The Bornhuetter-Ferguson reserving method uses R = Premium earned × Loss ratio ultimate losses × (1 - 1/Chain-ladder age to ultimate factor). Which option expresses this?

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The Bornhuetter-Ferguson method blends an a priori estimate of ultimate losses with what the development pattern suggests, using a weight that reflects how much of the ultimate has yet to materialize. In this formula, multiply the earned premium by the assumed ultimate loss ratio to get the expected ultimate losses in dollars. Then scale that by the portion still to develop, captured by (1 − 1/Chain-ladder age to ultimate factor). The chain-ladder age-to-ultimate factor grows as development progresses, so 1/that factor shrinks and the remaining fraction (1 minus that) grows toward 1. This weighting reflects that, with more development observed, you rely more on the a priori ultimate, yielding a larger reserve up to the full expected ultimate.

So the expression with the (1 − 1/Chain-ladder age to ultimate factor) factor is the correct one. The other forms omit this adjustment or use an incorrect sign, which would misstate how much of the ultimate losses remains to be realized.