Transcript Retention Based on a Survival Model

Retention Based on a Survival
Constant Force Model
A Life Actuary’s Approach
What is a survival model?
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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:
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3)
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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.
Survival Model Graph
Sample Survival Model
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Time
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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.
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.
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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6 Mth Obs 22.36
1 Mth Obs 22.85
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Time
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S(x)
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Things to consider in the Model
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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.
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Assume something besides constant force:
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Uniform distribution of lapse
Balducci assumption
Analytical laws of lapses (Gompertz, Makeham, etc)
Summary
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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:
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Survival Models by Dick London
Actuarial Mathematics by Bowers
www.soa.org, Society of Actuaries website.