With reference to Wikipedia, a special feature of non-life insurance especially in motor insurance is that very often the premium charged largely depends on the claims history. Merit-rating systems known as NCD in the UK and Bonus-Malus in the rest of Europe, Asia and some African countries, are used by insurers to adjust the premium paid by a customer according to his/her claim history. Bonus is most often a discount in the premium on the renewal of the policy if no claim is made in the previous year. Malus is an increase in the premium if there is a claim in the previous year. The basic principle of bonus-malus systems is that, the higher the claim frequency of a policyholder, the higher the cost that is charged on average to the policyholder. Most systems categorise policyholders by classes, where each class has its own discount or add-on that is applied to the basic premium. A claim-free year implies a decline of one or more degrees on the bonus-malus scale on the renewal of the policy. If one makes a claim, an increase of a given number of degrees on the bonus-malus scale is given. The entry class may depend on the driver's age, gender, claims experience, the car's catalogue value at the time of insurance, the number of kilometres driven per year or certain occupations. Countries have diverse rules in place, which determines how many levels an insurer may increase, or decrease, the maximum bonus or malus allowed and which statistics insurers can use to evaluate the entry class of a driver.
The Scope of this report is very narrow. We have used the most simple of calculations and made very strong assumptions to make our analysis as simple as possible. The report is divided into two parts, A and B. Part A involves the analysis of the Dutch system which is regarded as a Markov system since it follows the properties of a Markov chain(as discussed in chapter 2) and Part B involves analysing a Non-Markov system, the Luxembourg system.
We begin our report of by looking at the systems that have been or are in place in the Netherlands.
Chapter two involves analysing the BM-14 system. The tools for analysis include finding the stationary distribution in the various classes of the system, finding the expected premium income once stability has been reached by using the occupancy proportions and simulations and then variability.
In Chapter three we present the BM-20 system but we do not delve into very rigorous calculations.
In Chapter four we discuss the NC-07 system.
We then compare the three systems using criteria such as efficiency and occupancy frequencies. This is done in Chapter five.
We treat the Non-Markov case in Chapter 7. We find that the Luxembourg system has got a special rule that makes it Non-Markov. Here we convert the system into a Markovian one to make the technicalities simpler.
We conclude our report by checking the consequences of the assumptions made throughout the report and then perform some sensitivity analysis.
The Dutch System
There are three main bonus-malus systems which have been in force in the Netherlands. The systems are abbreviated as
the NC-07:
A large number of companies knew and used this system until the end of 1981 when a group of researchers known as the Working Group extensively researched and introduced new rating systems. This is a no-claims discount system with seven classes.
the BM-20 :
After consultation with the market, the Working Group came up with the bonus-malus system with 20 classes in 1982. This system designed was to give proper credit in terms of premium both to careful drivers and risky drivers. Also the Working group decided that extra attention needed to be paid to the length of the scale and the set of transition rules.
the BM-14:
After further consultation, the BM-20 was modified into the BM-14 which was introduced in the same year.
To analyse these systems we will use the theory of Markov Chains, which makes it possible to investigate asymptotic properties of the systems.
The BM-14 System
In this section we will analyse the BM-14 as it is the latest. The other Systems will be analysed later on in the other sections. This is a bonus-malus scale with 14 classes.
First a basic premium is determined using certain rating factors which are company specific and this is the premium that drivers without a known past claims experience have to pay. The bonus and malus for good and bad claims experience respectively, are implemented through the use of a bonus-malus scale. Policyholders move down one class or remain at the lowest class, paying a lower premium after a claim free year and move up several classes after having filed one or more claims. The bonus-malus scale, including the percentages of the basic premium charged and the transitions made after 0, 1, 2, and 3 or more claims is shown in Table 1.
