Function giving the share of the original premium attributable to a given deductible or layer, as a function of that layer relative to the insured value, allowing pricing without any loss history.
An industrial risk with 200M EUR insured value has never had a claim: no history supports pricing a layer between 20M and 50M EUR. The exposure curve answers by changing the question. It plots the ratio of a loss amount to insured value on the horizontal axis, and on the vertical axis the share of total premium consumed by covering all losses below that amount. Its derivative at a point gives the share of premium above, so a layer rate reads as a difference of two ordinates on the curve. The curve is concave, starts at zero and ends at one, and its curvature describes the relative severity of the risk: a curve rising very fast describes a risk of frequent small losses, a flat curve describes total destruction risk. It assumes the distribution of damage ratio does not depend on risk size, an acceptable assumption inside a homogeneous occupancy class and a false one between a warehouse and a refinery.
Industrial risk with 100M EUR insured value, original premium 400K EUR. Pricing a 20M EUR layer above 10M EUR. The selected exposure curve gives G(0.10) equal to 0.74 and G(0.30) equal to 0.93. The layer's premium share is 0.93 minus 0.74, that is 19% of 400K EUR, or 76K EUR before loadings, and that figure is obtained without a single claim ever having been reported on the risk.
exposure curve, courbe de Lieberman, courbe d'exposition normalisée, répartition de la prime par tranche