Long-Term Risk: A Martingale Approach

Likuan Qin, Vadim Linetsky

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


This paper extends the long-term factorization of the stochastic discount factor introduced and studied by Alvarez and Jermann (2005) in discrete-time ergodic environments and by Hansen and Scheinkman (2009) and Hansen (2012) in Markovian environments to general semimartingale environments. The transitory component discounts at the stochastic rate of return on the long bond and is factorized into discounting at the long-term yield and a positive semimartingale that extends the principal eigenfunction of Hansen and Scheinkman (2009) to the semimartingale setting. The permanent component is a martingale that accomplishes a change of probabilities to the long forward measure, the limit of T-forward measures. The change of probabilities from the data-generating to the long forward measure absorbs the long-term risk-return trade-off and interprets the latter as the long-term risk-neutral measure.

Original languageEnglish (US)
Pages (from-to)299-312
Number of pages14
Issue number1
StatePublished - Jan 1 2017


  • Stochastic discount factor
  • long bond
  • long forward measure
  • long-term factorization
  • long-term risk-neutral measure
  • pricing kernel
  • principal eigenfunction

ASJC Scopus subject areas

  • Economics and Econometrics


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