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Predicting Mortality Rates and Longevity Using Cains -Blake-
Dowd Model
Jonah Mudogo Masai, Wafula Isaac
Machakos University, Kenya
DOI : https://doi.org/10.51583/IJLTEMAS.2024.131023
Received: 26 October 2024; Accepted: 06 November 2024; Published: 17 November 2024
Abstract: Pension schemes and annuity providers frequently guarantee their retirement payouts until the retirees' deaths. As a
result of longer life expectancies and declining rates of death in old age, trends in mortality and longevity have become evident.
Academicians and actuaries have been forced to concentrate their research on mortality and longevity concerns in particular as a
result of this. Instead of a provident fund, the new National Social Security Fund Act Number 45 of 2013 established a pension
fund that is a requirement for every employee. Annuity service providers are exposed to the longevity risk when the scheme's
participants retire. For pricing and reserving, appropriate modeling tools or projected life tables are required. In comparison to
deterministic models, which were based on projected present values, stochastic models allow a variety of risk causes and
components as well as pertinent effect on portfolio performance. The long term mean level of Longevity has become more
uncertain exposing the annuity service providers such as assurance companies and states to the risk of uncertainty after
retirement. Most industrialized countries' national security systems, pension plans, and annuity providers have revised their
mortality tables to account for longevity risks due to decline mortality rates and rising life expectancy. Kenya is one of the
developing nations that has seen a drop-in death rates and a rise in life expectancy recently. Since developing nations choose to
take the longevity risk into account when pricing and reserving annuities because such long term mean level in mortality rates
declines and increases life expectancy, particularly at retirement age, pose risks to annuity service providers and pension plans
that has been pricing annuities based on mortality tables that do not take these trends into account. The stochastic aspect of
mortality was ignored by earlier actuarial models used to estimate trends. The actuary will therefore likely be interested in
knowing how the future mortality trend utilizing stochastic models affects annuity pricing and reserve. Demographers and
actuaries have since employed a variety of stochastic methods to forecast mortality while examining a variety of stochastic model
ranges. The CBD stochastic model, which was the first to take longer life expectancies into account, is now extensively used, and
a number of expansions and adjustments have been suggested to stop the major characteristics of mortality intensity. The CBD
model, developed by Andrew Cairn, David Blake, and Kevin Dowd, is being used in this study to fit mortality rates, forecast
mortality trends, using least square method and then calculate projections for life expectancy. Regarding the longevity risk, we
take into account the possibilities of computing annuity benefits by connecting the benefits to actual mortality and calculating the
present value on annuities. The results of the study showed that, the CBD model can be used to forecast mortality rates where
parameters estimating the CBD model are performed using the bivariate random walk (drift).
Keywords: Human Mortality database, defined Benefits, age-period-cohort, Defined contribution, Root Mean Square Error.
I. Introduction.
The long term mean level of Longevity has become more uncertain exposing the annuity service providers such as assurance
companies and states to the risk of uncertainty after retirement.
Most industrialized countries' national security systems, pension plans, and annuity providers have revised their mortality tables
to account for longevity risks due to decline mortality rates and rising life expectancy. Kenya is one of the developing nations that
has seen a drop in death rates and a rise in life expectancy recently. Since developing nations choose to take the longevity risk
into account when pricing and reserving annuities because such long term mean level in mortality rates declines and increases life
expectancy, particularly at retirement age, pose risks to annuity service providers and pension plans that has been pricing
annuities based on mortality tables that do not take these trends into account.
Longevity is a threat to pension funds and annuity service providers; it has been acknowledged. The mortality models were
divided by Booth and Tickle (2008) into extrapolative models, explanatory models, and expectancies models. There have been
more models proposed for explaining and estimating mortality as a result of recent improvements in actuarial methods,
particularly in pensions and life mathematics. The models were conveniently surveyed and explained by (Pitacco, Denuit,
Haberman, & Oliviera, 2009). It's still difficult to dynamically fit mortality rates and, thus, quantify longevity risk, especially in
emerging nations. Prior work was based on the Lee and Carter in the year 1992 being one-factor model. However, this model is
frequently used to give estimates and demographic projections that are quite accurate for academics and practitioners alike. This
model was examined and a new model was developed by Halzoupoiz (1996) and Renshaw and Heberman (2003).