Class
Premium
Class After
0
1
2
3+
Claims
14
120
13
14
14
14
13
100
12
14
14
14
12
90
11
14
14
14
11
80
10
14
14
14
10
70
9
13
14
14
9
60
8
12
14
14
8
55
7
11
14
14
7
50
6
10
14
14
6
45
5
9
13
14
5
40
4
8
12
14
4
37.5
3
8
12
14
3
35
2
7
11
14
2
32.5
1
7
11
14
1
30
1
6
10
14
Table : Dutch bonus-malus system (BM-14)
Analysis
Markov analysis
"Bonus-malus systems can be considered as special cases of Markov processes" [1]. A policyholder moves from one class to another in time, in such processes. The Markov property says that the process is memoryless, as the probability of such movements does not depend on how one arrived in a particular class but on which class one is currently occupying. Using Markov analysis, one will be able to determine the proportion of policyholders that is expected to be in each class of the bonus-malus scale eventually. It also gives a way to find out how effective the system is in influencing premiums representing the actual risks of policyholders.
Considering a simple example, we assume a policyholder's level next year depends only on:
The level this year
The number of claims of this year
Let pij = P [level j next year │level i this year].
We get the following matrix P of the transition probabilities pij to go from sate i to state j:
P =
The matrix P is a stochastic matrix: every row represents a probability distribution over classes to be entered, so all elements of it are non-negative. All row sums, are equal to 1, since from any state i, one has to go to some state j.
Let denote the proportion of policyholders in each class at time 0.
Let denote the vector of expected proportions at each level k years from now. Then P.
We assume there is no entry or exit out of the system and policyholders are homogeneous with some a prior characteristics.
These assumptions will be used throughout this report.
Tools for analysis
Stationary distribution
Now supposing the number of claims each year for each policyholder has a Poisson (0.1) distribution and the entry level for our system is class 10 (for simplicity), we calculate the probability of 0, 1, 2 and 3 or more claims in a year and write down the transition matrix.
The expected proportion of policy holders in each class is then calculated for a number of years. It is observed that after about 30 years, the distribution does not change and it is about the same for different entry classes. This is an expected result as a consequence of our Markov assumption. The stationary distribution remains the same for different entry classes as well and this can be shown using a simple spreadsheet.
A graph of the expected proportions of policyholders in each class once stability has been reached, shows that for a group of policyholders with low claim rates, there will be more policyholders in the lower classes and paying less premiums than there will be in higher classes, paying higher premiums. For policyholders with high claim rates it is the opposite
See Figure 1.
Figure : Expected Proportion of policyholders on stationarity
Expected Premium Income
Expected premium income per driver from a group of drivers with different assumed values of claim rates are calculated by multiplying the premium to be paid in each class with the expected proportion of policyholders in the various classes once stability has been achieved, assuming a full premium of 1.
Similar calculations are done but this time by generating a long chain of random variables from a U(0,1) distribution using a spreadsheet. Each of these random variables is translated into simulated number of claims by specifying conditions using the cdf of a Poisson distribution. See Table 2. The corresponding simulated mean premium for each claim frequency, depending on the level a policyholder is occupying, according to Table 1 is then found out.
Condition
Result
If X ≤ P[0 claims]
0 claims
If X≤ ( P[0] + P[1] )
1 claim
If X ≤ ( P[0] + P[1] + P[2])
2 claims
If X ≤ ( P[0] + P[1] + P[2] + P[3])
3+ claims
Table : Translation of random numbers generated into simulated number of claims
Where X denotes the random number generated.
A Graph of the expected premium income is plotted against a number of years. For lower claim rates (good drivers) the graph shows the premium income to be high in the early years because of the starting class but drops in subsequent years. It is the opposite for higher claim rates. See Figure 2. After about thirty years onwards the premium income approach and remain a constant. This constant depends on the claim rate used. It can be seen from Figure 2 that this constant is about 39% for λ = 0.1 and then about 90% for λ = 0.5.
Figure : Expected premium income
It is observed that as λ increases the premium income increases. This is also expected as higher λs are associated with worse drivers. Premium income at stationarity can be used by insurers as a planning tool. Now the expected premium income per driver from a group of drivers with λ = 0.1, 0.2, 0.3, 0.4 as a proportion of the expected income per driver from a group from which λ = 0.5 once stability has been reached are calculated. It is observed that the system does not really differentiate properly between good and bad drivers. For example, using a simple spreadsheet analysis, it is realised that drivers with a claim rate of 0.1 pay about 40% of the premium that drivers with claim of 0.5 pay whereas they should pay 20% (0.1/ 0.5). The effect of this is that it does not encourage good driving and this system does not penalize bad drivers but new policyholders instead. It can also be inferred that this system is in place mainly as a marketing tool.
