All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First and foremost it should be noted that a Muslim is obliged to seek the true Islamic jurisprudence in his dealings. So, he should not buy shares and bonds which include Riba (usury and interest) and he should not deal with (participate in) companies which deal with usury and interest.
As regards your question, the matter is of two types:
First, if the partner contributes to a company in order to benefit from the annual profit of the shares, and not with the purpose of trading in shares, then he should give only the Zakat of his profit, like one pays the Zakah of immovable property and rented non-agricultural lands, and the like.
In other words, he does not have to pay the Zakat regarding the origin of the share, but he has to pay Zakat for its profit, which is 2.5%, after a whole lunar year elapses from the day of the profit. This is so if the shares of the company are stable elements like real property, boats, factories etc.
If they are tradable goods, then the Zakat should be paid on the capital money and the profit.
Secondly, if the participant has acquired the shares for the purpose of trading, then the Zakat for these shares will be like tradable goods, be it elements or tradable goods. Experienced people will evaluate it at the end of the year from its beginning, and then 2.5% should be paid as Zakat from it.
If the bonds are fixed term loans which one expects to repay, then Zakat should be paid for it every year if they constitute a Nisaab (minimum amount liable for Zakat), even by adding them to other money or tradable goods that one possesses.
If it contains profits from usury and interest, then the Zakat should be paid on the capital money only and the profit (usury and interest) should be donated in charity, or should be spent on the general welfare of the Muslims.
To conclude, the Zakat should be paid yearly from the capital money and the profit if a person is engaged in buying and selling, but it is paid only on the profit if one is not selling and buying but he only rents his properties.
Allaah Knows best.