B.6. Aggregate excess loss cost estimation Flashcards
Advantages of using vertical or horizontal slices to calculate areas
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.
Definition and purpose of entry ratios
Entry ratios are ratios of actual to expected loss.
Allows us to apply aggregate distributions to risks of different sizes.
Definition of Table M charge and savings
Steps to build Table M
Why a separate Table M charge is needed for different risk size groups
Because aggregate loss distributions vary by risk size, with larger risks having more stable results.
How Table M can account for inflation
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.
Table M graph
Value of Table M and L charges and savings as entry ratios go to 0 and infinity
1st and 2nd derivatives of Table M charge and savings
Graph and formula for Table M charge for infinite risk size
Graph of Table M charge by risk size and entry ratio
How an overestimation of expected losse impacts insurange charge
- 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.
- 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.
In what cases are percentage and dollar error in insurance charges greatest?
- The percentage error in insurance charges is greatest for large policies with high entry ratios.
- The dollar error in insurance charges is the greatest for large policies with low entry ratios.
Graph of Table M savings by risk size and entry ratio
Formula and purpose of Adjusted Expected Loss for which curve to use
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.
Impact of NCCI move to Adjusted expected claim count
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.
General formula for retro premium without per occurence limit but with max and min ratable loss
Graph of expected ratable loss without occurence limit in retro rating
Relationship between expected ratable loss and expected ground-up loss with no occurence limit
Formula for Guaranteed cost premium in retro rating
Formula for basic premium using Table M (no occurence limit)
Table M balance equations
How an occurence limit impacts an aggregate distribution
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.
Two options for how to estimate limited aggregate losses
- 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.
- 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.
Formulas to create a Limited Table M and for Limited Table M savings
Limited Table M charge and savings formulas
Retro premium formula using Limited Table M
Formulas for net insurance charge and expected ratable loss for Limited Table M
Basic premium formula including occurence charge for a balanced plan using Limited Table M
Limited Table M balance equations
Purpose of and thinking behind ICRLL approximation
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.
Formula for adjusted expected loss using ICRLL
Table L charge and savings in words
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].
Formulas to create a Table L and for Table L savings
Graph of Table L charge and savings
Integral formulas for Table L charge and savings
Formula to relate Table L and M charge and savings
Graph and formula for Table L charge for infinite risk size
Table L charge by risk size and entry ratio graph
Retro premium formula using a Table L
Graph of expected ratable loss using Table L
Formulas for net insurance charge and expected ratable loss using a Table L
Basic premium formula for a balanced plan using a Table L
Table L balance equations
