In which type of insurance does the insurer have no right of recovery?

The correct answer is personal accident and sickness insurance because in this type of insurance, the benefits are typically paid directly to the insured without any subrogation rights for the insurer. This means that if a claim is made due to an accident or a sickness, the insurer does not have the right to recover the payouts from a third party responsible for the loss. This feature is designed to provide immediate financial support to the insured, often focusing on personal injuries or loss of income due to health issues without the complications of pursuing any potential recoveries from others.

In contrast, other types of insurance listed, such as fire insurance, professional indemnity insurance, and pet insurance, usually allow the insurer to seek recovery from third parties if they pay out a claim. For example, if a fire is caused by someone's negligence, the insurer may have the right to pursue that party for costs incurred. This key difference highlights the nature and intent of personal accident and sickness insurance in providing prompt and uncomplicated support to policyholders.