Your broker has a power to manage your trading account on your behalf but you would still be responsible for the trading outcome.

They can manage your account on your behalf.

Yup, your broker has a power to handles your account on your behalf.

If you allowed them to trade on your behalf, then they could have the power to manage your trading account. A broker won't touch your account without your consent, unless it is stated in the agreement.

Some brokers allowed their management to manage the account of their clients and I think the management fee is quiet high.