My HA is on my homeowners policy (apt insurance actually, but same thing).
Best talk with your insurance provider about it, but basically they take the replacement cost and add it to your "total insured amount" (ie actual house cost, plus "contents costs" etc). Many companies have "brackets" that you pay based on ... so if your total insured about is say $314K ... you may fall into a "bracket" that is "$300K-$325K". As long as you don't go over the $325 you pay the same premium.
However, if the CI pushes you into the next "bracket" you pay a bit more for your insurance premium.
If you need to make a "claim" because your CI is broken/stolen etc then you have to fill out the paperwork and then pay the deductible (it can be from $100 - thousands of $$, make sure you know what you're deductible is). Also if you make a claim, often your premium goes up for a number of years .. not always by a lot of money, but enough that many people don't put claims in for less than say $5thousand because when they take the deductible and then raised premiums on top account it's less money to just buy the new item, without going through insurance!
Talk to the insurance company though - because not all companies work the same way.
Make sure that ALL the CI components are listed with serial numbers if applicable on the policy itself !