C.3. Bernegger - Exposure Curves & MBBEFD

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EExam8
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307132
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C.3. Bernegger - Exposure Curves & MBBEFD
Updated:
2015-09-28 19:06:48
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Bernegger Exposure Curves Swiss Re MBBEFD Distribution
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Bernegger: Swiss Re Exposure Curves and the MBBEFD Distribution Class
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  1. 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 X0 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
  2. 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
  3. MBBEFD Class of Two-Parameter 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 Ga,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)
  4. Calculation of g and b
    • note that we must have μ2 ≤ E[x2] ≤ μ and p ≤ E[x2] in order to find the MBBEFD distribution
    • start with p* = E[x2] = μ2 + σ2 ≥ p as a first estimate (upper limit) for p, and calculate b* and g*
    • compare the second moment E*[x2] with the given moment E[x2] and find a new estimate for p*
    • repeat until E*[x2] is close enough to E[x2]
  5. 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
  6. How MBBEFD curves can be used to approximate the 4 Swiss Re exposure curves
    • estimate parameters bi and gi for each curve i
    • plot the values of bi and gi and noticed they’re on a smooth curve in the plane
    • model the MBBEFD curves as a function of a single parameter ci
    • 4 curves defined by c = {1.5, 2.0, 3.0, 4.0} coincide very well with Swiss Re curves {Y1, Y2, Y3, Y4}
    • also the MBBEFD curve with c = 5.0 coincides well with a Lloyd’s curve used for industrial risks

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