EDUCATION AND EXAMINATION COMMITTEE OF THE SOCIETY OF ACTUARIES CONSTRUCTION AND EVALUATION OF ACTUARIAL MODELS STUDY NOTE TOPICS IN CREDIBILITY THEORY by Curtis Gary Dean, FCAS, MAAA ©Copyright 2005 Curtis Gary Dean. Reproduced by the Casualty Actuarial Society and the Society of Actuaries with permission of the author. The Education and Examination Committee provides study notes to persons preparing for the examinations of the Society of Actuaries. They are intended to acquaint candidates with some of the theoretical and practical considerations involved in the various subjects. While varying opinions are presented where appropriate, limits on the length of the material and other considerations sometimes prevent the inclusion of all possible opinions. These study notes do not, however, represent any official opinion, interpretations or endorsement of the Society of Actuaries or its Education and Examination Committee. The Society is grateful to the authors for their contributions in preparing the study notes. C-24-05 Printed in U.S.A. Preface This study note was written to supplement the “Credibility” chapter of Foundations of Casualty Actuarial Science as a reading for the fourth CAS/SOA examination. It presents important topics not covered in the Foundations chapter including the Bühlmann-Straub Model and nonparametric and semiparametric estimation of credibility formula parameters. The author would like to thank Clive Keatinge, a member of both the CAS Examination and Syllabus Committees, for his suggestion to write the study note. Clive provided oversight during the writing and editing process, and made helpful suggestions about the content and clarity of exposition. He also chaired the committee that reviewed the study note. I would also like to thank the reviewers on the committee for their many valuable contributions including Joseph Boor, Russell Greig, Nasser Hadidi, Leigh Halliwell, Stuart Klugman, Walter Lowrie, Marjorie Rosenberg, and Gary Venter. Gary Venter is responsible for much of the material included in the section “An Intuitive Model for Credibility.” Ball State University students also provided valuable comments and should be recognized: Heather Adams, Chuket Ounjitti, Doug Pirtle, Lori Thompson, and Melany Tower. 2 Table of Contents 1. Credibility Models 1.1 Review of Bühlmann Model 1.2 Bühlmann-Straub Model 2. Estimation of Credibility Formula Parameters 2.1 Nonparametric 2.2 Semiparametric 3. Conclusion Appendix A: Bühlmann and Bühlmann-Straub Credibility Estimators Are Linear Least Squares Estimators Appendix B: Proof of Unconditional (Total) Variance Formula Appendix C: Nonparametric Estimators for the Expected Value of the Process Variance and the Variance of the Hypothetical Means in the Bühlmann-Straub Model are Unbiased References Exercises Solutions to Exercises 3 1. Credibility Models This study note supplements the “Credibility” chapter of Foundations of Casualty Actuarial Science as a reading for the fourth CAS/SOA examination. Several important topics not covered in the Foundations text are presented here, including the Bühlmann- Straub credibility model and estimation of credibility formula parameters. It is assumed that the student already has some familiarity with the material covered in the “Credibility” chapter before reading this study note. The credibility models that will be discussed are often referred to as greatest accuracy credibility or least squares credibility. As will be explained later, these methods attempt to produce linear estimates that will minimize the expected value of the square of the difference between the estimate and the quantity being estimated. Bühlmann credibility will be reviewed paying particular attention to the simplifying assumptions that distinguish it from the more general Bühlmann-Straub model that follows. The second half of the study note covers estimation of credibility formula parameters when underlying distributions are unknown. Before beginning a more rigorous study, an intuitive derivation of the useful Bühlmann credibility model will be presented. An Intuitive Model for Credibility The actuary uses observations of events that happened in the past to forecast future events or costs. For example, data that was collected over several years about the average cost to insure a selected risk, sometimes referred to as a policyholder or insured, may be used to estimate the expected cost to insure the same risk in future years. Because insured losses arise from random occurrences, however, the