I am foreign graduate student in USA. I recently completed my undergrad (from USA) and have a bit of money saved up. I know that F1 students cannot work off campus and other business activities such stock trading etc are prohibited as well. I wanted to know if opening an account in an investment management firm like Charles Schwabb or Betterment is allowed. I would not be working but I will still be earning a small interest on my deposits. If that is not allowed, are there any good and legal ways for foreign students to earn a little interest on savings?
There's no limitation on what you can invest in, including trading stocks (as long as trading is not a business activity, like day-trading or investing for others).
You just need to make sure you have a tax ID (either ITIN or SSN) and pay taxes on all the gains and dividends. Also, consider your home country tax laws, since you're still tax resident in your home country (most likely).