BAIC 6 - Advanced Ratemaking and Professionalism Flashcards
Probability Density Function (PDF)
Describes density of the outcome of a random variable X, theoretically equivalent of a histogram of empirical data
Ex: Loss severity distribution - skewed because large losses are fewer in number, but make up a significant amount of total dollars of loss
Cumulative Distribution Function (CDF)
Describes the probability that a random variable X takes on values less than or equal to x
The Lee Diagram
- Switching the axes of the CDF to imagine the loss distribution
S(x) or G(X) = 1-x - Has the same shape as a representative sample from the distribution, sorted by size and equally spaced
Expected Value
Mean, mu, first raw moment; average value of a random variable = integral of S(x) where S(X) = 1- F(x)
Basic Limits aggregation
- Losses are restated as if all policies were purchased at the basic limit.
- Basic limit is usually the financial responsibility limit or a commonly selected limit
- ALAE is not capped
Increased Limits data aggregation
- Losses are limited to a higher limit
2. ALAE remains unlimited
Limit Loss Cost Reviews and Pricing
Basic limit loss costs are reviewed and filed on a regular basis (perhaps annually)
1. A larger volume of losses capped at the basic limit can be used for a detailed experience analysis
2. Experience is more stable since large, volatile losses are capped and excluded from the analysis;
Higher limits are reviewed less frequently
1. Requires more data volume
2. Fewer policies are written at higher limits
3. Large losses are highly variable
*The two approaches are used together to price a book of business at various policy limits
Increased limits ratemaking process
Developing charges for expected losses at higher limits of liability
Usually results in a multiplicative factor to be applied to basic limits loss cost, increased limit factor (ILF)
Increased Limit Factor
ILF(k) = Expected Pure Premium at Policy limit k / Expected Pure Premium at Basic Policy Limit
or
E[X^k] / E[X^b]
Assumptions to modify the ILFformul
Frequency is assumed to be independent of severity and policy limit
Size of Loss Method
- Individual losses are grouped by size into redetermined intervals
- The aggregate loss within each interval is limited, if necessary, to the limit being reviewed
- ALAE is added to the aggregate limit loss
E[X^k] = (losses on claims up to k + k * number of claims above k + total ALAE) / total number of claims
Layer Method
- Individual losses are sliced into layers
- Fro each loss, the amount of loss corresponding to each layer is added to the aggregate for that layer
- The aggregate loss for each layer up to the limit is added together
- ALAE is added to the aggregate limited loss
E[X^k] = (sum of all losses in each layer up to k + ALAE) / total claims
Size Method advantages and disadvantages
Advantages: 1) Conceptually Straightforward; 2) Data can be used in calculations immediately; 3) More complicated integral is actually generally easier to calculate
Disadvantages: 1) Computationally intensive for calculating sets of increased limits factors
Layer Method advantages and disadvantages
Advantages: 1) Computationally simple for calculating sets of increased limits factors; 2) no integration disadvantage when data is given numerically (generally the case)
Disadvantages: 1) Unintuitive; 2) Data must be processed so that it can be used in calculation; 3) S(x) is generally a more difficult function to integrate
The Consistency Rule
- Each layer represents the additional marginal cost for higher limits and cannot be larger than any lower layers
- Marginal premium per dollar of coverage should decrease as the limit of coverage increases
- ILFs should increase at a decreasing rate
- Expected costs per unit of coverage should not increase in successively higher layers
- For example: change in ILF / change in Limit should decreases as limit grows
Loss Elimination Ratio (LER)
- Savings associated with use of deductible
- Equal to proportion of ground up losses eliminated
LER(j) = E[X^j] / E[X]
E[X^j] = Expected losses below deductible j (limited expected losses)
E[X] = Expected ground-up losses (full value property or total limits liability)
Deductible Relativities
- Deductible analogue of an increased limits factor
- Factor applied to the base premium to reflect a deductible
DR(j) = (1 0 LER( j ) ) / ( 1 - LER(base) )
Dependency factor for relativity
Depends on:
- Loss elimination ratio of the base deductible
- Loss elimination ratio of the desired deductible
Problems with construction - censorship and truncation
Censorship - amounts are known to equal or exceed the policy limit, but their values are capped at the policy limit; Right censorship (from above) occurs when a loss exceeds the policy limit; the loss is known, but its value is recorded as the policy limit amount
Truncation - events are undetected and their values are completely unknown; Left truncation (from below) occurs when a loss below the deductible is not reported
**Data is sparse at higher limits
Fitted distribution to severity function
Data can be used to fit the severity function to a probability distribution
Fitted distributions - addresses concerns
- ILFs can be calculated for all policy limits
- Empirical data can be smoothed
- Trend
- Payment lag
ISO uses these fitted distributions
Truncated Pareto
Mixed exponential
Policy Rating Formula
LC = Terr.LC x ILF x (PCF + SCF) x discounts LC = final loss cost Terr.LC = territorial base loss cost ILF = increased limits factor PCF = primary class factor SCF = secondary classs factor
Basic Policy Rating vs Extension of Exposures
The policy rating procedure is used to determine the premium for individual exposures
The same procedure is used in the extension of exposures ratemaking methodology
-Exposure detail must be collected through statistical reporting
-The statistical records are used in conjunction with the current manual rates to determine the current premium for the exposures in the experience period
-In practice, some modifications may be made
Modifications made between policy rating and extensions of exposures
- For basic limit ratemaking, substitute the basic limit for the actual limit used on the policy
- At ISO, we use the manual loss costs and factors instead of the rates and factors actually used by individual insurers
- Many insurers deviate from ISO loss costs
- The LCCL we use for ratemaking is not the reported premium
