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I was researching how short selling worked and I was wondering what are the advantages for the broker to allow this. Why would the broker lend out stocks with the expectation that they will get it back later at a lower value?

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To earn money. Brokers are not lending out "their" shares but the shares of their customers (that happen to lie around anyways). So lending this out for a fee is additional earnings.
Also people who sell short are likely to be more active traders. More transactions equals more earnings as well (through commissions, payment for order flow)

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Brokers do not lend shares for shorting with the "expectation that they will get it back later at a lower value." It's the borrower of the shares who shorts them with the hope that he will be able to buy to close the position at a lower price, thereby earning a profit.

Brokers facilitate the lending of shares from one person's account to another. If done in house, it involves one broker though it can involves two different brokerage firm if the shares are not available in house.

The borrower of the shares pays a borrow fee to the lending broker. Some brokers share a portion of this with the lending client.

The lending of the stock results in additional transactions. Since this involves additional bid/ask spreads, the broker may benefit if he makes a market in the securities or crosses orders between in house clients. More transactions also benefits the broker if he still charges commissions.

Some brokers route client orders to exchanges for payment for order flow. More orders means more PFOF.

The shorter receives a cash credit for selling and his broker earns interest on cash balances. This is a major source of revenue for all brokers.

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