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.• 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:

1) S(0) = 1

2) S(x) does not increase as x increases

3) As x gets large, S(x) goes to 0

• S(x) is a model of retention.

Survival Model GraphSample Survival Model

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Time

S(x)

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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.

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• 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.

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.

Retention curve based Constant force model

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Time

S(x) 6 Mth Obs 22.36

1 Mth Obs 22.85

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:1. Uniform distribution of lapse

2. Balducci assumption

3. Analytical laws of lapses (Gompertz, Makeham, etc)

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.