Again the data was sourced from Klugman et al.[1998] and details could be seen in Appendix A. This serves as an example of illustration how our hybrid model works.
Applying the model on the same dataset as we have used before yields the following results. Initially, by observation, the value of z and ρ are observed as explained above and they are z = 5784.47, ρ = 0.184. Substitute these into (3.11), ccan be easily obtained which is c= 0.02225763.
Subsequently, following the same steps to estimate the parameters in the Pareto distribution and combining (3.11) and (3.12), the estimates of m and s can be calculated as
(
m= 1475.0447, s= 1.0622451. with the help of the values of z and ρ.
Hence, figure3.7shows a hybrid distribution fitting to the dataset. Compared to our former results, this yields a better fit.
Premium Calculations Again we analysed several scenarios similar to Section
3.2. But now we need to specify the number of small and large claims and fix the corresponding total costs accordingly. Note that since we have a threshold z = 5784.47, a reasonable setting of the scenarios should satisfy several constraints.
3. PRICING A BM SYSTEM BY ADDRESSING CLAIM SEVERITY DISTRIBUTIONS
Distribution of Claim Severities
Claim Severities F re qu en cy 0 10000 20000 30000 40000 50000 60000 0 50 100 150 Weibull Fit Pareto Fit
Figure 3.7: Distribution of Claim Severities.
For instance, if K1 ranges from 1 to 5, M1 6z and similarly if K1 >2,M1 62z.
Thus, we will be looking at several special scenarios as described below.
Scenario 1 One simple scenario is that policyholders only make small claims and no big ones, i.e., K2 = M2 = 0. Fixing the total cost M1 = 5000 and
employing (5.4) would yield the following results as shown in Table 3.6.
Scenario 2 On the contrary, we also consider a case where a policyholder has only big claims and no small ones, i.e., K1 = M1 = 0. Keeping the total large
claim size fixed at M2 = 30000, we obtained Table 3.7. It is essential to mention
3. PRICING A BM SYSTEM BY ADDRESSING CLAIM SEVERITY DISTRIBUTIONS
Table 3.6: Premiums for Scenario 1
M1 = 5000, M2 =K2 = 0, t = 1,2
K1 = 1 2 3 4 5
t= 1 707.907 1061.181 1346.342 1530.187 1628.134 t= 2 617.699 906.294 1139.247 1289.434 1369.448
Table 3.7: Premiums for Scenario 2
M1 =K1 = 0, M2 = 30000, t= 1
K2 = 1 2 3 4 5
t= 1 1471.1 1665.3 1694.0 1653.0 1577.8 t= 2 1360.4 1543.4 1570.5 1531.8 1461.0 claim size for small claims using,
Π[Xw|M1 = 0] =
1 1−ρ ·
2 c2
Scenario 3 This is when a policyholder has £5000 worth small claims and
£30000 worth big claims in a single year. We analysed 25 different scenarios where the number of both the small and large claims vary from 1 to 5. And for simplicity, we only look at the first two years’ premium levels. Results are demonstrated in Table3.8and Table3.9respectively. A clearer comparison could be seen in Figure 3.8 and Figure3.9.
We have actually looked at premium levels for claimers only. Generally speak- ing, premiums decrease overtime. According to Table3.6 and Table3.7, it is also obvious that a policyholder with both small and large claims would definitely pay more than those who have small claims only knowing that their costs of small claims are the same. Furthermore, by comparing Scenario 2 and 3, each additional small claim adds quickly on the premium levels.
