Tail Factor Convergence in Sherman’s Inverse Power Curve Loss Development Factor Model

The infinite product of the age-to-age development factors in Sherman’s inverse power curve model is proven to converge to a finite number when the power parameter is less than ?1, and alternatively to diverge to infinity when the power parameter is ?1 or greater. For the convergent parameter values, a simple formula is derived, in terms of any finite product of age-to-age factors, for the endpoints of an interval containing the limit of the infinite product. These endpoints converge to the limit as the finite time cutoff point increases. For any finite time cutoff, the product of age-to-age factors lies below the interval, and thus the lower endpoint of the interval is always a better estimate of the limit than the finite product itself. Several numerical examples are included for illustration. The convergence condition and the interval formula are applicable to the selection of a finite cutoff age, review of the reasonability of the convergence rate, and actual numerical calculations of the tail factor.