The cohort effect was recently taken into account in longevity modeling, which Lee and Carter's model lacked. For example,
Currie (2006) offers an APC model after Renshaw and Haberman (2003) incorporated a cohort effect. The most recent proposals,
made by CBD (2006b), find that all the issues with the Lee and Carter model can be resolved by including both a cohort impact
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and a quadratic age effect in their cairns-Blake-Dowd model. In this Research paper, we examine the stochastic aspect of
mortality which was ignored by earlier actuarial models used to estimate trends. The actuary will therefore likely be interested in
knowing how the future mortality trend utilizing stochastic models affects annuity pricing and reserve. The Cairns-Blake-Dowd
(CBD) model take into account two factors to calculate the mortality rates. The two factors are employed in the CBD model to
compute mortality rates. While the first component has an impact on mortality rate dynamics at older ages compared to younger
ages, the second element has an equal impact on mortality rates at all ages. This model has been used in England and Wales,
Spain, and Italy, to compute and forecast death rates.
II. Methodology
Mortality Assumption.
These mortality rate forecasts are used to calculate annuity prices and calculate pension liabilities. The mortality tables form the
foundation for mortality assumptions. Since mortality rates and assumptions are essential to annuity pricing and reserving, the
insurance and retirement benefits regulator in the majority of nations offers guidelines on their use.
The mortality assumption is one of the most important elements taken into account when estimating the life expectancy at birth.
Estimating life expectancy will subsequently be used to compute the pension fund's and annuity providers' long-term liabilities.
Long-term liabilities for both pension fund and insurance business are exaggerated if mortality assumptions are low. If the
assumptions are too high, the pension plan's life expectancy will be overestimated, which will lead to an underestimation of the
pension plans and annuity providers' responsibilities.
Inter-age dependency and heterogeneity
The degree of heterogeneity varies from population to population for a particular population. Heterogeneity results from a variety
of observable elements, such as gender, age, occupation, and physiological parameters, as well as from aspects of the
environment in which an individual lives, such as the climate, population, and dietary norms. greater socioeconomic status
pensioners or insurance holders those with longer life expectancies or a propensity for experiencing lower mortality rates have
greater life expectancies. Since females often have lower mortality rates than males, considerable differences also exist within the
same socioeconomic status.
Closing the tables and blending
Empirical yearly death rates have erratic age distributions at advanced ages. Actuaries typically deduce the shape of the survival
functions S(x) at higher ages from a few exogenous assumptions, which involves closing the mortality tables. In the past, death
after 100 was not given much attention because it had a relatively minor effect on pensioners' residual life expectations (and thus,
pensions). This is no longer the case due to recent advancements in longevity, and it is crucial to have better understanding of
both mortality rate and risk of longevity for higher ages.
Initial Rate of Mortality
The mortality rate q
x
which is the conditional probability of person aged x dying in the next year of age.
=
where;
x
is the expected number of deaths at age x last b birthday among l
x
lives aged x over the next year.
lx - represent the radix at start of each year (People alive).
Central death Rate
It is denoted by mx which represents the probability of dying between exact ages and per person-year lived. Definition;
T
x
=
=
=
-
= T
x –
T
x+1
Hence mx, can be written as;
m
x
=
=
= µ
x+1/2
Force of Mortality
It is the death rate at exact time t for individuals aged x+t at time t. It is denoted by
x
m
x =
x =
For the small h,
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Expected Future Lifetime
This measures remaining time until death. It is given as [];
=
=
=
-
=
The curtate future lifetime of a life aged x is, Kx = Tx represents integer part of future lifetime. Kx is given by
E[ Kx ]=
Age Specific Death Rates
ASDR =
* 100,000
Cairns-Blake-Dowd Model
It is given as ;
logit(
) =
+
(2.1)
where logit(
)=
where x is the age group (x = x1, x2, ..., xp), t is the period (t = t1, t2, ..., tq),
is the mortality rate, which is the probability an
individual at age group x in period t will die at intervals of time t and t + 1, κt (1) represents the intercept and κt (2) represents the
slope, and x is the average of the age group. The parameters κt (1) and κt (2) can be estimated using the Least Square Method.
