Section 20, Bernegger, Exposure Curves they sum to unity.R(x) = ? EX - EX x EX= 1 - ? EX x EX= 1 - LER(x).The percentage of total losses in the layer from d to u is: LER(u) - LER(d) = R(d) - R(u).For a distribution with support starting at zero, the Limited Expected Value can be written as an integral of the Survival Function from 0 to the limit:EXx = ? S(t) dt 0x. 837 “Swiss Re Exposure Curves and the MBBEFD Distribution Class,” by Stefan Bernegger, ASTIN Bulletin, Vol. 27, No. 1, May 1997, pp. 99-111. CAS Learning Objectives C3 and C5.838 Added to the syllabus for the 2011 exam.839 As discussed in “Basics of Reinsurance Pricing” by David R. Clark, “Property per-risk excess treaties provide a limit of coverage in excess of the ceding companys retention. The layer applies on a “per risk” basis, which typically refers to a single property location. This is more narrow than a “per occurrence” property excess treaty which applies to multiple risks to provide catastrophe protection.” 840 See pages 16 to 18 of “Basics of Reinsurance Pricing,” by David R. Clark.841 Bernegger uses the notation L(d) for the limited expected value at d. Other syllabus readings use EX ; d.2014-CAS8 20 Bernegger, Exposure Curves HCM 5/13/14, Page 1197LER(x) = ? EX x EX= ? S(t) dt 0x EX= ? S(t) dt 0x S(t) dt 0 .Thus, for a distribution with support starting at zero, the Loss Elimination Ratio is the integral from zero to the limit of S(x) divided by the mean.Since R(x) = 1 - LER(x) = (EX - EXx) / EX, the Excess Ratio can be written as:R(x) = ? S(t) dt x EX= ? S(t) dt x S(t) dt 0 .So the excess ratio is the integral of the survival from the limit to infinity, divided by the mean.For example, for the Pareto Distribution, S(x) = (+x). So that: R(x) = ? (+ x)1- / (-1) / ( -1)= /(+x)1.LER(x) = ? S(t) dt 0x EX. ? d LER(x) dx= ? S(x) EX.Since ? S(x) EX 0, the loss elimination ratio is a increasing function of x.842 Also, if there is no point mass of probability for a loss of size zero, then S(0) = 1, and the derivative at zero of the LER is: LER(0) = 1/EX. For a distribution with support starting at zero:? d LER(x) dx= ? S(x) EX. ? d LER(0) dx= ? 1 EX. S(x) = ? d LER(x) dx/ ? d LER(0) dx.842 If S(x) = 0, in other words there is no possibility of a loss of size greater than x, then the loss elimination is a constant 1, and therefore, more precisely the loss elimination is nondecreasing.2014-CAS8 20 Bernegger, Exposure Curves HCM 5/13/14, Page 1198Therefore, the loss elimination ratios (or the excess ratios) determine the distribution function, as well as vice-versa.843 844 ? d LER(x) dx= ? S(x) EX. ? d2 LER(x) dx2= -? f(x) EX. Since ? f(x) EX 0, ? d2 LER(x) dx2 0; the loss elimination ratio is a concave downwards function of x.The loss elimination ratio as a function of x is increasing, concave downwards, and approaches one as x approaches infinity.For example, here is a graph of the loss elimination ratio for a Pareto Distribution with parameters = 3 and = 100,000:845 20040060080010000.20.40.60.81.0LERSize ($000) Since the loss elimination ratio is increasing and concave downwards, the excess ratio is decreasing and concave upwards (convex).843 “A distribution is characterized by its excess ratios and so there is no loss of information in working with excess ratios rather than with the size of loss density or distribution function.” Page 196 of “NCCIs 2007 Hazard Group Mapping” by John Robertson.844 Note that depending on the application, this could be either a distribution of size of loss or aggregate loss.845 As x approaches infinity, the loss elimination ratio approaches one. In this case it approaches the limit slowly. 2014-CAS8 20 Bernegger, Exposure Curves HCM 5/13/14, Page 1199Normalizing:Bernegger normalizes everything with respect to the Maximum Possible Loss, M.846 If X is the loss in dollars, and M is the Maximum Possible Loss, then the normalized loss is: x = X / M. 0 x 1. If D is the retention in dollars, then the normalized retention is: d = D / M. x has a distribution with support on 0 to 1, with a point mass of probability at 1, corresponding to a total loss or the Maximum Possible Loss (MPL). Note that since x is restricted to the interval 0 to 1, its limited expected value at one equals the mean.Exposure Curves:847 The exposure curve, G(x), is just the loss elimination ratio at x.G(d) = EX d / EX 1 = L(d) / L(1), 0 d 1.848 EX 1 = EX. G(d) = EX d / EX = ? S(t) dt0d / EX.G(d) = S(d) / EX. G(d) 0.G(0) = S(0) / EX . EX = 1/G(0).G(1) = S(1) / EX . S(1) = G(1)/G(0). The probability of having the Maximum Possible loss is: G(1) / G(0).G(d) = -f(d) / EX. G(d) 0.G(d) is an increasing and concave function on the interval 0, 1. In other words, G(d) 0 and G(d) 0. In addition, G(0) = 0 and G(1) = 1 by definition.849 846 Berneg