Mortality Trend–Model and Analysis

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Mortality Trend–Model and Analysis

May 21, 2013Lina Xu

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1. Methods to model Mortality Improvement

2. Lee Carter Model

3. Model fit/Analysis/Result

4. Industry Mortality Improvement Study

5. Conclusion

Overview

Methods Used for Mortality Improvement Factor

1. SOA2. CMIB3. Other Actuarial Method4. Lee-Carter

Methods Lee-Carter Model Model fit/Analysis/Result Industry Trend Study

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SOA Methods –Formula

This method will have a base rate q1994 and the reduction factor AAx for each age x.

SOA Group Annuity Valuation Table Task Force GAR94 TablesAAx was obtained: 1. Data Source

CSRS for age 25-65 for 1987-93add SSA for age 1-24 and 60-120 for 1977-93

2. Average TrendsLinear Regression of log(mx,t) 5-year age group for data CSRS and SSA for1987-

1993 and 1977-1993 respectively


nxx

nx AAqq )1(19941994

4Methods Lee-Carter Model Model fit/Analysis/Result Industry Trend Study

CMIB Methods –Formula

RF(x,t) - Exponential Decay Characterized by two age-dependent parameters. α(x) denotes the value to be asymptotically approached when t ends to infinite, while fn is the percentage of the total fall (1- α(x)) assumed to occur in n years.

Two set of tables 80 and 92 series (1979-82 and 1991-94 experiences, respectively) for annuitants and pensioners.

1. 80 Series - n was fixed at value of 20 and f20 at 0.6 for all ages. And α(x) is expressed as:


ntn xfxxtxRF /)](1[)](1[)(),(

),(0,, txRFqq xtx

0.5, 6010( ) , 60 110

1001, 110

xxx x

x


CMIB Methods –Formula

2. “92” Series - n remained fixed at 20 but f20 values linearly from 0.45 to 0.71 between ages 60 to 110, below 60 and above 110, constant values with the values already mentioned apply. And α(x) and fn are expressed as the following:


ntn xfxxtxRF /)](1[)](1[)(),(

),(0,, txRFqq xtx

0.13, 60110( ) 1 0.87 , 60 110

501, 110

xxx x

x

20

0.55, 60(110 ) 0.55 ( 60) 0.29( ) , 60 110

500.29, 110

xx xf x x

x


Other Actuarial Method Specifications1. Define Factor2. Determine “standard” rate for trend

a) Ratio of the mortality to the previous year’s mortalityb) Most Recent Mortality Ratec) Mean

3. Methodsa) Arithmetic averageb) Meanc) Geometric averaged) Weighted average



1. Data Source2. Lee-Carter Model3. Model Analysis

Lee-Carter Model


HMD: Human Mortality Database is a population data that it is administrated by UC Berkley

US population data from HMD period 1933 – 2006 is used to demonstrate Lee-Carter Method.

Mortality – Data Source


Region Source CalYr # of CalYr ProjPeriod # of ProjYrJapan HMD 1947-08 62 1979-08 30

Taiwan HMD 1970-08 39 1970-08 39USA HMD 1933-06 74 1977-06 30

KoreaKorean

Population Statistics

1983-03 21 1983-03 21

Lee-Carter’s Extrapolating Method1. Project age pattern and to measure uncertainty2. SVD to Solve the age-specific parameters as well

as the preliminary mortality index3. ARIMA (0,1,0) a random walk with a drift to

project mortality index

Mortality Trend – Methodology overview


General Model (1)

Where, is the central mortality rate for age x for year t; ax and bx are parameters dependent only on age x; kt is factor to be modeled as a time series; and the error term, is assumed to have mean zero and standard deviation .

Mortality Trend - Lee-Carter Method

, ,ln( )x t x x t x tm a b k ,x tm

,x t


Re-Write Model as (2)

Where, is the general shape across age of the mortality schedule; the bx profile tells us which rates decline rapidly and which rates decline slowly in response to changes in kt ( )


,x x ta b k

x tm e xae

,ln x t tx

d m dkbdt dt


The model (1) cannot be fitted by simple regression methods;

It allows for several solutions.

To deal the above, Use SVD, and bx and kt is normalized to sum to unity and

to zero respectively.



SVD Let A be order nxm, then there are unitary matrices U and V, of order n and m respectively, such that , where F is a rectangular diagonal matrix of order mxn,

With Fii = µi. The numbers µi are called the singular values of A. They are all real and positive, and they can be arranged so that

Where r is the rank of the matrix. V* is a conjugate transpose.

Lee-Carter Method – SVD overview

1 0

0 0rF

1 2 0r


SVD applied to ln(mx,t),(3)

Or (4)

Note that, V (and also U) is real number, so the conjugate transpose V* is equal to the transpose V’.