To illustrate the evolution of premium income over several years, a set of 130 random variables each with a U(0,1) distribution are generated using simulations on a spreadsheet, each of these random numbers is again translated into a simulated number of claims per year by specifying the same conditions as on Table 2.
Using an assumed claim rate and assuming an entry level for policyholders, the mean simulated premium income for successive years is calculated.
Let us consider an example, suppose
claim rate, λ = 0.1
entry level, class 10
full premium is 1
The simulated mean premium income over 130 years for a single policyholder is calculated using Excel and is found to be about 34% with a simulated standard deviation, 7%. The average premium for the stationary distribution using the same assumptions and calculated by multiplying the proportion of policyholders in each class once stability has been reached by the premium applicable in each class is found to be about 39%. These two figures are comparable, and this is expected. It is found that the figures are even closer as λ increases. A graph of the expected premium calculated for different claims rates for a single policyholder shows an increase in premium with increasing claim rates (See Figure 3).
This is also an expected result as it is assumed that the more a policyholder claims, the higher the premiums he/she pays. We note that the standard deviation calculated in this way is not really a proper standard deviation since the sequence of simulated premiums is not identically distributed. For example the first and second simulated premiums in the sequence heavily depend on the entry level whereas the 130th does not.
Figure : Expected Premium Income for different claim rates
Variability
Insurance involves the transfer of risk from policyholder to the insurer. If the premiums calculated for and paid by the policyholder do not involve any experience rating, the transfer of risk is total since the variability of premiums is zero. But using the bonus-malus systems in determining premiums involves using claims experience and hence variability in premiums is expected. The pseudo coefficient of variation (standard deviation divided by mean), is calculated for various claim frequencies by simulating various random variables representing the number of claims in a year, averaging the expected premium income for each claim number and then finding out the pseudo standard deviation from these premiums.
The pseudo coefficient of variation shows how variable the premiums paid by a policyholder are and how much risk he/she is carrying.
A graph of the pseudo coefficient of variation as a function of the claim frequencies shows that the pseudo coefficient of variation starts as zero since claims history is zero, increases to about 0.0015 at λ = 0.1 to about 0.0033 at λ = 0.25 and then decreases to about 0.0019 at λ = 0.5 until stationarity is achieved. See Figure 4.
As the claims frequency increases, the variability in premiums increases so it is expected that the coefficient of variation will increase as well, as standard deviation and coefficient of variation are directly related. The shape of the graph can possibly be related to the fact that as claim frequency increases, the probability of policyholders ending up in the lowest possible discount level increases, hence the deviation in premiums tends to zero although expected premium income continue to increase. This implies that after a number of years, the weight of risks that insurers carry remain at a constant even when claim rates are high.
Figure : Change in pseudo coefficient of variation as claim frequencies change
The coefficient of variation also determines the solidarity between drivers in terms of how variable the premiums they pay are. To illustrate this solidarity we calculate the coefficient of variation for a portfolio of policyholders over several years.
We suppose the number of policyholders, i = 1,..., 14, in each of the classes of the bonus-scale are as achieved on stationarity. Using a claim frequency of λ = 0.1, the expected premium income, i = 1, ..., 14, starting in each class is found by simulation.
The total expected premium for a number of policyholders can be defined as . The variations in the premium income for all 14 classes are also determined as , where, = the variations in the premium income in the various classes. Check!
The coefficient of variation is then calculated as
Supposing for example that there are 10000 policyholders i.e. we find the coefficient of variation, to be about 0.18 percent. It is observed that as the number of policyholders increase, the coefficient of variation decreases implying a stronger solidarity between policyholders. That is to say the expected premiums from policyholders are less variable if there are more policyholders.
The BM-20 System
This system was the first to be proposed and actually used by most insurers after the research as discussed previously. This system has 16 bonus steps, 1 level step and 3 malus steps. See Table 3. The same assumptions, calculations and analysis as was done with the BM-14 apply here. These will be discussed in depth in the next section where all three systems are compared.
