Bernegger Flashcards
Define exposure curve as per Bernegger
Based on Maximum Possible Loss instead of IV.
G(d) = expected losses capped at % d of MPL divided by expected unlimited losses = E(X;dM)/E(X)
Bernegger normalizes curve by letting Y = X/M (scale from 0 to 1):
G(d) = E(Y;d)/E(Y)
List 4 conditions for a valid normalized exposure curve
- G(0)=0
- G(1)=1
- G’(d) positive (increasing)
- G’‘(d) negative (concave)
State the relationship between F(d) and G(d)
G’(d) = (1-F(d))/E(Y)
F(d) = 1 - G’(d)/G’(0)
State the relationship between u and G(d)
u = 1/G’(0)
State the relationship between p and G(d)
p = G’(1)/G’(0)
Define the MBBEFD exposure curves
Can be used to approximate Swiss Re exposure curves.
G(x) = ln(a+b^x)-ln(a+1) / ln(a+b)-ln(a+1)
G’(x) = lnbb^x / (a+b^x)(ln(a+b)-ln(a+1))
F(x) = 1 - (a+1)b^x / (a+b^x)
p = (a+1)b/(a+b)
State the 3 conditions for MBBEFD curve to be valid
- p between 0 and 1
- b positive
- g at least 1
State the relationship between g and p
g = 1/p = (a+b) / (a+1)b
Briefly explain the problem in exposure rating
How to divide tot premiums for each risk size group between ceding company and reinsurer.
This is solved in 2 steps:
1. Estimate overall risk premium (expected loss) for each risk size group by applying ELR to gross premiums.
2. Split risk premiums into retained and ceded portions with distribution functions.
Calculate Expected Ceded Risk P
Expected policy risk premium = Gross P * ELR
Exposure factor = G(ret+lim / MPL) - G(ret/MPL)
Expected Ceded Risk P = E(risk premium)*Exposure Factor
Loss Cost = ELR*Exposure factor
Describe how to fit MBBEFD curve to existing data
If given p and u:
g = 1/p
u = ln(gb)(1-b) / lnb(1-gb) solve for b
If given u and sigma:
1. Start with p* = E(y^2)+u^2+s^2
Will be an upper bound for p (so first g* is a lower bound)
2. Use method when p and u are known to get g* and b*
3. Use these to recalculate E(Y^2) and update p* accordingly
if E(Y^2) < E(Y^2), increase P
4. Repeat steps 2 and 3 until E*(Y^2) is sufficiently close to E(Y^2)
Discuss how MBBFED curves can be used to approximate Swiss Re curves
- Estimated gi and bi for each curve I
- Plotted values of bi and gi, noticed they were on a smooth curve in plane
- Modeled MBBEFD curves as function of c
b(c) = exp(3.1-0.15(1+c)c)
g(c) = exp((0.78+0.12c)*c) - The 4 MBBEFD curves defined by c = (1.5, 2, 3, 4) coincide with Swiss Re curves (Y1, Y2, Y3, Y4).
MBBEFD curve with c=5 coincides with Lloyd’s curve used for rating industrial risks.
When c=0, we have the curve for total losses
Explain whether accident date underlying curve matters
Yes since severity distribution changes over time due to inflation.
Using a curve based on different aggregate average accident date could result in using inappropriate distribution.
Explain whether curves should be considered fully credible
Exposure curves are usually created based on data from many risks within a company or across industry.
They are often created because individual risk experience is not credible enough to use on its own.
Exposure curves should be used considering they may be based on limited data or data less relevant to risk being rated.
Describe 2 ways insurer can stabilize its results for sliding scale commission structure
- Introduce carry forward provision
- Calculate LR on result for group of years instead of single year.
- Profit sharing provision would incentivize insurer to manage losses before potential return in premiums.
- Reduce range of possible commission so they are more certain
- Reduce range of LR leading to commissions so comm are more stable
- Decrease sliding scale sensitivity to losses
