Suppose that the user has some unsavoury aspects to their life, such as pornography, or that their computer gets confiscated by the Police or HMRC in the course of their investigations. They would access the entire storage on the machine, not just the user / owner's personal data. This could compromise client sensitive data.
Not a fan of BYOD, although I have to say that I wouldn't mind bringing in my own 4K display to run from the company PC, if they came down a bit in price (my eyesight is to what it use to be). Would that count at BYOD?