2.6 Numerical Analysis
2.6.2 Empirical analysis
The empirical analysis is based on the insurance industry data8We use annual margin data ranging
from 1980 to 2005.
The margin on the insurance business is defined as
M = 1−CR= 1−LR−ER,
8The data is provided by the reinsurance company General Re-New England Asset Management (GR-NEAM),
Figure 2.9. The moments are µ = (0,1,0,3,0,15,0,105,0,945) – the same moments as the standard normal distribution. The support is(−∞,+∞) and the mode is 0. The graphs show the cases where the specified numbers of moments arek = 2,4,6,8, and10, withk = 2on the upper left and running to the right then down. The last figure shows all of the bounds. In each graph, the highest and lowest lines with
−o−are, respectively, the upper and lower bounds onPr(X ≤t)for any distribution on the same interval with the same set of moments. The solid lines within the arbitrary bounds represent the upper and lower bounds for any distribution under unimodal assumption with the same mode and the same moments as the standard normal distribution. The middle line with−∗−is the true standard normal distribution.
where CR is the combined ratio, LR is the loss ratio with LR = Losses Incurred
Earned premiums and ER is the expense ratio with ER = Expenses
Written premiums. It can be considered as the profit of insurance business per dollar premium earned (or written).
Below is a data summary of the three lines of business, Allied, PPauto, and Comp. There are
n= 26observations for each line.
Figure 2.12 draws the histograms of the business lines Allied, PPauto and Comp. For each lines, 6 bins are used. We used the histogram to estimate the mode of each line. The estimated modes are -5.0, -5.1, and -21.4, respectively. From the histograms, we can see that PPauto and Comp do not look unimodal.
Figure 2.10. Arbitrary and unimodal bounds on standard normal distribution. The left and right graphs show bounds given 12 and 14 moments, respectively. Each bound is drawn based on the piecewise linear interpolation of 21 points.
Figure 2.11. The moments are the same as the moments of the gross return on an asset with a lognormal price with parametersµ = 0.05andσ = 0.1. The moments areµ= (1.06,1.13,1.22,1.32). The support is[0,+∞]and the mode is 1.0408. The first graph shows bounds on distribution given the first two mo- ments. The second one shows bounds with four moments considered. They are shown together in the third graph. In each graph, the lines with−o−are bounds on arbitrary distribution. The solid lines within the arbitrary bounds are bounds for any distribution under unimodal assumption with the same mode and the same moments. The middle line with−∗−is the true lognormal distribution with parametersµ= 0.05and
σ= 0.1.
business, we calculated upper and lower bounds onF(t) = Pr(X ≤ t)for a range of values of t
running over the support ofXwhich we take to beI = (−∞,∞)using the two methods, the SOS approach and Smith’s approach.
As shown in Figure 2.13, the solid curves obtained by the SOS programming solver are bounds on the distributions given two or four moments. For any random variable X with the same mo- ments, its distributionPr(X ≤t)must fall within the interval formed by the bounds. The solutions of Smith’s method are plotted by the lines with−o−if only the first two moments are given and by
Figure 2.12.Histograms (left to right, top to bottom) of Allied, PPauto, and Comp.
Figure 2.13. Bounds onF(t) = Pr(X ≤ t)for three lines of business, Allied, PPauto and Comp (left to right, then down). The solid lines are solutions obtained by the SOS method. The lines with−o−represent the bounds of Smith’s approach, given only the first two moments. Smith’s bounds given 4 moments are denoted by the lines with−∗−.
Table 2.1. Descriptive Statistics of three lines of business from 1980 to 2005
Allied PPauto Comp
E(X) 0.30344 -7.3231 -9.7802 E(X2) 97.008 360.61 233.59 E(X3) 334.61 -10,011 -8,433.3 E(X4) 23196 496,000 402,000 Mode -7.5 -0.5 -12.5 Maximum 20.146 20.3 4.9 Minimum -19.203 -52 -55.717 Range 39.349 72.3 60.617
Figure 2.14.Comparison of arbitrary and unimodal bounds onF(t) = Pr(X≤t)for the line Allied given 4 moments.
the lines with−∗−if given four moments. The Smith’s bounds fall exactly on the SOS solutions, confirming each other. Adding more moment constraints tighten bounds on Pr(X ≤ t), but not uniformly witht.
When the unimodal assumption is added, the bounds are greatly improved. Again, for each line of business, we calculate upper and lower bounds onF(t) = Pr(X ≤t)for a range of values of t running over the support of X given 2 (for Comp) or 4 moments (for Allied and PPauto) and the estimated mode m. In Figures 2.14 and 2.15, we show the upper and lower bounds on distributions of Allied and PPauto respectively, comparing the bounds obtained by the first four raw moments with unimodal assumption and the arbitrary distribution bounds given only the same set of moments. The red lines with−o−are the bounds of arbitrary distributions and the solid blue lines represent the bounds of the unimodal distribution.
Figure 2.15. Comparison of arbitrary and unimodal bounds on F(t) = Pr(X ≤ t) for the line PPAuto given 4 moments. 0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 -20.000 -15.000 -10.000 -5.000 0.000 5.000 10.000 15.000 20.000 25.000 Two moments Four moments
Figure 2.16. The upper left plot shows the comparison of arbitrary and unimodal bounds on F(t) = Pr(X ≤t) for Comp given 2 moments and the estimated empirical modem = −5. The upper right plot shows unimodal bounds for Comp with 2 (in red) or 4 (in blue) moments. Maximum-entropy distributions for Comp given 2 (in red) or 4 (in black) moments are drawn in the bottom plot.
Figure 2.16 compares the arbitrary and unimodal bounds for Comp. Specifying the first four moments makes the unimodal bounds collapse. As a representative distribution with given mo- ments, the maximum-entropy distribution is analyzed to check the unimodality. When 4 moments are considered, the distribution is bimodal. Constraints with only the first 2 moments fail to cap- ture this bimodal characteristic. Therefore, if the data with a given number of moments is not unimodally distributed, adding the unimodal assumption to “force” the distribution to be unimodal makes the bounds estimation fail.