Fisher - Retro

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

calculations so far have assumed that risk we are pricing is exactly the same size as risks underlying the distribution

-if new risk is slightly bigger or smaller,

Answer 2

• want to make an adjustment for this otherwise expected losses will be off
• to deal with this, re-scale vertical axis by dividing actual losses by expected loss
• new values on y axis are entry ratios, r, which are just ratios of actual loss to expected loss
• area under curve is 1

Question 3

table M charge & table M savings

Answer 3

expected % of loss in excess of max ratio

expected % of loss short fall min ratio

Question 4

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

Answer 4

variance of aggregate loss distribution will vary by risk size

Question 5

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

Answer 5

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 6

Table M Charges and Risk Size

Answer 6

• 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 7

to summarize about errors in insurance charges

Answer 7

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 8

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

Answer 8

φ(r) = max(1-r,0)

ψ(r) = max(r-1,0)

Question 9

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

Answer 9

φ(r) -> 1

ψ(r) ->0

Question 10

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

Answer 10

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 11

basic premium formula derived contains net insurance charge I

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

Answer 11

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 12

when per occurrence limit is applicable,

Answer 12

table M distributions will no longer be appropriate as they don’t recognize overlap between occurrence limit and aggregate limit

Question 13

occurrence limit reduces

Answer 13

the variance of aggregate loss distribution; this is because variance of underlying severity distribution is reduced by occurrence limit

-in general, the smaller the limit, the less variance in severity distribution, so limited aggregate distribution will have less variance as well

Question 14

have 2 options for how to deal with estimating excess losses with per occurrence limit

Answer 14

1. can estimate per occurrence excess loss separately from limited aggregate excess loss; obtain expected limited aggregate excess loss using a limited table M which is identical to table M built using limited loss data instead of unlimited loss data
2. can estimate per occurrence excess losses and limited aggregate excess losses simultaneously; can obtain these amounts using a table L where table L charge will include charge for both per occurrence expected excess losses as well as limited aggregate excess losses

Question 15

limited table M

Answer 15

• is the same as creating a regular table M but you use limited losses for each policy instead of unlimited losses and a separate table is built for each occurrence limit
• diagram looks the same except now vertical axis represents limited losses or limited entry ratios
• all same methods for calculating the areas can be used and properties of tables are basically identical

Question 16

since presence of occurrence limit would normally require entirely new tables (varying by occurrence limit) compared to table M, approximation can be used to simulate limited table M by

Answer 16

adjusting column used from a regular table M

can use insurance charge reflecting loss limitation procedure

Question 17

insurance charge reflecting loss limitation procedure

Answer 17

• use procedure to change table M column used for risk with occurrence limit to be column that normally be used by larger risk in absence of occurrence limit
• to perform, add an extra term called loss group adjustment factor into calculation of adjusted expected loss for risk
• if you use procedure, you are still simulating a limited table M, you will use limited table M balance equations instead of regular table M balance equations to perform your table M search

Question 18

thinking behind insurance charge reflecting loss limitation procedure

Answer 18

occurrence limit reduces the variance of limited aggregate loss distribution compared to unlimited aggregate loss distribution which is the same thing that occurs as we move to larger risk sizes

Question 19

table L charge includes

Answer 19

charge for per occurrence limit in addition to charge for aggregate limit

so creating is slightly different from table M and limite table M

Question 20

separate table L would be built for

Answer 20

each occurrence limit

Question 21

table L entry ratios, charge, and savings

Answer 21

will be actual limited losses divided by expected unlimited losses

table L charge: average difference between risk’s actual unlimited loss and actual limited loss + risk’s expected % of limited losses excess of rE[A]

table L savings: average amount by which risk’s actual limited loss falls short of rE[A]

Question 22

Table L Charges and Risk Size

Answer 22

• as risk size goes to infinity, variance in entry ratio goes to 0 and curves will flatter
• graph is similar to Table M graph but asymptote is different

Question 23

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

Answer 23

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

Question 24

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

Answer 24

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

Question 25

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

Answer 25

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

Question 26

Briefly describe a procedure to construct a Table M.

Answer 26

First graph the aggregate loss distribution and draw horizontal lines at each entry ratio (equal to the actual losses divided by the expected losses). The Table M is constructed by summing the horizontal strips downward.

