Abstract
We derive a closed-form optimal dynamic portfolio policy when trading is costly and security returns are predictable by signals with different mean-reversion speeds. The optimal strategy is characterized by two principles: (1) aim in front of the target, and (2) trade partially toward the current aim. The aim portfolio is a weighted average of the current Markowitz portfolio and the expected Markowitz portfolios on all future dates. Predictors with slower mean-reversion (alpha decay) get more weight in the aim portfolio.
Summary
This is the foundational single-agent paper that introduced the quadratic transaction cost framework later used by Herdegen-Muhle-Karbe and subsequent equilibrium models. An investor with mean-variance preferences trades S securities over discrete periods, facing quadratic costs TC = (1/2) Delta_x^T Lambda Delta_x. Returns are predictable via K factors f_t with different mean-reversion speeds (alpha decays) phi^k.
The model is solved in closed form via dynamic programming (Proposition 1, equation 6). The optimal strategy is characterised by two elegant principles:
Principle 1 — Aim in front of the target (Proposition 3): The aim portfolio is not the current Markowitz portfolio, but a weighted average of the current and ALL expected future Markowitz portfolios: aim_t = z * Markowitz_t + (1-z) * E_t(aim_{t+1}), where z = gamma/(gamma + a) decreases with transaction costs. With multiple return predictors, each factor is weighted by 1/(1 + phi^k * a/gamma), so persistent signals (small phi^k) get more weight (Proposition 4, equation 15).
Principle 2 — Trade partially toward the aim (Proposition 2): The optimal portfolio is x_t = (1 - a/lambda) * x_{t-1} + (a/lambda) * aim_t, trading a fixed fraction a/lambda toward the aim each period.
The trading rate a = (-gamma(1-rho) + lambda*rho) + sqrt(…)) / (2(1-rho)) decreases with transaction costs lambda and increases with risk aversion gamma.
Key Contributions
- First closed-form solution for dynamic portfolio choice with quadratic costs AND multiple return predictors with different alpha decays
- “Aim in front of the target” principle: the aim portfolio tilts toward persistent signals
- “Trade partially toward the aim” principle: fixed fraction a/lambda of the gap is closed each period
- Trading rate formula (equation 9) determines the optimal speed of portfolio adjustment
- Persistent costs extension (Section IV): with persistent price impact (Kyle’s lambda with resiliency R), the aim portfolio also depends on the price distortion D_t
- Empirical application: 20% better Sharpe ratio than static strategies for commodity futures
Methodology
Dynamic programming with quadratic value function V(x, f) = -(1/2)x^T A_xx x + x^T A_xf f + (1/2)f^T A_ff f + A_0. Under the key assumption Lambda = lambda * Sigma (costs proportional to risk), the coefficient matrices A_xx, A_xf, A_ff are given by explicit algebraic Riccati equations (Appendix, equations A15-A22). The model extends to persistent transaction costs via an augmented state variable (x, y) = (x, f, D).
Key Findings
- The optimal trading rate a/lambda is constant (independent of current position or signals)
- More persistent signals get more weight in the aim portfolio: weight ratio (1 + phi^j a/gamma)/(1 + phi^i a/gamma) increases in lambda when phi^j > phi
- The static optimiser (equation 29) cannot replicate the dynamic solution with multiple predictors, regardless of how gamma and lambda are adjusted
- “Position homing” (Proposition 5): the optimal position is an exponentially weighted average of past aim portfolios
- Commodity futures application: the dynamic strategy achieves net Sharpe ratio 0.41 vs 0.34 for the best static strategy
Important References
- Portfolio Choice and Pricing in Illiquid Markets — Garleanu (2009), the search-based equilibrium precursor
- Equilibrium Returns with Transaction Costs — Herdegen, Muhle-Karbe (2018), the multi-agent equilibrium extension
- Optimal Execution of Portfolio Transactions — Almgren and Chriss (2000), the optimal execution literature