Is it permissible to utilize the money of my mentally ill brother—which is deposited in my name in the bank as a trust—for the purpose of investing it in the sheep trade, while taking a percentage of the profits in return for the effort?

If the mentioned amount was gifted by the father to the mentally ill brother, and it was written in the name of the questioner to be his guardian, then the guardian must dispose of the ward's money only in a way that serves his best interest.

Therefore, the guardian is not permitted to engage in purely harmful dispositions such as gifting or charity. However, he is allowed to engage in purely beneficial dispositions such as accepting a gift, and dispositions that involve both benefit and harm, such as buying and selling, provided that no apparent harm results from them.

As for investing the ward's money in a partnership or Mudarabah (definition), it is permissible if it is in his interest, meaning the guardian can give it to someone who will invest it for a share of its profit. However, the guardian is not permitted to invest the money himself. If he does, he guarantees the capital if it loses, and all the profit belongs to the ward if it gains, with the guardian receiving nothing from it.

Accordingly, the questioner should dispose of his brother's money only in a way that realizes his brother's interest, and he should not invest it himself. He may give it to someone he deems trustworthy to invest it for a share of the profit, without favoritism.