Clark

Question 1

What are the main objectives for a statistical loss reserving model?

Answer 1

• Have a tool to describe loss emergence mathematically that can aid in selecting carried reserves.
  → Expected loss emergence
• Have a model that estimates a range around the expected reserve.
  → Distribution of loss emergence around the expectation

Question 2

What are two underlying causes for reserve variablity?

Answer 2

• Process Variance
  → Uncertainty due to randomness
• Parameter Variance
  → Uncertainty in the estimate of expected value, aka estimation error

Question 3

Model for the expected loss emergence pattern

Answer 3

Question 4

Loglogistic G(x) formula

Answer 4

Question 5

Weibull G(x) formula

Answer 5

Question 6

Comparison of Weibull and Loglogistic Curves

Answer 6

• Weibull generally results in a smaller tail factor (“thinner” tail)
• With the Loglogistic curve, there is more extrapolation in the tail
  → Might consider using a truncation point

Question 7

When will a Weibull or Loglogistic claims emergence model NOT work?

Answer 7

• When there is real, expected negative development (e.g. salvage and subrogation)
  o This model still works if some data points show negative development

Question 8

Advantages to using parameterized curves to describe expected loss
emergence patterns

Answer 8

• Estimating unpaid losses is simplified (only need 2 parameters)
• Can also use data that’s not from a triangle with evenly spaced evaluation dates
• Payout pattern, G(x), is a smooth curve and doesn’t overfit like age-to-age factors might

Question 9

What are the underlying assumptions of the two Clark methods for
estimating ultimate losses?

Answer 9

LDF Method:
* Assumes the ultimate loss in each accident year is independent of losses in other accident years (this is like the chain ladder method)

Cape Cod Method:
* Assumes a constant expected loss ratio across all accident years

Question 10

Expected incremental loss emergence for the LDF method

Answer 10

Question 11

Truncated LDF formula

Answer 11

Question 12

Expected incremental loss emergence for the Cape Cod method

Answer 12

Question 13

Which expected loss emergence method is preferred according to Clark and
why?

Answer 13

The Cape Cod method is preferred.

When using a development triangle, data is summarized into relatively few
data points for a model.
-> This results in the problem of over-parameterization (overfitting) with the LDF method, which has n + 2 parameters to fit. The Cape Cod method only has 3 parameters.
-> Cape Cod method uses more information (premium exposure)
=> CC method has a smaller parameter variance (due to add’l info + fewer parameters)

• CC process variance can be higher or lower than the LDF method
• in general, CC produces a lower total variance than LDF method

Question 14

How does the Cape Cod method take advantage of more information?

Answer 14

• Cape Cod uses exposure base
  o This may lead to somewhat higher process variance, but usually results in much smaller estimation error (i.e. parameter variance)

Key Point:
Additional information reduces the variance in the reserve analysis, which also produces a better reserve estimate.

Question 15

Variance/Mean Ratio

Answer 15

Question 16

MLE for estimating best-fit parameters

Answer 16

Question 17

Advantage of the maximum loglikelihood function

Answer 17

it works in the presence of negative or zero incremental losses
(since we never actually take the log of c_i,t)

Question 18

Coefficient of Variation for Reserves

Answer 18

Question 19

Variance of Reserves

Answer 19

Question 20

Key Assumptions of LDF Curve Fitting and why they may not hold in
practice (1)

Answer 20

1 - Incremental losses are IID

• Independent → One period doesn’t affect surrounding periods, BUT:
  o There may be positive correlation due to inflation on all periods
  o Neg. correlation if a large settlement replaces later payment streams
• Identically Distributed → Emergence patterns are the same for all accident years, BUT:
  o Different risks and business mix would have been written in different accident years with different claims handling processes

Question 21

Key Assumptions of LDF Curve Fitting (2)

Answer 21

2 – Variance/Mean scale parameter σ^2 is fixed and known
* Impact → The model ignores the variance on the variance

3 – Variance estimates are based on an approximation to the Rao-Cramer lower bound

Question 22

Impact of the Key Assumptions about the LDF Curve Fitting Model

Answer 22

Future loss emergence potentially may have more variability than what the
model produces.

Question 23

Normalized Residual formula

Answer 23

Question 24

How to test if a fixed σ^2 is an appropriate assumption

Answer 24

Question 25

Exposure bases for the Cape Cod method

Answer 25

On-Level Premium
* Premium adjusted to a common rate level per exposure
→ Better than unadjusted so that market cycles don’t distort results
→ Cape Cod method assumes a constant ELR across accident years
* We can make additional adjustment for loss trend net of exposure trend to get all years on the same cost level

Other Options
Original loss projections by year, estimated claim counts, …

Question 26

Variance of Prospective Losses

Answer 26

Question 27

Variance of Calendar Year Development formulas

Answer 27

Question 28

what’s the main conclusion of the Clark paper and what does it imply?

Answer 28

parameter variance is generally larger than the process variance

implying that our need for more complete data (such as the exposure information in the CC method)
outweighs the need for more sophisticated models

Question 29

adjustments for different exposure periods

Answer 29