Class
Premium
Class After
0
1
2
3+
Claims
20
160
17
20
20
20
19
140
16
20
20
20
18
120
15
20
20
20
17
100
14
19
20
20
16
95
14
19
20
20
15
90
13
18
20
20
14
85
12
18
20
20
13
80
11
17
20
20
12
75
10
17
19
20
11
70
9
16
19
20
10
65
9
15
19
20
9
60
8
14
19
20
8
55
7
13
18
20
7
52.5
6
13
18
20
6
50
5
12
18
20
5
47.5
4
12
18
20
4
45
3
11
17
20
3
42.5
2
11
17
20
2
40
1
10
16
20
1
40
1
9
15
20
Table : The BM-20 system
The NC-07 System
This is a no-claim bonus system with no malus steps. All policyholders start at the same discount level, class 7 corresponding to 0% discount on premium paid. As policyholders progress through the years, their discount level either increase or decrease based on their claims experience. The premium one pays over years will never exceed the premium he/she entered the system with unless there are other factors in play like inflation. There are just seven classes and so the mathematics is easier. Table 4 shows this bonus scale with the transitions made after 0, 1 and 2 or more claims.
Class
Discount %
Class After
0
1
2+
Claims
1
50
1
3
7
2
40
1
4
7
3
30
2
5
7
4
20
3
7
7
5
15
4
7
7
6
10
5
7
7
7
0
6
7
7
Table : The no-claim bonus system for NC-07
4.1 Analysis
The expected proportion of policyholders in each class over a number of years is calculated on a spreadsheet. A graph of the expected proportion of policyholders at stability shows the usual pattern we expect; on stability, policyholders with high claim rates cluster at the low discount levels paying higher premiums and the opposite for policyholders with low claim rates. See Figure 5.
Figure : Expected proportion of policyholders on stationarity for NC-07
Expected premium income per policyholder on stationarity and then from simulations are calculated for the same number of years for various assumed claim frequencies. On stationarity the expected income was about 51% for λ=0.1. The simulated expected premium income was about 54% with pseudo standard deviation of 8% for the same assumptions. These figures are again as expected. Premium income increases with claim frequency. It is again found that the system does not differentiate properly between policyholders. New entrants pay the same premium as the worst of drivers (drivers with very high claim rates) since there are no malus steps to punish drivers with high claim rates. The proportion of premium that good drivers pay as compared to that of bad drivers is very high. For example, by using a simple spreadsheet, we calculate the expected premium income per driver from a group of drivers with a given claim rate as a proportion of the expected income per driver from a group with claim rate, 0.5 by:
It is realised from the spreadsheet calculation that drivers with claim rate of λ = 0.1 pay about 60% of the premium drivers with λ = 0.5 pay whereas they should pay 20% (i.e. 0.1 / 0.5) This is also a feature of how inefficient this system is.
Comparison Between the systems
The NC-07 System was used by a large number of insurance companies in the Netherlands before the Investigations / Research that brought about the bonus-malus systems by the end of 1981.Conclusions from the Research pointed out two major disadvantages of the NC-07 system:
Policyholders with high claim rates (worse risk policyholders) paid the same premium as new entrants and as analysed earlier, the differentiation of policyholders with respect to premiums was not properly done.
Some insurers gave policyholders a no claims bonus of more than 40%.
These caused commotion and a lack of discipline in the insurance market. The conclusion that was derived was that there was a need for a system which gives proper credit in terms of premiums paid to both the careful driver and the driver who is prone to accidents. As was analysed earlier, the bonus-malus scale that was designed did well at the risk differentiation as compared to the no claims systems, with the BM-14 doing better than the BM-20.
The expected premium income for the BM-14 over a number of years once stability has been reached for λ=0.1 was found to be about 34%, for BM-20 with the same assumption, 51% and for NC-07, about 51%. The difference in the BM-20 and the BM-14 may be attributed to the difference in the number of classes. But the difference in the no-claims systems as compared to the bonus-malus systems is a problem that came up during the research known as the premium crash [9]. This goes to show that a change from the no claims system to the bonus-malus system favours policyholders more.
