Does my 1st class medical give me 2nd and 3rd class medical privileges after it expires?
A first class medical issued to a 35 year old pilot on 15 Jan 2012 will let that pilot:


exercise privileges of an ATP certificate until 23:59 31 Jan 2013
exercise privileges of a commercial certificate until 23:59 31 Jan 2013
exercise privileges of a private, recreational, student or CFI certificate (if the CFI is acting PIC) until 23:59 31 Jan 2017


Your first class medical is always a first class medical -- you don't get private pilot privileges for 5 years because a first class medical grants third class privileges, rather you get private pilot privileges for 5 years because that is what a first class medical grants you.  

These durations are from the table of 61.23 (d).  You don't see this language in the text above the table because they are not concerned with the duration of the privileges but only the minimum required medical class for specific operations. 

For example, 61.23 (a)(2)(ii) says you must hold at least a second class medical to exercise the privileges of a commercial certificate and 61.23(d)(1)(iii) and 61.23(d)(2)(i) list the duration that each of an first and second class medical can be used to for this purpose.