Determination of Term Life Insurance Premiums with Varying Interest Rates Following Cox Ingersoll Ross Model and Varying Benefits Value


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(4) (2023), Pages 630-640 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: April 06, 2023 Reviewed: April 09, 2023 Accepted: April 16, 2023 
DOI: http://dx.doi.org/10.18860/ca.v7i4.20542  

Determination of Term Life Insurance Premiums with Varying 
Interest Rates Following Cox Ingersoll Ross Model and Varying 

Benefits Value 

Dian Puspita*, I Gusti Putu Purnaba, Donny Citra Lesmana  

Department of Mathematics, IPB University, Bogor, Indonesia  

 

Email : dianpuspita@apps.ipb.ac.id 

ABSTRACT  

In life will always be surrounded by various things that can happen, which can lead to the risk of death 
and financial loss. One way to overcome is insurance. One type of insurance is term life insurance. Term 
life insurance is an insurance that provides protection for a certain period that has been agreed in the 
policy. Insurance premiums and benefits in the long term will be affected by interest rates . One of the 
models that can be used is the Cox Ingersoll Ross model (CIR). This research purposes to simulate the CIR 
model that will be carried out to determine interest rates for calculating term life insurance premiums for 

five years, with premiums paid at the beginning of the 
1

m
 interval and benefits paid at the end of the 

1

m
 interval when the participant dies. The method to estimate parameters in CIR model is Ordinary Least 

Square.  The results of this research is the CIR model can be applied to calculate the term life insurance 
premiums for five years and the premium calculation results show that the amount of the premium 
increase every year with varying benefits. The contribution of this research as information to insurance 
companies regarding the amount of premiums paid monthly. 
 
Copyright © 2023 by Authors, Published by CAUCHY Group. This is an open access article under the CC 
BY-SA License (https://creativecommons.org/licenses/by-sa/4.0/) 
 
Keywords: premium; term life insurance; CIR model; varying interest rates 

INTRODUCTION 

Insurance is a risk transfer from a financial loss caused by the deaths of insurance 
participants to an insurance company based on a policy agreement. One type of 
insurance is life insurance. Life insurance is insurance to cover the financial costs of 
death. Life insurance consists of four types namely whole life insurance, term life 
insurance, pure endowment insurance and endowment insurance [1]. In insurance, 
there is an agreement called a policy between insurance participants and insurance 
companies that include participants' obligation to pay premium contributions to 
insurance companies and insurance liability to pay the benefits if something happens to 
insurance participants has been compromised in policy, and other agreements 
associated with insurance.  

Premiums and benefits are affected by interest rates. Normally the interest rates 
used are constant even though the payment of the premiums for long-term payments on 
which interest rates will change over time. The Cox Ingersoll Ross model is one of the 

http://dx.doi.org/10.18860/ca.v7i4.20542
mailto:dianpuspita@apps.ipb.ac.id
https://creativecommons.org/licenses/by-sa/4.0/


Determination of Term Life Insurance Premiums with Varying Interest Rates Following Cox 
Ingersoll Ross Model and Varying Benefits Value 

Dian Puspita 631 

stochastic models used for fluctuating rates in period of time. The CIR model has a 
characteristic that the movement of interest rates would lead to the average interest 
rate and  CIR model ensures that interest rates will be positive [2]. 

Research relating to the CIR model has been done, such as the moment for 
solution of the CIR model [3],[4] then installment joint life insurance actuarial models 
with the stochastic interest rates [5] , premiums calculation for life insurance [6],[7] and 
the Cox Ingersoll Ross estimate model at the Indonesian bank using the Maximum 
Likelihood Estimation method [8]. The CIR model involving life insurance premiums has 
been carried out, by determining long-term life insurance premiums with interest rates 
following Vasicek and the Cox Ingersoll Ross [9] and calculating joint life insurance 
premiums with Vasicek and CIR [10]. Research related to varying benefits and varying 
rates of interest has been researched regarding the annual net premium of term life 
insurance for cases of multiple decrements with varying interest rates [11] and single 
net premium of units linked term life insurance using the point to point method with 
varying benefits [12] . None of the literature states any limitations in using  
CIR model in research. In addition to the financial research related to insurance and 
interest rates, CIR model research is also in mathematic computation to solve problems 
related to method and formula in the CIR model such as research about applied the role 
of adaptivity in a numerical method for the Cox – Ingersoll – Ross model [13] and 
formula and full parameter in CIR model [14], [15]. 

