Forward transition rates

  • Kristian Buchardt
  • Christian FurrerEmail author
  • Mogens Steffensen


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.


Forward rates Doubly stochastic Markov models Life insurance Kolmogorov forward equations 

Mathematics Subject Classification (2010)

60J28 60J75 60J27 91B30 91G40 

JEL Classification

G22 G12 



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.


  1. 1.
    Bauer, D., Benth, F., Kiesel, R.: Modeling the forward surface of mortality. SIAM J. Financ. Math. 3, 639–666 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Buchardt, K.: Dependent interest and transition rates in life insurance. Insur. Math. Econ. 55, 167–179 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Buchardt, K.: Continuous affine processes: transformations, Markov chains and life insurance. Adv. Appl. Probab. 48, 423–442 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Buchardt, K.: Kolmogorov’s forward PIDE and forward transition rates in life insurance. Scand. Actuar. J. 2017, 377–394 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Buchardt, K., Møller, T.: Life insurance cash flows with policyholder behavior. Risks 2015, 290–317 (2015) CrossRefzbMATHGoogle Scholar
  6. 6.
    Christiansen, M., Niemeyer, A.: On the forward rate concept in multi-state life insurance. Finance Stoch. 19, 295–327 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dahl, M.: Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts. Insur. Math. Econ. 35, 113–136 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dahl, M., Møller, T.: Valuation and hedging of life insurance liabilities with systematic mortality risk. Insur. Math. Econ. 39, 193–217 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Heath, D., Jarrow, R., Morton, A.: Bond pricing and the term structure of interest rates. Econometrica 60, 77–106 (1992) CrossRefzbMATHGoogle Scholar
  11. 11.
    Henriksen, L.: Aspects of valuation and optimization in life insurance. Ph.D. thesis, University of Copenhagen (2014). Available online
  12. 12.
    Hoem, J.: Markov chain models in life insurance. Blätter DGVFM 9, 91–107 (1969) CrossRefzbMATHGoogle Scholar
  13. 13.
    Jacobsen, M.: Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Probability and Its Applications. Birkhäuser, Basel (2006) zbMATHGoogle Scholar
  14. 14.
    Kallenberg, O.: Foundations of Modern Probability, 1st edn. Probability and Its Applications. Springer, Berlin (1997) zbMATHGoogle Scholar
  15. 15.
    Kraft, H., Steffensen, M.: Bankruptcy, counterparty risk, and contagion. Rev. Finance 11, 209–252 (2007) CrossRefzbMATHGoogle Scholar
  16. 16.
    Milevsky, M., Promislow, D.: Mortality derivatives and the option to annuitise. Insur. Math. Econ. 29, 299–318 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Miltersen, K., Persson, S.: Is mortality dead? Stochastic forward force of mortality rate determined by no arbitrage. Working paper, Norwegian School of Economics and Business Administration (2005). Available online at
  18. 18.
    Møller, T., Steffensen, M.: Market-Valuation Methods in Life and Pension Insurance. International Series on Actuarial Science. Cambridge University Press, Cambridge (2007) CrossRefzbMATHGoogle Scholar
  19. 19.
    Norberg, R.: Reserves in life and pension insurance. Scand. Actuar. J. 1991, 3–24 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Norberg, R.: Forward mortality and other vital rates—are they the way forward? Insur. Math. Econ. 47, 105–112 (2010) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Kristian Buchardt
    • 1
  • Christian Furrer
    • 1
    • 2
    Email author
  • Mogens Steffensen
    • 2
  1. 1.PFA PensionCopenhagen ØDenmark
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagen ØDenmark

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