RESEARCH ARTICLE | NOVEMBER 04 2019

The Cairns-Blake-Dowd model to forecast Indonesian

mortality rates 
Y. R. Safitri; S. Mardiyati  ; M. Malik

AIP Conf. Proc. 2168, 020039 (2019)

https://doi.org/10.1063/1.5132466

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05 November 2023 11:46:53

 The Cairns-Blake-Dowd Model to Forecast Indonesian
Mortality Rates
Y. R. Safitri, S. Mardiyatia) and M. Malik

Department of Mathematics, Faculty of Mathematics and Natural Sciences (FMIPA), Universitas Indonesia,
Depok 16424, Indonesia.
a)
Corresponding author: sri_math@sci.ui.ac.id

Abstract. Forecasting mortality rates has become important, especially for the pension fund and life insurance
companies. If mortality rates for the next few years are known, then it will help the company to determine the amount of
premiums. In this paper, the Cairns-Blake-Dowd (CBD) model will be used to forecast mortality rates of the Indonesian
population. The least square method is used to estimate the parameters in the CBD model. Forecasted values of
parameters in the CBD model can be obtained for the next three periods using Bivariate Random Walk with Drift. The
final results show that forecasted Indonesian mortality rates for the next three periods have a downward trend.

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Keywords: Bivariate random walk with drift, Cairns-Blake-Dowd model, least square method, mortality rates

INTRODUCTION

Population growth is influenced by three demographic components, namely fertility, migration, and mortality
[1]. Mortality is a measure of the number of deaths in a population. In actuarial terminology, the mortality rate is the
probability of a person's death in a period [2]. Based on United Nations data, mortality rates in Indonesia have
decreased over the past few years. Declining mortality rates have a negative impact on insurance and pension fund
companies that are caused by losses due to costs which are to be paid in the future being greater than expected.
Insurance companies can plan and determine the amount of premium to get optimal benefits if they know the
mortality rates in the future. Therefore, a method is needed to be able to calculate the mortality rates in the future.
Several methods have been developed to forecast mortality rates using stochastic models. One of the stochastic
models is the Lee-Carter Model. The Lee-Carter model has been widely used to forecast mortality rates in several
countries. In 1992, Lee and Carter [3] forecast the mortality rates of the United States of America by using the
ARIMA method which is used to forecast one of the parameters in the model. In addition to the Lee-Carter model,
there is a stochastic model that can forecast mortality rates, namely Cairns-Blake-Dowd Model.
In 2006, Cairns et al. [4] introduced a new model known as the Cairns-Blake-Dowd (CBD) model. The CBD
model is used to calculate the mortality rates by using two factors. The first factor affects the mortality rate at all
ages in the same way, while the second factor affects the mortality rate dynamics at an older age than at a young age
[4]. This CBD model has been applied to calculate and predict mortality rates in several countries, such as England
and Wales [5], United States [6], Spain [7], and Italy [8].
This paper presents the forecasting mortality rate in Indonesia with the CBD model. Parameters in the CBD
model will be estimated using the Least Square Method and the parameter forecasting process of the CBD model
will be done by using the Bivariate Random Walk with Drift.

Proceedings of the 4th International Symposium on Current Progress in Mathematics and Sciences (ISCPMS2018)
AIP Conf. Proc. 2168, 020039-1–020039-10; https://doi.org/10.1063/1.5132466
Published by AIP Publishing. 978-0-7354-1915-5/$30.00

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 METHODS

Cairns-Blake-Dowd Model

Cairns et al. [4] introduced a stochastic model that contains two-time factors. The Cairns Blake Dowd model can
be used to calculate and forecast mortality rates in the coming period. The CBD model can be expressed as [6]:

logit , =
( )
+
( )
( − ̅ ), (1)

probability an individual at age group x in period t will die at intervals of time t and t + 1, κt(1) represents the
where x is the age group (x = x1, x2, ..., xp), t is the period (t = t1, t2, ..., tq), qx,t is the mortality rate, which is the

intercept and κt(2) represents the slope, and x̄ is the average of the age group. The parameters κt(1) and κt(2) can be
estimated using the Least Square Method.

Least Square Method

The Least Square method is one of the methods used to estimate parameters by minimizing the sum of squared
residuals. Here is the application of the least square method based on the CBD model:

"

= ln − − ( − ̅) .
, ( ) ( )
1− ,
(2)
! !

To minimize J, Equation 2 will be derived against κt(1) and κt(2) as follows:

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%
"

= 2 ln − − ( − ̅) = 0,
, ( ) ( )

%
( ) 1− ,
(3)
!

