates for trend ranging from 25% to 30%. By using a 30% severity trend the reserves estimates would range between $ to $ , a 15% trend would indicate that the reserves should be less than $ The judgmental selection of a severity trend has a significant impact on the reserves estimate. 60

61 Chart 1 Medical Malpractice Difference in Reserv es Estimates Amounts In ($000) y = 248,95e 3,6675x R² = 0, y = 370,65e 2,1943x R² = 0, % 15% 20% 25% 30% 35% 40% Thorne also pointed out that the severity trend selected in Berquist and Sherman paper was derived from a triangle of average paid losses per closed claims with or without payments and the closed claims triangle and is applied to a open claims triangle. This remark is interesting it implies that when applying the method it is important to know what type of data one is dealing with. Is a claim count triangle including claims without payments? Is it including trivial claims? Has there been a shift in the presence of those claims? If trivial claims were to account for a bigger proportion of the total claim count, it could give the illusion of a speed up in claims. As it was repetitively assessed in the Berquist-Sherman paper, reserving cannot be done mechanically by applying a method blindly. 61

62 5.3 Adjustment made to the paid triangle Berquist and Sherman method In a second example, Berquist and Sherman looked at a different segment: Automobile Bodily Injury liability. The assumption underlying the paid development method is based on the consistency in settlement pattern. A method is suggested to detect a shift in settlement pattern and a way to correct the paid triangle is given. Here are the suggested steps: - Calculate the claim disposed ratio by dividing the closed claim by the projected claim ultimate; - Find a mathematical curve to approximate the relationship between paid losses and the number of closed claims; - Use the mathematical curve and the indicated closed claims ratio to adjust the cumulative paid losses triangle. Step 1: claim disposed ratio pattern selection In the Appendix, in Exhibit 3, Sheet 1 a first estimate of the ultimate losses is obtained by developing the paid triangle using the Chain Ladder method. The data used in that example is the same as the one shown in Berquist and Sherman paper. In Exhibit 3, Sheet 2 the same is done to determine the ultimate number of reported claims. This time the Chain Ladder method is applied to the triangle of reported claim counts. In Exhibit 3, Sheet 3, the second triangle represents the ratio of closed to reported claim counts. By looking down each column of that triangle, a decrease in the rate of claim settlement over the experience period can be suspected. By consulting with the claims team or other department within the insurance company or the Third Party Administrator if the claims are not handled in house, the suspected change in case settlement pattern should be explained. Before applying a development method to the triangle of payment, some adjustments can be made to that triangle. 62

63 The third triangle of that exhibit shows the claim disposed ratios. It is determined by dividing the cumulative number of closed claims by the ultimate number of reported claims associated to that accident year. This triangle shows evidence of a decrease in the rate of claims settlement. Claims with accident year 1973 have a 80% disposal rate after two years of development compared to 77.4% for claims with accident year Berquist and Sherman selected the disposal rate pattern indicated by the latest diagonal of that triangle. Step 2: mathematical relationship between paid claims and the number of closed claims Berquist and Sherman used an exponential curve to approximate the relationship between the cumulative number of closed claims and the cumulative paid loss dollars. Exhibit 3, Sheet 4, shows the estimated coefficients for the following equation (Method 1): With the cumulative paid losses for accident year j and the cumulative number of closed claims for that same accident year. In a discussion (Schwartz, 2002), James Berquist, Richard Sherman and Joseph Thorne all agreed that the exponential formula was a good choice. They had the chance to test it on a large number of studies. Step 3: adjusting the paid triangle The adjusted triangle of cumulative closed claims is then obtained by multiplying the selected disposal rate for each development period by the selected ultimate reported claim. Since the selected disposal ratio was obtained from the last diagonal, the adjusted cumulative number of closed claims triangle has the same last diagonal as the unadjusted triangle. The adjusted cumulative paid loss dollars triangle is eventually obtained by applying the above formula using the previously estimated coefficients by accident year to the adjusted cumulative number of closed claims. There are two main judgment calls made in this case. The first one regards the selection of a disposal rate pattern used in the adjustment of the cumulative number of closed claims triangle. 63

