Some do. Some do not.

Read the label. If the target pest (in this case, cockroach or bed bug) is not listed on the label, do not use it to try to kill that pest. For a list of pesticides that are registered for use on bed bugs, use EPA's registered product search tool. For general pesticide questions, visit the National Pesticide Information Center.

Note: boric acid does not work for bed bugs. Other insecticidal dusts, such as diatomaceous earth, have been proven to work for bed bugs.