The answer depends on several factors, not the least of which is the terms of the insurance policy. That is, an insurance policy is a contract that specifies the rights and responsibilities of the parties (the insurer and the insured). Every insurance policy provides limits as to how much, under what circumstances, and for what payment will be made.
A good example is the contents portion of a homeowner's policy (this is the coverage for your furniture, clothing, etc. that may be damages or destroyed in a covered occurrence, such as a fire). If the policy is written on a "replacement cost" basis, all other things being equal, you will be able to be reimbursed for the cost of replacing the items that you lost. This is subject to the limitation that the replacements be of "like kind and quality" of the ones destroyed. This goes to the principle that insurance is not intended to be a money-making proposition for the insured--it is to indemnify. However, if the policy provides for "actual cash value" contents coverage, the insurer will pay the depreciated value of the lost or destroyed property. Factors considered in doing this include age and condition at the time of the loss.
Assuming that a life insurance policy is in force at the time of a the insured's death, the insurer will ordinarily pay the face value of the policy, less any money owing from policy loans that the insured may have taken (assuming that whole life insurance is involved). If it is a term policy, the face value is normally paid.
Health insurance works similarly. The policy will provide what is covered and what is not, what is excluded, and any deductibles or copayments that is the responsibility of the insured.
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