Does WITH SCHEMABINDING on a multi-statement TVF improve cardinality estimates? Based on http://blogs.msdn.com/b/psssql/archive/2010/10/28/query-performance-and-multi-statement-table-valued-functions.aspx and other articles, SQL Server assumes that a multi-line table valued function returns one row. This causes the selection of a poor execution plan for the calling statement if it actually returns many rows. 

Does adding WITH SCHEMABINDING to the RETURNS clause of the CREATE FUNCTION result in a more correct cardinality estimate for the return value of the function?

If we assume that we are passing a UserId to this function and getting back a table of RecordId values that the user is allowed to access, and that some users are only allowed to see a few records and that some are allowed to see many or even all records, would either the function or the calling statements (or the procedures that include them) benefit from using FORCE RECOMPILE? Does the use of WITH SCHEMABINDING in the function change this answer?

I realize that I could figure this out by experimentation, but I am hoping that someone has already figured out the answer. A pointer to someplace where this is well documented would be helpful.
In my tests, no, adding WITH SCHEMABINDING does not improve cardinality estimates. I created a simple table:

CREATE TABLE dbo.myobjects(id INT PRIMARY KEY);

INSERT dbo.myobjects SELECT [object_id] FROM sys.all_objects;


Then two functions:

CREATE FUNCTION dbo.noschemabinding(@UserID INT)
RETURNS @x TABLE (id INT)
AS
BEGIN
  INSERT @x SELECT id FROM dbo.myobjects;

  RETURN;
END
GO

CREATE FUNCTION dbo.withschemabinding(@UserID INT)
RETURNS @x TABLE (id INT)
WITH SCHEMABINDING
AS
BEGIN
  INSERT @x SELECT id FROM dbo.myobjects;

  RETURN;
END
GO


Comparing the actual plans, both show estimated rows = 1, actual rows = 2112 (this latter number may differ on your system depending on version/SP etc).

Comparing the speed:

SET NOCOUNT ON;
GO
SELECT SYSDATETIME();
GO
SELECT id INTO #x FROM dbo.noschemabinding(1);
DROP TABLE #x;
GO 1000
GO
SELECT SYSDATETIME();
GO
SELECT id INTO #x FROM dbo.withschemabinding(1);
DROP TABLE #x;
GO 1000
SELECT SYSDATETIME();


Results:

                    run 1               run 2
----------------    ------------------  ------------------
No schemabinding    14632 milliseconds  14079 milliseconds
Schemabinding       14251 milliseconds  13979 milliseconds


So, does it matter much? Nope.

SCHEMABINDING in this case is used for a more important goal: underlying schema stability. You will probably have much better optimization opportunities if you pursue converting your function to an inline TVF than to chase down obscure plan-affecting differences in a multi-statement TVF.