Mortality Product Development Symposium 2008
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Transcript of Mortality Product Development Symposium 2008
watsonwyatt.com
2008 Product Development Actuary Symposium
The Changing Face and Pace of Mortality: An Enlightened View of Annuitant Mortality
Yuhong (Jason) Xue, FSA, MAAAMay 5, 2008
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2
Agenda
Measurement of Current Mortality Experience– Apply predictive modeling techniques currently used in
the P&C industry Development of Mortality Improvement Trend
– Advanced mathematical models to project future trends based on historical trends of mortality improvement
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Measurement of Current Mortality Experience
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Current Approach (Experience Study)
Focus on limited risk factors that impact mortality – Age, Sex, may extend to other factors (i.e. amount, marital
status, and geographical location)– Calculate A/E ratio with slicing and dicing techniques to come
up with a set of weights (or multipliers) for each factor to be applied to a basic age/sex table
Limitations– Mortality is simultaneously impacted by all risk factors and has
to be analyzed with all factors together– “true” weights for a factor are influences by that factor alone,
assuming all else equal– Fails to capture the “true” weights for each factor
Calls for more sophisticated mathematical approach
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Predictive Modeling
Statistical model that relates an event (death) with a number of risk factors (age, sex, YOB, amount, marital status, etc.)
Amount
Y.o.B.
Age
etc.
Sex
Married
Expected mortality
Model
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Generalized Linear Models (GLMs)
Special type of predictive modelling A method that can model
– a number
as a function of – some factors
For instance, a GLM can model– Motor claim amounts as a function of driver age, car type, no
claims discount, etc …– Motor claim frequency (as a function of similar factors)
Historically associated with non-life personal lines pricing (where there was a pressing need for multivariate analysis)
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E[Y] = = g ( X )-1
Observed thing(data)
Some function(user defined)
Some matrix based on data(user defined)
as per linear models
Parameters to beestimated
(the answer!)
Generalised linear models
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8
Bedtime reading
CAS 2004 Discussion Paper Programme
Copies available atwww.watsonwyatt.com/glm
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9
Examples
Examples Using GLMs to Analyze Annuitant Mortality Based on a test dataset created to simulate a typical
company’s portfolio of retirees currently receiving benefits
Results are merely for illustration purposes
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How to Read the Graphs
All graphs show relative Qx of different categories of one factor against a base level identified by “0%” label. Qx for other levels are “x%” higher than the base level.
Colors– Green: GLM results– Orange: “One-way” relatives are
the relative death rates for the factor before considering other factors simultaneously.
– Blue: 95% confidence interval. Tight confidence interval indicates statistical significance.
Exposure– The amount of exposure for a
category is indicated by the bar on the x-axis.
Generalized Linear Modeling IllustrationAnnual Income Effect
-29%
-17%
-14%
-6%
0%
-0.36
-0.3
-0.24
-0.18
-0.12
-0.06
0
0.06
Income
Log
of m
ultip
lier
0
200000
400000
600000
800000
1000000
1200000
1400000
1600000
<= 30K <= 50K <= 75K <= 100K > 100K
Exp
osur
e (y
ears
)
Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate
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Example 1: Effect of Annuity Amount
Results show evidence of reduced mortality with increased benefits
Generalized Linear Modeling IllustrationIncome Effect
-29%
-18%
-15%
-6%
0%
-0.36
-0.3
-0.24
-0.18
-0.12
-0.06
0
0.06
Income
Log
of m
ultip
lier
0
200000
400000
600000
800000
1000000
1200000
1400000
1600000
<= 30K <= 50K <= 75K <= 100K > 100K
Exp
osur
e (y
ears
)
Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate
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Example 2: Calendar Year Trend
Mortality improvements 1% per annum over previous six years
Generalized Linear Modeling IllustrationRun 1 Model 2 - GLM - Significant
0%
1%
2%
4%4%
5%
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Calendar year
Log
of m
ultip
lier
0
100000
200000
300000
400000
500000
600000
700000
2002 2003 2004 2005 2006 2007
Exp
osur
e (y
ears
)
Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate
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Example 3: The Selection Effect
Selection effect is not conclusive
Generalized Linear Modeling IllustrationRun 1 Model 2 - GLM - Significant
0%
-3%
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
Duration
Log
of m
ultip
lier
0
500000
1000000
1500000
2000000
2500000
3000000
<=5 5+
Exp
osur
e (y
ears
)
Approx 95% confidence interval Smoothed estimate
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Example 4: Birth Cohort Effect
Generalized Linear Modeling IllustrationBirth Cohort
0%
4%
-1%
5%5%
7%
5%5%4%
3%
-1%
2%
-4%
0%-1%
-2%
-1%
1%
-2%-1%
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Log
of m
ultip
lier
0
100000
200000
300000
400000
500000
<= 1915 <= 1918 <= 1921 <= 1924 <= 1926 <= 1928 <= 1931 <= 1933 <= 1936 <= 1940
Exp
osur
e (y
ears
)
Smoothed estimate, Sex: M Smoothed estimate, Sex: F
No Cohort Effect for male and Female
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Example 5: Effect of Joint Life Status
Evidence of “broken heart syndrome” which may influence pricing
Generalized Linear Modeling IllustrationJoint Survivor Status
3%
-4%
0%
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Log o
f m
ulti
plie
r
0
500000
1000000
1500000
2000000
2500000
Single Life Joint Life Primary Joint Life Surviving Spouse
Exp
osu
re (
yea
rs)
Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate
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Questions we wish to answer
How do different factors influence mortality? What is the cohort effect in my portfolio? What trends are there? How can I be sure of an accurate portfolio cash flow
valuation?
