Enrique de Alba

Bayesian Estimation of Outstanding Claim Reserves

This paper presents a Bayesian approach to forecasting outstanding claims, either the total number of claims or the total amount, that is used for claims reserving. The assumption is made that there is complete information for one or two past years of origin and partial information for some development years of other years of origin. It also assumes payments are made annually and that the development of partial payments follows a stable payoff pattern from one year of origin to another. Two different models are presented: one for the number of claims (intensity) and one for claim amounts (severity). The advantage of using this procedure is that actuaries can derive the complete predictive distribution of the reserve requirements, from which, in turn, it is possible to obtain point estimates as well as probability intervals and other summary measures, such as mean, variance, and quantiles.Abstract This paper presents a Bayesian approach to forecasting outstanding claims, either the total number of claims or the total amount, that is used for claims reserving. The assumption is made that there is complete information for one or two past years of origin and partial information for some development years of other years of origin. It also assumes payments are made annually and that the development of partial payments follows a stable payoff pattern from one year of origin to another. Two different models are presented: one for the number of claims (intensity) and one for claim amounts (severity). The advantage of using this procedure is that actuaries can derive the complete predictive distribution of the reserve requirements, from which, in turn, it is possible to obtain point estimates as well as probability intervals and other summary measures, such as mean, variance, and quantiles.

Forecasting an Accumulated Series Based on Partial Accumulation: A Bayesian Method for Short Series With Seasonal Patterns

We present a Bayesian solution to forecasting a time series when few observations are available. The quantity to predict is the accumulated value of a positive, continuous variable when partially accumulated data are observed. These conditions appear naturally in predicting sales of style goods and coupon redemption. A simple model describes the relation between partial and total values, assuming stable seasonality. Exact analytic results are obtained for point forecasts and the posterior predictive distribution. Noninformative priors allow automatic implementation. The procedure works well when standard methods cannot be applied due to the reduced number of observations. Examples are provided.

Stationarity and structural breaks -- evidence from classical and Bayesian approaches

Abstract The purpose of this paper is to analyze and compare the results of applying classical and Bayesian methods to testing for a unit root in time series with a single endogenous structural break. We utilize a data set of macroeconomic time series for the Mexican economy similar to the Nelson–Plosser one. Under both approaches, we make use of innovational outlier models allowing for an unknown break in the trend function. Classical inference relies on bootstrapped critical values, in order to make inference comparable to the finite sample Bayesian one. Results from both approaches are discussed and compared.

Disaggregation and Forecasting: A Bayesian Analysis

The problem of temporal disaggregation of time series is analyzed by means of Bayesian methods. The disaggregated values are obtained through a posterior distribution derived by using a diffuse prior on the parameters. Further analysis is carried out assuming alternative conjugate priors. The means of the different posterior distributions are shown to be equivalent to some sampling theory results. Bayesian prediction intervals are obtained. Forecasts for future disaggregated values are derived assuming a conjugate prior for the future aggregated value.

Constrained forecasting in autoregressive time series models: A Bayesian analysis

Abstract A Bayesian approach is used to derive constrained and unconstrained forecasts in an autoregressive time series model. Both are obtained by formulating an AR( p ) model in such a way that it is possible to compute numerically the predictive distribution for any number of forecasts. The types of constraints considered are that a linear combination of the forecasts equals a given value. This kind of restriction is applied to forecasting quarterly values whose sum must be equal to a given annual value. Constrained forecasts are generated by conditioning on the predictive distribution of unconstrained forecasts. The procedures are applied to the Quarterly GNP of Mexico, to a simulated series from an AR(4) process and to the Quarterly Unemployment Rate for the United States.

An empirical Bayes approach to outliers

Abstract A formulation of the problem of detecting outliers as an empirical Bayes problem is studied. In so doing we encounter a non-standard empirical Bayes problem for which the notion of average risk asymptotic optimality (a.r.a.o.) of procedures is defined. Some general theorems giving sufficient conditions for a.r.a.o. procedures are developed. These general results are then used in various formulations of the outlier problem for underlying normal distributions to give a.r.a.o. empirical Bayes procedures. Rates of convergence results are also given using the methods of Johns and Van Ryzin (1971, 1972).

Measurement and Transfer of Catastrophic Risks. A Simulation Analysis

When analyzing catastrophic risk, traditional measures for evaluating risk, such as the probable maximum loss (PML), value at risk (VaR), tail-VaR, and others, can become practically impossible to obtain analytically in certain types of insurance, such as earthquake, and certain types of reinsurance arrangements, specially non-proportional with reinstatements. Given the available information, it can be very difficult for an insurer to measure its risk exposure. The transfer of risk in this type of insurance is usually done through reinsurance schemes combining diverse types of contracts that can greatly reduce the extreme tail of the cedants loss distribution. This effect can be assessed mathematically. The PML is dei ned in terms of a very extreme quantile. Also, under standard operating conditions, insurers use several “layers” of non proportional reinsurance that may or may not be combined with some type of proportional reinsurance. The resulting reinsurance structures will then be very complicated to analyze and to evaluate their mitigation or transfer effects analytically, so it may be necessary to use alternative approaches, such as Monte Carlo simulation methods. This is what we do in this paper in order to measure the effect of a complex reinsurance treaty on the risk profile of an insurance company. We compute the pure risk premium, PML as well as a host of results: impact on the insured portfolio, risk transfer effect of reinsurance programs, proportion of times reinsurance is exhausted, percentage of years it was necessary to use the contractual reinstatements, etc. Since the estimators of quantiles are known to be biased, we explore the alternative of using an Extreme Value approach to complement the analysis.

“Principal Applications of Bayesian Methods in Actuarial Science: A Perspective”, Udi E. Makov, October 2001

title “Credibility Procedures: Laplace’s Generalization of Bayes’ Rule and the Combination of Collateral Knowledge.” In this paper, he presents cogent criticisms of existing statistical methods, quotes Price’s introduction to Bayes’ paper and discusses Laplace’s extension. He also applies these ideas to credibility. Most of the credibility ideas developed in the 1960s and 1970s have visible roots here. For example, the use of least squares regression lines to approximate posterior means is a thread running through much of the analysis. The fact that these least squares are exact for several common examples is illustrated. All of this was put forth over 20 years before Ericson (1970) and Jewell (1974) established that, when a conjugate prior distribution (the prior and posterior distributions are of the same type) is used with a likelihood (data distribution) that is a member of the linear exponential family of distributions, then the least squares linear approximation to the posterior mean is exact. Because he was a self-taught Bayesian, and many of our present day concepts and notation were in the future, Bailey is hard to read. Nevertheless, his work demonstrates that Bayesian ideas were an important ingredient of credibility theory before 1967.