Late 2013 Macbook Pro not connecting via AirPlay. What can I do?

Specs:

• Spanking new Apple TV with all the latest software updates
• Spanking new Macbook Pro Retina with Mavericks

The issue I'm having is streaming via Airplay to anything. I can't stream audio or mirror or extend my display to my Apple TV, or my friend's Apple TVs. What happens when I try to connect is this:

I'll click the AirPlay button in the menu bar, select my Apple TV and the icon changes blue. That's it. About twenty seconds later I get a dialogue saying 'Couldn't connect to your Apple TV'. Interestingly, if I set a password or a code, I get prompted for it but then the same thing happens.

Both are on the same network, the Apple TV is cabled in and the MBP is using WiFi. I've tried putting the Apple TV on the WiFi but the same problem occurs. Judging by the fact I can't connect to any AirPlay devices, I'm assuming the issue is with my MBP; but can anybody help me to get it working?