Does my 1st class medical give me 2nd and 3rd class medical privileges after it expires? If a 35 year-old pilot were issued a 1st class medical on January 15, 2012, am I correct that that pilot would be able to exercise privileges requiring a 1st class medical until January 15, 2013? Or would they extend until January 31, 2013? The chart in 14 CFR 61.23 mentions something about it expiring on the last day, but I can't find that exact language anywhere but in the header row of the reference table.

Additionally, after the 1st class medical certificate expires, does it not afford the pilot the privileges of a 2nd class medical for the 60 months listed in 14 CFR 61.23? I'm looking for the part of the FAR that explains how medical certificates of a higher class afford the privileges of a lower class after their original designation period 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.