Siewert Flashcards
Excess Losses
Losses excess of a per-occurrence deductible or an aggregate limit
Once aggregate deductible losses reach the aggregate limit, the insured stops retaining losses and the insurer pays for everything else
Advantages of a high deductible program
- Achieves price flexibility while passing additional risk to larger insureds
- Reduces residual market charges and premium taxes
- Gives cash flow advantages to insured (insurer pays claim first and must seek reimbursement from insured)
- Provides incentive for insureds to control losses while protecting them from large losses
- Allows self-insurance without subjecting insureds to demanding state requirements
Loss Ratio Approach
Per Occurrence XS Losses = P * ELR * x
where P = premium and x = per occurrence charge
Per Aggregate XS Losses = P * ELR * (1-x) * phi
where phi = per aggregate charge
Sum both losses to obtain the total losses the insurer pays
Advantages:
- Can be used when no data is available or when data is immature
- LR estimates can be tied to pricing programs (uses inputs from pricing)
- Relies on a more credible pool of company and industry experience
Disadvantages:
- Ignores actual emerging experience
- May not properly reflect account characteristics
Implied Development Approach
- Develop full coverage losses to ultimate
- Develop losses limited by the deductible to ultimate by applying development factors that reflect various inflation indexed limits
- Determine ultimate XS losses by subtracting the limited ultimate losses from the full coverage ultimate losses
Advantages:
- Provides an estimate of XS losses at early maturities even when XS losses have not emerged
- Development factors for limited losses are more stable than those for XS losses
- Estimating deductible losses helps determine the asset value represented by service revenue
Disadvantage:
- Does not explicitly recognize XS loss development
Indexing Limits for Inflation
When calculating development factors for various deductibles, we need to index the limits for inflation
This keeps the proportion of deductible/XS losses constant about the limit from year to year
- Fit a line to average severities over a long term history
- Use an index that reflects the movement in annual severity changes
Current deductible should be deflated before applying it to the previous year losses
Direct Development Approach
Focuses on XS development directly
XS LDFs are calculated so that XS factors in conjunction with limited LDFs balance back to full coverage factors
Advantage:
- Explicitly recognizes XS loss development
Disadvantages:
- XS factors tend to be overly leveraged and extremely volatile
- If XS losses have not yet emerged, we can’t estimate IBNR
BF Approach
Credibility weighting between the direct development and the expected method (LR Approach)
Ult XS Losses = O_t * CDF_t * Z + ELR * (1 - Z) = O_t + Expected Loss * (CDF_t - 1) / CDF_t
where O_t = observed XS loss at time t and Z = 1/CDF_t
Advantages:
- We can determine liabilities either directly or indirectly
- Gives the ability to tie into pricing estimates for recent years where XS losses have yet to emerge
- Provides more stable estimates over time
Disadvantage:
- Ignores actual experience to the extent of the complement of credibility (may need to find weights that are more responsive to actual experience)
Tail Factors
Computed based on an inverse power curve to develop limited losses to ultimate
y = 1 + a * (t + c)^-b
where a, b, and c are given and t represents the beginning of the development period in years
- Fit the inverse power curve on unlimited age-to-age factors to project unlimited ultimate losses
- Select a time at which the projection should stop
- Once the ultimate age is chosen for unlimited losses, fit the curve to each deductible limit and extend that to a common maturity
Advantages:
- Consistent for each limit
- Produces uniformly decreasing tail factors
Disadvantage:
- Bias exists due to extending each limit to the same maturity (lower limits should fully develop much sooner than higher limits)
Severity Relativities
R_t(L) = Severity Limited to limit L at Age t / Unlimited Severity at Age t
Severity relativity should decrease as age increases because more losses are capped at the per-occurrence limit as age increases
Severity relativity should be higher for a larger limit because a higher limit means less of the loss is capped, so the relativity is higher
Consistent Development Factors
Unlimited LDF = U / U_t
Limited LDF(L) = LDF(L) = (U / U_t) * (R(L) / R_t(L)) = LDF * Change in R(L)
XS LDF(L) = XSLDF(L) = (U / U_t) * [(1 - R(L)) / (1 - R_t(L))] = LDF * Change in 1 - R(L)
