It seems that the Skype 220.127.116.11 adds new requirement for Windows 7 (not sure about 8 and 10, but most likely for them too). So, beginning with the the new version of Skype, on some computers trying to run installer the following error may occurs:
Cannot find or load Microsoft Installer. Please update your computer and try running Skype again
Unfortunately it does not say what exactly is outdated and which components should be updated. But do not worry, after a couple of tests I was able to find the right solution for this error.
To fix the "Cannot find or load Microsoft Installer" error you should install Internet Explorer 11. Even if you already have installed it, download and run the specified IE installer which will install all required updates for latest Skype version.
As another solution, you can install KB2533623 update, which also fixes this error. But please keep in mind that according to Skype minimal requirements on Windows 7 SP1 it’s required to use Internet Explorer 11.