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Exposure rating
- similar size risks from same risk category placed in bands
- risks in a band assumed to be homogeneous
- can use a single loss dist to model
- fit 1 exposure curve per band
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First loss scales / exposure curves
- gives proportion of P allocated to limited primary layers
- % value of imposing a deductible
- limits usually expressed as % of sum insured (SI), maximum probable loss (MPL) or estimated maximum loss (EML)
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Notes on exposure curve table
- can allow % > 100% of building value (other covg)
- implicit assumption that same exposure curve applies regardless of the size of the insured value
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Analytical exposure curve
- used when looking for values btwn 2 discrete curves
- (-) must fill certain conditions which restrict the range of param
- (-) practical issues w fctns w >2 param
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Deriving distribution function from exposure curve
- G(d) is increasing concave on [0,1]
- G'(d) = [1 - F(d)] / E(X)
- F(x) = 1 - G'(x) / G'(0) (1 if x = 1)
- μ = E(x) = 1 / G'(0)
- p = 1 - F'(1-) = G'(1) / G'(0)
- G'(0) ≥ 1 ≥ G'(1) ≥ 0
- 0 ≤ p ≤ μ ≤ 1
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Unlimited distribution
- normalize deductible to some other ref loss like sum insd
- G(d) still concave increasing on [0,1]
- M = E(x) = 1 / G'(0)
- G(∞) = 0 (no total loss)
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MBBEFD class of 2-parameter exposure curves
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Curve fitting
- there exists exactly 1 dist fctn belonging to MBBEFD class for each given pair of functional p and μ
 - if first 2 moments are known we can find g and b
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Exposure curves used by non proportional prop uwrs
- can be approximated using subclass of MBBEFD
- bi, gi evaluated for each curve i
- curve modeled as a fctn of single parameter c
- c{1.5,2.0,3.0,4.0} corresponds to Swiss Re curve Y1-4
- c = 5 corresponds to Lloyd's curve for industrial risks
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