Estimation by Least Square Method
The parameters of CBD model can be estimated by minimizing the sum of squared residuals as follows:
S =
(2.2)
Derving Equation (2) against κt (1) and κt (2) gives;
=
= 0 (2.3)
=
= 0 (2.4)
By solving both equations, the least square estimates of κt (1) and κt (2) are obtained as follows:
κ
t
(1) =
(2.5)
and
κ
t
(2) =
(2.6)
Substituting the results of equation (2.5) and (2.6) to equation (2.1) to compute the estimated mortality rate, will be compared
with actual mortality rate to see the stability of the model.
Error of the estimation process can be calculated with the Root Mean Square Error as follows:
RMSE =
(2.7)
where ŷi is the estimated value, yi is the actual value, and n is sample size. If the error is small enough, then predicting the
estimated value of parameters can be processed. We shall use the Bivariate random walk with drift to forecast the time series data.
Bivariate Random Walk (Drift)
Written as: κt(i) = κt-1(i) + (i) +t , t = t2, t3, …, tq ( 2.8)
Suppose i = 1, 2, then κt = (κt (1), κt (2))ʹ, μ = (μ(1), μ (2))ʹ, and ε = (εt (1), εt (2))ʹ, It can be given as follows;
kt – kt-1 = + t t = t2, t3, …, tq
where μ is drift parameter and εt ~ N(0, Σ). From Equation (2.8), it is obtained that
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So that, in general, can be written into:
Y= M +
Y= M +
Y =
M=
Y, M, and ε is 2 × (q - 1) matrix. Equation (2.9) will be estimated with Ordinary Least Square (OLS),where GG = [1 1 1] is a
constant
vector, then the estimate may be expressed as follow:: =
= (YG’)(GG’)
-1
Based on Equation 2. 8, it is obtained that:
=
+ m
so the forecasting of the parameters
and
with t = tq (as in Equation (2.8)) for m periods ahead are:
Ktq+m = Ktq + mµ
Error checking is performed to evaluate the accuracy of forecasting results. This process can be done by using the RMSE in
Equation (2.7).
Actuarial present Values
This
the present value of an annuity of 1 p.a annum payable monthly in advance.
=
( 2.9.1)
This is expressed in terms of
as follows;
Euler-McLaurin formula:
=
Woolhouse’s formula:
Assuming that ; f(t) 0 and f’(t) 0 as t tends to infinity
Using equation 3.5.1 to fit f(t) = v
t
=
then
f’(t) = -(
therefore f(0) = 1 and f’(0) =-
this gives
=
-
=
-
III. Main Results
Source of data
Data sources utilized in modeling longevity risk are deaths of individuals in pension plans and annuity services providers. The
UK's CMIB collects mortality data from those sources and information from pension schemes on insured lives. Although KRA
collects this data, it is not widely accessible because it is never made public or published every year. Furthermore, usage of this
data might lead to sampling issues because it is not the accurate representative of the whole population. HMD is where countries
publish their mortality data. The most appropriate data is the entire population since it includes large number of individuals. Our
analysis will be based on mortality data from United States obtained from the HMD via the demography package's dedicated
function. The data in HMD consist of sex , age and year.
Fitting the Model
U.S.A mortality rates data ranges from 1950-1955 to 2015-2020 with age groups 0, 1–4, …, 80–84 years old. Let age group (0)
years old be x1, (1–4) years old with x2, (5–9) years old with x3, ..., 80–84 years old with x18. Let again x be the midpoint value
of each age group that is 0, 2.5, 7, …, 82 years old.
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The value of (x - ) has a considerable impact on logit (
) in the CBD model, hence the age group utilized in the parameter
estimate method influences the mortality rates. As a result, simulations are run during the parameter estimation process to obtain
the best estimated results. To obtain best estimated results, simulation in the process parameter estimation divides 18 different age
groups into multiple groups. Parameter estimate simulations were performed using the age group x1-x18 and by dividing the 18
age groups into three different groups, that is x1-x6 (children and adolescents), x7-x13 (adults), and x14-x18 (elderly). Equations
2. 5 and 2.6 are used to calculate parameter estimations.
