Assets Under Administration - AUA

A company that provides financial services might report the value of the assets they have under administration using a statistic called "assets under administration," or AUA, (a name without much panache in our opinion).
A key comparison here is with assets under management. AUM, as it's also called, measures the value of assets a financial service company manages for its clients. In this case, the company actually controls the assets - it buys and sells the stocks or bonds or whatever, making direct investment decisions with capital provided by clients.
AUA works differently. The assets in an AUA situation are still controlled by the clients themselves. The financial service company simply provides what's called custodial services.
In most walks of life, "custodial services" responds to the stuff janitors do - taking out the trash, sweeping the floors, unclogging the toilet. Think of "custodial services" in the context of assets as the financial equivalent to that. It can include stuff like overseeing tax-related functions or providing accounting services.
It can also include a different context of "custodial," like having custody of something. The assets measured in the AUA figure includes assets that the company holds in custody for clients, though the firm doesn't actively manage the asset.
So, the company holds the stock certificates or the bonds for the client, and can sell them at the client's request. But the firm wouldn't make any decisions without the client instructing them to do so.