Exposure Rating Loss Layers: Unifying the Property Perspective of Severity with the Liability Perspective of Frequency

For problems such as rating excess of toss reinsurance and estimating deductible credits, actuaries frequently employ exposure rating factors. In the context of property insurance this takes the form of loss tables such as the Lloyds scale or Salzmann tables. These tables display the fraction of loss cost retained for layers expressed as fractions of insured value, or policy limit. In the liability insurance context, Increased Limits Factors (ILFs) or Excess Loss Factors (ELFs) tables are expressed in terms of actual dollar amounts for attachment points and limits. Implicit in the property tables is the assumption that an increase in policy limit or insured value corresponds to a proportional scale factor increase in the claim severity random variable, but other than the change in scale the distribution of claim sizes remains the same and any increase or decrease in loss cost per exposure is frequency based. Without a special adjustment to the loss cost or premium rate, the implied loss frequency is the same for the larger policy. Implicit in the liability tables is the assumption that larger policies produce the same distribution of claim severity. In summary, the property perspective generally assumes that all the extra exposure shows up as larger claims, and the liability perspective generally assumes that all the extra exposure shows up as more claims. This paper shows how both perspectives for claim severity, and additional considerations of frequency changes may easily be incorporated into a unified model. Additionally, such a unified approach allows for a compromise where increasing exposure for a given policy or risk may be partially reflected in the scale of claim size and partially in the frequency.