“assets under management” is money/assets you are in charge of but are not yours. (assets can be literally anything with monetary value. i.e. If you can sell it for money its a asset. For example my 401k is with vanguard. I do not decide what stocks bonds etc. get bought or sold, vanguard does. However much money I have with vanguard contributes to their assets under management as they control the assets but the assets belong to me.
vanguard decides how to best use the money/assets to maximize my returns and in exchange I pay vanguard a percent to do so.
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