A liquidity premium is the difference between the equilibrium expected return in a market with transaction costs and its frictionless counterpart. In the multi-agent equilibrium models of Herdegen et al. (2018) and Shelley (2023), the liquidity premium takes the form:
LiPr_t = mu_t^lambda - mu_t = sum_{n=1}^{N} (gamma^n - gamma_bar) * sigma^2 / N * (phi_t^{lambda,n} - phi_bar_t^n) + additional belief-dependent terms
where gamma_bar = (1/N) sum gamma_n is the average risk aversion and Delta_t^n = phi_t^{lambda,n} - phi_bar_t^n is agent n’s deviation from their frictionless target.
The key insight is that the liquidity premium is the sample covariance between the risk aversion vector (gamma^1, …, gamma^N) and the portfolio deviation vector. It is positive when more risk-averse agents hold larger positions relative to their frictionless targets — i.e., when the more risk-averse agents are net buyers, requiring a positive expected return adjustment to clear the market.
Under homogeneous risk aversions (gamma^1 = … = gamma^N), the liquidity premium vanishes: the frictionless equilibrium price clears the frictional market. This result holds regardless of the cost specification (quadratic, power, or proportional).
Key Details
- Vanishes with homogeneous agents: identical risk aversions ⇒ no liquidity premium (Corollary 5.3 in Herdegen et al.)
- Non-trivial with heterogeneity: heterogeneous risk aversions generate premia proportional to the degree of heterogeneity
- Mean-reverting dynamics: liquidity premia are mean-reverting (Ornstein-Uhlenbeck-type) even when endowments have independent increments, because portfolio sluggishness introduces autocorrelation
- Sign: positive when more risk-averse agents are net buyers; negative when they are net sellers
- Scaling: proportional to sigma^2 (asset volatility) and lambda (cost parameter)