Benefits of high deductible programs
- Price flexibility with additional risk passed to larger insureds
- Improved residual market charges and premium taxes in some states
- Cash flow advantages similar to those of paid loss retro policies
- Provides insureds a way to control losses while protecting against
random, large losses - Allows for easier “self-insurance” for insureds
How to estimate the overall reserve while reflecting
differing mixes of deductibles and limits
After selecting appropriate development factors, apply them at the account
level using each account’s deductibles & limits.
Then aggregate the estimated ultimate over all accounts.
Selecting the Loss Ratio for the Loss Ratio Method
- Use the company experience by state and calculate the full-coverage loss
ratio using an individual account’s premium distribution by state - We might blend that experience with industry experience using
credibility
Note: Loss experience should be developed to ultimate, brought on level
and trended to the appropriate exposure period for calculating loss ratios
Loss Ratio Method:
Estimate of per-occurrence excess losses
Loss Ratio Method:
Estimating the aggregate loss charge
Loss Ratio Method:
Advantages and Disadvantages
Advantages
* Useful when no data is available or data is very immature
* Loss ratio estimates can be consistently tied to pricing, initially
* Relies on a more credible pool of company and/or industry experience
Disadvantages
* Ignores emerging experience
→ Not very useful after several years of development
* May not properly reflect account characteristics and losses may develop
differently due to the type of exposures written
Implied Development Method
Implied Development Method:
Indexed LDFs
- To calculate limited LDFs for deductible loss, index limits for inflation:
o This keeps the proportion of deductible/excess losses constant
around the limit from year to year
o Otherwise, a constant deductible implies increasing excess losses
Possible ways to determine the index:
* Fit a line to average severities over the long-term history
* Use an index that reflects the change in annual severity
Implied Development:
Advantages and Disadvantages
Advantages
* Provides an estimate for excess losses at early maturities, even when
excess losses haven’t emerged
* LDFs for limited losses are more stable than LDFs for excess losses
Disadvantages
* Misplaced focus, because we would like to explicitly recognize excess
loss development
Direct Development Method
Credibility-Weighted Method formulas
Credibility-Weighted Method:
Advantages and Disadvantages
Advantages
* Ties with pricing estimates for immature years where excess losses
haven’t emerged
* Estimates are more stable over time compared to direct development
Disadvantages
* Ignores actual experience for the complement of credibility
→ Might use alternative credibility weights that are more responsive to
actual experience if desired
Limited Severity Relativity
Relationship between limited LDF,
unlimited LDF and severity relativities
Relationship between XSLDF,
unlimited LDF and severity relativities
Relationship between limited,
excess, and unlimited LDFs
Relationship between limited severity relativities over time
Distributional Model
Fit a model (e.g. Weibull) to severities in order to calculate consistent
severity relativities and LDFs.
→ This makes it easy to interpolate among limits and years
Distribution parameters vary over time by development period.
→ Parameters can be estimated by minimizing χ 2 between actual &
fitted severity relativities at the deductible size
→ Constrain parameters so that the model produces the actual unlimited
severity at maturity
Partitioning expected development around the deductible limit
Relationship of excess and limited development over time
As development age increases, an increasing proportion of development is
excess the deductible limit.
Aggregate Loss Limit
Limits the total losses below the deductible that are paid by the insured
Collective Risk Model approach to Aggregate Limits
1) Model frequency and severity separately
→ Siewert uses Weibull for severity and Poisson for claim counts
2) Combine frequency and severity into a collective risk model
3) Sample from collective model to calculate development factors
→ Might improve by including parameter risk in the model
4) Use the BF method with LDFs from model to estimate losses excess
the aggregate limit
Relationships regarding development of losses
excess of aggregate limits
- Development for losses excess aggregate limits decreases more rapidly
over time with smaller deductibles than larger ones
→ Because most later development is in layers of loss above the
deductible limit and not subject to the aggregate limit - Development is more leveraged for larger aggregate limits
→ Takes longer for losses to breach the aggregate limit
BF Method for Aggregate Limits
