Methods and apparatus for predicting the value of a stack pointer which store data when an instruction is seen which grows the stack. The information which is stored includes a size parameter which indicates by how much the stack is grown and one or both of: the register ID currently holding the stack pointer value or the current stack pointer value. When a subsequent instruction shrinking the stack is seen, the stored data is searched for one or more entries which has a corresponding size parameter. If such an entry is identified, the other information stored in that entry is used to predict the value of the stack pointer instead of using the instruction to calculate the new stack pointer value. Where register renaming is used, the information in the entry is used to remap the stack pointer to a different physical register.