I fail to understand reinvestment of coupons to calculate yield to maturity (YTM). I understand that YTM is the rate at which coupon payments and par value of the bond are discounted to today. i.e.
C = Coupon
T = Time
If a bond pays
C1 @ T1,
C2 @ T2,
C3 + par value @ T3
YTM is rate at which the price of the bond (determined by market) equals
the present value of (C1, C2, C3 + par value) at respective times
There is no reinvestment of C1, C2, C3 but reinvestment of the interest earned on these coupons at compounding intervals T1, T2 & T3.
So my assumption is that if a bond is bought @ x% YTM, it will always yield x% if held till maturity irrespective of YTM in the future when the coupons are paid.