Explanation of trade half life in Almgren-Chriss paper

External question — Quantitative Finance Stack Exchange Author: parky Original post: https://quant.stackexchange.com/questions/81211 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. In Section 2.3 of Almgren, Robert, and Neil Chriss. "Optimal execution of portfolio transactions." Journal of Risk 3 (2001): 5-40 the authors define the half-life of a trade as $\theta \equiv 1/\kappa$ and note that it is exactly the amount of time it takes to deplete the portfolio by a factor of $e$ . Eq(17) derives the amount of inventory remaining at a trading time $t$ to be $$ \frac{\sinh(\kappa(T - t))}{\sinh(\kappa T)} X. $$ However, setting $t = \theta \equiv 1 / \kappa$ in the above does not yield $X / e$ . I suspect I am misinterpreting the claim and seek some guidance to set me straight. Thank you!