It is true that the CGL form says that the insurer has the duty to defend the insured against any lawsuit. But, the policy also says that the insurer has no duty to defend the insured against any lawsuit seeking damages to which the insurance does not apply. And, the policy goes on to say that the insurance applies to bodily injury or property damage only if caused by an occurrence. An occurrence is defined in the policy as an accident. Also, the expected or intended injury exclusion prevents coverage for bodily injury expected or intended from the standpoint of the insured. So, by alleging that the insured intentionally caused the injury, the claimant allows the insurer the right to deny coverage and defense to the insured.

Most complaints allege a negligence claim against the insured, in addition to any allegation of intentional injury, in order for the insured's liability insurance policy to respond and become engaged in the claim activity. But, if this complaint against the insured is based only on intentional injury, the insurer does have the right to deny coverage and defense under the terms of the policy.