', ')ln( TTFVUm nntx

r

iiiitx tvxum

1, )()()ln(


Singular values – US 1933-2006

The first singular value is larger


Mu Male Female Both1 195.40 214.09 203.58 2 5.91 4.48 5.33 3 2.34 1.96 1.97 4 1.09 0.95 0.89 5 0.81 0.67 0.65 6 0.55 0.53 0.44 7 0.43 0.37 0.38 8 0.38 0.32 0.31 9 0.29 0.23 0.27 10 0.20 0.22 0.18


Use the first singular value component

(5)

The portion of the total temporal variances explained by the first SV component is all over 99%, that seemed captured important of data. The second singular value didn’t seemed add much value.


txtx tvxum ,111, )()()ln(

Mu Male Female Both1 0.9989 0.9994 0.9992 2 0.9998 0.9999 0.9999


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Mortality Trend -Lee-Carter Method


SVD to Solve {ax}, {bx} and {kt},(6)

(7)


txxtxtx kbaBAm )ln( ,

t

tt

xtxx T

BABA

Ta 1

T

BBAk t

t

tx

xt

xx

x

t

xtxx A

Ak

aBAb


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Fitted Values of {ax} and {bx} for 1933-2006 from equation (7) above,



A second stage of estimation of kt, whereby the kt’s are recalculated from the equation, (8)

Taking the estimated {ax} and {bx} as fixed from equation (7).

Note that: there is no closed form solution for equation (8) above. (Newton method is employed)


ˆˆ( ) [ ( , ) exp( )]x t xD t N x t a k b


Take an initial kt1 equation along with {ax} and {bx} from equation (7) above, the following vector is employed for the Newton’s method to obtained the 2nd stage k

Re-write the equation (8) above,

(9)The Jacobian matrix for equation (9) is,

The first order Taylor series becomes:

(10)


0)()()(),(),()( 1)()()()()()(

ii

XbTkXai

XBTkXaii TkTkeXbTXNeTXNTD ii

)()()(),()()( XbTkXaeTXNTDTF

)()(),()( )()()( TkeXbTXNTJ XbTkXa


After iterated equation (10) to find solution kt for equation (8). ARIMA(0,1,0) time series model that a random walk with drift is found to be a good fit, for the mortality index kt, That is,

(11)


1t t tk c k u


From the kt formula (11),

Summing up the above,


)1()1(

121

1

mtmtmt

ttt

ttt

uCkk

uCkkuCkk

1

01

1 m

jjtjt kk

mC


the projection of the kt into the i years from the current year t, (12)

Equation (12) implies, the quantity for the error term is,


1

0

i

jjttit uiCkk

iu

i

jjt

1

0


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Lee-Carter Model fit Example – US


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The error Term Analysis Recall that the error is assumed to follow a

normal distribution with mean 0 and standard deviation . The should not be too much different across ages.

,x t


29



30



31



32




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Age Male Female Both5-9 2.4414 2.6466 2.4852

10-14 2.4132 2.6748 2.478120-24 1.9795 2.3692 2.088830-34 1.9380 2.2244 2.020740-44 1.7267 1.9751 1.803350-54 1.4564 1.7044 1.540460-64 1.1943 1.4324 1.282470-74 0.9447 1.1412 1.023880-84 0.6848 0.8267 0.746790-94 0.4383 0.5204 0.4761

(1-Exp(bxC))*100


Singular values

The first singular value is larger for all three countries (Japan, Taiwan, and US)


SV Japan Taiwan US SV Japan Taiwan USµ1 189.985 147.231 195.399 µ1 206.401 161.933 214.087 µ2 6.329 3.600 5.912 µ2 5.858 2.766 4.483 µ3 2.758 2.092 2.341 µ3 3.831 1.490 1.956 µ4 1.168 1.262 1.090 µ4 1.166 0.765 0.946 µ5 0.687 0.672 0.806 µ5 0.780 0.639 0.672 µ6 0.626 0.517 0.549 µ6 0.462 0.521 0.526 µ7 0.561 0.389 0.428 µ7 0.388 0.443 0.367 µ8 0.428 0.342 0.382 µ8 0.289 0.368 0.322 µ9 0.327 0.268 0.286 µ9 0.238 0.311 0.227 µ10 0.271 0.264 0.197 µ10 0.205 0.254 0.219

Male Female


Use the first singular value component

(5)

The portion of the total temporal variances explained by the first SV component is all over 99%, that seemed captured important of data.


txtx tvxum ,111, )()()ln(

Japan Taiwan US Japan Taiwan US0.9986 0.9991 0.9989 0.9988 0.9995 0.9994

Male Female


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1947

1951

1955

1959

1963

1967

1971

1975

1979

1983

1987

1991

1995

1999

2003

2007

-20

-15

-10

-5

0

5

10

15

20

25

30 k1

k1

1947195

1195

5195

9196

3196

7197

1197

5197

9198

3198

7199

1199

5199

9200

3200

7

-0.800

-0.600

-0.400

-0.200

0.000

0.200

0.400

0.600

0.800

1.000 k2k2


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-8.0

-7.5

-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

1947

1949

1951

1953

1955

1957

1959

1961

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

Log D

eath

Rat

e

Japan Male Actual and Fitted Death Rate (log Scale)

