[Paper Review] Long Term Risk: A Martingale Approach

Likuan Qin, Vadim Linetsky|arXiv (Cornell University)|Nov 12, 2014

Stochastic processes and financial applications65 references4 citations

TL;DR

This paper extends the long-term factorization of the stochastic discount factor to general semimartingale environments using a martingale approach, showing that the long-term risk-return trade-off vanishes under the long forward measure—interpreted as the long-term risk-neutral measure—thereby unifying discrete-time and Markovian frameworks without requiring Markovian assumptions.

ABSTRACT

This paper extends the long-term factorization of the stochastic discount factor introduced and studied by Alvarez and Jermann (2005) in discretetime 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.

Motivation & Objective

• To generalize the long-term factorization of the stochastic discount factor beyond Markovian and discrete-time settings.
• To establish the long forward measure as the limit of T-forward measures in continuous-time semimartingale models.
• To show that the long-term risk premia for stochastically growing cash flows vanish under the long forward measure.
• To unify the discrete-time results of Alvarez and Jermann (2005) with the Markovian operator-based results of Hansen and Scheinkman (2009) and Hansen (2012).
• To provide a rigorous semimartingale foundation for long-term asset pricing without relying on Markovian or ergodicity assumptions.

Proposed method

• Uses semimartingale decomposition to model the stochastic discount factor in continuous-time, generalizing to non-Markovian environments.
• Constructs the long forward measure as the weak limit of T-forward measures as maturity T → ∞.
• Identifies the permanent component of the pricing kernel as a positive martingale that induces a change of measure to the long forward measure.
• Factorizes the transitory component into long-term yield λ and a positive semimartingale extending the Perron-Frobenius eigenfunction to semimartingales.
• Employs exponential ergodicity assumptions to ensure convergence of bond prices and forward measures in total variation and ucp topology.
• Applies semimartingale convergence theory to derive error bounds on long-term pricing approximations.

Experimental results

Research questions

• RQ1 How can the long-term factorization of the stochastic discount factor be extended beyond Markovian and discrete-time models?
• RQ2 What is the role of the long forward measure in absorbing long-term risk adjustments in general semimartingale models?
• RQ3 Under what conditions does the long forward measure exist and coincide with the recurrent eigen-measure?
• RQ4 How does the martingale component of the pricing kernel affect the long-term risk-return trade-off?
• RQ5 Can the long-term pricing formula be derived with explicit convergence rates in non-Markovian settings?

Key findings

• The long forward measure exists as the limit of T-forward measures and serves as the long-term risk-neutral measure, under which long-term risk premia vanish.
• The long-term yield λ is identified as the asymptotic yield on cash flows with bounded moments, even in non-Markovian settings.
• In the degenerate case λ = 0, an asymptotic power yield is defined, extending the long-term pricing framework to sub-exponential discounting.
• Under exponential ergodicity, the convergence of bond prices and forward measures to their long-term limits occurs at an exponential rate α > 0.
• The error in the long-term pricing formula for bounded payoffs is bounded by c‖f‖∞e−(λ+α)t, ensuring fast convergence.
• The martingale component drives the wedge between the data-generating and long forward measures, and is empirically significant in bond markets, as shown in Qin et al. (2016).