Final answer:
The least equitable method for allocating partial fills of aggregated trade orders is preferred client account allocation because it implies preferential treatment, as opposed to methods like pro-rata, rotational, or random allocation which offer a fairer distribution.
Step-by-step explanation:
The method that would be least equitable for allocating partial fills of aggregated trade orders is preferred client account allocation. This method involves giving preferential treatment to certain clients, which could be based on their relationship with the broker, the size of their investment, or other subjective criteria. As a result, this does not ensure a fair distribution of the limited number of shares among all participants.
In comparison, pro-rata allocation divides shares proportionally based on order size, ensuring that all participants receive an equitable portion. Rotational allocation assigns shares on a rotating basis which over time can average out to be fair, and random allocation ensures that all participants have an equal chance of receiving shares regardless of their order size or relationship with the broker.