Your password is between you, and your computer alone.
No one else.
Not your boss, his boss, the system administrator, your bank official, your insurance agent, your ISP support technician, or your cat. Well, your cat you can tell, if she promises not to share it.
There is NEVER a good reason to share a password.
There are many reasons NOT to. Mostly, because a password is YOUR authentication, and as soon as even ONE other person knows it, it can no longer prove your identity.
Any reason your admin comes up with, is bogus, either because he is malicious, lazy, misinformed, or incompetent.
That said, it may not be his fault, but the fault of his organization.
Either way, there is incompetence, ignorance and laziness abound.
If an admin, or ANY support technician asks for your password, the correct response is to LAUGH.
Because there's no way they're serious, right?
If your admin insists - explain to him that you will document sharing your password with him... and that, based on this, you are going to send nasty emails to all around - not about him, but you will claim that they came from him (using your account, in your name, using your password that you just shared with him). Of course he won't be able to prove that he didn't misuse your password... which is the point.
No, on second thought, just don't give him your password. It's yours, between you and the computer alone.