In developing countries such as Malaysia, the availability of reinsurance arrangements provides several advantages to the primary insurers such as keeping their risk exposures at prudent levels by having their large risk exposures reinsured by another company, meeting client requests for larger insurance coverage by having their limited financial sources supported by another company, and acquiring underwriting skills, experience and ability of handling complex claims by depending on another company for such services. This paper aims to model insurance claims and assess the insolvency probability of reinsurance treaties. Claims data was obtained from one of the leading insurers in Malaysia and R programming with actuar package is used to compute the probability of insolvency.Reinsurance, pricing, insolvency probability, excess-of-loss., Financial Economics, C10, C13,

Ahmad Anuar, Ansar Asnawi

Ismail, Noriszura

English

Research Papers in Economics

Perspectives of Innovations, Economics & Business, Volume 3, 2009          
  www.pieb.cz 
 
International Cross-Industry Journal  
62 
INSOLVENCY PROBABILITY IN REINSURANCE 
TREATY: A CASE STUDY IN MALAYSIA  
  NORISZURA ISMAIL, PH.D., 
ANSAR ASNAWI AHMAD ANUAR 
 
Faculty of Science and Technology 
Universiti Kebangsaan Malaysia, Malaysia 
JEL Classifications: C10, C13 
Key words:  Reinsurance, pricing, insolvency probability, excess-of-loss. 
Abstract:    In  developing  countries  such  as  Malaysia,  the  availability  of  reinsurance  arrangements  provides  several 
advantages to the primary insurers such as keeping their risk exposures at prudent levels by having their large risk exposures 
reinsured by another company, meeting client requests for larger insurance coverage by having their limited financial sources 
supported by another company, and acquiring underwriting skills, experience and ability of handling complex claims by 
depending on another company for such services. This paper aims to model insurance claims and assess the insolvency 
probability of reinsurance treaties. Claims data was obtained from one of the leading insurers in Malaysia and R programming 
with actuar package is used to compute the probability of insolvency. 
ISSN: 1804-0527 (online)  1804-0519 (print)    PP. 62-64 
 
Introduction 
The  volume  of  reinsurance  business  in  the  Malaysian 
general insurance industry may be observed from the amount 
of premiums ceded to companies abroad and within Malaysia. 
In  1965  and  1975  for  instance,  the  amount  of  reinsurance 
premiums ceded to companies abroad were RM12 million and 
RM60  million,  equivalent  to  17%  and  21%  of  written 
premiums  respectively.  The  amount  increased  to  RM296 
million  and  RM1223  million  each  in  1985  and  1995, 
deteriorating  to  24%  and  27%  of  written  premiums 
respectively,  but  decreased  to  RM957  million  in  2005, 
showing an improved percentage of 10% of written premiums 
(Lee,  1997;  BNM,  1995;  BNM,  2005).  Based  on  the 
proportions  of  written  premiums,  there  was  a  marked 
deterioration in the 1980s and 1990s in terms of the domestic 
retention capacity of premiums compared to the 1960s and 
1970s,  due  to  the  fact  that  Malaysia  has  never  imposed 
restrictions  on  foreign  exchange  ouflows  for  reinsurance 
purposes.  For  most  companies,  their  limited  financial 
resources  and  small  capability  in  underwriting  skills  and 
handling  complex  claims  have  enhanced  their  dependence 
upon outside reinsurers, leading to the issue of unsatisfactory 
domestic  retention  of  premium  (Lee,  1997).  The  level  of 
retention capacity improved however in the 2000s, largely due 
to the continuous efforts taken by the regulatory bodies and 
industry players, especially in encouraging domestic insurers 
and reinsurers to absorb higher proportions of large risks.  
Several  studies  focusing  on  reinsurance,  deductible  and 
policy limit have been carried out in the actuarial literature. 
Zhuang (2008) established optimal allocations of policy limits 
and deductibles with respect to the distortion of risk measures, 
Hua and Cheung  (2008) applied equivalent utility premium 
principle and study the worst allocations of policy limits and 
deductibles,  Dimitriyadis  and  Oney  (2008)  modeled  loss 
distributions through Allianz tool pack, derived premiums at 
different  levels  of  deductibles  and  computed  ruin 
probabilities, and Wang (1996; 1998) introduced the method 
of Proportional Hazard Transform for pricing risks, excess-of-
loss  coverages,  increased  limits,  risk  portfolios  and 
reinsurances  treaties.  This  study  aims  to  model  insurance 
claims  and  assess  insolvency  probability  of  reinsurance 
treaties. Claims data was obtained from one of the leading 
insurers in Malaysia and R programming with actuar package 
is used to compute the insolvency probability. 
Loss model 
Several parametric distributions were fitted on the claims 
amount using maximum likelihood method. The best model, 
each  for  one-parameter,  two-parameter  and  three-parameter 
distributions, was selected by choosing the largest value of log 
likelihood  function  and  is  shown  in  Table  1.  The  tests  of 
Kolmogorov-Smirnov  (K-S),  Anderson-Darling  (A-D)  and 
Bayesian Schwarz Criterion (BSC) were carried out to select 
the best model (Klugman et al. 2008). Table 2 shows that the 
best model selected is Burr distribution.  
Pricing of layers 
As a developing country, insurance industry in Malaysia 
seldom has a single local insurer able to cover a single large 
risk, especially in non-life insurance business. In practice, the 
coverage of large risks is usually divided into several excess-
of-loss  layers  shared  and  underwritten  by  several  insurers 
and/or  reinsurers.  Therefore,  the  pricing  of  layers  is  very 
crucial, especially in the process of dividing risks and pricing 
risks fairly for each insurer and/or reinsurer. 
Let  N  denotes the random variable for claim frequency. 
The expected claim frequency can be calculated as 
,... 1 , 0    , ) ( ) (
0
= =∑
¥
=
k k S N E
k
,   (1) Perspectives of Innovations, Economics & Business, Volume 3, 2009          
  www.pieb.cz 
 
