Premium comes from Latin praemium, meaning a reward or prize, built from prae- (before) and emere (to take or to buy). The original concept was something taken before others — the first share, the winner's prize, the profit gained by being first. The word entered English in the early 17th century, during the rapid expansion of commercial and insurance vocabulary.
The insurance meaning developed quickly. Maritime insurance, already established in Italian and Dutch commercial practice, required regular payments from policyholders to underwriters. These payments were called premiums — the price of buying future protection. This sense of paying above and beyond for something extra eventually generalized into the commercial meaning: a premium is any amount paid above standard price for superior quality or special access.
Latin emere (to take, to buy) produced a large and varied English family. Redeem means to buy back. Exempt means taken out (freed from an obligation). Example means something taken out to show (a specimen). Sample is a contracted form of example. Preempt means to take before someone else can. All preserve the core sense of taking or acquiring, filtered through
The adjective use of premium — premium quality, premium product, premium service — emerged in the 20th century marketing vocabulary and has become ubiquitous. In this usage, premium has drifted from meaning a specific extra payment to implying general superiority, often with little concrete justification. The inflation of the word in advertising has diluted its precision.
The phrase at a premium, meaning scarce and therefore expensive, preserves the older financial sense most accurately. Something available only at a premium costs more than its face value or standard price because demand exceeds supply. This usage connects directly to the Latin idea of praemium as an extra amount gained or paid beyond the ordinary.