%
"

= 2( − ̅ ) ln − − ( − ̅) =0∈
, ( ) ( )

%
( ) 1− ,
(4)
!

By solving Equation 3 and Equation 4, the least square estimates of κt(1) and κt(2) are obtained as follows:

∑ ln * ,
+− ̂
( )
∑ ( − ̅)
1−
" "

̂ =
( ) ! , !

,
(5)

and

,∑ ( − ̅ )ln * ,
+−∑ ln * ,
+∑ ( − ̅)
1− 1−
" " "

̂ =
( ) ! , ! , !

,∑ "
( − ̅) − ∑ "
( − ̅)
(6)
! !

Furthermore, the estimation results of parameters κt(1) and κt(2) will be substituted to Equation 1 to calculate the
estimated mortality rate. Then, the estimated mortality rate will be compared with the actual mortality rate to see the
suitability of the model. The error of the estimation process can be calculated with the Root Mean Square Error as
follows:

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 1
6

-./0 = 1 (34 − 354 ) ,

2
(7)
4

where ŷi is the estimated value, ŷi is the actual value, and n is sample size. If the error is small enough, then
forecasting the estimated value of parameters can be processed. Bivariate random walk with drift is one of the
methods that can be used to forecast time series data [4].

Bivariate Random Walk with Drift

Random walk with drift can be expressed as [9]:

(4)
=
(4)
7 + 8 (4) + 9
(4)
,: = : , :; , … , := . (8)

Suppose i = 1, 2, then κt = (κt(1), κt(2))ʹ, μ = (μ(1), μ (2))ʹ, and εt = (εt(1), εt(2))ʹ, so that bivariate random walk with
drift can be expressed as:

? =? 7 +@+A ,: = : , :; , … , := . (9)

where μ is drift parameter and εt ~ N(0, Σ). From Equation 8, it is obtained that

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? −? 7 = @+A ,: = : , :; , … , := .

So that, in general, can be written into:

C D E

where
…
( ) ( ) ( ) ( ) ( ) ( )

C F J,
G ! H G I!

…
( ) ( ) ( ) ( ) ( )
G ! H G I!

8 8 … 8
D K8 8 … 8 L,

Y, M, and ε is 2 × (q - 1) matrix. Equation 9 will be estimated with Ordinary Least Square (OLS), so [9]:

M = (CNO )(NNO )7 ,
@

where G = [1 1 ... 1] is a constant 1 × (q - 1) vector, then the estimate @

M can be expressed as follow:

? −?
@
M=
!

−1
. (10)

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Based on Equation 8, it is obtained that:

? PQ
? R@ A PS
.
U

So, the forecasting of the parameter ̂ and ̂ with t = tq (as in Equation 8) for m periods ahead are:

M
? PQ
? R@
M. (11)

Furthermore, an error check is performed to evaluate the accuracy of forecasting results. This process can be done
by using the Root Mean Square Error in Equation 7.

RESULTS AND DISCUSSION

Data Simulation on Parameter Estimates

The data used is the Indonesian mortality rates data from 1950–1955 to 2010–2015 with age groups 0, 1–4, 5–9,
and so on up to 80–84 years old. Indonesian mortality rates data are from the official United Nations website [10]. In
this study, age group 0 years old will be called x1, 1–4 years old with x2, 5–9 years old with x3, ..., 80–84 years old
with x18. Furthermore, the x value used is the middle value of each age group, namely 0, 2.5, 7, 12, and so on up to

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82 years old.
In the CBD model, the value of (x – x̄) has a significant impact on logit (qx,t), so that the age group used in the
parameter estimation process will influence the mortality rates. Therefore, simulations are carried out in the
parameter estimation process to get the best-estimated results. The simulation in the parameter estimation process is
done by partitioning 18 different age groups into several groups to get the best-estimated results. Parameter
estimation simulations were done by using the age group x1–x18 (using all age groups) and by partitioning 18
different age groups into three groups, namely, x1–x6 (children and adolescent), x7–x13 (adult), and x14–x18 (elderly).
The parameter estimates are calculated using Equation 5 and Equation 6. The results of parameter estimation for

A reduction in ̂ shows that the mortality rate decreases from year to year, while the change in ̂ represents
( ) ( )
each simulation are shown in Table 1.

the change in the steepness of the mortality curve. An increase in the steepness of the mortality curve means that
mortality rates at younger ages increase more rapidly than at older ages [5].

