longevity risk: recent development and actuarial implications

Longevity Risk: Recent Developments and Actuarial Implications

March 2, 2013

The Chinese University of Hong Kong

(A SOA Center of Actuarial Excellence)

Introduction to Stochastic Mortality Models

Wai-Sum Chan, FSA, CERA, Ph.D.

Introduction to Stochastic Mortality Models

Wai-Sum Chan, FSA, CERA, PhDProfessor of Finance

The Chinese University of Hong Kong

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Outline

Longevity Risk An OverviewSome TrendsThe Role of the Law of Large Numbers

Mortality Improvement ScalesScale AAThe VBT

Stochastic ModelingConsiderationsThe Lee-Carter ModelThe Cairns, Blake and Dowd ModelMeasuring Uncertainty

Some ApplicationsStochastic SimulationsOutliersEstimation of

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Longevity Risk An Overview

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Life Expectancy at Birth

69

71

73

75

77

79

81

83

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

Year

L if e

Ex p

e ct a

n cy

a t B

i r th

Value ofoe0, Canadian Females, 19502004

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Life Annuity at Age 65

14

15

16

17

18

19

20

21

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000Year

An n

u it y

Ra t

e

Values of a65 at i = 2%, Canadian Females, 19502004

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Crude & Graduated Death Rates, Males, Age 60

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Doesnt the Law of Large Numbers Solve theProblem?

Traditional mortality risk: Assumes the survival distribution is fixed and known. The risk is the random fluctuations around the fixed and known

distribution.

The law of large number works.

Longevity risk: The risk is the error in estimating future mortality. Affects all policies in force. The law of large numbers does NOT work.

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Mortality Improvement Scales

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What Are Improvement Scales?

A simple formula for projecting future mortality.

Applied to base mortality rates.

That is,qx ,t = qx ,0IS(x , t),

where s is the number of years from time-0.

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Scale AA

Developed by SoAs Group Annuity Valuation Task Force in 1995.

Assumes that each mortality rate are reduced by a fixed percentageeach year.

Mathematically,IS(x , s) = (1 AAx)s ,

where s is the number of years from now.

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Scale AA

Some values of AAx :Age (x) Males Females

55-59 1.6-1.9% 0.5-0.8%60-64 1.4-1.6% 0.5%65-69 1.3-1.4% 0.5%

E.g., Estimate of q69 for females in 2009 is

q69,2007 (1 0.5%)2.

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The VBT

An improvement scale is used in the development of the 2001valuation basic experience tables (VBT).

The scale is used to project 1990-95 mortality experience to 2001.

For males:

IS(x , s) =

0, x < 45(1 0.01(x45)10

)s, 45 x < 55

0.99s , 55 x < 80(1 0.01(90x)5

)s, 80 x < 90

0, x > 90

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The VBT

For females:

IS(x , s) =

0, x < 45(1 0.005(x45)10

)s, 45 x < 55

0.995s , 55 x < 85(1 0.005(90x)5

)s, 85 x < 90

0, x > 90

E.g., Estimate of q69 for females in 2009 is

q69,2007 (1 0.5%)2.

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How Accurate Are They?

Estimated/Actual reductions of female Canadian mortality:

Age group Scale AA VBT Actual (9605)

60-64 0.5% 0.5% 1.67%65-69 0.5% 0.5% 1.53%70-74 0.5-0.7% 0.5% 1.50%75-79 0.7-0.8% 0.5% 1.41%

A similar problem is also seen in the UK Continuous MortalityImprovement Bureau (CMIB) 92 Series improvement scales; soas in the US 2008 VBT studies.

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Stochastic Modeling

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Trends of Death Rates

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1950 1960 1970 1980 1990 2000

Year

Ce n

t ra l

De a

t h R

a te

i n L

o g S

c al e

Age 0Age 10Age 20Age 30Age 40Age 50Age 60Age 70Age 80Age 90

Central Death Rates, Canadian Females, 19502004

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What Do We Need to Consider?

In building a stochastic model, we should:

1. allow rates at different ages to move at different speeds;

2. consider the correlations between rates at different ages;

3. take account of the randomness in mortality reduction;

4. prevent the model from resulting in negative death rates.

A poorly built model may lead to an anti-intuitive forecast.

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The Lee-Carter Model

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The Lee-Carter Model: Specification

Invented in 1992 by Ronald Lee and Lawrence Carter.