Research that has been done before still calculates annual insurance premium 
payments and annual single premium payments paid at the beginning of the year. 
Therefore, this research aims to determine varying interest rates in the future using CIR 
model simulation to calculate term life insurance premiums 5 years with premiums paid 

at the beginning of 
1

𝑚
 year intervals with 𝑚 = 12 months or premiums payment every 

month and benefits paid at the end of  
1

𝑚
  year interval of death or at the end of each 

month of death with the amount varying benefits each year. 
 

METHODS  

Data  

This research used secondary data such as monthly history data BI interest rates 
period January 2015 until December 2020 which was accesed from website 
https://www.bps.go.id/indicator/13/379/12/bi rates.html and Indonesian Life Table 
2019. The data analysis process was done using Octave Software and Microsoft Excel.  

Estimation CIR Model 

 The Cox Ingersoll Ross model is one of the stochastic models used for fluctuating 
interest rates in period of time. The CIR model can be used to estimate interest rates that 
changes in period of time in the future. The algorithm in the process of determining CIR 
model following the steps: 

a. Input monthly history data BI interest rates period Januari 2015 until December 
2019. 

b. Estimate parameters CIR model using algorithm as following the equation [9] . 

𝜅 =
∑ 𝑟𝑖

𝑛−1
𝑖=1 ∑ (

1

𝑟𝑖
)𝑛−1𝑖=1 − (𝑛 − 1)

2 − ∑ (
1

𝑟𝑖
) ∑ 𝑟𝑖+∆𝑡 +

𝑛−1
𝑖=1

𝑛−1
𝑖=1 (𝑛 − 1) ∑

𝑟𝑖+∆𝑡

𝑟𝑖

𝑛−1
𝑖=1

(∑ 𝑟𝑖
𝑛−1
𝑖=1

∑ (
1

𝑟𝑖
)𝑛−1𝑖=1 − (𝑛 − 1)

2) ∆𝑡
       (1) 

https://www.bps.go.id/indicator/13/379/12/bi%20rate.html


Determination of Term Life Insurance Premiums with Varying Interest Rates Following Cox 
Ingersoll Ross Model and Varying Benefits Value 

Dian Puspita 632 

  𝜽

=
∑

𝒓𝒊+∆𝒕

𝒓𝒊

𝒏−𝟏
𝒊=𝟏  (∑ 𝒓𝒊

𝒏−𝟏
𝒊=𝟏 ) − (𝒏 − 𝟏)(∑ 𝒓𝒊+∆𝒕

𝒏−𝟏
𝒊=𝟏 )

∑ 𝒓𝒊
𝒏−𝟏
𝒊=𝟏 ∑ (

𝟏

𝒓𝒊
)𝒏−𝟏𝒊=𝟏 − (𝒏 − 𝟏)

𝟐 − ∑ (
𝟏

𝒓𝒊
) ∑ 𝒓𝒊+∆𝒕 +

𝒏−𝟏
𝒊=𝟏

𝒏−𝟏
𝒊=𝟏 (𝒏 − 𝟏) ∑

𝒓𝒊+∆𝒕

𝒓𝒊

𝒏−𝟏
𝒊=𝟏

        (𝟐) 

  𝝈

= √
𝟏

𝒏 − 𝟐
∑ (

𝒓𝒊+∆𝒕

√𝒓𝒊
− (

𝜿𝜽∆𝒕 

√𝒓𝒊
+

𝟏 − 𝜿∆𝒕𝒓𝒊 

√𝒓𝒊
))

𝟐
𝒏−𝟏

𝒊=𝟏

                                                         (𝟑) 

 where 𝜿 is the speed of mean reversion, 𝜽 is the long-term value, 𝝈 is the 
volatility interest rate, 𝒓𝒊 is the interest rates period of time, 𝒏 is a lot of data 
used, and ∆𝒕 is the interval of time. 

c. Fit Test CIR Model using the equation 
MAPE is one of the statistics method used for the level of forecasting accuracy. 
The better model can be used when MAPE has the lower value. MAPE is 
calculated following the equation: 

       𝑀𝐴𝑃𝐸 =
100

𝑛
∑ |

𝐴𝑡 − 𝐹𝑡
𝐴𝑡

|

𝑛

𝑡=1

%                                                                                         (4) 

where 𝐴𝑡  is an actual data by BI interest rates, 𝐹𝑡  is a data estimate by the CIR 
model, and 𝑛 is a lot of data used [13]. 

