Under Almgren-Chriss with linear temporary impact eta, volatility sigma and risk aversion lambda, the optimal trajectory decays at rate kappa = sqrt(lambda*sigma^2/eta). By what factor does the effective trading horizon change when risk aversion quadruples, and what does that cost?

Under Almgren-Chriss with linear temporary impact eta, volatility sigma and risk aversion lambda, the optimal trajectory decays at rate kappa = sqrt(lambda*sigma^2/eta). By what factor does the effective trading horizon change when risk aversion quadruples, and what does that cost?

Approach: The horizon is the reciprocal of the decay rate, so read off the exponent on lambda. Then use the scaling of impact cost and shortfall variance in the horizon.

1/2. The horizon scales as 1/kappa, which is proportional to 1/sqrt(lambda), so quadrupling risk aversion doubles kappa and halves the horizon. Expected temporary impact cost under a linear model is proportional to 1/T, so halving the horizon doubles the expected cost, while the variance of the shortfall is proportional to T and therefore halves. Those two scalings are the Almgren-Chriss efficient frontier of execution: the horizon is the single control, cost falls and timing risk rises as it lengthens, and lambda selects the point. A desk with zero risk aversion trades at a constant rate over the whole window, which is the volume weighted schedule, and a desk with unbounded risk aversion crosses immediately and pays the full spread and impact.

Follow-up: How does the optimal trajectory change when the impact function is a square root of the trading rate rather than linear?

Key concepts: almgren-chriss, temporary impact, timing risk, risk aversion.