I come from Hong Kong, a non-German citizen living outside the German and trading German stocks through a U.S. internet broker. Should I have to pay capital gains tax to German government on the money I earn?
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The Federal Central Tax Office says you may not have to pay taxes in germany for capital gains. You may have to apply for a tax relief to prevent the tax from being collected.
You very likely will have to pay whatever taxes there are in Hong Kong on capital gains. Since you use an US broker withholding tax may apply to you but this is a different question that has nothing to do with Germany or german stocks.
To be sure you should contact a local expert on this topic.
EDIT: I missed some informations that I found on the english site of the german Federal Central Tax Office homepage.
I don't have enough reputation to leave comments, so here comes some addendum to the first (and only) answer. It's very simplistic and any comments are more than welcome.
While the capital gain might be tax-free, you also have to consider the effect of any received dividends: they are subject to the "Abgeltungsteuer" and deducted at the source. Rate: 25%. On top of the Abgeltungsteuer amount, two other additional taxes will be applied: the Solidarity tax and a Church tax (the latter is optional).
You might be exempted if both countries have a double taxation treaty.