TABLE 1. Estimated results of κt(1) and κt(2) using all age groups.
̂ ̂
( ) ( )
t Periods
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 0.057702580
t11 2000-2005 -3.337483154 0.057702580
t12 2005-2010 -3.406147813 0.059173814
t13 2010-2015 -3.487173490 0.061725970

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 The estimated results of κt(1) and κt(2) in Table 1 and Table 2 will be substituted to Equation 1 to obtain an
estimated value of qx,t. The estimated value of qx,t will be compared to the actual value of qx,t to identify the
suitability of the model. Figure 1 and Fig. 2 are the comparison chart of the actual value of qx,t with an estimated
value of qx,t for some age groups that produce the best estimate of qx,t from each simulation.
From Fig. 1, the estimated qx,t including all age groups has not approached the actual qx,t. Furthermore, from Fig.
2, it can be seen that the estimated results of qx,t have approached the actual qx,t. Then the RMSE calculation will be
done using Equation 7 to find the error from the estimated results of qx,t. The RMSE values for each parameter
estimation simulation are obtained (Table 3).

TABLE 2. Estimated results of κt(1) and κt(2) with three group partitions.

V
MV
W MV
W MV
W MV
W MV
W MV
W
(X) (Y) (X) (Y) (X) (Y)
x1–x6 x7–x13 x14–x18
Periods

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

(a) (b)

(c)
FIGURE 1. The comparison of the actual value of qx,t and the estimated value of qx,t for age groups (a) 15–19 years old,
(b) 30–34 years old and (c) 75–79 years old with parameter estimates using all age group.

020039-5
 Based on Fig. 1 and RMSE values in Table 3 and Table 4, the parameter estimation by partitioning 18 different
age groups into three groups produces a better estimate of . , so that the parameter simulation results of each group
x1–x6, x7–x13 and x14–x18 is used in forecasting parameters with the Bivariate Random Walk with Drift.

(a) (b)

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(c)

FIGURE 2. The comparison of the actual value of qx,t and the estimated value of qx,t for age groups (a) 15-19 years old, (b)
30-34 years old and (c) 75-79 years old with parameter estimates using three group partitions.

TABLE 3. RMSE value of estimated qx,t using all age groups.

Age Age Age Age
RMSE RMSE RMSE RMSE
Group Group Group Group
0 0.093301 20–24 0.011492 45–49 0.036859 70–74 0.099627
1–4 0.047992 25–29 0.016627 50–54 0.038031 75–79 0.193946
5–9 0.000729 30–34 0.021807 55–59 0.035057 80–84 0.300001
10–14 0.008453 35–39 0.026874 60–64 0.006523
15–19 0.009628 40–44 0.032019 65–69 0.030089

TABLE 4. RMSE value of estimated qx,t with partition into three groups.
Age Age Age Age
RMSE RMSE RMSE RMSE
Group Group Group Group
0 0.047181 20–24 0.006866 45–49 0.003039 70–74 0.007628
1–4 0.014864 25–29 0.001829 50–54 0.000558 75–79 0.003775
5–9 0.015614 30–34 0.000213 55–59 0.008138 80–84 0.009236
10–14 0.01004 35–39 0.001601 60–64 0.004931
15–19 0.000655 40–44 0.002672 65–69 0.005231

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 Forecasting Parameters and Mortality Rates

The next step in forecasting mortality rates is to forecast the parameters κt(1) and κt(2) by using the Bivariate
Random Walk with Drift. From the estimated results of κt(1) and κt(2) in Table 2, using Equation 10 obtained @M for
each group x1–x6, x7–x13 and x14–x18 as follows:

0.160122469 0.06065725 0.04439084

@
MZ [ a, @
Mb [ a, @
Md [ a
0.00706812 0.001818 0.00032

Then, the forecasting parameters κt(1) and κt(2) will be calculated for the period 1955–1960 to 2010–2015 using
Equation 11. The results of forecasting parameters for each group are shown in Table 3.
Next, based on the forecasted value of parameters in Table 5 and estimated parameters in Table 2 for the period

forecasted values of κt(1) and κt(2).

