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Retention Based on a Survival Constant Force Model

Retention Based on a Survival Constant Force Model. A Life Actuary’s Approach. What is a survival model?. Retention, which is interchangeable with survival, has been modeled by life actuaries since the profession was born. Many techniques exist to build and model the survival function.

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Retention Based on a Survival Constant Force Model

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  1. Retention Based on a Survival Constant Force Model A Life Actuary’s Approach

  2. What is a survival model? • Retention, which is interchangeable with survival, has been modeled by life actuaries since the profession was born. • Many techniques exist to build and model the survival function. • Definition: A function S(x), which represents the probability a policy is in force at time x is a survival function if it satisfies these three properties: • S(0) = 1 • S(x) does not increase as x increases • As x gets large, S(x) goes to 0 • S(x) is a model of retention.

  3. Survival Model Graph

  4. Sample model design • The force of mortality u(x) is defined as –S’(x)/S(x) and can be thought of as the instantaneous measure of mortality at time x. X • It can be shown that S(x) = exp[-∫ou(y)dy]. • Assuming u(x) = ux over an entire period, then the MLE estimator of ux = lapses/exact exposures in period x. • Exact exposures = 1 for policies which retain the entire period plus the proportion of survival for each lapse. i.e. .25 for a policy which lapsed ¼ into the period. • Nonrenewal rates are considered separately since they are points of discontinuity in S(x) • MLE for nonrenewal rates is nonrenews/exposure. • Using monthly periods, S(x)=S(i)*exp[-ux*(x-i)] for x between integers i and i+1.

  5. Is Retention Improving? • Compare key values of S(x), S(4), S(6), S(6)/S(6-δ)(i.e. probability of renewal). Many statistics to consider. • Comparison of Forces. Difficult to communicate meaning. • Policy Life Expectancy = ∫S(x)dx = ∑iS(i-1)*q(i)/u(i) over each period i where q(i) is the probability of non-survival in period i given survival to i-1. • Policy Life Expectancy gives a measure of retention in one number, but can give incomplete information.

  6. Retention curve based Constant force model

  7. Things to consider in the Model • Length of period of constant force. • Length of observation period to estimate force. • Other periods of discontinuity. • Credibility, Raw data can be graduated into smoother, more reasonable curve using prior opinion of retention curve properties. • Assume something besides constant force: • Uniform distribution of lapse • Balducci assumption • Analytical laws of lapses (Gompertz, Makeham, etc)

  8. Summary • The constant force survival model gives a simple to use, but statistically sophisticated model, that models the nature of retention. • More information on life tables and survival models can be found in: • Survival Models by Dick London • Actuarial Mathematics by Bowers • www.soa.org, Society of Actuaries website.

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