actual costs of paying insurance losses in past years may be a poor estimator of future costs. Consider a risk that is a member of a particular class of risks. Classes are groupings of risks with similar risk characteristics, and though similar, each risk is still unique and not quite the same as other risks in the class. In class rating, the insurance premium charged to each risk in a class is derived from a rate common to the class. Class rating is often supplemented with experience rating so that the insurance premium for an individual risk is based on both the class rate and actual past loss experience for the risk. The important question in this case is: How much should the class rate be modified by experience rating? That is, how much credibility should be given to the actual experience of the individual risk? Intuition says that two factors appear important in finding the right balance between class rating and individual risk experience rating: (1) How homogeneous are the classes? If all of the risks in a class are identical and have the same expected value for losses, then why bother with individual 4 experience rating? Just use the class rate. On the other hand, if there is significant variation in the expected outcomes for risks in the class, then relatively more weight should be given to individual risk loss experience. Each risk in the class has its own individual risk mean called its hypothetical mean. The Variance of the Hypothetical Means (VHM) across risks in the class is a statistical measure for the homogeneity or vice versa, heterogeneity, within the class. A smaller VHM indicates more class homogeneity and, consequently, argues for more weight going to the class rate. A larger VHM indicates more class heterogeneity and, consequently, argues for less weight going to the class rate. (2) How much variation is there in an individual risk’s loss experience? If there is a large amount of variation expected in the actual loss experience for an individual risk, then the actual experience observed may be far from its expected value and not very useful for estimating the expected value. In this case, less weight, i.e., less credibility, should be assigned to individual experience. The process variance, which is the variance of the risk’s random experience about its expected value, is a measure of the variability in an individual risk’s loss experience. The Expected Value of the Process Variance (EPV) is the average value of the process variance over the entire class of risks. Let X represent the sample mean of n observations for a randomly selected risk i. i Because there are n observations, the variance in the sample meanX is the variance in i one observation for the risk divided by n. Given risk i, this variance is PV / n where PV i i is the process variance of one observation. Because risk i was selected at random from the class of risks, an estimator for its variance is E[ PV / n ] = E[ PV ] / n = EPV / n. i i This is the Expected Value of the Process Variance for risks in the class divided by the number of observations made about the selected risk.1 It measures the variability expected in an individual risk’s loss experience. Letting µ represent the overall class mean, a risk selected at random from the class will have an expected value equal to the class mean µ. The variance of the individual risk means about µ is the VHM, the Variance of the Hypothetical Means. There are two estimators for the expected value of the ith risk: (1) the risk’s sample meanX , and (2) the class mean µ. How should these two estimators be weighted i together? A linear estimate with the weights summing to 1.00 would be Estimate = wX +(1−w)µ . i An optimal method for weighting two estimators is to choose weights proportional to the reciprocals of their respective variances. This results in giving more weight to the 1 The expectation is taken over all risks in the class. 5 estimator with smaller variance and less weight to the estimator with larger variance. In many situations this will result in a minimum variance estimator. (Please see the first problem in the exercises at the end of the study note.) The resulting weights are 1 1 EPV /n VHM w= and (1−w) = . 