To make the comparison of the BM-20 with the BM-14 scale, in terms of premium levels possible or more realistic, we relate a 100% level to the highest step and the basic premiums by multiplying all percentages in the BM-20 system with 0.75(120/160, 1.e the highest premium level in BM-14 divided the highest premium level in the BM-20), so as to make the two systems equivalent. This has been done in Table 4. We start the comparison from class 17. From this table, one can conclude that the premium percentages in the BM-20 are slightly lower than in the BM-14 but this difference should not be overstated since in the limit, the majority of policyholders will be in classes with higher discounts.
BM-20
Class
Premium %
B4-14
Class
Premium %
Difference
17
75
11
80
+5
14
64
10
70
+6
12
56
9
60
+4
10
49
8
55
+6
9
45
7
50
+5
8
41
6
45
+4
7
39.5
5
40
+0.5
6
37.5
4
37.5
0
5
35.5
3
35
-0.5
4
33.75
2
32.5
-1.25
3
32
1
30
-2
2
30
1
30
0
1
30
1
30
0
Table : Classes with premiums adjusted for comparison
Other testing criteria we could use for comparison are occupational frequencies in the different classes once stability has been achieved, the efficiency, the minimum variance bonus scale etc. We would just focus on the efficiency and occupational frequencies.
Asymptotic occupational frequency
This is the proportion of policyholders in the various classes once stability has been reached. For λ=0.1, we find the frequencies to be as in Table 5.
Class
NC-07
BM-14
BM-20
1
0.772188
0.529908
0.389133
2
0.081212
0.055731
0.040925
3
0.089753
0.061592
0.04523
4
0.021973
0.06807
0.049987
5
0.016163
0.075229
0.055244
6
0.008888
0.083141
0.061054
7
0.009823
0.038894
0.067475
8
0.031252
0.074571
9
0.020209
0.082414
10
0.01402
0.026148
11
0.008956
0.02602
12
0.006186
0.024805
13
0.004099
0.019235
14
0.002271
0.015784
15
0.007054
16
0.003946
17
0.005257
18
0.003235
19
0.001554
20
0.00093
Table : Occupational frequency on stability
We find that for all the systems, policyholders cluster in class 1, reasons being the same as discussed earlier. At NC-07 we find that in the highest bonus class, an ultimate occupation of about 77%. It is considerably lower for the BM-14 and BM-20 which are about 53% and 39% respectively. Also the deviation over the classes for the BM systems seems to be more balanced than for the no claims system.
Efficiency
Let p (λ) denote the average premium for a claim frequency, λ, the efficiency e (λ) of a system is given by the formula:
e (λ) = x , Lemaire (1993)
and reflects the sensitivity of the average premium changes in the claims frequency. Table 6 (source: New motor rating structure in the Netherlands, Astin-groep Nederland) shows the efficiency of the different systems with respect to different values of λ. This table proves that the efficiency of the BM-14 has the highest values of all three for all the values of λ chosen. A major reason for this is that the ratio between the highest and lowest premium is large; 4 for both the BM-14 and BM-20 and 2 for the NC-07. Another reason affecting the efficiency is the number of steps backwards into the scale after a claim. For the BM-20 and BM-14, 1 claim leads to a 50% increase for the policyholders who occupy the top of the scale from 40% to 60% and 30% to 45% respectively. For the NC-16 this increase is considerably lower. See Tables 1, 2 and 3.
Whichever criteria one uses for the analyses and comparison of these systems, we can conclude that the BM-14 scores the best, based on the calculations done in this report.
Source: New motor rating structure in the Netherlands
λ
NC-07
BM-14
Bm-20
0.1
0.118
0.304
0.250
0.12
0.153
0.407
0.299
0.14
0.188
0.512
0.342
0.16
0.223
0.608
0.380
0.18
0.256
0.686
0.411
0.2
0.286
0.742
0.437
Table 6: Efficiency of the different systems
Analysis of a Non-Markov system
The Luxembourg System
So far we have analysed systems that use the knowledge of current level and claim experiences. These systems are Markovian as mentioned earlier. Markov systems have a nice mathematical property and the Markov property generalises more complicated problems ensuring easy calculations. Unfortunately the Luxembourg System is not Markovian due to some special bonus rules. The Luxembourg System consists of a scale with 22 classes each with premium levels associated as described in Table 6.