Table 3.0.The estimated values of K
t(1) and K
t(2)
T
Periods
K
t(1)
K
t(2)
t1
1950-1955
-2.415646996
0.037325948
t2
1955-1960
-2.507819312
0.039018362
t3
1960-1965
-2.599202081
0.040700171
t4
1965-1970
-2.689945927
0.042376603
t5
1970-1975
-2.789380200
0.044222561
t6
1975-1980
-2.913925914
0.046647877
t7
1980-1985
-3.034071316
0.048944461
t8
1985-1990
-3.114601821
0.051105996
t9
1990-1995
-3.205308913
0.053500361
t10
1995-2000
-3.297892093
0.05770258
t11
2000-2005
-3.337483154
0.05770258
t12
2005-2010
-3.406147813
0.059173814
t13
2010-2015
-3.478742245
0.06175235
t14
2015-2020
-4.056432455
0.06246754
To generate an approximated value of
Table 2 will be replaced into Equation 2. 1. The anticipated value of
will be
compared against the actual value of
to determine the model's appropriateness.
The RMSE computation is performed using Equation (2.7) to determine the error from the estimated
findings.
Table3.1. The estimated values of K
t(1) and K
t(2) with three partition groups
x1–x6
x7–x13
x14–x18
t
period
K
t(1)
K
t(2)
K
t(1)
K
t(2)
K
t(1)
K
t(2)
t1
1950-1955
-3.143545145
-0.11302412
-3.128116411
0.048260644
-0.544712036
0.114542622
t2
1955-1960
-3.293842153
-0.10660832
-3.196490027
0.049781847
-0.600452903
0.114680792
t3
1960-1965
-3.442797063
-0.10024799
-3.264521541
0.051301960
-0.655440860
0.114845829
t4
1965-1970
-3.590694945
-0.09388190
-3.332311770
0.052827236
-0.709734924
0.115020959
t5
1970-1975
-3.752810960
-0.08685924
-3.406916885
0.054498474
-0.768711931
0.115218461
t6
1975-1980
-3.945040132
-0.08564213
-3.531508946
0.058461945
-0.811972607
0.116492851
t7
1980-1985
-4.129440187
-0.08439366
-3.651459539
0.062164793
-0.855285159
0.117658517
t8
1985-1990
-4.284653652
-0.07803920
-3.704331129
0.064204521
-0.884918592
0.117266797
t9
1990-1995
-4.460861394
-0.06942212
-3.758352374
0.066049331
-0.924385090
0.116666379
t10
1995-2000
-4.640922339
-0.05945270
-3.808333383
0.067620892
-0.971637992
0.115677701
t11
2000-2005
-4.764786467
-0.04540144
-3.782162694
0.067286918
-1.002167823
0.113028112
t12
2005-2010
-4.912600833
-0.03526591
-3.803541879
0.068413398
-1.042052494
0.111368643
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t13
2010-2015
-5.065014774
-0.02820669
-3.856003369
0.070077092
-1.077402119
0.110681393
t14
2015-2020
-5.165225531
-0.02920669
-3.896003369
0.072077092
-1.97402119
0.100681393
Table 3.2 Shows RMSE values of estimated
for entire age groups.
Age Group
RMSE
Age Group
RMSE
Age Group
RMSE
Age Group
RMSE
0
0.093301
20–24
0.011492
45–49
0.036859
70–74
0.099627
1–4
0.047992
25–29
0.01662
50–54
0.038031
75–79
0.193946
5–9
0.000729
30–34
0.02180
55–59
0.035057
80–84
0.300001
10–14
0.008453
35–39
0.02687
60–64
0.006523
15–19
0.009628
40–44
0.03201
65–69
0.030089
Table 3.3. RMSE value of estimated
with partition into three groups.