Simulation The Future Interest Rates   

 The Cox Ingersoll Ross model is one of the stochastic models used for fluctuating 
interest rates in period of time. The CIR model can be used to simulate changes in 
interest rates in the future. The algorithm in the interest rates simulation process for the 
future using the CIR model following the steps: 

a. Input parameter values such as r0, κ, θ, σ, ∆t, m dan 𝑡. 
where r0 is interest rates in BI rates, κ is the speed of mean reversion, θ is the 
long-term value, σ is the volatility interest rate, m is a lot of simulations, and ∆t is 
the interval of time, and 𝑡 is the due date monthly. 

b. Generate random variable εt~ N (0,1) used to calculate interest rates r (t + ∆t).  
where εt is a random number which is normally distributed N (0,1), r (t + ∆t) an 
interest rates period of time. 

c. Calculate future interest rates with octave software as follow the equation  

  𝒓𝒕+∆𝒕 =  𝒓𝒕 +  𝜿 (𝜽 − 𝒓𝒕)∆𝒕 + 𝛔√𝒓𝒕 √∆𝒕 𝜺𝒕                   

 (5) 
 where 𝐫𝐭 is an interest rates in the CIR model 
d. Monte Carlo simulation for interest rates which is calculated following the 

equation 

 �̅�𝒊 =
𝟏

𝒎
∑ 𝒓𝒊𝒋

𝒎

𝒋=𝟏

                                                                                                                        (𝟔)  

Determination Term Life Insurance Premium  

 This research to determine term life insurance premium paid at the beginning of 
each month. The steps of research are carried out as follow: 

a. Calculate monthly interest rates in the CIR model using the equation 



Determination of Term Life Insurance Premiums with Varying Interest Rates Following Cox 
Ingersoll Ross Model and Varying Benefits Value 

Dian Puspita 633 

𝒋𝒏 = (𝟏 + 𝒊𝒏)
𝟏/𝟏𝟐 − 𝟏                                                                                                  (𝟕) 

where 𝑗𝑛 is monthly interest rates, 𝑖𝑛 is an interest rates in the CIR model 
b. Calculate the discount factor of interest rates in the CIR model using the 

equation 

(𝒗∗)𝒌 =
𝟏

(𝟏 + 𝒋𝟏) × (𝟏 + 𝒋𝟐) × … × (𝟏 + 𝒋𝒌)
𝐰𝐢𝐭𝐡 𝟏 ≤ 𝐤 ≤ 𝟔𝟎                  (𝟖) 

where (𝑣∗)𝑘  is the discount factor in 𝑘 month 
c. Calculate the probability of death for age fraction in Indonesian Life Table 2019 

𝑞𝑥+𝑡𝑦 =
𝑦𝑞𝑥

1 − 𝑡 𝑞𝑥
                                                                                                              (9) 

where 𝑞𝑥+𝑡𝑦  is probability of death at fractional age with y =
1

12
 and 𝑡 =

1

12
, . . ,

11

12
 

d. Calculate the actuarial present value of the annuity of term life insurance using 
the equation 

�̈�
𝑥∶𝑛|
∗(𝑚)

=
1

𝑚
∑ (𝑣∗)𝑘  

𝑚𝑛−1

𝑘=0

𝑝𝑥𝑘
𝑚

                                                                                       (10) 

where �̈�
𝑥∶𝑛|
∗(𝑚)

 is the actuarial present value of the annuity monthly, (𝑣∗)𝑘  is the 

discount factor, 𝑝𝑥𝑘
𝑚

 is the probability of survival at fractional age with 
𝑘

m
=

1

12
 

, . . ,
60

12
 

e. Calculate the actuarial present value of the varying benefits of term life insurance 
using the equation  

