This paper focuses on modeling surrender time for policyholders in the context of life insurance. In this setup, a large lapse rate at the first months of a contract is often observed, with a decrease in this rate after some months. The modeling of the time to cancelation must account for this specific behavior. Another stylized fact is that policies which are not canceled in the study period are considered censored. To account for both censoring and heterogeneous lapse rates, this work assumes a Bayesian survival model with a mixture of regressions. The inference is based on data augmentation allowing for fast computations even for datasets of over millions of clients. Moreover, frequentist point estimation based on Expectation–Maximization algorithm is also presented. An illustrative example emulates a typical behavior for life insurance contracts, and a simulated study investigates the properties of the proposed model. A case study is considered and illustrates the flexibility of our proposed model allowing different specifications of mixture components. In particular, the observed censoring in the insurance context might be up to $50\%$ of the data, which is very unusual for survival models in other fields such as epidemiology. This aspect is exploited in our simulated study.

The objective of this work was to estimate genetic parameters for a measure of persistency of milk yield and to evaluate its association with 305-d cumulative milk yield and lactation length. 12 346 records from 8202 dairy Gyr cows including lactations up to fifth calving were used. The measure of persistency was obtained from one of the parameters of a quadratic model that describes the cumulative yield across lactation as a function of days in milk. A three-trait multivariate analysis was done. Heritability and repeatability for persistency were 0·08 and 0·21, respectively. Deviance Information Criterion provided evidence that the additive genetic covariance between the measure of persistency studied and 305-d cumulative yield is zero. Genetic correlations between persistency and lactation length were 0·50 and 0·27 for first or all lactations, respectively. Milk yield persistency as measured in this study has low heritability. Selection for persistency can increase lactation length. The measure of milk yield persistency studied here is genetically independent of total milk yield and can be included in routine genetic evaluations of dairy cattle.

Twelve spring-calving and twelve winter-calving cows were managed for extended lactation cycles of 18-months duration, with the former group then completing a second extended lactation. Half of the cows were fed according to standard management practice for the herd; the other half received supplementary concentrate from week 9 of lactation onwards. Commencing at the same time, half of the udder of each cow was subjected to increased milking frequency (thrice daily rather than twice daily). Lactation persistency (and hence total milk yield) was significantly increased by frequent milking. Winter calving cows and supplemented cows also exhibited better persistency, but this was only evident up until the point of re-breeding, at around lactation week 33. Milk composition was measured in the spring-calving cows in both their first and second extended lactations. Composition altered during the course of the lactation, protein and fat percentages increasing and lactose percentage decreasing, irrespective of treatment. The quality of the milk for processing into cheese, fermented products, heat-treated products and cream liqueurs was assessed by calculation of casein number (casein protein as a proportion of total protein). Processing quality declined across the course of lactation in those groups that showed poor persistency but not in those that maintained a persistent lactation. Milk hygienic quality (somatic cell counts) showed parallel changes. Body condition score increased during the course of lactation but was not affected by supplementation; none of the cows became excessively fat. All cows remained healthy throughout the extended lactations and the majority (33/36) re-bred successfully. By demonstrating that lactation persistency is plastic and can be improved by simple management interventions, the results lend support to the economic arguments in favour of extended lactation cycles. The likely welfare benefits of extended lactation are also discussed.

Policyholders often decide to buy, renew, or cancel insurance based on the premium charged by the insurer compared with what they expect their claims will be. It is important for actuaries to consider the persistency of policyholders because the financial well-being of the insurer depends on spreading its risk over a large book of business. We use statistical decision theory to develop premium formulas that account for the past experience of a given policyholder, the experience of the entire collection of policyholders, and the likelihood of the policyholder renewing with or buying from a given insurer, that is, persistency.

We assume that the persistency of policyholders depends on the arithmetic difference between the premium charged and their anticipated claims. We extend the work of Taylor (1975) in which he obtains linear credibility formulas by minimizing loss functions that incorporate the persistency of policyholders. We consider Taylor's loss functions and other objective functions, including those that account for the amount of business the insurer writes or renews.

We consider an infinite Markov chain with states E0, E1, …, such that E1, E2, … is not closed, and for i ≧ 1 movement to the right is limited by one step. Simple algebraic characterizations are given for persistency of all states, and, if E0 is absorbing, simple expressions are given for the probabilities of staying forever among the transient states. Examples are furnished, and simple necessary conditions and sufficient conditions for the above characterizations are given.