What is the correct way to sign a business letter for someone else?
p.p. or per pro. = "per procurationem". This means "by/through the agency of". The agent is the person who is signing the letter on someone else's behalf.

Thus if a PA is signing on behalf of an absent person, the correct form is as follows:

Yours sincerely,

p.p. [signature of PA]

Name of Absent Person