C.3. Bernegger  Exposure Curves & MBBEFD
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Exposure Curves
 define d = D/M and x = X/M = normalized deductible & loss
 M = maximum possible loss
 if F(X) is defined on the interval [0, ∞] we use an arbitrary reference loss X_{0} in lieu of M
 L(d) = limited expected value function, so that M * L(d) is the expected value of losses retained
 exposure curve = G(d) = L(d) / L(1) = ∫_{0 to d} (1  F(y))dy / E[x]
 F(x) = 1 for x = 1, and F(x) = 1  G’(x)/G’(0) for 0 ≤ x < 1

Total Loss Probability and Expected Value
 probability p for a total loss = 1  F(1) = G’(1)/G’(0)
 expected loss = μ = E[x] = 1/G’(0)
 1 ≥ μ ≥ p ≥ 0

MBBEFD Class of TwoParameter Exposure Curves
 G(x) = [ln(a + bx)  ln(a + 1)] / [ln(a+b)  ln(a + 1)]
 F(x) = 1 for x = 1, and F(x) = 1  (a + 1)bx/(a+b) for 0 ≤ x < 1
 in order for G_{a,b}(x) to be real, increasing and concave on [0,1], we need g ≥ 1, b ≥ 0
 special cases are MB distribution (bg = 1), BE distribution (bg > 1), FD distribution (bg < 1)

Calculation of g and b
 note that we must have μ^{2} ≤ E[x^{2}] ≤ μ and p ≤ E[x^{2}] in order to find the MBBEFD distribution
 start with p^{*} = E[x^{2}] = μ^{2} + σ^{2} ≥ p as a first estimate (upper limit) for p, and calculate b^{*} and g^{*}
 compare the second moment E^{*}[x^{2}] with the given moment E[x^{2}] and find a new estimate for p^{*}
 repeat until E^{*}[x^{2}] is close enough to E[x^{2}]

Exposure Rating
 problem is how to divide the total premiums of each risk size group between ceding and reinsurer
 first step is to estimate to overall risk premium for each risk size group E[x] = ELR * GWP
 then use the exposure curves to allocate losses accordingly

How MBBEFD curves can be used to approximate the 4 Swiss Re exposure curves
 estimate parameters b_{i} and g_{i} for each curve i
 plot the values of b_{i} and g_{i} and noticed they’re on a smooth curve in the plane
 model the MBBEFD curves as a function of a single parameter c_{i}
 4 curves defined by c = {1.5, 2.0, 3.0, 4.0} coincide very well with Swiss Re curves {Y_{1}, Y_{2}, Y_{3}, Y_{4}}
 also the MBBEFD curve with c = 5.0 coincides well with a Lloyd’s curve used for industrial risks