## Abstract

The idea of forward rates stems from interest rate theory. It has natural connotations to transition rates in multi-state models. The generalisation from the forward mortality rate in a survival model to multi-state models is non-trivial and several definitions have been proposed. We establish a theoretical framework for the discussion of forward rates. Furthermore, we provide a novel definition with its own logic and merits and compare it with the proposals in the literature. The definition turns the Kolmogorov forward equations inside out by interchanging the transition probabilities with the transition intensities as the object to be calculated.

## Keywords

Forward rates Doubly stochastic Markov models Life insurance Kolmogorov forward equations## Mathematics Subject Classification (2010)

60J28 60J75 60J27 91B30 91G40## JEL Classification

G22 G12## Notes

### Acknowledgements

Christian Furrer’s research is partly funded by the Innovation Fund Denmark (IFD) under File No. 7038-00007B. We should like to thank Lars Frederik Brandt for fruitful discussions.

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