This paper compares the tail dependence and risk spillovers from the oil and gas to high-yield (HY) and investment grade (IG) bond markets. We use time-varying optimal copula framework to examine the dependence and further quantify upside and downside risk spillovers. We also explore how energy futures can be used to hedge risk of HY and IG bond portfolios. Our results show that the bond returns are more sensitive to risk shocks in the oil market compared to gas market. We find both negative and positive tail dependence between the bond and energy pairs and the relationship is stronger during the oil-crunch period. The dependence however is asymmetric across the tails. Finally, compared to oil futures, gas futures are found to be better hedge for the bond investment. These results can help in managing portfolio risk and designing optimal asset allocation strategies. These might also assist in formulating policies and regulations to manage the effects of cross-market risk transmissions.