No, there is no such thing. One could probably create a complicated setup with a text animator, possibly with an expression, but otherwise you will have to adjust it manually.
You may be able to do this with scripting. The new this.sourceRectAtTime() expression can be used to find the size of a text layer and may be able to be used to automatically center a single line of text horizontally and vertically but it doesn't seem to work when you draw a text box and then enter multiple lines of text. In my quick test a single line of text could easily be centered but if I typed in a text box the centering would get fouled up unless there was enough text to fill the entire text box. IOW, if the text box could contain 4 lines of text and I only typed in 3 the centering didn't work. It might be an interesting project to try when I have some time.
The align tool will center a text box horizontally and vertically no matter the size of the text box or the layer. I'm not sure if the align tools are available through scripting but if they are that would be my best guess at automating the task.
I did a quick search of the usual script sources and didn't see any that would automatically horizontally and vertically center any sized text box. There are several that will bring in multiple lines of text from a text file and create a horizontally centered block of text. You might want to dig around for more options at AE Scripts and other resources.
Rick, thanks for the scripting ideas. I will give them a try.
At AE scripts I did find a Mamoworld script advertised that does something similar but it lacks the precision I was after.