If you are a college student living off-campus, you need your own renter's insurance policy to cover your belongings and any damage to your apartment or the premises.
According to the National Association of Insurance Commissioners (NAIC),
"College students renting an off-campus apartment or house while away at school should consider purchasing renters insurance to protect their personal property, such as a computer, television, stereo, bicycle or furniture, in the event that it is damaged, destroyed or stolen.
Even if a student is a dependent under his or her parent’s insurance, the student’s personal property, in many cases, is not covered if the student lives off-campus. Parents should check their policy or contact their insurance agent to see if renters insurance is right for their son or daughter who is away at school."
However, often parents and/or students assume or are misled to believe that the parents' homeowner's insurance will cover any loss or damage to an off-campus house or apartment. This can be a very costly mistake. The NAIC notes that "...if a college student is under 26 years old, enrolled in classes and living in on-campus housing, the student may be covered under his or her parents’ homeowners or renters insurance policy."
For the potential benefits and coverage, purchasing renter's insurance is not costly at all.  The NAIC says that "the premiums for renters insurance average between $15 and $30 per month depending on the location and size of the rental unit and the policyholder’s possessions."
Additionally, students living in off-campus housing should consider purchasing both personal property and liability coverage. "Most renters insurance policies provide two basic types of coverage: personal property and liability. Personal property coverage pays to repair or replace personal belongings if they are damaged, destroyed, or stolen. This is the most commonly purchased renter's policy.
Liability insurance provides coverage against a claim or lawsuit resulting from bodily injury or property damage to others caused by an accident while on the policyholder’s property."