The Table M charge at a given entry ratio is the area of all colored horizontal strips above that entry ratio divided by the total area under the curve (the sum of all horizontal strips).

Question 27

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

Answer 27

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 28

why an iterative procedure is used for retro prem

Answer 28

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 29

table L differs from table M

Answer 29

Table L incorporates a per accident limit and accounts for the overlap between the charge for accident limit and the charge for the aggregate limit, whereas Table M only reflects the charge for the aggregate limit.

Question 30

how NCCI’s Insurance Charge Reflecting Loss Limitation procedure reflects the
selection of a loss limitation on a retrospectively rated policy.

Answer 30

NCCI uses the ICRLL procedure to shift the Table M column used to that of a larger risk as an approximation to using a Limited Table M.

This is done by adjusting the expected losses of the risk used to lookup the expected loss group (Table M column) by an adjustment factor

Question 31

entry ratio curve - Table M

Answer 31

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 32

entry ratio curve - Table L

Answer 32

y=A_D/E[A];

area under limited loss curve = 1-k

k=excess ratio due to per occurrence limit=1-E[A_D]/E[A]

φ_D*(r) = area between horizontal line r and limited loss + k

φ_D*(0) = 1, φ_D*(inf) = k

ψ_D*(r) = area between limited loss and horizontal line r

ψ_D*(0) = 0, ψ_D*(inf) = inf

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

r=ψ_D*(r) + 1 - φ_D*(r)

Question 33

entry ratio curve - Table M with min and max

Answer 33

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 34

final premium for a retrospectively-rated workers compensation policy compared to a large deductible workers compensation policy.

Answer 34

final premium for a retrospectively-rated workers compensation policy is typically greater than for a large deductible workers compensation policy.

Retrospectively rated policies provide coverage for ground-up losses, even if there are limits on the amount of actual losses used in premium calculation.

LDD policies provide coverage in excess of the deductible(s), so the premium will be lower, since much less coverage is provided.

Question 35

aggregate deductible relationship with table M charge

Answer 35

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 36

Advantages of using vertical slices/disadvantages of horizonal slices

Answer 36

• 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 37

Advantages of using horizontal slices/disadvantages of vertical slices:

Answer 37

• 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 38

If the aggregate loss limits are set too low (based on incorrect expected losses), then

Answer 38

then there is a higher probability of losses exceeding these limits, and there should be an additional charge for this.

The loss component of the policies will be priced as expected losses in excess of the per occurrence limit + expected limited losses × limited table M insurance charge. If the expected losses are too low, then both expected excess losses and expected limited losses will be too low, and the policies will be under-priced.

Question 39

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

Answer 39

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.

Question 40

how the impact of the estimation error (expected losses 20% too low) on per occurrence excess loss adequacy varies between smaller and larger sized risks.

Answer 40

The per occurrence charges would be 20% too low for all risk sizes if based on expected ground-up losses that were 20% too low (and we assume the XS ratios don’t change). In dollar terms this will be larger for larger risks.

Question 41

overlap in a retrospective rating plan with a maximum
premium and per-accident limit.

Answer 41

The overlap reflects that the presence of a per-accident limit makes it less likely that the aggregate amount of losses that correspond with the maximum premium will be hit with limited losses.

Question 42

how the NCCI Retrospective Rating Plan accounts for the maximum premium,
per-accident limit, and overlap.

Answer 42

NCCI uses the ICRLL procedure to shift the expected loss group to a bigger size to
approximate the effect of a per-accident limit on aggregate losses.

i. Obtain the excess ratio for the occurrence limit from NCCI tables.
ii. Calculate the Adjusted Expected Loss as the unlimited expected losses times the
state/hazard group adjustment factor times the loss group adjustment factor.
iii. Lookup the Expected Loss Group based on the Adjusted Expected Loss.
iv. Lookup the Table M charge for the Expected Loss Group, with the entry ratio being the aggregate loss limit divided by expected limited losses. This will be the aggregate charge for limited losses, and will approximate the charge derived from a Limited Table M.

Question 43

how a Table L accounts for the maximum premium, per-accident limit, and overlap.

Answer 43

Table L is based on limited loss data and varies by per-accident limit, so there will be no overlap when looking at aggregate charges. The Table L charge includes the charge for both the per-accident limit and the maximum premium (aggregate limit).