Policyholders enter the system in class 11. The rules are as follows:
for each claim free year, a one-class bonus is given
each claim is penalised by 2 classes
a policyholder with four consecutive claim-free years cannot be above class 11
(Special rule)
policyholders occupying higher classes are sent to class 11 after four consecutive claim free years because of the special bonus rule.
The motivation behind that special rule is that policyholders with high claim rates (bad drivers) who would suddenly improve should not be penalized for too long a period. It is possible to make up some classes in order to meet the Markovian property. We split classes 21 to 15 depending on the number of consecutive claim free years. A class j.i is to be understood as class, j and i consecutive claim free years.
Class
Premium
Class After
0
1
2
3
4
5
6
Claims
22
250
21
22
22
22
22
22
22
21
225
20
22
22
22
22
22
22
20
200
19
22
22
22
22
22
22
19
180
18
21
22
22
22
22
22
18
160
17
20
22
22
22
22
22
17
140
16
19
21
22
22
22
22
16
130
15
18
20
22
22
22
22
15
120
14
17
19
21
22
22
22
14
115
13
16
18
20
22
22
22
13
110
12
15
17
19
21
22
22
12
105
11
14
16
18
20
22
22
11
100
10
13
15
17
19
21
22
10
100
9
12
14
16
18
20
22
9
90
8
11
13
15
17
19
21
8
85
7
10
12
14
16
18
20
7
80
6
9
11
13
15
17
19
6
75
5
8
10
12
14
16
18
5
70
4
7
9
11
13
15
17
4
65
3
6
8
10
12
14
16
3
60
2
5
7
9
11
13
15
2
50
1
4
6
8
10
12
14
1
50
1
3
5
7
9
11
13
Table : The Luxembourg system
The transition rule for classes 21 to 15 are as follows:
Class 21 is split into 2 classes:
Class 21.0: the policyholder had at least one claim in the previous year and hence was in class 19, 17, 15, 13, 11 or 9 last year.
Class 21.1: the policyholder had no claims last year and hence arrives from class 22.
Class 21 is split into just 2 classes because one can only come from class 22 into class 21 with a claim free year since there are no classes above 22. All the other ways of getting into class 21 are included in class 21.0, for simplicity and more because it is just claim free years we are interested in for the splits.
Class 20 is split into 3:
Class 20.0: the policyholder had at least one claim in the previous year and hence was in class 18, 16, 14, 12, 10 0r 8 last year.
Class 20.1: the policyholder has had one claim free year and has arrived from class 20.0. Class 20.2: the policyholder arrives from class 21.1 and hence has 2 consecutive claim free years.
There is no class 20.3 because again it is just the claim free years we are interested in and for claim free years a policyholder can get into class 20 only from classes 20.0 and 21.1.
Class 19 down to class 15 have all been split into 4 classes each and they are as follows:
Class 19.0: the policyholder had at least one claim in the previous year and hence was in class 17, 15, 13, 11, 9, or 7 last year.
Class 19.1: the policyholder arrives from class 20.0 and hence has 1 claim free year.
Class 19.2: the policyholder arrives from class 20.1 and hence has 2 consecutive claim free years.
Class 19.3: the policyholder arrives from class 20.2 and hence has 3 consecutive claim free years.
Class 18.0: the policyholder had at least one claim in the previous year and hence was in class 16, 14, 12, 10, 8, or 6 last year.
Class 18.1: the policyholder arrives from class 19.0 and hence has 1 claim free year.
Class 18.2: the policyholder arrives from class 19.1 and hence has 2 consecutive claim free years.
Class 18.3: the policyholder arrives from class 19.2 and hence has 3 consecutive claim free years.
Class 17.0: the policyholder had at least one claim in the previous year and hence was in class 15, 13, 11, 9, 7 or 5 last year.
Class 17.1: the policyholder arrives from class 18.0 and hence has 1 claim free year.
Class 17.2: the policyholder arrives from class 18.1 and hence has 2 consecutive claim free years.
Class 17.3: the policyholder arrives from class 18.2 and hence has 3 consecutive claim free years.
Class 16.0: the policyholder had at least one claim in the previous year and hence was in class 14, 12, 10, 8, 6 or 4 last year.
