Yes, agreeing to pay the first x dollars in a loss is assuming an obligation. When a policy does not define a term, courts generally refer to a standard desk reference since that is what the average insured has access to. Merriam Webster Online defines obligation as “a commitment (as by a government) to pay a particular sum of money; also : an amount owed under such an obligation; the action of obligating oneself to a course of action; something (as a formal contract, a promise, or the demands of conscience or custom) that obligates one to a course of action. Assume is defined as: to take to or upon oneself : <assume responsibility>; : to take over (the debts of another) as one's own.”

By agreeing to make payment, the insured has taken it upon himself to pay funds that are due; he has assumed the responsibility, and therefore the obligation, of the other party. Generally, policies are worded this way so that the insured does not inadvertently obligate the carrier to cover something that is not covered in the policy— the insurer is saying that if the insured makes some sort of agreement without the company's knowledge, it is the insured's agreement alone and the company will not pay for it.