International Cross-Industry Journal  
63 
TABLE 1. BEST MODELS 
  Parametric distribution  # of parameters  Estimated parameters 
Log likelihood  Exponential  1  0.000025 = l   -2 207.0 
Gamma  2  4637 . 1 = a   57 . 26279 = q   -2 199.9 
Burr  3  43 . 426 , 86 = q   5169 . 1 = g   7783 . 3 = a   -2 197.0 
 
TABLE 2. RESULTS OF KOLMOGOROV-SMIRNOV  (K-S), ANDERSON-DARLING  (A-D) AND BAYESIAN 
SCHWARZ CRITERION (BSC) TESTS 
Parametric distribution  # of parameters  K-S Test  A-D Test  BSC 
Exponential  1  0.18655  389.31  - 
Gamma  2  0.11098  384.68  -2205.16 
Burr  3  0.09454  383.87  -2204.40 
         
where  (.) S   denotes  the  survival  function.  Under  PH-
Transform assumption, the expected claim frequency is equal 
to (Wang 1996; 1998), 
[ ] ∑
¥
=
=
0
) ( ) (
k
r k S N H ,                  (2) 
where  r  denotes the index of ambiguity degree. Let  X  
denotes the random variable for claim severity. The expected 
claim severity is,  
∫
¥
=
0
) ( ) ( dx x S X E ,                   (3) 
whereas  under  the  PH-Transform  assumption,  the 
expected claim severity is (Wang 1996; 1998),  
[ ] ∫
¥
=
0
) ( ) ( dx x S X H
r .                   (4) 
If  the  amount  of  loss  follows  Burr  distribution  with 
parameters  ( ) g q a , , ,  Wang  (1996;  1998)  showed  that  the 
calculation of  ) (X H  also follows Burr, but with parameters 
( ) g q a , , r .  By  implementing  the  frequency  and  severity 
approach, the expected aggregate claims can be calculated as 
) ( ) ( ) ( X E N E S E =   whereas  under  the  assumption  of  PH-
Transform,  the  expected  aggregate  claims  is  equal  to 
) ( ) ( X H N H . The same approach may also be implemented 
for calculating the price of a layer. The average loss or mean 
severity of a layer  ] , ( u d d +  may be written as, 
( ) ( )dx x S M E
u d
d
  ∫
+
= ,                           (5) 
whereas under the PH-Transform, the average loss of the 
same layer is (Wang 1996; 1998), 
( ) [ ] dx x S M H
u d
d
r  ) ( ∫
+
= ,                   (6) 
where  M  is  the random variable for the loss of a layer 
] , ( u d d + , 





£ +
+ < £ -
< £
=
X u d u
u d X d d X
d X
M
,
,
0 , 0
. 
Therefore,  the  expected  aggregate  claims  is 
) ( ) ( ) ( M E N E S E = ,  whereas  under  the  PH-Transform,  the 
expected aggregate claims is  ) ( ) ( M H N H .     
Table 3 shows the mean severity, mean frequency, burning 
cost,  loaded  rate  and  relative  loading  under  PH-Transform 
assumption  for  several  excess-of-loss  layers  assuming  N  
follows  Poisson( 6 = l )  and  X   is  distributed  as 
Burr( 43 . 426 , 86 = q ,  5169 . 1 = g ,  7783 . 3 = a ).  The  burning 
cost is calculated as 
SEP
S E ) ( , where SEP denotes the subject 
earned premium, assumed to be RM10 000 000. The loaded 
rate is calculated as 
SEP
M H N H ) ( ) ( ,  which can also be written 
in the function of burning cost, 
SEP
S E
SEP
M H N H ) (
) 1 (
) ( ) (
x + = , 
where x  denotes the relative loading. It is worth to note that 
the  relative  loading,  x ,  under  PH-Transform  assumption 
increases as the excess-of-loss layer,  ] , ( u d d + , increases.  
Table  4  shows  the  premium  and  relative  loading  under 
several  assumptions  of  PH-Transform 
( 9 . 0 = r , 8 . 0 = r , 6 . 0 = r ). It should be noted that the lower 
the  r ,  the  higher  the  premium,  implying  that  the  relative 
loading is higher when ambiguity increases. In addition, the 
premium is lower when the layer,  ] , ( u d d + , increases. 
Insolvency probability 
If the claim frequency follows Poisson(l ), the aggregate 
claims,  S , follows compound Poisson where the variance of 
aggregate claims can be calculated as  ) ( ) (
2 M E S Var l = . 
The distribution of aggregate claims,  S , by applying Central 
Limit Theorem, may be estimated by the Normal distribution. 
Therefore, the insolvency probability, i.e. the probability of 
having  the  aggregate  claims  larger  than  the  aggregate 
premiums, under PH-Transform assumption is, 
( ) 