60-64Approx 60-6440-44Approx 40-4420-24Approx 20-24

Lee-Carter Model fit Example – Japan Male


Mortality Improvement – Selected Age

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Age Japan Korea Taiwan US Japan Korea Taiwan US5-9 4.28 5.79 3.67 3.64 6.75 6.25 5.08 2.32

20-24 3.63 5.15 3.08 2.82 6.14 5.91 4.61 2.0430-34 3.54 4.79 2.90 2.76 5.81 5.64 4.37 1.9240-44 3.14 4.13 2.58 2.50 5.23 5.10 3.93 1.7150-54 2.66 3.53 2.24 2.15 4.58 4.46 3.40 1.4770-74 1.74 2.23 1.43 1.40 3.20 2.86 2.15 1.0080-84 1.20 1.38 1.02 1.02 2.23 1.68 1.50 0.74

FemaleMale


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Korean has the highest improvement however the mortality rates are mostly higher than other regions for most ages;

US mortality has the lowest improvement.

Mortality Improvement for Some Regions

0 5-915-

1925-

2935-

3945-

4955-

5965-

6975-

7985-

8995-

99 1050.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0 Male Mortality Improvement by Age

JapanKoreaTaiwanUS

0 5-915-

1925-

2935-

3945-

4955-

5965-

6975-

7985-

8995-

99 1050.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0 Female Mortality Im-provement by Age

JapanKoreaTaiwanUS


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a. Leading Cause of Deathb. Mortality Trend for cause of death Datac. Mortality Rates change by calendar year,

type of underwriting, age, sex, Cause of Death etc.

Short Term Mortality Trend


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a. Japan: mortality and % of the number of deaths by cause were provided by age, sex, and type of underwriting for each year from 2001 to 05. The mortality and the number of death for leading cause of death by sex and type of underwriting for years from 1992 to 2005.

b. Korea: Mortality Rates by age for leading cause of day by age and sex; and the mortality for study period

c. Taiwan: mortality by sex and medical examination from year 1982 to 2000.

Short Term Mortality Trend


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Leading Cause of Death and the additional Data Used in this analysis

Leading Cause of Death

Region Current Prior Current PriorJapan Neoplasm Neoplasm 2001-05 1997-98Korea Neoplasm Accident 2000-02 1996-98China Accident N/A 2000-03 N/ATaiwan N/A Accident N/A 1995-99

Leading Cause of death Study Period


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Leading Cause of DeathCause of Death - Japan

2001-05Cause Med NonMed Med NonMed Male FemaleMalignant Neoplasm 42.61 36.14 47.52 45.62 41.70 47.06 Heard Disease 10.39 11.26 9.05 8.61 10.51 8.94 Cerebrovascular 6.89 7.36 8.54 8.71 6.95 8.58 Suicide 7.00 8.53 4.34 5.64 7.22 4.65 Accident 4.39 6.55 3.26 4.41 4.70 3.54 Subtotal 71.28 69.84 72.72 72.99 71.08 72.78

1997-98Cause Med NonMed Med NonMed Male FemaleMalignant Neoplasm 41.28 38.51 44.40 42.21 40.61 43.53 Heard Disease 11.93 13.86 10.85 12.17 12.40 11.38 Cerebrovascular 8.18 9.17 10.66 11.13 8.42 10.85 Suicide 5.39 4.20 3.95 4.68 5.10 4.24 Accident 6.60 6.52 4.49 4.55 6.58 4.52 Subtotal 73.38 72.26 74.35 74.74 73.11 74.52

Male Female Combined

Male Female Combined


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Number of Death Age distributionCause of Death - Japan

0

2

4

6

8

10

12

14

16

18

20

0-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79

Japan Cause of Death Percent by Age

M.N.H.D.C.D.S.C.A.D.


45

0

2

4

6

8

10

12

2005 Male Med Mortality by Age

M.N.

0.0

0.1

0.2

0.3

0.4

0.5

0-4

5-9

10-1

415

-19

20-2

425

-29

30-3

435

-39

40-4

445

-49

50-5

455

-59

60-6

465

-69

70-7

475

-79

0-79


S.C.

0.0

0.5

1.0

1.5

2.0

2.52005 Male Med Mortality by Age

H.D.C.D.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0-4

5-9

10-1

415

-19

20-2

425

-29

30-3

435

-39

40-4

445

-49

50-5

455

-59

60-6

465

-69

70-7

4


A.D.

