B.6. Aggregate excess loss cost estimation

Question 1

Advantages of using vertical or horizontal slices to calculate areas

Answer 1

The vertical slice approach is more intuitive, but the horizontal slice approach is faster if we need to calculate expected XS losses for multiple limits.

Question 2

Definition and purpose of entry ratios

Answer 2

Entry ratios are ratios of actual to expected loss.

Allows us to apply aggregate distributions to risks of different sizes.

Question 3

Definition of Table M charge and savings

Answer 3

Question 4

Steps to build Table M

Answer 4

Question 5

Why a separate Table M charge is needed for different risk size groups

Answer 5

Because aggregate loss distributions vary by risk size, with larger risks having more stable results.

Question 6

How Table M can account for inflation

Answer 6

After risks become larger due to inflation, they can be mapped to different existing Table M columns that are more appropriate for their new risk size. The table M values themselves don’t necessarily need to be recalculated.

Question 7

Table M graph

Answer 7

Question 8

Value of Table M and L charges and savings as entry ratios go to 0 and infinity

Answer 8

Question 9

1st and 2nd derivatives of Table M charge and savings

Answer 9

Question 10

Graph and formula for Table M charge for infinite risk size

Answer 10

Question 11

Graph of Table M charge by risk size and entry ratio

Answer 11

Question 12

How an overestimation of expected losse impacts insurange charge

Answer 12

1. We might incorrectly use the Talbe M curve for a larger risk size, which would result in a lower charge for a fixed entry ratio.
2. We might incorrectly use too low an entry ratio (same aggregate limit / overestimated expected loss), which would result in a higher charge for a fixed Table M curve.

Fisher’s example assumes the Table M curbe doesn’t change, in which case only the 2nd item above happens.

Question 13

In what cases are percentage and dollar error in insurance charges greatest?

Answer 13

1. The percentage error in insurance charges is greatest for large policies with high entry ratios.
2. The dollar error in insurance charges is the greatest for large policies with low entry ratios.

Question 14

Graph of Table M savings by risk size and entry ratio

Answer 14

Question 15

Formula and purpose of Adjusted Expected Loss for which curve to use

Answer 15

Some policies of even the same size are riskier than other policies, so we can adjust the expected losses for a risk to match those of a risk with a different size but similar variance in their aggregate loss distribution (the same distrubtion shape).

Adjusted expected loss = E[A] x State HG differential

The state HG differentiel adjusts for :

• RIskiness: Some classes have a higher proportion of larger claims
• Location: Differences in medical care costs, benefits, court verdicts, etc.

Question 16

Impact of NCCI move to Adjusted expected claim count

Answer 16

This will mean the NCCI will no longer need to adjust the mapping between expected losses and risk size group (Table M column) due to inflation.

The NCCI will continue to apply a state / HG factor to the claim counts, such that riskier states / HG’s will lower the risk’s expected claim counts, which means the variance of the aggregate loss distribution will be higher. This new approach should work well if severity distributions differ in size but not in shape, otherwise we would also need to adjust for differences in the severity distribution. This need to adjust for severity distributions is more likely to exist for General Liability (products liability in particular) and excess of loss reinsurance.

Question 17

General formula for retro premium without per occurence limit but with max and min ratable loss

Answer 17

Question 18

Graph of expected ratable loss without occurence limit in retro rating

Answer 18

Question 19

Relationship between expected ratable loss and expected ground-up loss with no occurence limit

Answer 19

Question 20

Formula for Guaranteed cost premium in retro rating

Answer 20

Question 21

Formula for basic premium using Table M (no occurence limit)

Answer 21

Question 22

Table M balance equations

Answer 22

Question 23

How an occurence limit impacts an aggregate distribution

Answer 23

In general, the smaller the occurence limit, the less variance in the severity distribution, and as a result, the limited aggregate loss distribution will have less variance as well.

Question 24

Two options for how to estimate limited aggregate losses

Answer 24

1. We can estimate the per-occurence excess losses seperately from the limited aggregate excess losses. We can obtain the expected limited aggregate excess losses using a Limited Table M, which is identical to a Table M but built using limited loss data instead of unlimited loss data.
2. We can estimate the per-occurence excess losses and limited aggregate excess losses simultaneously. We can obtain these amounts using a Table L, where the Table L charge will include a charge for both the per-occurence expected losses as well as the limited aggregate excess losses.

Question 25

Formulas to create a Limited Table M and for Limited Table M savings

Answer 25

Question 26

Limited Table M charge and savings formulas

Answer 26

Question 27

Retro premium formula using Limited Table M

Answer 27

Question 28

Formulas for net insurance charge and expected ratable loss for Limited Table M

Answer 28

Question 29

Basic premium formula including occurence charge for a balanced plan using Limited Table M

Answer 29

Question 30

Limited Table M balance equations

Answer 30

Question 31

Purpose of and thinking behind ICRLL approximation

Answer 31

Purpose is to approximate a Limited Table M using a regular Table M by adjusting the column used to be for a larger risk size. The thinking behind the approximation is that an occurence limit reduces the variance of the limited aggregate loss distribution compared to the unlimited aggregate loss distribution, which is the same thing that occurs as we move to larger risk sizes.

Question 32

Formula for adjusted expected loss using ICRLL

Answer 32

Question 33

Table L charge and savings in words

Answer 33

The charge represents the average difference between a risk’s actual unlimited loss and actual limited loss plus the risk’s expected percent of limited losses excess of r E[A].

The savings represent the average amount by which the risk’s actual limited loss falls short of r E[A].

Question 34

Formulas to create a Table L and for Table L savings

Answer 34

Question 35

Graph of Table L charge and savings

Answer 35

Question 36

Integral formulas for Table L charge and savings

Answer 36

Question 37

Formula to relate Table L and M charge and savings

Answer 37

Question 38

Graph and formula for Table L charge for infinite risk size

Answer 38

Question 39

Table L charge by risk size and entry ratio graph

Answer 39

Question 40

Retro premium formula using a Table L

Answer 40

Question 41

Graph of expected ratable loss using Table L

Answer 41

Question 42

Formulas for net insurance charge and expected ratable loss using a Table L

Answer 42

Question 43

Basic premium formula for a balanced plan using a Table L

Answer 43

Question 44

Table L balance equations

Answer 44