Hattendorff's theorem for a continuous-time Markov model

Abstract

For a continuous-time Markov model with finite number of states Hattendorff's theorem is proved for two different aggregate loss functions. Hattendorff's theorem asserts that the variance of the aggregate loss for the whole duration of a life policy equals the sum of the variances of random losses for the individual policy years. For the aggregate and single-year loss functions that take account of all random variation it is proved that the variance of the loss function is equal to the expected value of the squared discounted sum at risk plus the expected value of the variance of the discounted value of random losses in states that can be reached from the considered state.