B6. Aggregate excess loss cost estimation

Question 1

Differences between using vertical vs horizontal slices to calculate areas of a Lee diagram

Answer 1

Vertical slices / size method(per risk view):

• Advantage: more intuitive, corresponds to the way data is naturally shown
• Quick way to compute insurance charge for one entry ratio
• Disadvantage: slower for multiple limits/risks since need to repeat each time
• process becomes cumbersome when dealing with thousands of risk in a sample

Horizontal slices (per layer view):

• Advantage: faster for multiple limits/risks, method used in practice
• Disadvantage: less intuitive, requires more data preparation

The horizontal slicing method is more convenient for producing an entire Table M, but it is not as accurate as the vertical slicing method.
In this example, 0.1 is too wide a gap to produce accurate insurance charges. With 0.01 as the gap, the horizontal approach is much closer the the more accurate vertical approach.

Question 2

Why use entry ratios (actual loss at k /expected loss) instead of actual loss amounts directly

Answer 2

ALLOWS YOU TO OBTAIN XS RATIOS FOR A DIFFERENT BOOK OF BUSINESS WITH A SIMILAR SEVERITY DISTRIBUTION SHAPE BUT A DIFFERENT EXPECTED LOSS LEVEL

Question 3

Why a separate Table M is needed for different risk sizes

Answer 3

-when risk sizes differ too much they have different severity distributions since larger risks have more stable results

**variance of aggregate loss distribution will vary by risk size

Question 4

Why using a competitor basic premium may not be appropriate

Answer 4

Competitors may have different:

• expenses => directly impact basic prm
• mix of business => different aggregate distribution curve => different insurance charges

Question 5

How Table M can account for inflation

Answer 5

If there is inflation:

• risks become larger
• their new risk size is mapped to higher groups / columns of Table M
• therefore the Table M values do not necessarily need to be recalculated

Question 6

Why use Adjusted Expected Loss (Expedted loss * HG differential) instead of expected loss to determine which aggregate distribution curve to use

Answer 6

Some policies have the same size but have different risks than others:

• riskiness: some classes have a higher proportion of larger claims
• location: some states have higher medical costs, benefits, court verdicts

Therefore these policies are adjusted to be moved to a different group with a different size but with similar variance in their aggregate distribution curve

Question 7

How an occurrence limit impacts the aggregate distribution curve

Answer 7

Table M is no longer appropriate since it does not recognize the overlap between occurrence limit and aggregate limit:

-per occurrence limit => less variance in severity distribution => less variance in aggregate distribution => insurance charges should be lower

Question 8

2 options to estimate aggregate distribution curves with a per occurrence limit

Answer 8

1. Limited Table M: estimate separately
   - first estimate the per occurrence loss elimination ratio
   - then estimate the aggregate insurance charges using a Table M built with limited losses instead of unlimited losses
   - only reflects charge for aggregate limit
2. Table L: estimate simultaneously
   - estimate the aggregate insurance charges using a Table M built with entry ratios defined as (actual limited)/(expected unlimited) instead of all unlimited losses
   - can estiamte the per occurence xs loss and limtied aggregate excess losses simultaneously.

Question 9

Table L caracteristic

Answer 9

Built using california taxes so not appropriate for use in other state.

less flexible since based on a pre-determined per occurrence limit ( cannot be used for alternate accident limits)

true number since does not rely on any simplifying assumption ( accurate estimation for the insurance charge)

does not account for inflation, so Table L needs to be updated each year

since there is a fixed loss limit there is not a need for a large number of tables to accomodate changing limits

Question 10

Table M Charges and Risk Size

Answer 10

• 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* ( table M charge will approach 1-r if r < 1

Question 11

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

Answer 11

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 12

Disadvantages of continuous approximation for TAble M ( instead of ICRRL)`

Answer 12

1. Lack of data, the distribution estimated can be difficult or far away from reality due to sparse data
2. Need more calculation = time consuming
3. Sensitive to input parameter, the distribution can be changed significatly due to the initial set of parameter and it can be significantly different than the inital expectation
4. Data is thin especially for the highst claim amounts, the cherge for highest entry ratio will have high standard errors

Question 13

validate if table M appropriate for other business

Answer 13

1. same expected loss
2. same aggregate loss distribution / variance
3. subject to same limit structure

Question 14

what does the net insurance charge reflect

Answer 14

reflect the charges to compensate for the possibility that the policy will exceed the maximum premium amount. It also reflect the savings resulting from the possibility that the policy will be less than the minimum premium amount.

Question 15

charges depend on whtat ?

Answer 15

max subject loss
min subject loss
size of employer ( bc expected variation in losses is lower for larger employers)

Question 16

ALF on demand

Answer 16

• relies on more recent data
• **elimitnates some approximationg adjustement
• it increase accuracy and ***equity of insurance charges within each state and across the country
• real time calcualtion instead of obtaining value via lookups within a precomputed table
• by directly calculating the limited aggregate loss distribution, it eliminates the need for the adjustment to account for overlap between the liss limit and the aggregte loss limitation.

• reflect the ** state and HG specific loss distribution **of the policy entered by the user ( improve the accuracy of insurance charges)
• simplifies calculations that carriers need to perform for every retro rated policy
• easier to perform annual updates
• net insurance charge more responsive to to the latest available data

possibility to genereate graph based on policy info entered

• policy specific estimated claim count distribution
• policy specific estimated severity distribution
• projected ultimate distribution of claim by state hg and claim group

Question 17

NCCI ALF table ( not on demand)

Answer 17

• account for inflation in claim size over time automatically by mapping a given policy to the appropriate subtable based on the policy xs ratio
• eliminates updates to HG differential and Expected loss groups range
• more accurate thant the current table ( rely on more recent data)

***However, does not reflect the state and hg differences in severity distribution that are incorporated in ALF on demand bc of its countrywide nature

Question 18

why use policy excess ratio to subdivide table of aggregate loss factors

Answer 18

Aggregate xs loss facotrs vary by loss limit, increasing the los limit ( and thus decreasing the xs ratio) for a policy places upward pressure on aggregate exs loss factors.

Question 19

why use expected claim cout to subdivide table of aggregate loss factors

Answer 19

for a given loss limit increasing the expected number of claims put downward pressure on AELF at each entry ratio. this is because there is less loss ratio variation as risk size increase

Riskier groups will lower expected # of claim counts, so variance of aggregate distribution would be higher

Question 20

benefit to use expected claim count not in $ term

Answer 20

doesn’t require to be updated on a regular basis

Question 21

desirable properties of AELF

Answer 21

1. should be monotonically decreasing with respect to entry ratio
2. the rate of decrease of AELF should be monotonically decreasing with respect to entry ration
3. AELF sould be monotonically increasing with respect to loss limit
4. AELF at a given entry ratio should be monotonically decreasing with respect to risk size
5. The rate of decrease of AELF at a given entry ratio is expected to be monotonically decreasing with respect to risk size

Question 22

single set of AELF is included for use with all policies, no matter if loss limit is claim made or per occuence basis, no matter if loss include or not ALEA .
Why ?

Answer 22

The relative change in AELF based on included/exlcuding ALEA or apply per claim / per occ limit is immaterial. The small relative differences are further offset within the calculation of the net aggregate loss factors.