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

Copyright © Watson Wyatt Worldwide. All rights reserved

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


3

Measurement of Current Mortality Experience


4

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


8

Bedtime reading

CAS 2004 Discussion Paper Programme

Copies available atwww.watsonwyatt.com/glm


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

)



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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

)



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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)



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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

Mortality Product Development Symposium 2008

Documents

pace of mortality

impact mortality age

expected mortality model

evidence of reduced

different factors

limited risk factors

function of similar

data user