For annual policies issued uniformly throughout a year, where is the effective midpoint relative to the policy start?

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Multiple Choice

For annual policies issued uniformly throughout a year, where is the effective midpoint relative to the policy start?

With annual policies issued uniformly over a year, the halfway point for the total earned premium from all those policies occurs not at six months after an individual start, but at the end of the year. Think of time t measured from the start of the calendar year. For t in [0,1] (the first year), the cumulative earned premium grows as t^2/2; for t in [1,2] (into the next year), it grows as t − 1/2. The total premium is fully earned by t = 2, and the halfway point occurs when the earned amount equals 0.5 of the total, which happens at t = 1 year. So the effective midpoint relative to the calendar-year start is one year after that start—the end of the year. Note this differs from a single policy’s midpoint (which is six months after its own start); the question concerns the aggregate timing across uniformly issued policies.

With annual policies issued uniformly over a year, the halfway point for the total earned premium from all those policies occurs not at six months after an individual start, but at the end of the year. Think of time t measured from the start of the calendar year. For t in [0,1] (the first year), the cumulative earned premium grows as t^2/2; for t in [1,2] (into the next year), it grows as t − 1/2. The total premium is fully earned by t = 2, and the halfway point occurs when the earned amount equals 0.5 of the total, which happens at t = 1 year. So the effective midpoint relative to the calendar-year start is one year after that start—the end of the year. Note this differs from a single policy’s midpoint (which is six months after its own start); the question concerns the aggregate timing across uniformly issued policies.

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