Developing models to represent and manage the two outcome variables, frequency and severity, is the focus of the early chapters of this text. However, when actuaries and other financial analysts use those models, they do so in the context of externally available variables. In general statistical terminology, one might call these explanatory or predictor variables; there are many other names in statistics, economics, psychology, and other disciplines. Because of our insurance focus, we call them rating variables as they will be useful in setting insurance rates and premiums.
This chapter considers a sample of 1,110 observations which may seem like a lot. However, as we will seen in our forthcoming applications, because of the preponderance of zeros and the skewed nature of claims, actuaries typically yearn for more data. One common approach that we adopt here is to examine outcomes from multiple years, thus increasing the sample size. We will discuss the strengths and limitations of this strategy later but, at this juncture, just want to show the reader how it works.
Specifically, Table 1.4 shows that we now consider policies over five years of data, years 2006, …, 2010, inclusive. The data begins in 2006 because there was a shift in claim coding in 2005 so that comparisons with earlier years are not helpful. To mitigate the effect of open claims, we consider policy years prior to 2011. An open claim means that all of the obligations are not known at the time of the analysis; for some claims, such an injury to a person in an auto accident or in the workplace, it can take years before costs are fully known.
Table 1.4 shows that the average claim varies over time, especially with the high 2010 value due to a single large claim. The total number of insureds is steadily declining yet, conversely, the coverage is steadily increasing. The coverage variable is the amount of coverage of the property and contents. Roughly, you can think of the coverage as the maximum possible payout of the insurer. For our immediate purposes, it is our first rating variable. Other things being equal, we would expect that insureds with larger coverage will have larger claims. We will make this vague idea more precise as we proceed.
Table 1.4. Building and Contents Claims Summary| Year | Average Frequency | Average Severity | Average | Number of Policyholders |
|---|---|---|---|---|
| 2006 | 0.951 | 9,695 | 32,498,186 | 1,154 |
| 2007 | 1.167 | 6,544 | 35,275,949 | 1,138 |
| 2008 | 0.974 | 5,311 | 37,267,485 | 1,125 |
| 2009 | 1.219 | 4,572 | 40,355,382 | 1,112 |
| 2010 | 1.241 | 20,452 | 41,242,070 | 1,110 |
For a different look at this five-year sample, Table 1.5 summarizes the distribution of our two outcomes, frequency and claims amount. In each case, the average exceeds the median, suggesting that the distributions are right-skewed. In addition, the table summarizes our continuous rating variables, coverage and deductible amount. The table also suggests that these variables also have right-skewed distributions.
Table 1.5. Summary of Claim Frequency and Severity, Deductibles, and Coverages| Minimum | Median | Average | Maximum | |
|---|---|---|---|---|
| Claim Frequency | 0 | 0 | 1.109 | 263 |
| Claim Severity | 0 | 0 | 9,292 | 12,922,218 |
| Deductible | 500 | 1,000 | 3,365 | 100,000 |
| Coverage (000's) | 8.937 | 11,354 | 37,281 | 2,444,797 |
Table 1.6 describes the rating variables considered in this chapter. To handle the skewness, we henceforth focus on logarithmic transformations of coverage and deductibles. To get a sense of the relationship between the non-continuous rating variables and claims, Table 1.7 relates the claims outcomes to these categorical variables. Table 1.7 suggests substantial variation in the claim frequency and average severity of the claims by entity type. It also demonstrates higher frequency and severity for the Fire5 variable and the reverse for the NoClaimCredit variable. The relationship for the Fire5 variable is counter-intuitive in that one would expect lower claim amounts for those insureds in areas with better public protection (when the protection code is five or less). Naturally, there are other variables that influence this relationship. We will see that these background variables are accounted for in the subsequent multivariate regression analysis, which yields an intuitive, appealing (negative) sign for the Fire5 variable.
Table 1.6. Description of Rating Variables| EntityType | Categorical variable that is one of six types: (Village, City, County, Misc, School, or Town) |
|---|---|
| LnCoverage | Total building and content coverage, in logarithmic millions of dollars |
| LnDeduct | Deductible, in logarithmic dollars |
| AlarmCredit | Categorical variable that is one of four types: (0%, 5%, 10%, or 15%), for automatic smoke alarms in main rooms |
| NoClaimCredit | Binary variable to indicate no claims in the past two years |
| Fire5 | Binary variable to indicate the fire class is below 5. (The range of fire class is 0~10) |
| Variable | Number of Policies | Claim Frequency | Average Severity |
|---|---|---|---|
| EntityType | |||
| Village | 1,341 | 0.452 | 10,645 |
| City | 793 | 1.941 | 16,924 |
| County | 328 | 4.899 | 15,453 |
| Misc | 609 | 0.186 | 43,036 |
| School | 1,597 | 1.434 | 64,346 |
| Town | 971 | 0.103 | 19,831 |
| Fire5=0 | 2,508 | 0.502 | 13,935 |
| Fire5=1 | 3,131 | 1.596 | 41,421 |
| NoClaimCredit=0 | 3,786 | 1.501 | 31,365 |
| NoClaimCredit=1 | 1,853 | 0.31 | 30,499 |
| Total | 5,639 | 1.109 | 31,206 |
Table 1.8 shows the claims experience by alarm credit. It underscores the difficulty of examining variables individually. For example, when looking at the experience for all entities, we see that insureds with no alarm credit have on average lower frequency and severity than insureds with the highest alarm credit (15%, with 24/7 monitoring by a fire station or security company). In particular, when we look at the entity type School, the frequency is 0.422 and the severity 25,257 for no alarm credit, whereas for the highest alarm level it is 2.008 and 85,140. This may simply imply that entities with more claims are the ones that are likely to have an alarm system. Summary tables give us insights but do not examine multivariate effects; for example, Table 1.7 ignores the effect of size (as we measure through coverage amounts) that affect claims.
Table 1.8. Claims Summary by Entity Type and Alarm Credit Category| No Alarm Credit | Alarm Credit 5% | |||||
| Entity | Claim | Avg. | Num. | Claim | Avg. | Num. |
| Type | Frequency | Severity | Policies | Frequency | Severity | Policies |
| Village | 0.326 | 11,078 | 829 | 0.278 | 8,086 | 54 |
| City | 0.893 | 7,576 | 244 | 2.077 | 4,150 | 13 |
| County | 2.14 | 16,013 | 50 | - | - | 1 |
| Misc | 0.117 | 15,122 | 386 | 0.278 | 13,064 | 18 |
| School | 0.422 | 25,523 | 294 | 0.41 | 14,575 | 122 |
| Town | 0.083 | 25,257 | 808 | 0.194 | 3,937 | 31 |
| Total | 0.318 | 15,118 | 2,611 | 0.431 | 10,762 | 239 |
| Alarm Credit 10% | Alarm Credit 15% | |||||
| Entity | Claim | Avg. | Num. | Claim | Avg. | Num. |
| Type | Frequency | Severity | Policies | Frequency | Severity | Policies |
| Village | 0.5 | 8,792 | 50 | 0.725 | 10,544 | 408 |
| City | 1.258 | 8,625 | 31 | 2.485 | 20,470 | 505 |
| County | 2.125 | 11,688 | 8 | 5.513 | 15,476 | 269 |
| Misc | 0.077 | 3,923 | 26 | 0.341 | 87,021 | 179 |
| School | 0.488 | 11,597 | 168 | 2.008 | 85,140 | 1,013 |
| Town | 0.091 | 2,338 | 44 | 0.261 | 9,490 | 88 |
| Total | 0.517 | 10,194 | 327 | 2.093 | 41,458 | 2,462 |