Age Group
RMSE
Age Group
RMSE
Age Group
RMSE
Age Group
RMSE
0
0.047181
20–24
0.00686
45–49
0.003039
70–74
0.007628
1–4
0.014864
25–29
0.00182
50–54
0.000558
75–79
0.003775
5–9
0.015614
30–34
0.00021
55–59
0.008138
80–84
0.009236
10–14
0.01004
35–39
0.00160
60–64
0.00493
15–19
0.000655
40–44
0.00267
65–69
0.00523
Forecasting Parameters and Mortality Rates
Parameters
and
are forecasted using the Bivariate Random Walk (Drift). From the results in Table 2, the estimated
values of
and
by using Equation (2.9.1) we obtain µ for each group x1–x6, x7–x13 and x14–x18 as follows:
=
,
= (-0.06065725, 0.001818),
= (-0.04439084, -0.00032)
Then, the forecasting parameters
and
will be calculated for the period 1955–1960, …, 2015–2020 using Equation
(2.9.1). As can be shown in Table 3.2.
Next, based on the forecasted value of parameters in Table 3.4 and estimated parameters in Table 3.2 for the period 1955–1960
…, 2015–2020, the RMSE computation will be done using Equation (2.7) in order to find out the error from predicted results of
and
.As shown in table (3.3)
Clearly the it can be seen that values of
and
are quite small for each group which means that predicting method with
bivariate random walk (Drift) is better enough to be use to predict
and
. The predicting results of this parameter will be
then replaced in Equation (2.1) to get the predicted value of
and then it will be further examined with the actual value of
to determine the suitability of forecasting method performed
Table 3.4. Predicted values of parameters κt(1) and κt(2) for the period 1955-1960 …, 2015-2020.
x1–x6
x7–x13
x14–x18
t
period
K
t(1)
K
t(2)
K
t(1)
K
t(2)
K
t(1)
K
t(2)
t2
1955-1960
-3.30366761
-0.10595600
-3.188773658
0.0500787
-0.58910288
0.11422085
t3
1960-1965
-3.45396462
-0.09317987
-3.325178787
0.0531200
-0.69983170
0.11452406
t4
1965-1970
-3.60291953
-0.09317987
-3.325178787
0.0531200
-0.69983170
0.11452406
t5
1970-1975
-3.75081741
-0.08681378
-3.392969017
0.0546453
-0.75412576
0.11469919
t6
1975-1980
-3.91293343
-0.07979112
-3.467574131
0.0563165
-0.81310277
0.11489669
t7
1980-1985
-4.10516260
-0.07857401
-3.592166193
0.0602800
-0.85636345
0.11617108
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t8
1985-1990
-4.28956266
-0.07732554
-3.712116786
0.0639828
-0.89967600
0.11733675
t9
1990-1995
-4.44477612
-0.07097108
-3.764988375
0.0660226
-0.92930943
0.11694503
t10
1995-2000
-4.62098386
-0.06235400
-3.819009621
0.0678674
-0.96877593
0.11634461
t11
2000-2005
-4.80104481
-0.05238458
-3.868990630
0.0694389
-1.01602883
0.11535593
t12
2005-2010
-4.92490894
-0.03833332
-3.842819941
0.0691050
-1.04655866
0.11270634
t13
2010-2015
-5.07272330
-0.02819779
-3.864199126
0.0702314
-1.08644333
0.11104687
t14
2015-2020
-5.17271380
-0.02830780
-3.87419915
0.0716319
-1.1860435
0.11004713
Table 3.5. RMSE values of forecasted κt(1) and κt(2).
x1–x6
x7–x13
x14–x1
period
K
t(1)
K
t(2)
K
t(1)
K
t(2)
K
t(1)
K
t(2)
RMSE
0.01871021
0.003403717
0.03800260
0.00106784
0.00956901
0.001064359
Table 3.7. Predicted values of κt (1) and κt (2) for the period 2020-2025, and 2025-2030.
