When a bank says that the interest rate is 4.5% with interest compounded monthly, that means that after each month your investiment grows by (=4.5/1200 or 0.045/12), so that it becomes . After another month, the interest is computed from what you now have (i.e. ), thus after two months you have and so on, after month you have .

Note that after one year you have , thus the annual rate is a bit more than 4.5%. If I were a bankier, I would define the monthly rate to be , it would make more sense! (especially if I am the bankier)