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Development of Mortality Improvement Trend
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Introduction
Current demographic trends should lead to accelerating growth in payout annuities
– Aging population
– Longevity Potential rates of mortality improvement are key
drivers in the profitability of this business Current industry experience of mortality improvement
is limited and dated
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The Potential Impact of Longevity
•General Perception that investment assumption more critical than mortality
•Consider the impact of differences in assumption in terms of basis points
10-year Annuity Life AnnuityRP2000_AA 0% 0.00%RP2000_LCmean -0.07% -0.11%RP2000_LC95th -0.01% 0.00%RP2000_LC5th -0.13% -0.22%
Basis Point Impact for Male Age 65
Mortality Table
Male Age 65
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
2007
2008
2009
2010
2011
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
LC_95th Percentile LC_Mean LC_5th Percentile Scale AA
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Existing Improvement Projection
Historical US approach - single vector scale, deterministic
Published Scales– Scale AA: Experience from 1977-1993
– Scale G, Scale H, etc.
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Mortality Models Characteristics
Regression method traditionally used
– Endpoint/Slope Method
Stochastic Analysis of Historical Trends
– Lee-Carter Method
Two dimensional regression apply to historical data, particularly useful for the measurement of cohort effect
– P-Spline Method
Predetermined mathematical formulas fit to historical data
– Relational models: Logistic, Weibull
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Mortality Models Characteristics - Continued
CMIB (UK based mortality research group) recommends for mortality projection
– Lee-Carter, P-spline Well recognized in the U.S., has given plausible results for
various countries (U.S., G7, Norway, Finland, Sweden, Denmark, Chile etc.).
– Lee-Carter Widely used in life data analysis
– Weibull Has been applied to mortality experience at older ages in
different developed countries– Logistic
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Data Sources for Projecting Rates of Mortality Improvement
No suitable insured data available in public domain Potential to base projections on insurance company data Results of projection based on insurance company data may be
modified based on population data U.S. population data
– Population exposure data from the Census Bureau– Population death records from the Centers for Disease
Control and Prevention– Per Capita Income Level Data from the Census Bureau– Education Level Data from the Department of Agriculture
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Typical Data Requirements: Insurance Company
P-Spline & Lee-Carter– Minimum period of 20 calendar years– Minimum age-range of 40 years– Minimum exposure of 1000 lives and deaths of 30 in each data cell
by age and year
Logistic & Weibull– Minimum 20 years of data
In the final analysis, the ultimate data requirements will depend upon the historical trend of the underlying data.
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Appendix
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GLM Results: Multipliers for Each Factor
We are modelling Probability of death in year =
Base level for observed population ×Factor 1 (based on age) × Factor 2 (based on sex) × Factor 3 (based on amount) …
So the model results will be:– One number (the base level – everything else will be relative to this
– eg the mortality of a 65 yr male with pension $1000)– A series of multiplicative coefficients for Factor 1 (age)– A series of multiplicative coefficients for Factor 2 (sex)– A series of multiplicative coefficients for Factor 3 (amount) …
We prefer to look at these multiplicative coefficients via graphs! The factor results are only valid in their totality – we cannot take results
for one factor and use those in isolation
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Multipliers are Expressed as Relativities
Normal (absolute)
Age x qx65 0.00877466 0.01021667 0.01186268 0.01373769 0.01586370 0.01826771 0.02097672 0.02401773 0.02741874 0.03120975 0.035418
We are used to seeing mortality rates (etc) presented as ‘absolute’ numbers (qx etc)
With GLMs, results are shown as multiplicative relativities
Multiplicative relativities
Age x qx Relativity65 0.008774 100%66 0.010216 116%67 0.011862 135%68 0.013737 157%69 0.015863 181%70 0.018267 208%71 0.020976 239%72 0.024017 274%73 0.027418 312%74 0.031209 356%75 0.035418 404%
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Factors “true” Weights Applied to Standard Mortality Table to Arrive at Mortality Assumption
Age Qx Age Qx
22 0.000599 IncomeMarrital Status 22 0.000719
23 0.000627 23 0.00075224 0.000657 24 0.00078825 0.000686 25 0.00082326 0.000714 <=10k 195.97% Single 120.00% 26 0.00085727 0.000738 <=20k 167.17% Married 100.00% 27 0.00088628 0.000758 <=30k 142.35% Divorced 140.00% 28 0.0009129 0.000774 <=40k 124.58% Widowed 160.00% 29 0.00092930 0.000784 <=50k 111.80% 30 0.00094131 0.000789 >50k 100.00% 31 0.00094732 0.000789 32 0.00094733 0.00079 33 0.00094834 0.000791 34 0.00094935 0.000792 35 0.0009536 0.000794 36 0.00095337 0.000823 37 0.00098838 0.000872 38 0.00104639 0.000945 39 0.00113440 0.001043 40 0.001252
Factor level
WeightsFactor
levelWeights
Standard Table
Table for Income>50k &
Single