Age Group
RMSE
Age Group
RMSE
Age
Group
RMSE
Age
Group
RMSE
0
0.04483099
20-24
0.00631994
45-49
0.00325275
70-74
0.007919307
1-4
0.01098748
25-29
0.00172644
50-54
0.00139707
75-79
0.004234615
5-9
0.01353896
30-34
0.00069549
55-59
0.00779937
80-84
0.009890479
10-14
0.00895576
35-39
0.00185672
60-64
0.00490045
15-19
0.00065824
40-44
0.00289712
65-69
0.00595395
Table 3.9. The outcomes of forecasted mortality rates
Age
Actual Value
Forecasted Value
Group
1950-1955
…2005-2010
2010-2015
2015-2020
2020-2025
2025-2030
15-19
0.01795
… 0.00641
0.00588
0.004626
0.004141
0.003707
20-24
0.02238
… 0.00854
0.00790
0.004164
0.003861
0.00358
25-29
0.02338
… 0.00924
0.00857
0.006726
0.006163
0.005647
30-34
0.02631
… 0.01087
0.01013
0.009607
0.008885
0.008217
35-39
0.03133
… 0.01421
0.01335
0.013706
0.012794
0.011943
40-44
0.03858
… 0.01957
0.01857
0.019519
0.018391
0.017328
45-49
0.04782
… 0.02872
0.02762
0.027728
0.026371
0.025079
50-54
0.06641
… 0.04304
0.04163
0.039252
0.037681
0.036171
55-59
0.09348
… 0.06526
0.06347
0.055293
0.053574
0.051907
60-64
0.16136
… 0.10828
0.10584
0.097494
0.093937
0.090484
65-69
0.23886
… 0.16533
0.16069
0.157946
0.152339
0.146896
70-74
0.36295
… 0.25078
0.24386
0.245679
0.237545
0.229599
75-79
0.50507
… 0.37540
0.36593
0.361238
0.350693
0.340292
80-84
0.65154
… 0.53017
0.52033
0.495451
0.483555
0.471677
Based on Table 3.4 and Table 3.7, all groups have the best estimation and forecasting results of
. So, predicting mortality rates
will be carried for all ages by replacing the predicted values of κt (1) and κt (2) from Table 3.8 to Equation (2.1).
Executing Actuarial Projection
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I used the following formula calculate life expectancy and the actuarial present values;
=
+
and
=
We computed the actuarial present value of
for the total population as can be shown in the table below.
Table 3.9.1: the actuarial present value of
for the total population
Cohort for Year
Life expectancy
Present values
1950
55.62
6.10
1960
65.03
6.53
1970
72.49
6.98
1980
77.68
7.48
1990
80.01
7.68
2000
82.88
8.20
2010
84.39
8.58
2020
86.02
8.78
2030
89.99
9.20
There is an increase of annuity due to lower mortality rates as well as increase of life expectancy actuarial present value results
show that annuities have increased through time as a result of lower mortality rates and increased life expectancy. As a result, the
amount that annuity providers should pay to individuals must be reduced in order to avoid overpaying annuitants. Longevity risk
arises because general degree of mortality change is uncertain at the time annuities are purchased.
IV. Conclusions and recommendations
Conclusions
It is evident that the best estimate of mortality rate is by using the CBD model, which is obtained by breaking down eighteen age-
groups(18) into three different groups. Parameter predicting results using the bivariate random walk(drift) shows the long term
mean level as the actual parameter value. Forecasting qx,t is carried out for ages 15–19, 20–24, 25–29, ..., 80–84 years old
because it has the least Root Mean Square Error(RMSE) value as shown in (Table 3.4 and Table 3.7). The mortality rates for the
next two periods have a downward long term mean level as shown in table 9 which is consistent from the period from the 1950–
1955 …, 2010–2015. Thus, the CBD model can be used to forecast mortality rates where parameters estimating the CBD model
are performed using the bivariate random walk(drift).
Recommendations
Since the risk of longevity exists in both pension schemes and annuity service providers that is assurance companies, therefore, I
urge the management of longevity risk to implement these opinions. Therefore, individuals are encouraged to work for longer
periods in order to save more.
Longevity risk being a major concern for both annuity service providers and pension schemes, we advise future academicians to
use other models in place of the CBD to model and predict mortality rates, subsequently they measure longevity risks. This will
address CBD's limitations.
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