With mine there is a "named storm" endorsement.
Basically, the excess goes up.
That is with my custom written policy (no one of the insurers who advertise in the airplane mags will insure airplanes down here).
My previous insurer has a very simple rule if the storm gets a name you have no cover.
Oddly policies written in the Caribbean give no cover in US airspace and no US insurer will insure a US registered airplane in the US if one has a non-US address. Mine was up in Florida for nearly three years no cover even though it was just sitting on the ground not being flown.
When I bought it back down my insurer gave me two weeks to get it out of the US (my last US airspace was St. Croix) and during that time I was not permitted to carry a US citizen in the airplane. Yes, US people sue for anything - but my insurance on my 40 year old PA28 cost $12,000, a policy with a US insurer would have been $4,000 if I was a US address holder operating in the US and with lots of US citizens flying in the airplane.
I cannot understand it - I have asked for a reason, only answer I ever got was "computer say no" or I need a zip code for the computer.
I spoke with a top broker in Florida, he just gave up. In the end I got my professional indemnity insurance broker from Trinidad to speak with a Lloyds syndicate in London to get it. So a custom written policy, agreed hull value (like classic cars), the named storm thing reduced i.e. I have cover unlike before.
Airplane insurance is very weird.