I'm trying to wrap my head around pricing a Constant Maturity Swap (CMS). Let's imagine the following deal: 6m LIBOR in one direction, 10y swap rate in the other. The discount curve is derived from OIS.
Naively I would just price this by taking the difference between the present value of cash flows from the forward 6m LIBOR rates and the present value of the cash flows from the forward 10y swap rates. I assume the cash flow from the swap leg is:
10y swap rate * notional
But apparently this is not right, as quoting from here "the expected swap rate $\not=$ the forward swap rate" and this is the origin of the famous convexity adjustment.
But why does the expected rate not equal the forward rate and how might one compute the difference?