Abstract
We develop a tractable risk-sharing model that allows to study how trading costs are reflected in expected returns. In a continuous-time model populated by heterogenous mean-variance investors, we characterize the unique equilibrium by a system of coupled but linear FBSDEs. This system can be solved in terms of matrix power series, and leads to fully explicit equilibrium dynamics in a number of concrete settings.
Summary
This paper introduces the foundational multi-agent equilibrium model with quadratic transaction costs that serves as the basis for Shelley’s thesis and subsequent work by Muhle-Karbe et al. N agents with heterogeneous risk aversions receive random endowments and trade a risky asset to maximise discounted mean-variance utility, penalised by quadratic costs on their trading rates. The model admits a unique equilibrium characterised by a coupled system of linear FBSDEs (Lemma 4.1), solved explicitly via matrix power series (Theorem A.2/A.4).
The key economic insight is that transaction costs create endogenous liquidity premia: the equilibrium expected return differs from its frictionless counterpart by an amount that depends on the sample covariance between agents’ risk aversions and their portfolio deviations from frictionless targets (equation 5.5). With homogeneous agents, transaction costs do not affect equilibrium prices (Corollary 5.3), but heterogeneous risk aversions generate positive liquidity premia when more risk-averse agents are net buyers.
Key Contributions
- First fully explicit multi-agent equilibrium with quadratic transaction costs in continuous time
- Coupled linear FBSDE characterisation (Lemma 4.1, equations 4.1-4.2)
- Equilibrium return formula (Theorem 5.2, equation 5.2): mu_t^Lambda = sum of risk-aversion-weighted portfolio deviations + endowment terms
- Liquidity premium characterised as sample covariance of risk aversions and portfolio deviations (equation 5.5)
- Explicit Ornstein-Uhlenbeck dynamics for equilibrium returns in concrete examples (Corollaries 5.5-5.6)
- Mean-reverting autocorrelation in equilibrium returns arising endogenously from portfolio sluggishness
Methodology
Individual optimality via calculus of variations leads to a forward-backward SDE for each agent’s optimal portfolio and trading rate. Market clearing couples these into a system of N-1 FBSDEs. For quadratic costs, the system is linear and solved via matrix exponentials (T = infinity, Theorem A.2) or hyperbolic matrix functions (T < infinity, Theorem A.4). Existence and uniqueness follow from the positive definiteness of the coupling matrix B.
Key Findings
- Homogeneous agents: transaction costs do not affect equilibrium prices (but do affect strategies)
- Heterogeneous risk aversions: positive liquidity premia arise when more risk-averse agents hold larger positions
- Portfolio sluggishness from transaction costs introduces autocorrelation in equilibrium returns
- Equilibrium returns mean-revert around frictionless counterparts with speed sqrt(Delta) where Delta depends on costs and risk aversions
- Liquidity premium is proportional to the degree of heterogeneity (gamma^n - gamma_bar)
Important References
- Investing with Liquid and Illiquid Assets — Garleanu and Pedersen, the seminal single-agent quadratic cost model
- Dynamic Trading with Predictable Returns and Transaction Costs — Garleanu and Pedersen, partial equilibrium with quadratic costs
- Risk sharing equilibria with heterogeneous risk aversions — Muhle-Karbe, Liang, related risk-sharing setup