| Abstract: | Upon differentiating the Thiele differential equations and the equivalence condition with respect to some parameter appearing in the equations, one obtains differential equations for the derivatives of the state-wise reserves and the premium level with respect to the parameter. The solution to these equations measures the impact on premiums and reserves of a change in the parameter. Typically only numerical results can be obtained, but the method applies quite generally to multi-state policies and to virtually any parameter, and so represents a panacea in (the vast majority of) situations where analytical results are out of reach. Extensions to higher order derivatives and higher order conditional moments are straightforward. A difference method for computation is devised, and numerical results are reported for some practical cases. |