64 The second regards the selection of the equations used to approximate the relationship between cumulative paid losses and cumulative closed claims Sensitivity of the reserves to the selected parameters Impact of the disposal rate pattern selection Exhibit 3, Sheet 6 was created to measure the impact of the selection of a disposal rate pattern on the estimation of the reserves. Estimate 1 was obtained by taking the minimum of the claims disposed ratio for each column of the claims disposed ratio triangle displayed in Exhibit 3, Sheet 3. Conveniently enough this case corresponds to the base scenario(the selection made in Berquist and Sherman paper) since for each column the minimum ratio is on the latest diagonal. The following estimates were obtained by increasing the base scenario ratio by 1% until it reaches the maximum ratio of the column. Scenario 10 conveniently corresponds to selecting the disposal ratio pattern indicated for accident year Scenario 10 leads to the calculation of a reserve estimate that is 40% smaller than the one indicated by scenario 1. Impact of the curve fitting In Exhibit 3, Sheet 4, the impact of the curve fitting method on the estimation of the reserves is assessed. Three cases are tested. Those tests are run with the same selection of disposal rate pattern (it corresponds to scenario 1 in the above paragraph). The first corresponds to the estimation of the relationship between the cumulative closed claims count and the cumulative paid losses by using an exponential fitting as it was done in Berquist and Sherman paper. The second correspond to a linear regression and the third to a power fitting. The goodness of fit seems to be a good metric to evaluate the best model since the number of parameters is the same in all of them and they can all be associated to a linear regression model. 64

65 The goodness of fit (R²) is closer to 1 for Method 1 than for Method 2 and 3. This is true for accident year 1969 to This would indicate that an exponential fitting is more adequate than the two other methods. Another interesting element is that the use of the linear regression leads to an adjusted triangle of cumulative paid losses with negative value for the first development period of accident years 1969 and 1970 event though the goodness of fit for method 2 is higher than 92%. This result would look irrelevant especially if the paid loss triangle does not include salvage and subrogation. Therefore the goodness of fit should not be the only criteria in the selection of a mathematical relationship between the cumulative paid loss and the closed claims count. Exhibit 3, Sheet 7 shows the estimates of the reserves for the different scenarios and methods. Twenty two out of the thirty estimates are below the one obtained with unadjusted data Influence of a change in development pattern on the case outstanding triangle As mentioned in point 4.2.2, a change in payment settlement pattern can also impact a change in case outstanding adequacy. In their paper (Fleming & Mayer, 1988), Kirk Fleming and Jeffrey Mayer addressed that problem. They noticed that Berquist and Sherman paper did not treat the question of whether the case outstanding triangle should also be adjusted for changes in claim closing patterns before reviewing a change in average case reserves. Their main point is that a change in claim settlement pattern affects the incurred development technique as well as the paid development technique. They therefore propose a method to adjust the incurred triangle for changes in settlement patterns. And show that a change in settlement pattern can mask or lead the actuary to believe that there was a change in case reserve adequacy. Fleming and Mayer noted that there is a relationship between claim reserves affected to a claim and the time remaining until the claim is closed. As the time to settlement approaches the reserve carried on a claim should be closer to the ultimate settlement value as the uncertainty of the ultimate value is reduced with time. They further explain the relationship between the claim 65

66 settlement rate and the reported triangle by describing the life of the incurred value. Before settlement it is equal to the case outstanding for the claim and as it settles it is equal to the paid value. Therefore there is a relationship between the incurred triangle and the settlement pattern. One of the assumptions is that the claim reporting pattern remains constant over time. The new adjustment method consists in the following steps: - Adjusting the paid triangle for changes in claims settlement rate by using Berquist and Sherman method as described above. - Finding a mathematical formula that links the incurred triangle to the paid triangle. - Applying that formula to the adjusted paid triangle to obtain the adjusted incurred triangle. 5.4 Other adjustments The adjustments described in the previous chapter are meant to address overall changes in claims handling techniques that might impact case outstanding adequacy or / and settlement rates. But has it was described in the last part of chapter 4, changes in the claim department are made to specific type of claims. Management might implement shifts so that small claims are dealt with more rapidly, fraudulent claims might be detected earlier or claims that might not involve an attorney evaluated with a more accurate technique than before. One change might only impact one type of claim and not the others. Berquist and Sherman already started to address this problem by mentioning that the claim s department could implement shifts in claim handling that might impact small claims and not big claims. Therefore they recommended that the actuarial analysis would be separated for big and small claims. In the discussion mentioned above (Schwartz, 2002), Joseph Thorne noted that over 90% of reserve studies do not require any of the adjustments described in the Berquist and Sherman paper. He was none the less surprised by the fact that many reserve reports do not address whether changes in case adequacy or in settlement rates have been at least tested for especially because the distortion in reserve estimates due to shifts in case reserve adequacy or settlement 66

