Abstract
The paper by Stelljes [1] the subject of this discussion is a welcome addition to the Casualty Actuarial Society literature on nonlinear regression for loss reserving. This discussion will predominantly concern a key assumption made in [1]. In particular, on page 361:
“Based on the assumption that the incremental pure premiums for different development intervals are independent, the variance of IBNR pure premium is the sum of the variances of the incremental pure premiums for the remaining development intervals.”
It may be true that the historical incremental pure premiums can be considered independent, but it does not follow that the future fitted incremental pure premiums are independent. An analogous situation exists for ordinary linear regression, where the hat matrix provides for the covariance of the fitted values. Since the variance of the sum of random variables depends on covariance between the random variables, the variance of the reserve will depend on the covariance of the incremental IBNRs.
After providing a brief review on traditional nonlinear regression in section 2, the bulk of this discussion is concerned with two issues. First, modifying the methods of [1] to reflect covariance among the fitted values and is described in section 3. Second, there are times when a reliable insurance trend factor is not available. In such circumstances the actuary needs to derive the trend as part of the model, as in the model on page 359 of [1]. [1] succinctly describes the problems with such an approach. Section 4 discusses this latter model and shows simulation is not required to calculate confidence intervals. The last section, section 5 will discuss some miscellaneous issues.
“Based on the assumption that the incremental pure premiums for different development intervals are independent, the variance of IBNR pure premium is the sum of the variances of the incremental pure premiums for the remaining development intervals.”
It may be true that the historical incremental pure premiums can be considered independent, but it does not follow that the future fitted incremental pure premiums are independent. An analogous situation exists for ordinary linear regression, where the hat matrix provides for the covariance of the fitted values. Since the variance of the sum of random variables depends on covariance between the random variables, the variance of the reserve will depend on the covariance of the incremental IBNRs.
After providing a brief review on traditional nonlinear regression in section 2, the bulk of this discussion is concerned with two issues. First, modifying the methods of [1] to reflect covariance among the fitted values and is described in section 3. Second, there are times when a reliable insurance trend factor is not available. In such circumstances the actuary needs to derive the trend as part of the model, as in the model on page 359 of [1]. [1] succinctly describes the problems with such an approach. Section 4 discusses this latter model and shows simulation is not required to calculate confidence intervals. The last section, section 5 will discuss some miscellaneous issues.
Volume
Summer
Year
2007
Categories
Financial and Statistical Methods
Statistical Models and Methods
Regression
Actuarial Applications and Methodologies
Reserving
Reserving Methods
Actuarial Applications and Methodologies
Reserving
Uncertainty and Ranges
Publications
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