𝐴
  𝑥∶𝑛|
1∗(𝑚)

= ∑ 𝑏𝑛  (𝑣
∗)𝑘+1 

𝑚𝑛−1

𝑘=0

𝑝𝑥𝑘
𝑚

𝑞
𝑥+

𝑘

𝑚
 

1

𝑚

                                                                   (11) 

where 𝐴
  𝑥∶𝑛|
1∗(𝑚)

 is the actuarial present value of the varying benefits, 𝑏𝑛 is the 

varying benefit each year, (𝑣∗)𝑘   is the discount factor, 𝑝𝑥𝑘
𝑚

 is the probability of 

survival at fractional age with 
𝑘

m
=

1

12
 , . . ,

60

12
, 𝑞

𝑥+
𝑘

𝑚
 

1

𝑚

is the probability of death at 

fractional age. 
f. Calculate the value of monthly net premiums term life insurance using the 

equation  

 𝑃
  𝑥∶ 𝑛|
1(𝑚)

=
𝐴∗𝑥:𝑛|

1(𝑚)

�̈�
𝑥∶𝑛|
∗(𝑚)

                                                                                                              (12) 

where 𝑃
  𝑥∶ 𝑛|
1(𝑚)

 is the net premiums monthly , 𝐴
  𝑥∶𝑛|
1∗(𝑚)

 is the actuarial present value 

of the varying benefits, and  �̈�𝑥∶𝑛|
∗(𝑚)

 is the actuarial present value of the annuity 

monthly.  

RESULTS AND DISCUSSION  

Data  

 This research used data monthly interest rates from the BI 7-Day Repo Rate from 
January 2015 until December 2020 which was accessed from website 
https://www.bps.go.id/indicator/13/379/12/bi rate.html. The data monthly interest 
rates from BI as shown in Figure 1. 

https://www.bps.go.id/indicator/13/379/12/bi%20rate.html


Determination of Term Life Insurance Premiums with Varying Interest Rates Following Cox 
Ingersoll Ross Model and Varying Benefits Value 

Dian Puspita 634 

 
Figure 1.  Interest Rates BI 7-Day Repo Rate 

 
Estimation Parameter CIR Model  
 Estimation parameters in the CIR Model calculates using equation (1) to (3) and 
solved with octave software. Estimate parameters CIR model as shown in Table 1.  

Table 1. Parameters CIR Model 
Parameter Value 

𝜅 0,5309 
𝜃 0,047218 
𝜎 0,7679 

Based on Table 1, the parameter values are κ is the speed of reversion as 0,5309, 
 θ is the long-term value as 0,047218 and σ is the volatility as 0,7679. The error value of 
the parameter values in the CIR model using MAPE in equation 4 is 3,9014%. Error 
MAPE <10% indicates that the estimated data is excellent for describing the actual data 
in BI interest rates. The comparison between data interest rates in BI and estimated data 
interest rates in the CIR model is shown in Figure 2. 

 

       Figure 2.  Interest Rates BI and Estimate Interest Rates CIR Model 
 
Simulation The Future Interest Rates  

Interest rates in the CIR model for January 2022 to December 2026 with numeric 

variable 𝑟0 = 0,035, ∆𝑡 =
1

12
, 𝑛 = 60, simulated 100 times and calculated using equation 

(5) with octave software. The results to estimate future interest rates as shown in Table 
2. 

 
  

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70

In
te

re
st

 R
a

te
s 

B
I 

7
-D

a
y
 R

e
p

o
 R

a
te

Time (month)

0

0,01

0,02

0,03

0,04

0,05

0,06

1 2 3 4 5 6 7 8 9 10 11 12

V
a

lu
e

 o
f 

In
te

re
st

 R
a

te
s

Time (month)

BI Interest Rates Interest Rates CIR model



Determination of Term Life Insurance Premiums with Varying Interest Rates Following Cox 
Ingersoll Ross Model and Varying Benefits Value 

Dian Puspita 635 

Table 2. Interest Rates in the CIR Model 
Month Interest 

Rates 
(%) 