 -
> = >
) (
) ( ) ( ) (
Pr ) ( ) ( Pr
S Var
S E M H N H
Z M H N H S   (7) 
Table  5  shows  the  insolvency  probability  under  several 
linear  loading  assumptions,  i.e.  premium= ) ( ) 1 ( S E x + ,  and 
several  PH-Transform  assumptions,  i.e. 
premium= ) ( ) ( M H N H .  
 Perspectives of Innovations, Economics & Business, Volume 3, 2009          
  www.pieb.cz 
 
International Cross-Industry Journal  
64 
TABLE 3. MEAN SEVERITY, MEAN FREQUENCY, BURNING COST, LOADED RATE AND RELATIVE LOADING 
Layer 
 
( ) M E  
 
( ) M H   ( ) 9 . 0 = r  
( ) N E   ( ) N H   ( ) 9 . 0 = r  
Burning 
Cost 
 
Loaded Rate  Relative 
Loading 
( ] k k 300 , 100   1 652.40  2 509.38  6  6.247  0.0009914  0.0015676  1.58 
( ] k k 500 , 300   30.10  70.84  6  6.247  0.0000181  0.0000443  2.45 
( ] k k 700 , 500   2.91  8.75  6  6.247  0.0000017  0.0000055  3.13 
( ] k k 900 , 700   0.56  2.00  6  6.247  0.0000003  0.0000012  3.71 
( ] k k 900 , 100   1 685.97  2 590.97  6  6.247  0.0010116  0.0016185  1.60 
 
TABLE 4. PREMIUM AND RELATIVE LOADING 
Layer  Premium 
( ) 9 . 0 = r  
Relative 
loading 
Premium 
( ) 8 . 0 = r  
Relative 
loading 
Premium 
( ) 7 . 0 = r  
Relative 
loading 
( ] k k 300 , 100   15 676.10  1.58  25 177.84  2.54  41 193.44  4.16 
( ] k k 500 , 300   442.54  2.45  1 095.53  6.07  2 743.96  15.20 
( ] k k 700 , 500   54.66  3.13  172.62  9.90  550.16  31.54 
( ] k k 900 , 700   12.49  3.71  46.59  13.88  175.61  52.32 
( ] k k 900 , 100   16 185.79  1.60  26 492.58  2.62  44 663.17  4.42 
 
TABLE 5. INSOLVENCY PROBABILITY 
Insolvency probability 
Linear loading  PH-Transform 
Layer 
1 . 0 = x   15 . 0 = x   2 . 0 = x   9 . 0 = r   8 . 0 = r   7 . 0 = r  
( ] k k 300 , 100   0.49  0.48  0.47  0.42  0.29  0.13 
( ] k k 500 , 300   0.50  0.50  0.50  0.48  0.42  0.29 
( ] k k 700 , 500   0.50  0.50  0.50  0.49  0.46  0.37 
( ] k k 900 , 700   0.50  0.50  0.50  0.49  0.48  0.41 
( ] k k 900 , 100   0.49  0.48  0.47  0.42  0.29  0.12 
             
It is worth to note that the insolvency probability under 
PH-Transform  assumption  is  lower  than  the  linear  loading 
assumption for all layers, and the difference is lower when the 
layer,  ] , ( u d d + ,  increases.  Therefore,  PH-Transform 
assumption  may  be  used  as  an  alternative  in  reducing  the 
insolvency  probability  of  the  excess-of-loss  layers  in 
reinsurance treaties. 
 Conclusion 
In this paper, we have modeled the amount of insurance 
claims,  selected  the  best  model  using  the  tests  of 
Kolmogorov-Smirnov,  Anderson-Darling  and  Bayesian 
Schwarz, priced several layers of  ] , ( u d d +  using frequency 
and severity approach, and computed insolvency probability 
for  several  layers  of  ] , ( u d d + .  PH-Transform  assumption 
may  be  used  as  an  alternative  in  reducing  the  insolvency 
probability of the excess-of-loss layers in reinsurance treaties.  
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Insolvency probability in reinsurance treaty: a case study in Malaysia

Insolvency probability in reinsurance treaty: a case study in Malaysia

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