Class 16.1: the policyholder arrives from class 17.0 and hence has 1 claim free year.
Class 16.2: the policyholder arrives from class 17.1 and hence has 2 consecutive claim free years.
Class 16.3: the policyholder arrives from class 17.2 and hence has 3 consecutive claim free years.
Class 15.0: the policyholder had at least one claim in the previous year and hence was in class 13, 11, 9, 7, 5 or 3 last year.
Class 15.1: the policyholder arrives from class 16.0 and hence has 1 claim free year.
Class 15.2: the policyholder arrives from class 16.1 and hence has 2 consecutive claim free years
Class. 15.3: the policyholder arrives from class 16.2 and hence has 3 consecutive claim free years.
Class
Premium
Class After
0
1
2
3
4
5
6
Claims
22
250
21
22
22
22
22
22
22
21.0
225
20.1
22
22
22
22
22
22
21.1
225
20.2
22
22
22
22
22
22
20.0
200
19.1
22
22
22
22
22
22
20.1
200
19.2
22
22
22
22
22
22
20.2
200
19.3
22
22
22
22
22
22
19.0
180
18.1
21.0
22
22
22
22
22
19.1
180
18.2
21.0
22
22
22
22
22
19.2
180
18.3
21.0
22
22
22
22
22
19.3
180
11
21.0
22
22
22
22
22
18.0
160
17.1
20.0
22
22
22
22
22
18.1
160
17.2
20.0
22
22
22
22
22
18.2
160
17.3
20.0
22
22
22
22
22
18.3
160
11
20.0
22
22
22
22
22
17.0
140
16.1
19.0
21.0
22
22
22
22
17.1
140
16.2
19.0
21.0
22
22
22
22
17.2
140
16.3
19.0
21.0
22
22
22
22
17.3
140
11
19.0
21.0
22
22
22
22
16.0
130
15.1
18.0
20
22
22
22
22
16.1
130
15.2
18.0
20.0
22
22
22
22
16.2
130
15.3
18.0
20.0
22
22
22
22
16.3
130
11
18.0
20.0
22
22
22
22
15.0
120
14
17.0
19.0
21.0
22
22
22
15.1
120
14
17.0
19.0
21.0
22
22
22
15.2
120
14
17.0
19.0
21.0
22
22
22
15.3
120
11
17.0
19.0
21.0
22
22
22
14
115
13
16.0
18.0
20.0
22
22
22
13
110
12
15.0
17.0
19.0
20.0
22
22
12
105
11
14
16
18.0
20.0
22
22
11
100
10
13
15.0
17.0
19.0
21.0
22
10
100
9
12
14
16.0
18.0
20.0
22
9
90
8
11
13
15.0
17.0
19.0
21.0
8
85
7
10
12
14
16.0
18.0
20.0
7
80
6
9
11
13
15.0
17.0
19.0
6
75
5
8
10
12
14
16.0
18.0
5
70
4
7
9
11
13
15.0
17.0
4
65
3
6
8
10
12
14
16.0
3
60
2
5
7
9
11
13
15.0
2
50
1
4
6
8
10
12
14
1
50
1
3
5
7
9
11
13
Table : The Luxembourg system with special bonus rules
Analysis
After the special rule has been applied, the number of classes increases to 40. See Table 7. The system now meets the memoryless property that we desire and analysis of the system becomes easier. We find the proportion of policyholders at stationarity in each level over time, in the first instance by ignoring the special rule and then later by using the system with the special rule. Considering an example, suppose the number of claims each year for a policyholder has a Poisson distribution, we calculate the probability of 0, 1, 2, 3, 4, 5 and 6 or more claims in a year and write down the 40 x 40 and 22 x 22 transition matrices for the system with and without the special bonus rules respectively. We calculate the asymptotic occupational frequencies in each level as in the early chapters. See figure 6.
We can see that if we use the special bonus rule, we have less policyholders in classes above 11 (increasing as the claim frequency increases) and more policyholders in classes below 11 than without the special rule. This is only logical from the definition of the special rule. This feature is illustrated on Figure 6, for λ=0.5. We also find that whereas without the special rule the proportion of policyholders keep increasing (except for class 2 which has a lower proportion than class 1), with the special rule, the proportion increases until it get to class 11 and then starts decreasing until class 20 and then increase sharply to the maximum at class 22.
