Learning about Risk and Return: A Simple Model of Bubbles and Crashes

Abstract

This paper demonstrates that an asset pricing model with least-squares learning can lead to bubbles and crashes as endogenous responses to the fundamentals driving asset prices. When agents are risk-averse they generate forecasts of the conditional variance of a stock's return. Recursive updating of the conditional variance and expected return implies two mechanisms through which learning impacts stock prices: occasional shocks may lead agents to lower their risk estimate and increase their expected return, thereby triggering a bubble; along a bubble path recursive estimates of risk will increase and crash the bubble.JEL Classifications: G12; G14; D82; D83

Behavioral learning equilibria

“…There is a continuum of firms indexed by i ∈ [0, 1]. Each firm produces a differentiated good, but they all use the same technology which uses labor as the only 13 We also simulated SAC-learning with a constant gain parameter (not shown here) and, similar to factor of production. The demand curve for the good produced by firm i is given by …”

Imperfect knowledge, liquidity and bubbles

“…These nearly self-fulfilling random walk beliefs lead to recurrent bubbles and crashes in stock prices. While there are similarities in the mechanism generating bubbles in Branch and Evans (2011) and in this paper, there are important differences. The present environment includes a liquidity services role for assets and this liquidity demand is essential for generating bubbles.…”

“…For fixed belief parameters (a, b), the law of motion in (7) is conditionally linear and the analysis in Branch and Evans (2011) can be used to gain analytic insight into the nature of real-time learning dynamics. When σ = 1 there is a unique SCE that corresponds to the fundamentals equilibrium.…”

“…The framework employed here is related to an extensive literature that employs adaptive learning in macroeconomics. Most closely related are papers that incorporate constant gain learning in studies of monetary policy and asset pricing: see, for example, Branch and Evans (2011);Sargent (1999); Adam, Marcet, and Nicolini (2010); Orphanides and Williams (2005) ;Cho, Williams, and Sargent (2002); Williams (2004); Cho and Kasa (2008);McGough (2006).) Branch and Evans (2011), in particular, find that risk-averse agents in an OLG asset pricing model forecast both the risk and return of stock prices using a forecasting model whose parameters are updated using constant gain least squares then traders may also come to believe that stocks follow a random walk.…”

“…The present environment includes a liquidity services role for assets and this liquidity demand is essential for generating bubbles. While Branch and Evans (2011) emphasizing variations in the perceived riskiness of assets.…”