One of the most popular stochastic mortality models.

Used in projections of the U.S. social security system and the U.S.federal budget.

Mathematically,

ln(mx ,t) = ax + bxkt + x ,t ,

where mx ,t is the central death rate at age x and in year t.

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ln(mx ,t) = ax + bxkt + x ,t

ax an age-specific parameter; the set of {ax , x = 0, 1, ...} reflectsthe general shape of the mortality schedule.

kt a time-varying parameter; the time-trend of kt signifies thegeneral speed of mortality improvement.

bx an age-specific parameter which characterizes the sensitivityof to kt at age x .

x ,t the error-term, which has no long-term trend.20 of 53


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Model Parameters

Estimates of model parameters ax , bx , and kt .

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The Lee-Carter Model: Forecasting

The pattern of kt is highly linear.

To make a forecast, we extrapolate kt using an AutoregressiveIntegrated Moving Average (ARIMA) model.

In most cases, an ARIMA(0,1,0) model,

kt = c + kt1 + x ,t ,

where c is the drift term and x ,t is the error term, gives anadequate fit.

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A Lee-Carter Forecast

1960 1970 1980 1990 2000 2010 2020 2030 2040 20503.2

3

2.8

2.6

2.4

2.2

2

Year

Mor

talit

y ra

te (in

log s

cale)

Central forecast of the central death rate at age 85.

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Lee-Carter Modelling: A Beginners Guide

A Freeware for Lee-Carter Modelling lcfit.demog.berkeley.edu

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Lee-Carter Modelling: A Beginners Guide

Some Sample Datasets lcfit.demog.berkeley.edu

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The Cairns, Blake and Dowd Model

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The Cairns, Blake and Dowd Model: Specification

Invented in 2006 by Andrew Cairns, David Blake and Kevin Dowd(CBD).

Model specification:

qx ,t =eA1(t)+A2(t)(x)

1 + eA1(t)+A2(t)(x),

where qx,t is the death probability at age x and time t, {A1(t)} and {A2(t)} are discrete-time stochastic processes.

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Alternatively, the model can be written as

lnqx ,t

1 qx ,t = A1(t) + A2(t)(x).

A1(t): affects all ages by the same amount; represents the overall mortality level at time t.

A2(t): affects different ages differently; the steepness of the mortality curve at time t.

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1960 1980 2000 202010.4

10.2

10

9.8

9.6

9.4

9.2

9

8.8

Year

A1

1960 1980 2000 20200.082

0.084

0.086

0.088

0.09

0.092

0.094

0.096

Year

A2

Estimated values of A1(t) and A2(t).

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The Cairns, Blake and Dowd Model: Forecasting

The pattern of A1(t) and A2(t) are highly linear.

To make a forecast, we extrapolate A(t) = (A1(t),A2(t)) with abi-variate random walk with drift:

A(t + 1) = A(t) + + CZ (t + 1),

where is a constant 2 1 vector, C is a constant 2 2 upper triangular matrix, Z (t) is a 2-dimensional standard normal random variable.

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A Cairns, Blake and Dowd Forecast

1960 1970 1980 1990 2000 2010 2020 2030 2040 2050

3

2.8

2.6

2.4

2.2

2

Year

Mor

talit

y ra

te (in

log s

cale)

Central forecast of the central death rate at age 85.

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Quantifying Uncertainty

Stochastic uncertainty Can be estimated by simulating the random error terms.

Parameter uncertainty Can be estimated by the bootstrap or Markov Chain Monte Carlo.

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Some Applications

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Application 1: Stochastic Simulations

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Sample Paths

2010 2015 2020 2025 2030 2035 2040 2045 2050 20553.3

3.2

3.1

3

2.9

2.8

2.7

2.6

Year

Mor

talit

y ra

te (in

log s

cale)

Path 1Path 2

Sample paths of the central death rate at age 85.

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An Interval Forecast

1960 1970 1980 1990 2000 2010 2020 2030 2040 2050

3.2

3

2.8

2.6

2.4

2.2

2

Year

Mor

talit

y ra

te (in

log s

cale)

Interval forecast of the central death rate at age 85.