Moving into details, clearly the premium levels jump upwards column-wise. In a practical language, when the aggregate costs of claims are fixed, the higher
3. PRICING A BM SYSTEM BY ADDRESSING CLAIM SEVERITY DISTRIBUTIONS
Table 3.8: Premiums for Scenario 3t = 1 M1 = 5000, M2 = 30000, t= 1 K2 = 1 2 3 4 5 K1 = 1 1658.7 1852.8 1881.6 1840.5 1765.4 2 2011.9 2206.1 2234.9 2193.8 2118.7 3 2297.1 2491.3 2520.0 2479.0 2403.9 4 2480.9 2675.1 2703.9 2662.8 2587.7 5 2578.9 2773.1 2801.8 2760.8 2685.6
Table 3.9: Premiums for Scenario 3t = 2 M1 = 5000, M2 = 30000, t= 2 K2 = 1 2 3 4 5 K1 = 1 1513.7 1696.7 1723.7 1685.1 1614.3 2 1802.3 1985.3 2012.3 1973.7 1902.9 3 2035.2 2218.2 2245.3 2206.6 2135.8 4 2185.4 2368.4 2395.5 2356.8 2286.0 5 2265.4 2448.4 2475.5 2436.8 2366.0
their frequency is the cheaper is each claim. The increase in premiums suggests that this system punishes severely on people who frequently make small claims. It is also noticeable that the increase of premiums with respect to K1, i.e., the
number of small claims, is faster than that with regard to K2, i.e., the number
of large claims. In fact, premiums almost stay quite stable with the change in K2. There is even a decreasing trend starting from the 4th column in Table 3.7-
3.9. However, this does not mean the rise in claim frequency would lead to lower premium level. Notice that the total claim size is fixed. So when the counts of large claims move towards right, it only implies that each individual claim actually costs less. Such reduction in premiums could be understood as when it comes to large claims, the system punishes less if the frequency is high. Figure
3.8 and 3.9 reinforces this statement and notice that the premium does not drop much.
3. PRICING A BM SYSTEM BY ADDRESSING CLAIM SEVERITY DISTRIBUTIONS
Figure 3.8: Premiums for Scenario 3t = 1.
3. PRICING A BM SYSTEM BY ADDRESSING CLAIM SEVERITY DISTRIBUTIONS
This means that our model makes more emphasise on the total claim severity rather than the claim frequency component for large claims and vice versa for small ones. In other words, the proposed model punishes more on someone making a lot of small claims while such punishment is not obvious when large claims are made. On the other hand, when the frequency of large sized claims is raised, the per-claim cost actually is smaller which might fall into our smaller size claims category. For instance, the K2 cannot increase further after a sum of 5 claims
worth £30,000 in total (Table3.7,3.8,3.9), because otherwise the average size of the claims would fall belowzand will be reconsidered as small claims. We penalise less because the drivers having claims valued near the threshold are actually not affecting the income of a company too much and in addition they have informed the insurer about their claims. Such information is valuable in estimation and forecast. Technically speaking, this is due to the fact that M2+m
M2+m+z is between
(0,1) as can be seen in Π[Xp] in Corollary 3.4.
However, for small claims, the premium still rises as the frequency increases when the total costs are kept constant. These behaviours can also be seen from Figure 3.8 and Figure 3.9, the colour gets warmer when moving upwards on the K1 axis. Practically speaking, one reason would be that frequently dealing with
small claims would probably induce more administrative costs which should be offset by forcing higher premiums. In addition, it is very likely that people with many small claims would create a big loss in the future. They are potential risks and likely to cause a sudden loss to the insurer.
That is to say, this model assigns more attention on potential risks and is relatively milder in penalising those who already reported a larger claim. In fact, it is often the case that these people would be more careful in the future, while those constantly filing small claimers possibly have a potential to create an unexpected attack to the insurance company.
Another interesting question to ask is how this model compares to the previous one in Section3.2. So let us look at the same example where a policyholder reports one claim worth £7500 and then £2500 in the subsequent year. That means a ’big’ claim in the first and a ’small’ in the second year using the hybrid model and the resulting premium for the second year should be £554.7 and £646.3 for the third year. These figures are much smaller than those of both the Weibull and
3. PRICING A BM SYSTEM BY ADDRESSING CLAIM SEVERITY DISTRIBUTIONS
Pareto models. It could be the reason that the fitting of this hybrid distribution performs better for small claims and also the data does not contain many large claims.