Is it permissible to rent out a kilogram of gold for a monthly sum to be used for trade?

It is not permissible to lease gold because leasing (ijarah (definition)) is a contract that applies to usufructs, not to the underlying assets themselves. Gold cannot be used while preserving its essence. Therefore, the aforementioned transaction is a forbidden usurious loan (riba) from which one must repent. This is because a loan involves giving money to someone to benefit from it, with the expectation of returning an equivalent amount. If an increase is stipulated upon repayment, it becomes a usurious loan. Allah Almighty has restricted the lender's right to the principal amount and has forbidden any increase upon it.