Fisher Retro2

Question 1

aggregate loss distribution:

area under curve, area above max ratable loss, area below min ratable loss

Answer 1

• total area under curve is equal to expected losses per policy
• area above max ratable limit asymptote represents expected aggregate losses in excess of max ratable limit
• area below min ratable limit asymptote represents expected shortfall of aggregate losses below min ratable limit

Question 2

table M charge & table M savings

Answer 2

expected % of loss in excess of max ratio

expected % of loss short fall min ratio

Question 3

entry ratio curve - Table M with min and max

Answer 3

charge @ rG = D

charge @ rH = C+D

expected retro prem = A+B+C

in balanced plan, E[R]=GCP=(e+E[A])T

minimum prem = A+B

Question 4

entry ratio curve - Table M

Answer 4

y=A/E[A]

the area under the unlimited loss curve must equal 1

φ (r) = area between horizontal line r and unlimited loss

φ (0) = 1, φ (inf) = 0

ψ (r) = area between unlimited loss and horizontal line r

ψ (0) = 0, ψ (inf) = inf

area from (0,0) to (1,r) = r

r=ψ(r) + 1 - φ(r)

Question 5

in practice, separate tableM will be built for different risk size groups since

Answer 5

variance of aggregate loss distribution will vary by risk size

Question 6

with tables specified in terms of entry ratios and charges, tables are less vulnerable to

Answer 6

inflation

-as risk increases in size due to inflation, it can simply be mapped to different existing table M charge column this is more appropriate for its new risk size

Question 7

Table M Charges and Risk Size

Answer 7

• for smaller risk sizes, majority or risks have no claims at all but small number or risks can have 1 or 2 large claims
• for very large risks, all risks will have claims and experience across all risks becomes more similar as there is less variance in loss experience between risks
• as risk size goes to infinity, variance in entry ratios goes to 0 and curve will flatten to look like all risks have exact same amount of losses

Question 8

to summarize about errors in insurance charges

Answer 8

1. % error in insurance charges is greatest for large policies with high entry ratios
2. $ error in insurance charge is greatest for large policies with low entry ratios

Question 9

asymptote approached by very large risks for table M charge and savings

Answer 9

φ(r) = max(1-r,0)

ψ(r) = max(r-1,0)

Question 10

as risk size goes to 0 for table M charge and savings

Answer 10

φ(r) -> 1

ψ(r) ->0

Question 11

• some policies of even same size are riskier than other policies and should account for this
• to do this,

Answer 11

adjust expected losses for risk to match those of a risk with different size but similar variance in aggregate loss distribution

example of this, historically NCCI has adjusted for riskiness differences between states and hazard groups

Question 12

basic premium formula derived contains net insurance charge I

when max and min premiums are explicitly selected, net insurance charge depends on

Answer 12

max and min premium selected

• max and min premiums also depend on basic premium
• so trial and error procedure called table M search is needed to determine correct Table M rows for rating a policy

Question 13

If the minimum premium is increased but the maximum premium and the loss conversion factor remain the same, then the basic premium

Answer 13

will decrease because the insurance savings will increase (which is
subtracted as part of the net insurance charge).

Question 14

If the maximum premium is decreased but the minimum premium is increased by an equal amount, then the basic premium will

Answer 14

the impacts depend on the shape of the aggregate distribution underlying the Table M.

Question 15

If the basic premium and loss conversion factor remain the same but the minimum premium increases, then the maximum premium will

Answer 15

decrease -> this has to be true for the basic premium to remain constant.

Question 16

the difference between the guaranteed cost premium and the retro premium will be driven by

Answer 16

the net insurance charge I, which will depend on the selected maximum and
minimum premium.

R-GCP = cIT

If the max and min are selected to result in a zero net charge, then the retro premium will equal the guaranteed cost premium. If the max and min are selected such that there is a positive net charge, the retro premium will be higher than the guaranteed cost premium.

Question 17

why an iterative procedure is used for retro prem

Answer 17

The iterative procedure is needed because the insured selects H (minimum premium) and G (maximum premium), instead of rG and rH. Since H and G depend on B, which in turn, depends on H and G, an iterative procedure is needed.

Question 18

Advantages of using vertical slices/disadvantages of horizonal slices

Answer 18

• More natural since it corresponds to the way the data is presented.
• Easier to understand.
• Quicker if just need φ(r) for 1 entry ratio.

Question 19

Advantages of using horizontal slices/disadvantages of vertical slices:

Answer 19

• More efficient when calculating φ(r) for multiple values of r.
• Less time consuming when there are many risks, since vertical slices require dealing with each risk individually.

Question 20

aggregate deductible relationship with table M charge

Answer 20

The higher the aggregate deductible, the higher the entry ratio. The higher the aggregate deductible, the lower the probability of losses exceeding that aggregate deductible, which means a lower Table M charge.

Question 21

how the impact of the estimation error on aggregate excess loss adequacy varies between smaller and larger sized risks.

Answer 21

The percentage error and dollar error will be greater for the large policies.

For large risks, the curves are flatter, so a change in entry ratio will cause the percentage error in charges to be larger compared to small risks with steeper curves.