1955–1960 to 2005–2010, the RMSE calculation will be done using Equation 7 to find out the error from the

Table 6 are the RMSE values for forecasted values of κt(1) and κt(2). From Table 6, the forecasted RMSE values
of κt(1) and κt(2) for each group are quite small. It means that the forecasting method with the Bivariate Random
Walk with Drift is good enough to be applied to forecast κt(1) and κt(2). The forecasting results of this parameter will
be substituted in Equation 1 to get the forecasted value of qx,t and then it will be compared with the actual qx,t to
determine the suitability of the forecasting method performed. Figure 3 shows a comparison graph of the actual qx,t
with forecasted . for some age groups that produce the best qx,t forecast. Error from forecasting qx,t is calculate
using RMSE in Equation 7 (Table 7).

parameter κt(1) and κt(2) are good enough. Next, forecasting parameters κt(1) and κt(2) will be calculated for the next
From Fig. 3 and Table 7, the forecasted qx,t has approached the actual qx,t. It means that the forecasted values of

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three periods using Equation 11. The forecasting results of parameters κt(1) and κt(2) in the period 2015–2020, 2020–
2025 and 2025–2030 are shown in Table 8.

TABLE 5. Forecasted value of parameters κt(1) and κt(2) for the period 1955-1960 to 2005-2010.

V
x1–x6 x7–x13 x14–x18
MV
W MV
W MV
W MV
W MV
W MV
W
Periods (X) (Y) (X) (Y) (X) (Y)

t2 1955-1960 -3.30366761 -0.10595600 -3.188773658 0.0500787 -0.58910288 0.11422085

t3 1960-1965 -3.45396462 -0.09954020 -3.257147274 0.0515999 -0.64484374 0.11435902
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
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

TABLE 6. RMSE value of forecasted κt(1) and κt(2).

x1–x6 x7–x13 x14–x18
MV
W MV
W MV
W MV
W MV
W MV
W
Periods (X) (Y) (X) (Y) (X) (Y)

RMSE 0.01871021 0.003403717 0.03800260 0.00106784 0.00956901 0.001064359

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 (a) (b)

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(c)

FIGURE 3. Comparison of the actual value of qx,t with forcasted qx,t for age groups (a) 15-19, (b) 30-34 and (c) 75-79 years
old.

TABLE 7. RMSE value of forcasted qx,t.

Age Age
Age Group RMSE Age Group RMSE RMSE RMSE
Group Group
0 0.044830996 20-24 0.006319944 45-49 0.00325275 70-74 0.007919307
1-4 0.01098748 25-29 0.001726446 50-54 0.001397076 75-79 0.004234615
5-9 0.013538964 30-34 0.000695495 55-59 0.007799377 80-84 0.009890479
10-14 0.008955761 35-39 0.001856721 60-64 0.004900458
15-19 0.000658241 40-44 0.002897125 65-69 0.00595395

TABLE 8. Forecasted value of κt(1) and κt(2) for the period 2015-2020, 2020-2025, and 2025-2030.

V
x1–x6 x7–x13 x14–x18
MV
W MV
W MV
W MV
W MV
W MV
W
Periods (X) (Y) (X) (Y) (X) (Y)

t14 2015-2020 -5.2251372 -0.02113857 -3.916660615 0.0718951 -1.12179296 0.110359624

t15 2020-2025 -5.3852597 -0.01407045 -3.977317862 0.0737132 -1.16618380 0.110037855
t16 2025-2030 -5.5453822 -0.00700233 -4.037975108 0.0755312 -1.21057464 0.109716085

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 TABLE 9. The results of forecasting Indonesian 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 0.09393 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 4 and Table 7, ages 15–19, 20–24, 25–29, ..., 80–84 years old have the best estimation and
forecasting results of . . So, forecasting Indonesian mortality rates will be carried out for ages 15–19, 20–24,
25–29, ..., 80–84 years old by substituting the forecasted values of κt(1) and κt(2) from Table 8 to Equation 1. Table 9
are the results of forecasting Indonesian mortality rates for the periods 2015–2020, 2020–2025 and 2025–2030.

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CONCLUSION

Based on data simulations in the parameter estimation process, the best estimate of Indonesian mortality rates
using the CBD model is obtained by partitioning 18 age groups into three groups. Parameter forecasting results
using the bivariate random walk with drift shows the same trend as the actual parameter value. Forecasting
Indonesian mortality rates is carried out for ages 15–19, 20–24, 25–29, ..., 80–84 years old because it has the
smallest RMSE value (Table 4 and Table 7). From Table 9, Indonesian mortality rates for the next three periods
have a downward trend. It is consistent with the trend of Indonesian mortality rates from the period 1950–1955 to
2010–2015. Thus, the CBD model can be used to forecast Indonesian mortality rates where parameter forecasting in
the CBD model is done using the bivariate random walk with drift.

ACKNOWLEDGMENTS

This research is supported by PITTA Universitas Indonesia research grant 2018.

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