Single Payment Model
The single-payment model approximates the development pattern of coverages such as General Liability, where each occurrence is followed by one reporting, one or more valuations, zero or one payment, and zero or one recovery or other adjustment to the amount originally paid.
The mean and standard deviation for severity describe the size of the actual loss independent of responsibility for payment. The minimum and maximum may further describe the actual loss size or may describe the relevant layer of coverage. The deductible modifies the loss from the perspective of an insurer and the zero spike ("P(0)") expresses the probability that a claim will be closed without payment, for reasons other than being less than the deductible. This allows the use of continuous distributions with support on the positive axis for the size of the claimant's actual loss, while still allowing for the possibility that the insurer or the insured will not be liable for payment. The severity means are adjusted for trend, and the user may request a different trend between occurrence and payment of a particular claim than the trend up to the moment of occurrence, modeling this difference via Butsic's "alpha" parameter.
Case reserves are generated from adequacy factors applied to the actual severities, with the result modified by the minimum, maximum, and deductible from the severity distribution, and finally adjusted to reflect the case reserver's estimates that the claim will close without payment. Case reserve errors are reflected in the distribution of the case reserve factors, in the assumed probabilities of closure without payment, and in an assumed fast track reserve that applies from time of reporting to time of the next valuation.
Note that bias in case reserves may be introduced by adjusting the mean of the case reserve factor, or the assumed probability of closure without payment, or both. There are two assumed probabilities of closure without payment, conditional on whether or not the actual claim will close without payment. In practice this is not known to the case reserver, but there usually is some relevant evidence.
To model changes in case reserve adequacy as information about a loss accumulates, the user specifies a mean adequacy factor as of the report date and as of 40%, 70%, and 90% of the time between the report date and the payment date. The system interpolates the mean of the case reserve factor between these values and between the "90%-date" value and 1.00 at the payment date, and the system adjusts the standard deviation in the same proportions, except for reducing it linearly to zero between the "90% date" and the payment date. However, the case reserve itself only changes at discrete (and random) points in time determined by the distribution of inter-valuation waiting times.
The "P(2 sig dig)" entries represent the probability that a claim will be settled, or a case reserve estimated, to a nearby "round" number -- in this case rounding to two significant digits -- rather than its exact value.
Recoveries are modeled as one-time adjustments to correct errors in the original amount paid. For this purpose the amount paid is treated as an adequacy factor times the actual severity after application of minimum, maximum, deductible, and the Boolean variable representing closure without payment. Payment errors are reflected in the distribution of this factor less 1.00. In particular, if the adequacy factor is greater than 1.00, the initial payment will be too great and will produce a future recovery, represented as a negative payment. The simulator generates both the original excess payment and the later recovery.