Problem
The long-run risk (LRR) model of Bansal and Yaron (2004) explains asset pricing puzzles by combining a small persistent component in expected consumption growth with recursive (Kreps-Porteus) preferences. However, the existence of this predictable component in consumption growth is debated, and log-linearization techniques cannot handle the non-differentiable kink in generalized disappointment aversion (GDA) preferences. The paper asks whether persistent volatility risk alone, combined with GDA preferences, suffices to explain the equity premium and return predictability.
Key idea
Combine GDA preferences (Routledge and Zin 2010) with a Markov-switching representation of the Bansal-Yaron endowment process to derive closed-form solutions for asset valuation ratios, return moments, and predictability regressions. This avoids the Campbell-Shiller log-linearization entirely. The Markov-switching structure (4 states from 2 independent binary chains for mean and volatility) allows the price-payoff ratios to be written as linear functions of the regime indicator vector, yielding exact analytical formulas.
Method
- Match the continuous AR(1) processes in the Bansal-Yaron LRR model to a 4-state Markov-switching process for consumption and dividend growth via moment matching.
- Exploit the finite state space: any nonlinear function of the Markov state is a linear function of the regime indicator vector in R^N, so asset valuation ratios (P/D, P/C, risk-free bond price) are characterized by N-vectors solved from Euler equations.
- Derive analytical formulas for population moments of equity premia, risk-free rates, price-dividend ratios, and R^2 of predictability regressions.
- Benchmark model: random walk consumption (no LRR in mean) with LRR in volatility and GDA preferences (gamma=2.5, alpha=0.3, kappa=0.989, psi=1.5).
Results
- The benchmark random walk model with GDA preferences matches the equity premium (~7.2%), equity return volatility (~19.3%), low risk-free rate (~0.9%), and return predictability patterns (R^2 increasing with horizon).
- Persistent consumption volatility alone drives the equity premium through GDA’s endogenous overweighting of disappointing outcomes when uncertainty is high.
- Results do NOT depend on EIS > 1; values of psi below 1 still produce realistic asset pricing moments (main effect: higher risk-free rate level and volatility).
- Adding LRR in mean (full Bansal-Yaron model) changes very little for GDA preferences, in sharp contrast to Kreps-Porteus preferences where the mean channel is essential.
- Pure disappointment aversion (kappa=1, Gul 1991) reproduces return predictability reasonably well.
- The SDF has a kink at the disappointment threshold, generating counter-cyclical effective risk aversion.
Limitations
- The Markov-switching approximation is a discretization of the continuous LRR process; while moment matching is verified, higher-order dynamics may differ.
- Risk-free rate volatility is too low compared to data (~2.3% vs 4.1%).
- The model assumes a representative agent with known preferences; empirical identification of GDA parameters (alpha, kappa) is difficult.
- Only consumption and dividend growth are modeled; no term structure implications derived.
Open questions
- Can GDA preferences generate realistic term structure dynamics when embedded in a macro-finance model?
- How do GDA preferences interact with regime-switching monetary policy in a production economy?
- What is the empirical evidence for the disappointment threshold parameter kappa in micro data?
My take
This paper provides an elegant analytical framework that sidesteps the log-linearization debate entirely by using Markov-switching processes. The key insight that volatility risk alone suffices with GDA preferences (while LRR in mean is essential for Kreps-Porteus) is important for model selection. The closed-form solutions make sensitivity analysis tractable. However, the 4-state discretization is coarse, and the paper does not address whether the Markov-switching approximation introduces its own biases (a question later studied by higher-order-effects-asset-pricing-models). The connection to multiplicative-factorization-stochastic-discount-factor is implicit: GDA creates state-dependent SDF dynamics that interact with the permanent/transitory decomposition of the pricing kernel.