What are the main objectives for a statistical loss reserving model?
- 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
What are two underlying causes for reserve variablity?
- Process Variance
→ Uncertainty due to randomness - Parameter Variance
→ Uncertainty in the estimate of expected value, aka estimation error
Model for the expected loss emergence pattern
Loglogistic G(x) formula
Weibull G(x) formula
Comparison of Weibull and Loglogistic Curves
- 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
When will a Weibull or Loglogistic claims emergence model NOT work?
- When there is real, expected negative development (e.g. salvage and subrogation)
o This model still works if some data points show negative development
Advantages to using parameterized curves to describe expected loss
emergence patterns
- 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
What are the underlying assumptions of the two Clark methods for
estimating ultimate losses?
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
Expected incremental loss emergence for the LDF method
Truncated LDF formula
Expected incremental loss emergence for the Cape Cod method
Which expected loss emergence method is preferred according to Clark and
why?
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
How does the Cape Cod method take advantage of more information?
- 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.
Variance/Mean Ratio
MLE for estimating best-fit parameters
Advantage of the maximum loglikelihood function
it works in the presence of negative or zero incremental losses
(since we never actually take the log of c_i,t)
Coefficient of Variation for Reserves
Variance of Reserves
Key Assumptions of LDF Curve Fitting and why they may not hold in
practice (1)
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
Key Assumptions of LDF Curve Fitting (2)
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
Impact of the Key Assumptions about the LDF Curve Fitting Model
Future loss emergence potentially may have more variability than what the
model produces.
Normalized Residual formula
How to test if a fixed σ^2 is an appropriate assumption
