Word tables do not recalculate automatically. You can insert a field: { =SUM(ABOVE) } to sum the values, but you would have to update it following changes. You can do that easily enough with a macro e.g.
Code:
Sub UpdateAllFields()
Dim oStory As Range
For Each oStory In ActiveDocument.StoryRanges
oStory.Fields.Update
If oStory.StoryType <> wdMainTextStory Then
While Not (oStory.NextStoryRange Is Nothing)
Set oStory = oStory.NextStoryRange
oStory.Fields.Update
Wend
End If
Next oStory
lbl_Exit:
Set oStory = Nothing
Exit Sub
End Sub