Money Laundering
Assuming your story is set in the past...
Prior to 1982, bearer bonds could be used for money laundering and/or tax evasion. In traditional detective stories, possession of bearer bonds was often used to imply dubious if not outright fraudulant business practices.
If you made $2 million from a jewel heist, you could invest the money in stocks without leaving a paper trail by buying bearer bonds. The bonds gave you the ability to own stock anonymously. If you put the money in the bank, you'd likely be convicted of tax fraud, at the very least.
Bearer bonds could also be used to transport large amounts of money. A single sheet of paper could be worth $1 million if it was a bearer bond for $1 million in stocks. Much easier to move and hide than $1 million in cash.
The bonds were also attractive targets for theft. By definition, a bearer bond doesn't list the owner's name. So, if you steal the piece of paper, you own the stock -- sometimes thousands of shares of stock. A society debutante who swiped the jewel thief's bearer bond would now own $2 million in stock.