Figure : Expected proportion of policyholders for the two systems for λ=0.5
NB: we use a logarithmic scale for the vertical axis to reduce the range to a more manageable range to be able to show everything more clearly.
We calculate the expected premium income from a group of policyholders with different assumed claims rates as was than in earlier chapters again for the with special rule and without special rule systems. See Table 9.
λ
Without
With
Approximate
Special rule
Special rule
Decrease (%)
0.5
1.77
1.48
16
0.4
1.37
1.08
21
0.3
0.88
0.72
18
0.2
0.62
0.57
8
0.1
0.55
0.53
4
Table : Expected premium income over time for different claim rates
We observe that the premium income increases as the claim frequency increases, a feature which is expected. The premium income for the with special rule is lower than for the without special rule since policyholders who take advantage of the special rule pay lesser premiums than they would actually pay. We can therefore infer that the special rule clearly helps bad drivers. We note an interesting feature in Table 8, we would expect that the difference in premium income will be highest for the highest claim rates but we find that it is actually highest for λ= 0.4 and not λ=0.5. A possible explanation might be that λ=0.5 is a very unlikely claim rate; it is not very realistic for a policyholder to claim at a rate of 0.5 within a year.
These differences might not affect the insurer's finances that much compared to the decrease in premium income due to other causes such as increased competition, implying the special rule is just in place again as a marketing tool to attract customers. Insurers can always recover these losses from some other aspect of the policy like for example charging policyholders a premium before being allowed on the special rule.
Conclusion
This report is intended to be very simple and concise and the scope is very narrow. We have used very simple spreadsheet models in the analysis of both systems and have made many assumptions. We would discuss some questions that could be asked based on the analysis done in this report and our findings.
Some of these questions include the following:
Why did we choose the assumptions used?
What other assumptions could have been made?
How would our results and analysis be different if we used some other assumptions instead?
We have analysed both systems using Markov chain analysis and also assumed that there is no entry or exit into and out of the systems. These assumptions are to make the analysis of the properties of the systems in the limit very easy and avoid complications.
We have assumed policyholders are not very sophisticated and are homogeneous in making decisions. This also makes analysis easy but in reality we recognise that policyholders may be heterogeneous.
In modelling the experience of the portfolio of policyholders, each policy in a class is assumed to be subject to the same claims frequency, and the Poisson distribution is used. We chose the Poisson distribution since it is the easiest to use and manipulate and it is relatively straightforward. The Binomial distribution could have been used since it makes things even simpler but it would be too crude an assumption to make since multiple claims on an individual policy for a given year cannot be modelled. Another alternative is the Negative binomial distribution which gives a better fit but it requires an extra parameter so it is harder to estimate results accurately. We perform some sensitivity analysis using the Geometric distribution (a special case of the negative binomial). We find for example that assuming a claim frequency of 0.1, the expected premium income is 40% whereas for Poisson (0.1) it is 39%. All the other analysis done with Poisson distribution exactly holds for the Geometric distribution and so we can conclude that the impact of the simplifying distributional assumption is not great.
Based on our findings we can conclude that both the Dutch and Luxembourg systems are not very efficient in risk differentiation and the systems are in place merely as marketing tools.
Nevertheless, the notion of a personal discount seems popular with the policyholders. It does not seem to matter that high premiums have been put in place to offset these high discounts, the appearance of a high bonus figure on the proposal form still appeals to the policyholder. The message is that the policyholders like large bonus figures, and they are prepared to pay extra to keep them.
We can also conclude that using the Bonus-Malus scale as an acceptation tool in Luxembourg may not be very optimal since for example a driver who maybe systematically refused by an insurer because he is at level 14 based on his claims experience is less dangerous than some of the policyholders in level 11.
The systems the Bonus-Malus systems that we have analysed are based only on claims frequency and policyholders might bear the small losses themselves and just report the expensive losses to insurers in order to avoid next year penalties. This can bring about the bonus hunger problem. A more lengthy report could look into this problem and its solutions.