1 1 1 1 + + EPV /n VHM EPV /n VHM Note that a denominator was chosen so that the weights add to one. A little algebra produces n n w= and (1−w) =1− . EPV EPV n+ n+ VHM VHM Setting K = EPV / VHM , the weight assigned to the risk’s observed mean is n w= . n+ K This is the familiar Bühlmann credibility formula with credibility Z = n / (n + K ).2 In this section, a risk selected from a rating class was used to illustrate the concept of credibility. In general, an individual risk or a group of risks comes from a larger population and the goal is to find the right balance between using the data for the smaller group and the larger population. Many other examples are possible. Example An actuary calculated indicated rate changes by territory for automobile insurance. The rate change indication for the ith territory was R. Combined data for the i entire state indicated that a rate change of +2.0% was required. From these values, credibility weighted rate change indications were calculated: Credibility weighted rate change = Z x R + (1 − Z ) x (+2.0 %) . i i i indication for territory i The credibility weights Z were calculated from the formula Z = n / (n + K ) where i i i i n was the number of insured vehicles in the territory during the three-year data collection i period. ▐▐ 2 A rigorous derivation of the Bühlmann credibility formula is provided in Appendix A. 6 Preliminaries and Notation The actuary uses observations for a risk or group of risks to estimate future outcomes for that same risk or group. In this study note, although the term “a risk” is often used, the same comments can generally be applied to a group of risks where the group is a collection of risks with some common characteristics. The actual observation during time t for that particular risk or group will be denoted by x, which will be the t observation of corresponding random variable X, where t is an integer. For example, X t t may represent the following: • Number of claims in period t • Loss ratio in year t • Loss per exposure in year t • Outcome of the tth roll of a die. An individual risk is a member of a larger population and the risk has an associated risk parameter θ that distinguishes the individual’s risk characteristics. It is assumed that the risk parameter is distributed randomly through the population and Θ will denote the random variable. The distribution of the random variable X depends upon the value of θ: t f (x│θ). For example, θ may be a parameter in the distribution function of X. In the X |Θ t t case of a Poisson claims process, θ might be the expected number of claims. Although the examples in this study note will use θ’s that are scalars, one can also build models with θ as a multidimensional vector with each component of the vector describing some aspect of the individual’s risk characteristics. If X is a continuous random variable, the mean for X given Θ = θ, is the conditional t t expectation, E [X│Θ = θ] =∫x f (x θ)dx = µ(θ) , X | Θ t t XΘ t t where the integration is over the support of f (x│θ). If X is a discrete random X | Θ t t variable, then a summation should be used: E [X│Θ = θ] = ∑ x f (x |θ). X | Θ t t X Θ t allxt The integral notation will be used in general cases, but the reader should be aware that a summation is called for with discrete random variables. It will be assumed that µ(θ) = E [X│Θ = θ] is constant through time for the models considered in this study note.3 X | Θ t The risk parameter represented by the random variable Θ has its own probability density function (p.d.f.): f (θ). The p.d.f. for Θ describes how the risk characteristics are Θ 3 This is a major assumption that is easily violated in practice. Risk characteristics can change for a variety of reasons: a young driver becomes a better driver with experience; a business may institute risk control procedures that reduce losses; traffic densities may increase in an area leading to increased probabilities of auto accidents; and inflation will increase the costs of loss payments. 7 distributed within the population. If two risks have the same parameter θ, then they are assumed to have the same risk characteristics including the same mean µ(θ). The unconditional expectation of X is t E[X ]= ∫∫x f (x ,θ)dx dθ= ∫∫x f (x |θ)f (θ)dx dθ 4 t t X,Θ t t t X Θ t Θ t = ∫[∫x f (x |θ)dx ] f (θ)dθ= E [E [X |Θ]]= E [µ(θ)]=µ. t XΘ t t Θ Θ X Θ t Θ The conditional variance of X given Θ = θ is t Var [X Θ=θ]= E [(X −µ(θ))2 Θ=θ] = ∫∫(X −µ(θ))2 f (x |θ)dx = σ2(θ) . XΘ t X Θ t t X Θ t t This variance is often called the process variance for the selected risk. The unconditional variance of X, also referred to as the total variance, is given by the Total Variance t formula: Var[X ]=Var [E [X Θ]]+E [Var [X Θ]], or t Θ X Θ t Θ X Θ t Variance of the Expected Value of the Total Variance = + Hypothetical Means Process