Bernegger

Question 1

Exposure Curves in Bernegger

G(d), G’(d), G’‘(d)

Formula and conditions

Answer 1

X = Loss | M = max possible loss
G(d) = expected losses capped at d% of M / expected unlimited loss
y = X / M
G(d) = int (from 0 to d) [1-F(y)] dy / int (from 0 to 1) [1-F(y)] dy = int (from 0 to d) [1-F(y)] dy / E(y)
G’(d) = [ 1 - F(d) ] / E(y)
G’‘(d) = - f(d) / E(y)

Conditions
1. G(0) = 0, if max possible loss = 0, expected capped loss is also 0
2. G(1) = 1, if max possible loss = unlimited loss, expected loss capped is 100%
3. G’(d) ≥ 0 [0,1] , when max possible loss increases, the loss capped also increases
4. G’‘(d) ≤ 0 [0,1]

Question 2

Exposure Curves in Bernegger: Deriving the CDF, Total Loss Prob and Mean

E(y), F(d), prob total loss (p)

If needed: start with G’(d) Formula

Answer 2

G’(d) = [ 1 - F(d) ] / E(y)
CDF = F(d) = 1 - G’(d) * E(y)
F(0) = 0
G’(0) = 1 / E(y) = 1 / μ ≥ 1
E(y) = 1 / G’(0)
CDF = F(d) = 1 - G’(d) / G’(0)
p = prob total loss = 1 - F(1-) = G’(1) / G’(0)

p ≤ μ ≤ 1

Question 3

Different Cases of the MBBEFD

Gb,g(x)

Easy ones

Answer 3

Gb,g(x) = x if g = 1 OR b = 0

= (1 - b^x) / (1- b) if bg = 1 and g > 1

Question 4

Different Cases of the MBBEFD

Gb,g(x)

Hard ones - starts with ln

Answer 4

Gb,g(x) = ln[1+(g-1)x] / ln(g) if b = 1 g > 1

= ln[ [ (g-1)b + (1-gb)b^x ] / (1-b) ] / ln(gb) if b > 0 and b != 1 and bg != 1 and g > 1

Question 5

Swiss Re Curves, Gb,g

b(c), g(c), c

Formulas

Answer 5

b(c) = exp[ 3.1 - 0.15(1+c)c ]
g(c) = exp[ (0.78 +0.12c)c ]
c = {1.5, 2, 3, 4} with Swiss Re curves {Y1, Y2, Y3, Y4}
c = 5 with Lloyd’s curve