Construction of the German Annuity Table DAV 2004 R

Dr. Ralf Krüger

Agenda

Dr. Ralf Krüger – DAV 2004 R

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Jeanne Calment (1875 –1997)

 Sold her house to a 46-year-old notary for an annuity when she was 90  The buyer died at the age of 77  Jeanne Calment survived him and became the oldest person of the world. She died at the age of 122!

Dr. Ralf Krüger – DAV 2004 R

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Longevity – International comparison Source: http://www.un.org/esa/population

44 46 Dr. Ralf Krüger – DAV 2004 R 64 66 68 69 69 69 84 84 83 83 82 88

1950/55 2045/50 4

Longevity - Germany

80 60 39 40 36 20 1870 1905 Life expectancy at birth Census data Females Males 1940 Dr. Ralf Krüger – DAV 2004 R 1975 82 76 2010 Year

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The German annuity market Growing market

 46% of premiums of new business (2003)  Number of policies in force: 5% (1996)  16% (2003)

Two basic product concepts

 Immediate annuities  Deferred annuities (often including guaranteed annuity rates at outset and lump-sum payment option) Dr. Ralf Krüger – DAV 2004 R

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Starting point for DAV 2004 R (1) Mortality tables

 Valuation tables also used for pricing purposes  4 th German annuity valuation table since the 1950s

Data sources

 Munich Re’s and Gen Re’s data pools (13.7 million years’ exposure)  Population mortality tables  Data from social insurance Dr. Ralf Krüger – DAV 2004 R

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Starting point for DAV 2004 R (2)

100% 75% Mortality rate of 65-year-old male (in % of rate for 1970) previous annuity table population data 50% 1970 1975 1980 1985 1990 Dr. Ralf Krüger – DAV 2004 R 1995 2000

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Agenda

Introduction Construction of DAV 2004 R  Base tables  Mortality projections International Comparisons Dr. Ralf Krüger – DAV 2004 R

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Two-dimensional approach Base tables

 Insured lives mortality in 1999 weighted by amounts  Separate tables for deferment and payout period

Trend

 Appropriate model  Stronger for upper socio-economic group

Safety margins

Dr. Ralf Krüger – DAV 2004 R

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Agenda

Introduction Construction of DAV 2004 R  Base tables  Mortality projections International Comparisons Dr. Ralf Krüger – DAV 2004 R

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Base tables Payout period

 Enough data for ages 60 - 99  Selection effect: see extra slides  Extrapolation to ages 59- using population mortality rates  Extrapolation to ages 100+: see extra slide

Deferment period

 Enough data for ages 65  Extrapolation to ages 65+ using mortality rates of payout period Dr. Ralf Krüger – DAV 2004 R

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Selection Reasons for selection

 Annuities bought mainly from people who consider themselves to be healthy and long-lived  Self-selection worsened by the lump-sum payment option  Higher socio-economic status = lower mortality Main client group for annuities are people with above-average socio economic status Dr. Ralf Krüger – DAV 2004 R

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Selection effect

100% 80% 60% Males Females 1 67% 71% 2 Mortality rates by selection phase relative to ultimate mortality Males Females 3 4 88% 80% Dr. Ralf Krüger – DAV 2004 R 5 6+ 100% 100%

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Extrapolation for ages 100+

 Insured lives data insufficient  No population data available  Following the method in [TKV], we examine six approaches:  Fit all six models to the actual mortality rates at ages 85 to 95  Assess accuracy of the models at ages 96 to 99  Evaluate extrapolation  Compare with data from Japan Logistic model used for extrapolation [TKV] - Thatcher, Kannisto, Vaupel: The force of mortality at Ages 80 to 120 Dr. Ralf Krüger – DAV 2004 R

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Safety margins for the base tables Margin for level parameter risk

 Differences between the observed and actual portfolio (structure, mortality level)  Structural differences between observed portfolio and future new business  Statistical fluctuation of the observed portfolio  Reduction by 10%

Margin for risk of random fluctuation

 Protection against a maximum loss at a defined prognosis level  6.3% (males) and 7.2% (females) Dr. Ralf Krüger – DAV 2004 R

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Risk of random fluctuation Reduction s for the risk of random fluctuation

based on a model portfolio of 100,000 males resp. females with an age distribution typical for a German insurance portfolio so that