Variance A proof of this formula is shown in Appendix B. These concepts are best demonstrated with an example. Example The number of claims X during the tth period for a risk has a Poisson t x −θ θ e distribution with parameter θ: P[X = x] = . The risk was selected at random t x! from a population for which Θ is uniformly distributed over the interval [0,1]. (This simple distribution for Θ was chosen to make the integration easy.) It will be assumed that θ is constant through time for each risk. (1) Hypothetical mean for risk with parameter θ is µ(θ) =E [X Θ=θ]=θ X Θ t because the mean of the Poisson random variable is the parameter θ. (2) Process variance for risk with parameter θ is σ2(θ)=Var [X Θ=θ]=θ X Θ t because the variance equals the parameter θ for the Poisson. (3) Variance of the Hypothetical Means (VHM) is 4 Note that a substitution for the joint density function f (x ,θ)was made using the relationship X,Θ t f (x ,θ)= f (x θ)f (θ) . X,Θ t X Θ t Θ 8 2 1 ⎛1 ⎞ Var [E [X Θ]]=Var [Θ]= E [Θ2]−(E [Θ])2 = ∫θ2(1)dθ−⎜∫θ(1)dθ⎟ =1/12 . Θ X Θ t Θ Θ Θ ⎜ ⎟ 0 ⎝0 ⎠ (4) Expected Value of the Process Variance (EPV) is 1 E [Var [X Θ]]= E [Θ]= ∫θ(1)dθ=1/2 . Θ X Θ t Θ 0 (5) Unconditional Variance (or total variance) is Var[X ]=VHM +EPV =1/12 +1/2=7/12 . ▐▐ t 1.1 Bühlmann Model The Bühlmann model assumes that for any selected risk, the random variables { X , 1 X , ..., X , X , …} are independently and identically distributed. For the selected 2 N N+1 risk, each X has the same probability distribution for any time period t, both for the X , t 1 X , ..., X random variables in the experience period, and future outcomes X , X , …. 2 N N+1 N+2 As Hans Bühlmann described it, “homogeneity in time” is assumed. The characteristics that determine the risk’s exposure to loss are assumed to be unchanging and the risk parameter θ associated with the risk is constant through time for the risk. The means and variances of the random variables for the different time periods are equal and are labeled µ(θ) and σ2(θ), respectively, as shown in the table below: Assumptions of Bühlmann Credibility Hypothetical Mean: µ(θ) = E [X | θ] = … = E [X │θ] = E [X │θ] = … X |Θ 1 X |Θ N X |Θ N+1 Process Variance: σ2(θ) = Var [X │θ] = … = Var [X │θ] = Var [X │θ] = … X |Θ 1 X | Θ N X | Θ N+1 Of course the hypothetical means and process variances will vary among risks, but they are assumed to be unchanging for any individual risk in the Bühlmann model. To apply Bühlmann credibility, the average values of these quantities over the whole population of risks are needed, along with the variance of the hypothetical means for the population: (1) Population mean: µ = E [µ(Θ)] = E [E [X | Θ]] Θ Θ X | Θ t (2) Expected Value of Process Variance: EPV = E [σ2(Θ)] = E [Var [X | Θ]] Θ Θ X | Θ t (3) Variance of Hypothetical Means: VHM = Var [µ(Θ)] = E [(µ(Θ) − µ)2] . Θ Θ The population mean µ = E [E [X│ Θ]] provides an estimate for the expected value Θ X | Θ t of X in the absence of any prior information about the risk. The EPV indicates the t 9 variability to be expected from observations made about individual risks. The VHM is a measure of the differences in the means among risks in the population. N ⎛ 1 ⎞ Because µ(θ) is unknown for the selected risk, the mean X =⎜ ⎟∑X is used in t ⎝ N ⎠ t=1 the estimation process. It is an unbiased estimator for µ(θ), ⎛ 1 ⎞N ⎛ 1 ⎞N ⎛ 1 ⎞ N E [X θ]= E [⎜ ⎟∑X θ]=⎜ ⎟∑ E [X θ]=⎜ ⎟∑µ(θ) =µ(θ) . X Θ X Θ ⎝N ⎠ t ⎝ N ⎠ X Θ t ⎝N ⎠ t=1 t=1 t=1 The conditional variance of X , assuming independence of the X given θ, is t ⎛ 1 ⎞N ⎛ 1 ⎞2 N ⎛ 1 ⎞2 N σ2(θ) Var [X θ]=Var [⎜ ⎟∑X |θ]=⎜ ⎟ ∑ Var [X θ]=⎜ ⎟ ∑σ2(θ) = . X Θ X Θ ⎝N ⎠ t ⎝N ⎠ X Θ t ⎝N ⎠ N t=1 t=1 t=1 The unconditional variance of X is 2 E [σ (Θ)] EPV Θ Var[X]=Var [E [XΘ]]+E [Var [X Θ]]=Var [µ(Θ)]+ =VHM + . Θ XΘ Θ XΘ Θ N N Bühlmann credibility assigned to estimatorX is given by the well-known formula N Z = , N +K where N is the number of observations for the risk and K = EPV / VHM. Multiplying the numerator and denominator by (VHM / N) gives an alternative form: VHM Z = . EPV VHM + N Note that the denominator is just Var[X]as derived a few lines earlier. Therefore Z = N / (N+K) can be written as Variance of the Hypothetical Means Var [µ(Θ)] Θ Z = = . TotalVariance of the Estimator X Var[X] The numerator is a measure of how far apart the means of the risks in the population are, while the denominator is a measure of the total variance of the estimator. The credibility weighted estimate for µ(θ) = E [X│θ], for t =1, 2, ..., N, N+1, ... X | Θ t is 10