I have a Schengen visa that is valid from 13 Nov 2016 through 20 Nov 2016 with a duration of stay: 7 days. I entered the Schengen zone on the 13th.
Can I exit the Schengen zone on 20th (8th day) or is it mandatory to exit on the 19th?
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The difference between the duration and validity might sound stupid, but it isn't. When people apply for their visa the consulates often "round up" regarding validity so that the application doesn't have to be renewed just because there are minor changes to the travel date.
Example: You apply for a visa from May 14th to May 16th for a weekend trip to Europe. You are granted a visa with 3 days duration, but the validity could be May 1st to May 31st. If you change plans to travel one week earlier or one week later, you need no new visa. If you change plans to travel one week longer, the premise of your trip has changed and you need a new application.
In your case, you had one day more validity than duration, so you could have changed a little bit regarding the first and last day. Perhaps take a flight in the early morning of the 14th rather than the last flight in the evening of the 13th.
But by arriving on the 13th you started your first day on the 13th. You have to leave on the 7th day of your stay.