P(

Σ z

T

z Σ z

L

z

 q )

z

 95%

where L z T z reserve of insured persons aged z in the model portfolio random variable of the released reserve of deaths aged z within a year Dr. Ralf Krüger – DAV 2004 R

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Base mortality table

ln(q(z)) 1 0,1 0,01 0,001 0,0001 0,00001 0 Mortality rates - Year 1999 payout beginning at 65 Selection period Males Females 20 40 Deferment period 60 80 100 Payout period Dr. Ralf Krüger – DAV 2004 R 120 Age

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Agenda

Introduction Construction of DAV 2004 R  Base tables  Mortality projections International Comparisons Dr. Ralf Krüger – DAV 2004 R

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Models (1)

 Mortality decrease in the past due to  Medical reasons  Changes in nutrition  Better hygiene  Changes in style of living  Improvement in general living conditions  Nobody knows future mortality exactly  Attained age model mostly used Dr. Ralf Krüger – DAV 2004 R

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Models (2) Attained age model

 Trend F(x) dependant on attained age x  q(x,t+1)/q(x,t) = exp(-F(x))

Cohort model

 Trend G(t+1-x) dependant on cohort t+1-x  q(x,t+1)/q(x,t) = exp(-G(t+1-x))

Synthesis model

 Combination of attained age and cohort model Dr. Ralf Krüger – DAV 2004 R

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Models (3)

 Likelihood ratio test: Synthesis model is the best for modeling German population data of the past  However synthesis model not suitable for projecting mortality to the future  Trend of cohort model for late cohorts increasingly uncertain

Attained age model

chosen for projecting mortality Dr. Ralf Krüger – DAV 2004 R

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Attained age model – description (1)

Mortality rates - Males German census data ln(qx) -1 -3 -5 -7 1870 1910 1950 Dr. Ralf Krüger – DAV 2004 R 80 60 40 1990 Year

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Attained age model – description (2)

Mortality rates - Females German census data ln(qy) -1 -3 -5 -7 1870 1910 1950 Dr. Ralf Krüger – DAV 2004 R 80 60 40 1990 Year

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Attained age model – description (3) Assumptions

 Mortality decrease depends on sex  Mortality decrease depends on attained age  The percentage of annual mortality decrease for fixed attained age and fixed sex is time-independent Dr. Ralf Krüger – DAV 2004 R

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Attained age model – formula (1)

q(z,t) = q(z,t 0 ) * exp( –F(z) * (t – t 0 )) q(z,t) exp( –F(z)) t 0 mortality rate for a person aged z in calendar year t annual decrease factor base year q(z,t 0 ) q(z,t) t 0 Dr. Ralf Krüger – DAV 2004 R t time

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Projection of the mortality trend Trend function F(z)

 Estimated by using linear regression for ln(q(z,t)) = –F(z) * t + B(z)  Based on method of least squares  West German population trends are considered:  Short-term: based on 10 tables 1989/91 to 1998/2000  Medium-term: based on 28 tables 1971/73 to 1998/2000  Long-term: based on 12 tables 1871/80 to 1998/2000 Dr. Ralf Krüger – DAV 2004 R

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Population mortality trends

8% 6% Annual mortality decrease - Males Crude population trends Short-term Medium-term Long-term 4% 2% 0% 0 10 20 30 40 50 60 Dr. Ralf Krüger – DAV 2004 R 70 80 Age

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Trend loading for insured persons

Various international studies have shown:  Mortality decrease of insured persons greater than mortality decrease of the population  Mortality decrease of upper socio-economic groups greater than mortality decrease of lower socio-economic groups

Loading for insured persons:

Increase in annual mortality decrease of 0.2% Dr. Ralf Krüger – DAV 2004 R

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Trend extrapolation for ages 90+

 Based on data from Japan: Annual mortality decrease of 1% for ages ≥ 100  Annual mortality decrease in transition age band 90 to 99 defined by suitable quadratic polynomial Dr. Ralf Krüger – DAV 2004 R

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Linear trend reduction

Trend function F(z,t) with linear reduction F(z,t) Initial trend (short-term level) 1999 1999+ T1 Target trend (75% of medium term level) 1999 + T2 t Dr. Ralf Krüger – DAV 2004 R

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Safety margins for the trend Margin for model risk

