premium income rate

Money does not arrive at an insurer in one lump — it trickles in steadily as policyholders pay their premiums month after month. In the simplest models we smooth this trickle into a constant flow, like a tap left running at a fixed rate filling a tub. The premium income rate is the speed of that flow: how much premium money comes in per unit of time.

In the surplus process U(t) = u + c*t - S(t), the premium income rate is the constant c, measured in money per unit time (say dollars per year). After time t, the premium collected is simply c*t — the rate multiplied by how long the tap has run. The natural starting point for c is the expected claims per unit time: if claims are expected to cost 2,000,000 a year, then collecting exactly 2,000,000 a year would, on average, break even. In practice c is set a bit higher than this break-even level so the surplus has an upward drift; how much higher is the security loading.

The premium income rate is the lever that connects pricing to survival. If c is set right at or below the expected claims rate, the surplus line has no upward drift and is essentially certain to be ruined eventually, no matter how much initial surplus you start with — a striking and important result. Only when c strictly exceeds expected claims does the surplus tend to climb and ruin become merely possible rather than inevitable. A caveat: in this clean model c is a single smoothed number; real premium income is lumpy, seasonal, and reduced by expenses and commissions before any of it can absorb claims.