Month Interest 
Rates 
(%) 

Month Interest 
Rates 
(%) 

Month Interest 
Rates 
(%) 

Month Interest 
Rates 
(%) 

1 0,025255 13 0,050327 25 0,053169 37 0,052515 49 0,036882 
2 0,027668 14 0,056575 26 0,051294 38 0,047182 50 0,029854 
3 0,025565 15 0,056146 27 0,044594 39 0,057194 51 0,040682 
4 0,018886 16 0,055893 28 0,041811 40 0,053692 52 0,027046 
5 0,016036 17 0,053001 29 0,037798 41 0,056603 53 0,048051 
6 0,023475 18 0,045527 30 0,032737 42 0,061089 54 0,050728 
7 0,028245 19 0,049414 31 0,038453 43 0,051467 55 0,048738 
8 0,034898 20 0,052352 32 0,036794 44 0,031064 56 0,050071 
9 0,029248 21 0,056978 33 0,044810 45 0,036524 57 0,043225 

10 0,035791 22 0,047014 34 0,062456 46 0,038016 58 0,053388 
11 0,040769 23 0,052004 35 0,072659 47 0,036822 59 0,065475 
12 0,056432 24 0,052581 36 0,068307 48 0,032409 60 0,062451 

The Effective Interest Rates Monthly CIR Model 

This research will calculate the value of term life insurance premiums for 5 years 

with premiums paid at the beginning of 
1

𝑚
 with 

1

𝑚
 = 

1

12
 years or premiums paid every 

month, so the effective interest rates in the CIR model are required every month. 
Calculate the effective interest rates every month using equation (7). The results of 
interest rate every month in the CIR model are shown in Table 3.  

Table. 3 Interest Rates Monthly in the CIR Model 
Month Interest 

Rates 
(%) 

Month Interest 
Rates 
(%) 

Month Interest 
Rates 
(%) 

Month Interest 
Rates 
(%) 

Month Interest 
Rates 
(%) 

1 0,002081 13 0,004100 25 0,004326 37 0,004274 49 0,003023 
2 0,002277 14 0,004597 26 0,004177 38 0,003849 50 0,002454 
3 0,002106 15 0,004563 27 0,003642 39 0,004646 51 0,003329 
4 0,001560 16 0,004543 28 0,003419 40 0,004368 52 0,002226 
5 0,001327 17 0,004313 29 0,003097 41 0,004599 53 0,003919 
6 0,001936 18 0,003717 30 0,002688 42 0,004954 54 0,004132 
7 0,002324 19 0,004027 31 0,003149 43 0,004191 55 0,003974 
8 0,002863 20 0,004261 32 0,003016 44 0,002553 56 0,004080 
9 0,002405 21 0,004629 33 0,003660 45 0,002994 57 0,003533 

10 0,002935 22 0,003836 34 0,005061 46 0,003114 58 0,004344 
11 0,003336 23 0,004234 35 0,005862 47 0,003018 59 0,005299 
12 0,004585 24 0,004280 36 0,005521 48 0,002661 60 0,005061 

From the monthly effective interest rates in the CIR model, the value of the discount 
factor is calculated using equation (8). The results of calculating the discount factor with 
the monthly effective interest rate in the CIR model are shown in Table 4. 

 
 
 

  



Determination of Term Life Insurance Premiums with Varying Interest Rates Following Cox 
Ingersoll Ross Model and Varying Benefits Value 

Dian Puspita 636 

Table 4. The Value of Discount Factor in the CIR Model 
Mont

h 
Discoun
t Factor 

(%) 

Month Discoun
t Factor 

(%) 

Mont
h 

Discoun
t Factor 

(%) 

Month Discoun
t Factor 

(%) 