L is the deductible limit, U is unlimited losses, and R is the severity relativity
Ensures that smaller limits have lower development factors than higher limits
Used for the Direct Development Approach
Unlimited LDF as a weighted average of limited and XS LDFs
LDF_t = R_t(L) * LDF_t(L) + (1 - R_t(L)) * XSLDF_t(L)
Doesn’t need the change in the severity relativity, only need the severity relativity at time t
Partitioning Development
The development factor formulas shown earlier ensure that loss development is partitioned between limited and excess development in a consistent fashion
The % unreported can be partitioned into limited and XS portions using the weighted average formula:
% unreported = (LDF_t - 1) / LDF_t = [(R_t(L) * LDF_t(L) + (1 - R_t(L)) * XSLDF_t(L)) - 1] / [R_t(L) * LDF_t(L) + (1 - R_t(L)) * XSLDF_t(L)] = [(R_t(L) * (LDF_t(L) - 1) + (1 - R_t(L)) * (XSLDF_t(L) - 1)] / [R_t(L) * LDF_t(L) + (1 - R_t(L)) * XSLDF_t(L)]
This can be decomposed into 2 numbers in the numerator - if we divide by the denominator, we obtain the limited and XS % unreported, respectively
% unreported below = R_t(L) * (LDF(L) - 1) / LDF
% unreported above = (1 - R_t(L)) * (XSLDF(L) - 1) / LDF
As age increases, a larger portion of the expected development will be due to excess development above the limit
Distributional Model
To prevent instances of limited development exceeding XS development, which can occur if R(L) increases from one development period to the next
- Models the development process by determining severity distribution parameters that vary over time (i.e., separate severity distributions for each development period)
- Once the parameters are determined, we can calculate the severity relativities - comparing those relativities over time results in development factors
Common model is Weibull for WC - can estimate the parameters using method of moments, maximum likelihood, or Siewert’s approach of minimize the chi square between actual and expected severity relativities
Advantages:
- Helps tie the relativities to the severities and provides consistent loss development factors
- Allows for interpolation among limits and years
Aggregate Limits
Difficult to determine loss development factors for losses excess of aggregate limits (data tends to be sparse and not very credible)
To estimate per-aggregate excess losses:
1. Collective Risk Modeling
2. NCCI Table M - can be more practical
Collective Risk Modeling
We can use collective risk modeling techniques (i.e., modeling frequency & severity separately) to determine the loss development factors, relying on the loss distributions described for deductible limits in conjunction with claims frequency distributions
Siewert uses a Weibull distribution to model severity and a Poisson distribution to model claim counts
Development for losses excess of aggregate limits decreases more rapidly over time with smaller deductibles than larger ones - most of the later development occurs in the layers of loss above the per-occurrence deductible, which is not covered by the aggregate
Higher aggregate limits have more leveraged LDFs because there are fewer losses excess the aggregate limit when the aggregate limit is high
Due to the volatility of losses excess of aggregate limits, Siewert recommends using the BF method to determine the final estimate of ultimate aggregate excess of loss
Service Revenue
The amount paid by the insured to the insurer for handling claims below the deductible
In general, a factor (loss multiplier) is applied to deductible losses, limited by any applicable aggregate limit, to cover expenses that vary with those losses
Service Revenue = (Limited Ultimate Loss - XS of Agg Ultimate Loss) * Factor
To determine service revenue asset:
1. Determine the ultimate deductible losses at the account level
2. Subtract ultimate losses excess of aggregate limits from ultimate deductible losses - use BF Approach to develop XS of Agg Ultimate losses
3. Apply the loss multiplier to the difference in the previous step to determine ultimate recoverables
4. Determine the total asset by subtracting any known recoveries from the ultimate recoverables, and aggregate results for all accounts
Allocated Claim Expense (ALAE)
- Account manages expense itself (i.e., ALAE not covered) - development patterns reflect loss only
- ALAE is treated as loss and subjected to applicable limits - development patterns reflect a combination of loss and ALAE
Improvements to Siewert’s method
- Obtain longer histories of experience
- Derive parameters that provide better fits to the actual data
- Determine better tail factors
- Develop more advanced approaches to index loss limits