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Application of Sample Paths

17.2 17.4 17.6 17.8 18 18.2 18.4 18.60

0.5

1

1.5

2

2.5

3

Annuity value

Den

sity

Simulated distribution of the life annuity value for a person aged 65i.e., values of a65 at i = 2%, Canadian Females.

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Application 2: Outliers in the Lee-Carter Model

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Outliers in Mortality Trend

Interrupting phenomena are commonly encountered in time-seriesdata analysis with the study of mortality trends being noexception.

Pandemics, for instance, can bring about an immeasurable numberof deaths.

The so-called Spanish flu in the early 20th century caused anestimated 40 to 50 million deaths worldwide, and was followed bythe Asian flu in the 1950s, the Hong Kong flu in the 1960s, andmore recently SARS and the H5N1 avian flu.

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Outliers in Mortality Trend

These relatively infrequent shocks (or interrupting phenomena)often create discrepant observations in the time-series of mortalitytrends which, in the statistical literature, are usually referred to asoutliers.

It is important that actuaries be aware of the frequency, timing andpersistence of outliers and their effect in terms of vacillations inhuman mortality trends.

Outliers have adverse effects on the Lee-Carter modelling, actuariesmight wish to clean the data (Li and Chan, 2005, ScandinavianActuarial Journal).

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Outliers in UK Mortality Trend

-9

-8

-7

-6

-5

-4

-3

-2

-1

1841 1851 1861 1871 1881 1891 1901 1911 1921 1931 1941 1951 1961 1971 1981 1991

Year

- l n( m

x)

Age 0

Age 2

Age 65

Age 75

lnmx for the England and Wales mortality data, 1841-2000

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Outliers in UK Mortality Trend

1

0

100,000

200,000

300,000

400,000

500,000

600,000

700,000

1901 1911 1921 1931 1941 1951 1961 1971 1981 1991

75 and over65-7455-6445-5435-4425-3415-245-141-4< 1

1918

1915 1929

1940

1942

1948

19511980

1987

2000

Number of deaths per year (thousands), by age group, England and Wales,1901-2000

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Application 3: Estimation of

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Growing oldest-old populations

Stochastic mortality models can be used to predict the highestattained age, which is commonly referred to as omega or in theactuarial literature.

At the time of the 2006 Census, there were 3,154 centenarians(i.e., age > 100) in Australia.

This number is expected to increase to 17,408 by 2028. This increase poses different challenges to actuaries, economists

and policy planners.

Prime examples include life annuities, defined-benefit pension plansand reverse mortgages.

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Australia's oldest citizen celebrates her 112th birthday

Andra Jackson October 14, 2008

Piece of cake: Bea Riley, (centre) celebrates her 112th birthday with (left to right) her niece Bid Riley, nursing home manager

Andreas Kazacos, son Cliff Riley and his wife Jueno. Photo: Penny Stephens

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Results Li, Ng and Chan (2009, MATCOM)

Table 1. Predicted limiting agesCountry Central estimate 95% Confidence intervalAustralia 112.20 (108.09, 116.34)

New Zealand 109.43 (105.46, 113.40)

Table 2. Validated supercentenarians in Australia and New Zealandas of Nov. 22, 2008

Country Name Current status Current age /age at death

Australia E. Beatrice Riley Alive 112Australia Myrtle Jones Alive 111Australia Doreen Washington Alive 110Australia Myra Nicholson Died 2007 112

New Zealand Florence Finch Died 2007 113

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Results (Updated)

Table 1. Predicted limiting agesCountry Central estimate 95% Confidence intervalAustralia 112.20 (108.09, 116.34)

New Zealand 109.43 (105.46, 113.40)

Table 2. Validated supercentenarians in Australia and New Zealandas of Nov. 1, 2011

Country Name Current status Age at deathAustralia E. Beatrice Riley Died May 15, 2009 112.58Australia Myrtle Jones Died Jan 12, 2009 111.74Australia Doreen Washington Died Feb 4, 2009 110.70Australia Myra Nicholson Died 2007 112

New Zealand Florence Finch Died 2007 113

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Summing up...

Two types of stochastic mortality models are introduced.

They give both central and interval mortality projections.

Available software and datasets for the Lee-Carter modelling.

We need these models for pricing longevity securities.

The models are also useful for estimating hedge effectiveness.

Three applications of stochastic mortality models have beendiscussed.

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Thank you!

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Cover PageWai-Sum Chan

longevity risk: recent development and actuarial implications

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