 Mortality decrease will not decline in the future  No trend reduction

Margin for trend parameter risk

 Risk of an increase in the mortality improvement trend  Additional 0.25% annual mortality decrease for all ages Dr. Ralf Krüger – DAV 2004 R

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Projection of the mortality trend

Annual mortality decrease DAV 2004 R F(z) 4% 3% 2% 1% 0% 0 Females Males 20 40 60 80 Dr. Ralf Krüger – DAV 2004 R 100 120 Age

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Effect of mortality decrease

Remaining life expectancies according to DAV 2004 R Year of birth Age 0 Age 40 Age 60 Age 80 1960 87

Males

1970 1980 90 93 50 32 15 52 34 16 54 36 18 1960

Females

1970 1980 93 95 97 55 36 18 57 38 19 58 39 21 Dr. Ralf Krüger – DAV 2004 R

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Summary

Portfolio Population Base tables 1999 Level parameter risk Risk of random fluctuation Trend function Model risk Trend parameter risk DAV 2004 R Dr. Ralf Krüger – DAV 2004 R

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Agenda

Introduction Construction of DAV 2004 R  Base tables  Mortality projections International Comparisons Dr. Ralf Krüger – DAV 2004 R

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International comparisons - trends

 Annuity tables with mortality decrease projection for whole future (q(x,t+1)/q(x,t) < 1 for all future t):  Switzerland: ER 2000  UK: IA 92 mc  Austria: AVÖ 2005 R  Swiss table without trend reduction  UK and Austrian table with trend reduction Dr. Ralf Krüger – DAV 2004 R

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International comparisons - trends

4% Annual mortality decrease 2006 - Males Germany DAV 2004 R Switzerland ER 2000 Austria AVÖ 2005R UK IA 92 mc 3% 2% 1% 0% 50 60 70 80 Dr. Ralf Krüger – DAV 2004 R 90 100 Age

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International comparisons - trends

Annual mortality decrease 2006 - Females 4% 3% 2% 1% 0% 50 Germany DAV 2004 R Switzerland ER 2000 Austria AVÖ 2005R UK IA 92 mc 60 70 80 Dr. Ralf Krüger – DAV 2004 R 90 100 Age

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International comparisons – premiums (1) Immediate annuity

 Payment beginning of year  Interest rate 2.75%  Age 60  Year of birth 1945

Net single premiums

(in % of DAV 2004 R) Males Females

Ch

ER 2000 104% 102%

D A

DAV 2004 R AVÖ 2005 R 100% 100% 100% 98%

UK

IA 92 mc

B

MRFR1992 (incl. age shift) 97% 94% 96% 96% Dr. Ralf Krüger – DAV 2004 R

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International comparisons – premiums (2) Deferred annuity

 Deferment period 20 years  Payment beginning of year  Interest rate 2.75%  Age 40  Year of birth 1965

Net single premiums

(in % of DAV 2004 R) Males Females

Ch

ER 2000 105% 100%

D A

DAV 2004 R AVÖ 2005 R 100% 100% 97% 97%

UK

IA 92 mc

B

MRFR1992 (incl. age shift) 93% 85% 93% 89% Dr. Ralf Krüger – DAV 2004 R

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Jeanne Calment (1875 –1997)

Dr. Ralf Krüger – DAV 2004 R

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Acknowledgements

The new German annuity valuation table DAV 2004 R was derived by a DAV committee consisting of Holger Bartel, Marcus Bauer, Bärbel Michaeli, Werner Mörtlbauer, Eberhard Münzmay, Gabriele Nagel, Kornelia Nolle, Catherine Pallenberg, Ulrich Pasdika, Volker Priebe, Michael Rösgen, Esther U. Schütz and Jürgen Wolff. Without the substantial contributions made by every single member of the committee the new table would not have come into being. We are also grateful for the guidance of the steering committee, the “DAV-Arbeitsgruppe Biometrische Rechnungsgrundlagen”.

Parts of this presentation were previously published in a paper, "Coping with Longevity —The New German Annuity Valuation Table” by Ulrich Pasdika and Jürgen Wolff. Copyright 2005 by the Society of Actuaries, Schaumburg, Illinois. Reprinted with permission.

Dr. Ralf Krüger – DAV 2004 R

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Thank you for your interest.

Dr. Ralf Krüger