how much current bond, ie most recently issued bond (on-the-run bond is auctioned on a yield basis and priced at par), needs to pay depends on the current market situation. when folks are in a panic mode, they buy t-bond while accepting very little interest payment in return. this is called flight to safety. when folks are comfortable and want to take more risks in stock market, or worry about inflation, they require a higher interest rate to buy bond.
so when current market rate (also called yield to maturity) is high, old bond that is locked to a low coupon rate will have to trade at a discount to offer comparable yield over the life time of the bond.
so when current market rate is low, old bond that is locked to a high coupon rate will have to trade at a premium to offer comparable yield over the life time of the bond.
the key is to note that the coupon rate is locked, ie "FIXED" when bond is issued. when current market environment changes, old bond will be trading at a premium or discount. most of the time treasury bonds move in the opposite direction of the stocks. however when over the dominating theme in the market is monetary policy concern (say timing of QE or first raise of FF rate), stock and bond can demonstrate postive correlation.
bonds with built-in options are more complex due to uncertainties in the timing of par payment (callable/convertible bonds). credit-sensitive bonds also demonstrate more stock-like features.
just my 2c.