Bond values will, definitionally, fall when interest rates rise. However, different types of bonds have differing characteristics.
The chart below shows the annual performance of the S&P 500 Bond Index and the S&P/BG Cantor 7-10 US Treasury Bond Index. (The S&P/BG Cantor 7-10 US Treasury Bond Index is the treasury index most similar to the S&P 500 Bond Index in duration and yield.) Notably, the performance of corporate debt and Treasuries diverged in the 2008 financial crisis and the recovery thereafter.
As between the S&P 500 Bond index and the S&P/BG Cantor 7-10 US Treasury Bond Index, the former is much more correlated to the S&P 500, 0.256 versus -0.194, respectively. This is not surprising, since corporate debt carries a higher risk than Treasuries. When crisis is such a threat that corporations’ survival is at peril then naturally debt issued by corporations will also be in danger of default.