67 pattern can be huge. According to him This potentially critical effect invites at least testing for these types of shifts. One other issue that is brought to light in that paper is the a good history of size of loss data is not always available even thought the improvement in technology would lead us to believe that such data would be easy to gather. As revealed in the PwC survey (PricewaterhouseCoopers, 2013), over 60% of the respondents acknowledge that reserving team had no dedicated IT teams to help with the extraction and gathering of data. One other aspect is that through mergers and acquisitions, multiple data system are coexisting leading to frequent data issues. 67

68 Chapter 6. Testing the estimation In the reserving process, the actuary produces an estimate of the claim liabilities by: - Relying on a selection of actuarial methods; - Using a dataset; - Using his / her actuarial judgement. 6.1 Selection Criteria To measure the performance of different actuarial estimate some criteria can be checked. - Best Least Square error: The method should produce estimates that have minimum expected error, in the least squares sense. - Unbiased: There should be no bias. The expected error should be equal to zero. - Responsive: The selected method should respond quickly and accurately to changes in underlying trend or to operational techniques. This is a key criteria in this study. - Stability: The method should work well in different cases. 6.2 Pre-selection test Testing the forecasting abilities of the model The general approach consists into using all the data available but one part. The model is calibrated using the first part of the data and then it is tested on the second part of the data to evaluate its forecasting abilities by comparing the estimations to actual values. The process can be repeated a few times by excluding a different part of the data each time. Back-testing enables to compare the forecasting skills of different competing models. This method tests the predictive capacities of a certain model. It does not necessarily test the 68

69 parameters of the model but the model itself. Once a preferred model has been selected, all the data can be used to run the parameterization Testing the reasonableness implied by the selected loss ultimate Apart from measuring the forecasting ability of the model selected, the actuary can also apply reasonableness tests. Indeed in the Statement of Principles Regarding Property and Casualty Loss and Loss Adjustment Expense Reserves, as well as in the actuarial Standards of Practice recommend that the actuary tests for the reasonableness of loss reserves. Here are some measures that can be looked at: - the implied loss ratio for each accident year; - the IBNR to case reserve ratio for each accident year; - the implied severity and frequency. In each case, the reasonableness of each measure can be assessed by comparing the ratios for each accident year. The measures can also be compared to industry benchmarks. The loss ratio is obtained by dividing the estimated ultimate loss for an accident year by the earned premiums. If for one accident year the loss ratio looks particularly high compared to the other accident years, this might lead the actuary to look at the data again. For example a large unusual loss was included in the analysis and could distort the estimate of ultimate loss. 6.3 Post- selection test Another type of performance evaluation can be done over time. An estimation of the unpaid claims ultimate losses is made at a certain point in time. At the next evaluation point more data is available. It can therefore be used to assess the quality of the model by comparing the actual emergence of payment or of incurred losses to the predicted one. This process insures that the actuarial estimates of unpaid claims and the assumptions underlying the calculation of it. It enables to test the development factors chosen for the loss development method, to test the loss ratio selected for the Bornhuetter-Ferguson method. It also enables to test the method selected. 69

70 6.3.1 Actual versus Expected using the Chain Ladder method The Chain Ladder method enables to have an estimation of the amount that will be paid for accident year at development period. estimated today can be compared to the actual, once this value is available at time. The tables show to different triangles one evaluated at t1 which is the end of 2013, the other one evaluated at time t2 which is the end of At the end of 2014, an additional diagonal is available. evaluated at time t1 can now be compared to the real value of obtained at time t2. The formula used to obtain the estimation was at time t1.. This technique enables to test the validity of the development factors selected 70