Month Discoun
t Factor 

(%) 
1 0,99792

4 
13 0,96678

0 
25 0,91851

4 
37 0,87593

3 
49 0,83832

5 
2 0,99565

7 
14 0,96235

7 
26 0,91469

3 
38 0,87257

4 
50 0,83627

2 
3 0,99356

4 
15 0,95798

6 
27 0,91137

3 
39 0,86853

9 
51 0,83349

8 
4 0,99201

6 
16 0,95365

4 
28 0,90826

8 
40 0,86476

2 
52 0,83164

6 
5 0,99070

2 
17 0,94955

8 
29 0,90546

4 
41 0,86080

4 
53 0,82840

0 
6 0,98878

8 
18 0,94604

2 
30 0,90303

7 
42 0,85656

1 
54 0,82499

1 
7 0,98649

6 
19 0,94224

7 
31 0,90020

2 
43 0,85298

6 
55 0,82172

6 
8 0,98368

0 
20 0,93824

9 
32 0,89749

5 
44 0,85081

4 
56 0,81838

7 
9 0,98132

0 
21 0,93392

6 
33 0,89422

3 
45 0,84827

4 
57 0,81550

6 
10 0,97844

8 
22 0,93035

8 
34 0,88971

9 
46 0,84564

1 
58 0,81197

9 
11 0,97519

5 
23 0,92643

5 
35 0,88453

4 
47 0,84309

7 
59 0,80769

9 
12 0,97074

4 
24 0,92248

8 
36 0,87967

7 
48 0,84085

9 
60 0,80363

2 

The Actuarial Present Value of the Initial Annuity Term Life Insurance 

The actuarial present value of the initial annuity of term life insurance is used to 
obtain 5 years term life insurance premium will be calculated the actuarial present value 
of the annuity using equation (10) as follow. 

 �̈�
30∶5|

∗(12)
=

1

12
∑ (𝑣∗)𝑘  59𝑘=0 𝑝𝑥𝑘

12

          

=
1

12
 (1 +

1

(1 + 0,002081) 
𝑝301

12

+
1

(1 + 0,002081)(1 + 0,002277)
𝑝302

12

+
1

(1 + 0,002081)(1 + 0,002277) … (1 + 0,005299) 
𝑝3048

12

𝑝3411
12

)     

=
1

12
 (1 + (0,9999375)(0,997924) + (0,999875)(0,995657) + ⋯

+ (0,807699)(0,9990700)(0,9990925) 

�̈�
30∶5|

∗(12)
=  4,5220188 

 The results of calculating the initial annuity of 5 years term life insurance with 
varying interest rates in the CIR model for ages 30 to 75 years are shown in Figure 3. 



Determination of Term Life Insurance Premiums with Varying Interest Rates Following Cox 
Ingersoll Ross Model and Varying Benefits Value 

Dian Puspita 637 

 
Figure 3.  The Actuarial Present Value of the Initial Annuity Term Life Insurance 

 
The initial annuity value for a 30 years man is 4,522019, and the initial annuity 

value for a woman is 4,523243. The annuity value for a 31 years man is 4,521676, and 
the annuity value for a 31 years woman is 4,522986. The annuity value decreases every 
year because the older individual, the annuity value will decrease. 

The Actuarial Present Value of the Varying Benefit Term Life Insurance 

It is assumed that the number of benefits varies each year as in the first year is 𝑏1 =
1  unit, in the second year is 𝑏2 = 1,5  unit, in three years is 𝑏3 = 2  unit, in four years is 
𝑏4 = 2,5  unit, and in five years is 𝑏5 = 3  unit. Calculate the actuarial present value of 

the varying benefits paid at the end of  
1

12
 years of death for an individual aged 30 years 

using equation (11) as follow. 

𝐴
  30∶5|

1∗(12)
=  ∑ 𝑏𝑛 (𝑣

∗)𝑘+159𝑘=0  𝑝30𝑘
12

𝑞
30+

𝑘

12
 

1

12

          

𝐴
  30∶5|

1∗(12)
= ∑ 𝑏1 (𝑣

∗)𝑘+111𝑘=0  𝑝30𝑘
12

𝑞
30+

𝑘

12
 

1

12

+ ∑ 𝑏2 (𝑣
∗)𝑘+123𝑘=12  𝑝30𝑘

12

𝑞
30+

𝑘

12
 

1

12

+

∑ 𝑏3 (𝑣
∗)𝑘+135𝑘=24  𝑝30𝑘

12

𝑞
30+

𝑘

12
 

1

12

+ ∑ 𝑏4 (𝑣
∗)𝑘+147𝑘=36  𝑝30𝑘

12

𝑞
30+

𝑘

12
 

1

12

+

∑ 𝑏5 (𝑣
∗)𝑘+159𝑘=48  𝑝30𝑘

12

𝑞
30+

𝑘

12
 

1

12

  

 = 1(0,00073966) + 1,5(0,00075942) + 2(0,00077835) + 2,5(0,00079175) +
 3(0,0080956) 

𝐴
  30∶5|

1∗(12)
= 0,00784355 

The results of calculating the actuarial present value of 5 years term life insurance 
with varying benefits for ages 30 to 75 years are shown in Figure 4. 