Japan Cause of Death – Mortality Per 1,000


Leading Cause of DeathCause of Death - Korea

Cause of Death 5thEMT 4thEMT 5thEMT 4thEMT 5thEMT 4thEMTNeoplasm 29.9 21.5 28.8 28.6 32.4 23.7Injury & Poisoning 29.1 34.3 14.8 23.6 26.7 31Circulatory 19.3 16 21.2 19.5 20.2 17.1Digestive 8.7 9.7 7.8 4.7 8.7 8.1Subtotal 87 81.5 72.6 76.4 88 79.9

CombinedFemaleMale


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Number of Death Age DistributionCause of Death - Korea

0 1-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-59 80-84 85+0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

Korea 5th EMT Major Cause of Death Distribution

Neoplasm Injury & Poisoning


Taiwanese 5 leading Cause of Death

Cause of Death - Taiwan

Cause Male Female BothMalignant Neoplasm 24.93 30.51 26.55 Accident 26.15 20.15 24.41 Circulatory 10.23 10.24 10.23 Digestive System 8.95 4.99 7.80 Mental and Nervous 7.97 7.12 7.72 Subtotal 78.23 73.01 76.72


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1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Japan Mortality Per 1000 by Fiscal Year

Male Med Male NonMed Female Med Female NonMed

Mortality Trend – Japan


50

0.50

1.05

1.60

2.15

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Malignant Neoplasm - Mortality Per 1,000


Japan Mortality – Cause of Death


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0.65

0.85

1.05

1.25

1.45

1.65

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Malignant Neoplasm - Mortality Per 1,000

Male Female Med NonMed


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0.100.150.200.250.300.350.400.450.500.550.60

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Heart Disease - Mortality Per 1,000



53

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Cerebrovascular - Mortality Per 1,000




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0.05

0.10

0.15

0.20

0.25

0.30

0.35

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Suicide - Mortality Per 1,000



55


0.00

0.05

0.10

0.15

0.20

0.25

0.30

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Accident - Mortality Per 1,000



Improvement/Deteriorate (+/-) Percent

Mortality Trend – Japan

1997-05Cause Med NonMed Med NonMed Male FemaleM.N. (1.91) 5.79 (1.85) 2.80 (0.06) (0.04) H.D. (1.61) 6.91 (0.75) 5.92 0.48 1.67 C.D. 0.32 8.07 1.26 5.67 2.22 3.18 S.C. (0.07) (0.28) (0.34) 2.90 0.42 1.03 A.D. 2.44 4.78 1.53 5.34 3.18 3.26 Total (1.63) 5.10 0.94 2.33 0.06 0.94

CombinedMale Female


Improvement comparison to Population:

Mortality Improvement – Japan

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Industry Population (Age 20-69)1998-05 1933-08 1979-08

Male 3.27% 3.28% 2.95%Female 3.61% 3.85% 4.99%


Korea EMT Improvement Factor

Mortality Trend – Korea

From To Male Female EMT Period Male Female3rd EMT 4thEMT 4.88 6.36 3rd 1988-1992 2.972 1.624 4th EMT 5th EMT 1.14 1.08 4th 1996-1998 2.093 1.025 5th EMT 6th EMT 1.25 4.21 5th 2000-2002 2.000 0.982 4th EMT 6th EMT 1.19 2.43 6th 2003-2005 1.926 0.863

Period Factor Percent Korea EMT Mortality Per 1,000


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1.00

1.50

2.00

2.50

3.00

3.50

4.00

198219831984198519861987198819891990199119921993199419951996199719981999

Taiwan Mortality Per 1,000 by Calendar YearMale Female

Mortality Trend –Taiwan


Mortality Trend - Taiwan

Figure2 Taiwanese Aggregate Mortality by Type of Underwriting

Taiwan Aggregate Mortality

0

1

2

3

4

5

6

1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

Calendar Year

Mor

talit

y Pe

r 1,0

00

Medical Non-Med


Compare Industry Improvement to Population

*Population result is for Age 20-69

Mortality Improvement Comparison

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Industry Population*Japan 1992-05 1998-05 1979-08

Male 1.18% 3.27% 2.95%Female 1.69% 3.61% 4.99%

Korea 1988-05 1996-05 1970-08Male 3.05% 1.19% 4.16%Female 4.42% 2.43% 5.02%

Taiwan 1982-99 1988-99 1970-08Male 3.45% 3.74% 2.18%Female 4.28% 4.16% 2.72%


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1. Lee-Carter Method modeled the uncertainty of the reduction factor while other methods didn’t;

2. The reduction factor is vary by age and sex;3. The reduction factor is also vary by projection

year;4. The reduction factor vary by region;5. The industry mortality improvement is usually

smaller than the population’s improvement.

Conclusion

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Questions??

Mortality Trend–Model and Analysis

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