7.4.1. Diversification Benefit on the Incurred Losses
To visualize that our definition for diversification benefit can reflect the degree diversification benefit, we plot the diversification benefit on the incurred losses by each individual company. The company is sorted by the number of lines written, i.e., the company which is further to the right of the x-axis indicates the larger number of lines in their portfolio. As shown in figure 3, the diversification benefit appears to increase with the number of business lines. The correlation between the number of lines and the diversification benefit under this definition is equal to 73.62%
when all companies are included in the calculation and is equal to 56% if only multiple-line insurers are considered.
Furthermore, we compute the average and median of the diversification benefit by the number of product lines. According to table 10, the diversification
benefit is increasing with the more number of lines in the insurers’ portfolio.
However, the decrement of diversification benefit is observed if the insurers write the business in more than six lines. The result imply that the diversification effect from risk pooling seems to be beneficial if the insurers have less than six lines of product, but the diminishing knowledge can take an effect and reduce the diversification benefit if the insurers do the business in too many lines.
Tables 10.1-10.3 exhibit mean and median of the diversification benefit on the incurred losses for midsize, large and giant insurers respectively. The results support the hypothesis that large companies gain more benefit from product diversification.
The averages of diversification benefit are 0.078, 0.2, and 0.214 for class of midsize insurer, large insurer, and giant insurer respectively. Note that almost every small insurer has single line in their portfolio and thus have zero diversification benefit under our approach.
We include the diversification credit under BCAR definition to represent the current model used by regulators. The BCAR formula applies the net premium written in determining the diversification factor. If an insurer has the net premium written less than a specified amount, the BCAR assigns zero diversification credit to that company. Otherwise, the calculation is based on the line of business with the highest percentage of the net premium written. Unlike the diversification benefit under our definition, the BCAR approach fails to consider the risk in every line in the portfolio.
The diversification factor and diversification benefit have explicit formulas as follows.
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and
Diversification Benefit = 1 – Diversification factor.
However, we use the diversification benefit derived from BCAR model as a benchmark. Instead of the net premium written, the net premium earned in a given accident-year is applied in the BCAR formula for the comparison purpose.
Table 10 illustrates that the diversification credit under BCAR increases as more lines of product are written. This suggests that the BCAR tends to give more credit to the companies that write the business in more of product lines. The correlation between the number of lines and the diversification benefit under this definition is 70.39% when all companies are included in the calculation and is equal to 12.50% if only multiple-line insurers are included.
According to tables 10.1-10.3, large companies reap more product diversification benefit than the small sizes. The averages of diversification benefit are 0.022, 0.118, and 0.161 for class of midsize insurer, large insurer, and giant insurer respectively.
Comparing our definition and BCAR formula, our definition shows the higher correlation between the diversification benefit and number of product lines, especially when only multiple-line insurers are included in the correlation computation (56% vs.
12.5%). The greater correlation under our definition implies that our approach can reflect the product diversification benefit, which tends to tie to the number of lines, better than the BCAR definition. Another advantage of our approach is that it is a predictive model that includes the historical years of data in the calculation, while BCAR employs the present year data for the calculation.
According to table 10, mean and median of the correlation in each group of the number of lines suggest that the diversification credit is higher under our
definition relative to those derived from the BCAR formula. For example, the average of diversification benefit for the five-line insurers is 0.309 under our approach, while it is 0.124 under BCAR definition. Moreover, 62.09% of the insurers have the higher diversification factor if our definition is used rather than the BCAR formula.
Therefore, the BCAR formula tends to undervalue the benefit from product diversification for some companies if our proposed measure were more effective.
7.4.2. Diversification Benefit on the Reserving Risk
Figure 5 exhibits insurers’ diversification benefit on the loss development. The result suggests that diversification benefit generally increases with the number of business lines. The correlation between the number of lines and the diversification benefit on the loss development is equal to 64.22% when all companies are included in the calculation and is equal to 34.76% if only multiple-line insurers are included.
Next, we look into the diversification benefit when the insurers are grouped by their number of lines of business. The average and median of the diversification benefit are shown in table 11. With exception of the group of insurers who have four lines of product in their portfolio, the diversification benefit on the loss development is increasing as the insurers have more lines of business in their portfolio.
Nonetheless, the diversification benefit in the four-line group of insurers is greater than groups of insurers who write the business in less than seven lines. The result implies that the product diversification, under our definition, does not show clear beneficial effect to the reserving risk. However, when we consider the diversification benefit in different size of insurers, we find that large companies benefit from product diversification compared with small insurers, according to table 12.
CHAPTER 8