 
Figure 4.  The Actuarial Present Value of the Varying Benefits Term Life Insurance 

 

4,300000

4,350000

4,400000

4,450000

4,500000

4,550000

30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74

V
a

lu
e

 o
f 

In
it

ia
l A

n
n

u
ty

Initial Annuity in Man Initial Annuity in Woman

0,0000000

0,1000000

0,2000000

0,3000000

30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74

B
e

n
e

fi
t 

V
a

lu
e

Age (year)

Benefit value for man Benefit value for woman



Determination of Term Life Insurance Premiums with Varying Interest Rates Following Cox 
Ingersoll Ross Model and Varying Benefits Value 

Dian Puspita 638 

Figure 4 shows the value of the varying benefits of term life insurance with various 
interest rates in the CIR model from ages 30 to 75 years for man and woman. The 
benefits value for a man aged 30 is 0,0078436, and the benefit value for a woman aged 
30 is 0,0058254. The value of benefits for a man aged 31 is 0,0084167, and the value of 
benefits for a woman aged 31 is 0,0062626. If one unit of benefits value is Rp. 
10.000.000 so benefit value for man aged 30 is 0,0078436 × Rp. 10.000.000 = Rp 
78.436. The results of the varying benefits in term life insurance for men and women 
increase every year because the older a person is the greater the possibility of 
someone's death, so the varying benefits obtained will increase yearly. 

Net Premium Term Life Insurance 
The monthly net premium value of term life insurance with interest rates in the CIR 

model and varying benefits using the equivalence principle in equation (12). The results 
of calculating the monthly net premium for five years term life insurance paid at the 
beginning of each month for individuals aged 30 years are as follows 

𝑃
 30: 5|

1(12)
=

𝐴∗
30:5|

1(12)

�̈�
30∶5|
∗(12)

=
0,0078436

4,522019
= 0,00173452 

The results of calculating the net premium term life insurance with various interest 
rates in the CIR model for ages 30 to 75 years are shown in Figure 5. 

 
Figure 5.  The Net Premium Term Life Insurance 

Figure 5 shows that the value of term life insurance premiums monthly for a man 
aged 30 is 0,0017345 and for a woman aged 30 is 0,0012879. The premium for a man 
aged 31 is 0,0018614 and for a woman aged 31 is 0,0013846. The unit of premium value 
is rupiah, if one unit premium value is Rp. 10.000.000 so premium value for man aged 
30 is 0,0017345 × Rp. 10.000.000 = Rp 17.345. The value of the premium paid to men 
and women aged 30 years to 75 years increases every year. Because of the greater 
chance of someone’s death  so the premium payments will increase. 

 

CONSLUSIONS 

 The Cox Ingersoll Ross model simulation can be applied to obtain variable interest rates 
in the future with parameter values κ=0,5309, θ=0,047218, σ=0,7679 and an error value 
in MAPE of 3,9014%, indicating that the estimated CIR model value is very good to 
describe the actual data. The results of the premiums obtained from varying interest 
rates in the CIR model show that the older individual is the greater premiums paid with 
the varying benefits obtained increasing every year. For further research can be 
suggested to use another stochastic model as a simulation to obtain interest rates that 
will be used to calculate premiums in whole life insurance or other life insurance. 

0,0000000

0,0200000

0,0400000

0,0600000

0,0800000

30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74

P
re

m
iu

m
 v

a
lu

e

Ages (year)
Man Premium Woman premium



Determination of Term Life Insurance Premiums with Varying Interest Rates Following Cox 
Ingersoll Ross Model and Varying Benefits Value 

Dian Puspita 639 

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